Transcript Basic Concepts - Oakland University
Circuit Theorems
Discussion D2.5
Sections 2-9, 2-11 1
Circuit Theorems
•
Linear Circuits and Superposition
• Thevenin's Theorem • Norton's Theorem • Maximum Power Transfer 2
Linear Circuits
• A linear circuit is one whose output is directly proportional to its input. • Linear circuits obey both the properties of homogeneity (scaling) and additivity. 3
4
Superposition Principle
Because the circuit is linear we can find the response of the circuit to each source acting alone, and then add them up to find the response of the circuit to all sources acting together. This is known as the
superposition principle
.
The superposition principle states that the voltage across (or the current through) an element in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone.
5
Turning sources off
Current source:
i
i s
a
i s
We replace it by a current source where
i s
0 b An open-circuit Voltage source: DC
v s
+
v
v s
We replace it by a voltage source where
v s
0
i
An short-circuit 6
Steps in Applying the Superposition Principle
1. Turn off all independent sources except one. Find the output (voltage or current) due to the active source.
2. Repeat step 1 for each of the other independent sources.
3. Find the total output by adding algebraically all of the results found in steps 1 & 2 above.
In some cases, but certainly not all, superposition can simplify the analysis.
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Example: In the circuit below, find the current
i
by superposition 24V DC Turn off the two voltage sources (replace by short circuits).
DC 12V
i
3A 12V
v
1
i
1
v
2 3A 8
1 4
12V
v
1
i
1
i
1 1
1 4
v
1 5 6
v
1 1 4
v
2 0
v
2 3A
v
2 10 3
v
1
v
1 3 1 4
v
1 3 8
v
2 3
v
1 10 8 2 8 3 9
Example: In the circuit below, find the current
i
by superposition 24V DC Turn off the 24V & 3A sources: DC 12V
i
3A
i
1 DC 12V
i
2
i
2 O.C.
10
DC 12V
i
2 O.C.
16 3 DC 12V O.C.
i
2
i
2 12 6 2 DC 12V O.C.
i
2 11
Example: In the circuit below, find the current
i
by superposition 24V DC Turn off the 3A & 12V sources: DC 12V
i
3A 24V DC
i
2
i
3
i
3 O.C.
12
4
4
i
2
24 0
i
3
i
2
i
3 24V DC 16
i
2 4
i
3 24 4
i
2 7
i
3 0 O.C.
i
2 7 4
i
3
i
3
i
3 1 24 13
DC 12V
i
24V DC 3A
i
1
i
2
i
3 12V
v
1
i
1
i
1 1
v
2 3A
i
1 DC 12V
i
2
i
2
i
2 2 O.C.
2A
i
2 24V DC
i
3
i
3
i
3 1 14 O.C.
Circuit Theorems
• • Linear Circuits and Superposition
Thevenin's Theorem
• Norton's Theorem • Maximum Power Transfer 15
Thevenin's Theorem
In many applications we want to find the response to a particular element which may, at least at the design stage, be variable. Linear Circuit V b a + Variable R Each time the variable element changes we have to re-analyze the entire circuit. To avoid this we would like to have a technique that replaces the linear circuit by something simple that facilitates the analysis. A good approach would be to have a simple
equivalent circuit
to replace everything in the circuit except for the variable part (the load). 16
Thevenin's Theorem
Thevenin’s theorem
states that a linear two-terminal resistive circuit can be replaced by an equivalent circuit consisting of a voltage source
V Th
in series with a resistor
R Th
, where
V Th
open-circuit voltage at the terminals, and
R Th
is the is the input or equivalent resistance at the terminals when the independent sources are all turned off.
i
a
i
a Linear Circuit
R L
DC
V Th R Th R L
b
R in
b
R in
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Thevenin's Theorem
Thevenin’s theorem states that the two circuits given below are equivalent as seen from the load
R L
that is the same in both cases.
i
a
i
a
R Th
Linear Circuit
R L
DC
V Th R L
b
R in
b
R in V Th
= Thevenin’s voltage =
V ab
open-circuit voltage =
V OC
with
R L
disconnected (= ) = the 18
Linear Circuit
Thevenin's Theorem
i
a
i
a
R L
DC
V Th R Th R L
b
R in
b
R in R Th
= Thevenin’s resistance = the input resistance with all
independent
sources turned off (voltage sources replaced by short circuits and current sources replaced by open circuits). This is the resistance seen at the terminals
ab
when all independent sources are turned off. 19
DC 10V DC 10V b
Example
a
v OC
2 10V 5V
V Th
b a
i SC
2 10 2 2 3 3 10 4 2.5A
R Th
V Th i SC
5 2.5
R Th
a DC
V Th
5V a b
R Th
b 20
Circuit Theorems
• Linear Circuits and Superposition • Thevenin's Theorem •
Norton's Theorem
• Maximum Power Transfer 21
Norton's Theorem
Norton’s equivalent circuit
can be found by transforming the Thevenin equivalent into a current source in parallel with the Thevenin resistance. Thus, the Norton equivalent circuit is given below.
i
a
I N
V Th R Th R N
R Th R L
b Formally, Norton’s Theorem states that a linear two terminal resistive circuit can be replaced by an equivalent circuit consisting of a current source
I N
in parallel with a resistor
R N
, where
I N
short-circuit current through the terminals, and
R N
is the is the input or equivalent resistance at the terminals when all independent sources are all turned off.
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Circuit Theorems
• Linear Circuits and Superposition • Thevenin's Theorem • • Norton's Theorem
Maximum Power Transfer
23
Maximum Power Transfer
In all practical cases, energy sources have non-zero internal resistance. Thus, there are losses inherent in any real source. Also, in most cases the aim of an energy source is to provide power to a load. Given a circuit with a known internal resistance, what is the resistance of the load that will result in the maximum power being delivered to the load?
Consider the source to be modeled by its Thevenin equivalent.
i
a
R Th R L
DC
V Th
b 24
DC
V Th R Th i
a
R L
b The power delivered to the load (absorbed by
R L
) is
p
2
i R L
V Th
R Th
R L
2
R L
This power is maximum when
R L
0
p
R L
V Th
2
R Th
R L
2 2
R L
R Th
R L
3 0 25
dp dR L
V Th
2
R Th
R L
2
R Th
R L
2
R L
2
R L
R Th
R L
3 0
R L
R Th
Thus, maximum power transfer takes place when the resistance of the load equals the Thevenin resistance
R Th
. Note also that
p
max
V Th
R Th
R L
2
R L R L
R Th p
max
V Th
2
R Th
2
R Th
V Th
2 4
R Th
Thus, at best, one-half of the power is dissipated in the internal resistance and one-half in the load.
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