Transcript network theorems
Slide 1
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 2
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 3
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 4
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 5
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 6
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 7
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 8
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 9
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 10
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 11
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 12
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 13
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 14
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 15
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 16
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 17
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 18
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 19
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 20
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 21
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 22
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 23
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 24
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 25
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 26
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 27
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 28
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 29
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 30
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 31
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 32
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 33
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 34
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 35
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 36
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 37
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 38
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 39
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 40
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 41
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 42
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 43
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 44
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 45
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 46
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 47
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 48
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 49
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 50
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 51
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 52
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 53
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 54
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 55
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 56
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 57
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 58
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 59
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 60
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 61
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 62
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 63
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 64
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 65
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 66
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 67
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 68
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 69
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 70
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 71
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 72
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 73
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 74
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 75
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 76
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 77
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 78
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 79
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 80
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 81
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 82
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 83
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 84
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 85
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 86
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 87
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 88
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 89
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 90
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 91
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 92
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 93
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 94
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 95
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 96
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 97
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 98
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 99
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 100
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 101
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 102
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 103
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 104
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 105
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 106
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 107
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 108
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 109
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 110
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 111
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 112
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 113
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 114
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 115
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 116
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 117
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 118
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 119
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 120
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 121
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 122
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 123
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 124
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 125
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 126
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 127
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 128
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 129
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 2
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 3
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 4
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 5
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 6
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 7
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 8
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 9
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 10
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 11
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 12
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 13
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 14
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 15
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 16
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 17
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 18
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 19
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 20
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 21
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 22
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 23
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 24
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 25
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 26
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 27
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 28
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 29
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 30
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 31
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 32
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 33
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 34
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 35
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 36
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 37
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 38
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 39
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 40
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 41
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 42
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 43
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 44
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 45
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 46
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 47
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 48
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 49
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 50
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 51
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 52
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 53
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 54
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 55
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 56
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 57
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 58
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 59
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 60
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 61
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 62
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 63
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 64
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 65
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 66
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 67
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 68
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 69
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 70
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 71
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 72
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 73
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 74
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 75
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 76
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 77
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 78
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 79
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 80
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 81
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 82
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 83
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 84
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 85
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 86
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 87
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 88
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 89
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 90
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 91
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 92
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 93
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 94
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 95
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 96
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 97
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 98
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 99
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 100
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 101
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 102
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 103
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 104
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 105
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 106
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 107
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 108
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 109
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 110
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 111
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 112
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 113
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 114
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 115
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 116
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 117
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 118
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 119
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 120
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 121
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 122
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 123
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 124
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 125
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 126
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 127
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 128
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5
Slide 129
CHAPTER-2
NETWORK THEOREMS
CONTENT
1. Kirchhoff’s laws, voltage sources and current sources.
2. Source conversion, simple problems in source conversion.
3. Superposition theorem, simple problems in super position theorem.
4. Thevenin’s theorem, Norton’s theorem, simple problems.
5.Reciprocity theorem, Maximum power transfer theorem, simple
problems.
6. Delta/star and star/delta transformation.
Gustav
Robert
Kirchhoff
Definitions
• Circuit – It is an interconnection of electrical elements in a closed
path by conductors(wires).
• Node – Any point where two or more circuit elements are connected
together
• Branch –A circuit element between two nodes
• Loop – A collection of branches that form a closed path returning to
the same node without going through any other nodes or branches
twice
Example
• How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
Example-Answer
• Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
Example
b
How many nodes, branches & loops?
n 5
7
1
DC
2
3
l
6
4
9
5
2A
5
Kirchhoff's Current Law (KCL)
Also called Kirchhoff's Point law and Kirchhoff's First rule..
Kirchhoff's Current Law (KCL)
Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 +
total volume of water flowing through pipe 3)
Kirchhoff's Current Law (KCL)
Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 +
total current leaving the node through the wire 3)
Kirchhoff's Current Law (KCL)
"The algebraic sum
of all currents entering and leaving a node
must equal zero"
∑ (Entering Currents) = ∑ (Leaving Currents)
Established in 1847 by Gustav R. Kirchhoff
Kirchhoff's Current Law (KCL)
It states that, in any linear network the algebraic sum of the
current meeting at a point (junction) is zero.
∑ I (Junction) = 0
Kirchhoff's Current Law (KCL)
∑ I (Entering) = ∑ I (Leaving)
∑ I (Entering) - ∑ I (Leaving) =0
Kirchhoff's Current Law (KCL)
Assign positive signs to the currents entering the node and
negative signs to the currents leaving the node, the KCL can be reformulated as:
S (All currents at the node) = 0
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
Example
I1= 1 A
I2= 3 A
I3= 0.5 A
Find the current I4 in A
Kirchhoff's Voltage Law (KVL)
Also called Kirchhoff's loop rule and Kirchhoff's second rule..
Kirchhoff's Voltage Law (KVL)
“The algebraic sum of voltages around each loop is zero”.
Σ voltage rise - Σ voltage drop = 0
Or
Σ voltage rise = Σ voltage drop
Kirchhoff's Voltage Law (KVL)
It states that, in any linear bilateral active closed network the
algebraic sum of the product of the current and resistance in each of
the conductors in any closed path (mesh) in the network plus the
algebraic sum of e.m.f in the path is zero.
∑ IR + ∑ e.m.f = 0
Kirchhoff's Voltage Law (KVL)
Sign Convention
The sign of each voltage is the polarity of the terminal first encountered in
traveling around the loop.
The direction of travel is arbitrary.
I
+
R1
A
V0
R2
V 0 V1 V 2 0
V1
Clockwise:
+
Counter-clockwise:
V 2 V1 V 0 0
V2
-
V 0 V1 V 2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
Example
• Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
Source Conversion
ia '
ia
+
Rs
DC
vab
vs
-
+
vs
Rs
Rs v
ab
-
Voltage Source to Current Source
Current Source to Voltage Source
Proof
Convert to Current Source
Answer-1
Convert to Voltage Source
Answer-2
Superposition Theorem
STATEMENTIn a network of linear resistances containing more than one
generator (or source of e.m.f.), the current which flows at any point is
the sum of all the currents which would flow at that point if each
generator were considered separately and all the other generators
replaced for the time being by resistances equal to their internal
resistances.
Superposition Theorem
STATEMENTIn a linear circuit with several sources the voltage and
current responses in any branch is the algebraic sum of the voltage
and current responses due to each source acting independently with
all
other
sources
replaced
by
their
internal
impedance.
Superposition Theorem
Replace a voltage source with a short circuit.
Superposition Theorem
Replace a current source with an open circuit.
Superposition Theorem
Step-1:
Select a single source acting alone. Short the other voltage
source and open the current sources, if internal impedances are not
known. If known, replace them by their internal resistances.
Superposition Theorem
Step-2:
Find the current through or the voltage across the required
element, due to the source under consideration, using a suitable
network simplification technique.
Superposition Theorem
Step-3:
Repeat the above two steps far all sources.
Superposition Theorem
Step-4:
Add all the individual effects produced by individual sources, to
obtain the total current in or voltage across the element.
Explanation
Superposition Theorem
Consider a network, having two voltage sources V1 and V2.
Let us calculate, the current in branch A-B of network, using
superposition theorem.
Step-1:
According to Superposition theorem, consider each source
independently. Let source V1 is acting independently. At this time,
other sources must be replaced by internal resistances.
Superposition Theorem
But as internal impedance of V2 is not given, the source V2 must
be replaced by short circuit. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the
current through branch A-B i.e. IAB due to source V1 alone.
Superposition Theorem
Step 2:
Now Consider Source V2 volts alone, with V1 replaced by short circuit,
to obtain the current through branch A-B. Hence circuit becomes, as shown.
Using any of the network reduction techniques, obtain the current
through branch A-B i.e. IAB due to source V2 alone.
Superposition Theorem
Step 3:
According to the Superposition theorem, the total current
through branch A-B is sum of the currents through branch A-B
produced by each source acting independently.
Total IAB = IAB due to V1 + IAB due to V2
Example
Find the current in the 6 Ω resistor using the principle of
superposition for the circuit.
Solution
Step-1:Replace Current Source with open circuit
Step-2:Replace Voltage Source with Short circuit
Step-3:Current through 6 Ω resistor is
Thevenin’s theorem
Thevenin’s theorem
Statement
“Any linear circuit containing several voltages and
resistances can be replaced by just a Single Voltage VTH in series with
a Single Resistor RTH “.
Thevenin’s theorem
Thevenin’s Equivalent Circuit
Req or RTH
VTH
Thevenin’s theorem
Steps to be followed for Thevenin’s Theorem
Step 1:
Remove the branch resistance through which current is to be
calculated.
Step 2:
Calculate the voltage across these open circuited terminals, by
using any of the network simplification techniques. This is VTH.
Steps to be followed for Thevenin’s Theorem
Step 3:
Calculate Req as viewed through the two terminals of the branch
from which current is to be calculated by removing that branch
resistance and replacing all independent sources by their internal
resistances. If the internal resistance are not known, then replace
independent voltage sources by short circuits and independent current
sources by open circuits.
Steps to be followed for Thevenin’s Theorem
Step 4:
Draw the Thevenin’s equivalent showing source VTH, with the
resistance Req in series with it, across the terminals of branch of interest.
Step 5:
Reconnect the branch resistance. Let it be RL. The required current
through the branch is given by,
Example- Find VTH, RTH and the load current flowing through and load
voltage across the load resistor in fig by using Thevenin’s Theorem.
Step 1- Open the 5kΩ load resistor
Step 2-Calculate
/ measure the Open Circuit Voltage. This is the
Thevenin's Voltage (VTH).
Step 3-Open Current Sources and Short Voltage Sources
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Thevenin's Resistance (RTH)
Step 5-Connect the RTH in series with Voltage Source VTH and reconnect the load resistor. i.e. Thevenin's circuit with load resistor. This
the Thevenin’s equivalent circuit.
RTH =
Thevenin’s Equivalent Circuit
=VTH
Step 6- Calculate the total load current & load voltage
Norton’s theorem
Norton’s theorem
STATEMENTAny Linear Electric Network or complex circuit with Current and
Voltage sources can be replaced by an equivalent circuit containing of
a single independent Current Source IN and a Parallel Resistance RN.
Norton’s theorem
Norton’s Equivalent Circuit
IN
RN
Norton’s theorem
Steps to be followed for Norton’s Theorem
Step 1:
Short the load resistor
Step 2:
Calculate / measure the Short Circuit Current.
This is the Norton Current (IN)
Steps to be followed for Norton’s Theorem
Step 3:
Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Calculate /Measure the Open Circuit Resistance.
This is the Norton Resistance (RN)
Steps to be followed for Norton’s Theorem
Step 4
Now, Redraw the circuit with measured short circuit Current (IN)
in Step (2) as current Source and measured open circuit resistance (RN)
in step (4) as a parallel resistance and connect the load resistor which
we had removed in Step (3).
This is the Equivalent Norton Circuit.
Steps to be followed for Norton’s Theorem
Example 1-Find RN, IN, the current flowing through and Load Voltage
across the load resistor in fig (1) by using Norton’s Theorem.
Step 1-Short the 1.5Ω load resistor
Step 2-Calculate / measure the Short Circuit Current. This is the
Norton Current (IN).
Step 3-Open Current Sources, Short Voltage Sources and Open Load
Resistor.
Step 4-Calculate /measure the Open Circuit Resistance. This is the
Norton Resistance (RN)
Step 5- Connect the RN in Parallel with Current Source INand reconnect the load resistor.
Step 6-Now apply the last step i.e. calculate the load current through
and Load voltage across load resistor
Maximum Power Transfer Theorem
Maximum Power Transfer Theorem
Statement:
In an active resistive network, maximum power transfer to the
load resistance takes place when the load resistance equals the
equivalent resistance of the network as viewed from the terminals of
the load.
Steps to be followed for MPTT
Maximum Power Transfer Theorem
Example-In
the network shown, find the value of RL such that
maximum possible power will be transferred to RL. Find also the value
of the maximum power.
Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH
Thevenin’s Equivalent Circuit
Reciprocity Theorem
Statement:
In any linear bilateral network, if a source of e.m.f E in any
branch produces a current I in any other branch, then the same e.m.f
E acting in second branch would produce the same current I in the
first branch.
Reciprocity Theorem
Example-In
the network given below, find (a) ammeter current
when battery is at A and ammeter at B and (b) when battery is at B and
ammeter at point A.
What is STAR Connection?
If the three resistances are connected in such a manner that one
end of each is connected together to form a junction point called STAR
point, the resistances are said to be connected in STAR.
Star or Y or T Network
What is DELTA Connection?
If the three resistances are connected in such a manner that one
end of first is connected to first end of second, the second end of
second to first end of third and so on to complete a loop then the
resistances are said to be connected in DELTA.
Delta or π Network
STAR to DELTA
DELTA to STAR
To convert a STAR to DELTA
To convert a DELTA to STAR
Example 1-Convert given DELTA into STAR
Answer
Example 2-Convert given STAR into DELTA
Answer
R31 =
=R12
=R23
Example 3-Calculate the effective resistance between
points A & B
Answer-Step 1
Answer-Step 2
Answer-Step 3
Answer-Step 4 & 5
RAB = 3.69 Ω
Example 4-Find the equivalent resistance between
P & Q in the ckt
P
Q
Solution
Solution
Solution
Solution
Req =14.571Ω
Example 5-In the circuit shown, find the resistance between M and N.
Solution- Step 1
Solution- Step 2
Solution- Step 3
Solution- Step 4 & 5