Transcript File - Electric Circuit Analysis

Chapter 8 – Methods of Analysis
Lecture 10
by Moeen Ghiyas
02/11/2015
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 Nodal Analysis (General Approach)
 Super Nodes
 Nodal Analysis (Format Approach)
 Mesh Analysis employs KVL
 While Nodal Analysis uses KCL for solution
 A node is defined as a junction of two or more branches
 Define one node of any network as a reference (that is, a
point of zero potential or ground), the remaining nodes of the
network will all have a fixed potential relative to this reference
 For a network of N nodes, therefore, there will exist (N – 1)
nodes with a fixed potential relative to the assigned reference
node
 Steps
 Determine the number of nodes within the network
 Pick a reference node, and label each remaining node with a
subscripted value of voltage: V1, V2, and so on
 Apply Kirchhoff’s current law at each node except the reference
 Assume that all unknown currents leave the node for each
application of KCL.
 Solve resulting equations for nodal voltages
 Apply nodal analysis to the network of Fig
 Step 1 – The network has two nodes
 Step 2 – The lower node is defined as the
reference node at ground potential (zero
volts), and the other node as V1, the
voltage from node 1 to ground.
 Step 3: Applying KCL -
I1 and I2 are defined as leaving node
------- eq (1)
 By Ohm’s law,
where
 and
 .
Putting above in KCL eq (1)
 Putting above in KCL eq (1)
 Re-arranging we have
 .
Substituting values
 Now
 But from Ohm’s law we already know
 In nodal analysis technique, if voltage source is found
in the circuit, it is better to convert it to current source
and apply nodal analysis method
 Concept of super node becomes applicable when
voltage sources (without series resistance) are present
in the network
 Steps
 Assign a nodal voltage to each independent node, including the
voltage sources, as if they were resistors or voltage sources
 Remove the voltage sources (replace with short-circuit )
 Apply KCL to all the remaining independent nodes
 Relate the chosen node to the independent node voltages of the
network, and solve for the nodal voltages
 Any node including the effect of elements tied only to other
nodes is referred to as a super-node (since it has an additional
number of terms)
 Example – Determine the nodal voltages V1 and V2 of Fig (using the
concept of a super-node)
 Step 1 - Assign Nodal Voltages
 (All unknown currents leave node)
• Step 2 – Replace Voltage source
with short circuit
 Step 3 – Apply KCL at all nodes (here only one remaining super-node)
 Note that the current I3 will leave the super-node at V1 and then enter
the same super-node at V2.
0.25V1 + 0.5V2 = 2
 Step 4 – Relating the defined nodal voltages to the independent
voltage source (initially removed), we have
V1 – V2 = E = 12 V (Note why not V2 – V1 ??)
 Step 5 – Solve resulting two equations for two unknowns:
0.25V1 + 0.5V2 = 2
V1 – V2 = 12
 Step 5 – Solve resulting two equations for two unknowns:
0.25V1 + 0.5V2 = 2
&
 Here by substitution method,
V1 – 1V2 = 12
 Now,
 The currents can be determined as
and
 This technique allows us to write nodal eqns rapidly
 A major requirement, however, is that all voltage
sources must first be converted to current sources
before the procedure is applied
 Quite similar to mesh analysis (format approach)
 Choose a reference node and assign a subscripted voltage label to
(N - 1) remaining nodes of the network
 Column 1 of each eqn is summing the conductances with node of
interest and multiplying the result by that node voltage
 Each mutual term is the product of the mutual conductance and the
other nodal voltage and are always subtracted from the first column
 The column to the right of the equality sign is the algebraic sum of
the current sources tied to the node of interest. A current source is
assigned a positive sign if it supplies current to a node and a
negative sign if it draws current from the node
 Solve the resulting simultaneous equations for the desired voltages
 Example – Write the nodal equations for the given network
 Step 1 – Choose ref node & assign voltage labels
 Step 2 to 4 as below
 Example – Write the nodal equations for the given network
 Similarly for V2 ,
 Example – Using nodal analysis, determine the potential across the
4Ω resistor
 Step 1 – Choose ref node & Assign voltage labels, and redraw the
network
 Steps 2 to 4 as below:
 Check Solution
 Nodal Analysis (General Approach)
 Super Nodes
 Nodal Analysis (Format Approach)
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