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Chapter 3
Methods of Analysis
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So far, we have analyzed relatively simple circuits by applying Kirchhoff’s laws in combination with Ohm’s law. We can use this approach for all circuits, but as they become structurally more complicated and involve more and more elements, this direct method soon becomes cumbersome. In this chapter we introduce two powerful techniques of circuit analysis: Nodal Analysis and Mesh Analysis.
These techniques give us two systematic methods of describing circuits with the minimum number of simultaneous equations. With them we can analyze almost any circuit by to obtain the required values of current or voltage.
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Nodal Analysis• Steps to Determine Node Voltages:1. Select a node as the reference node(ground), define the node voltages
1, 2,… n-1 to the remaining n-1nodes . The voltages are referenced with respect to the reference node.
2. Apply KCL to each of the n-1 independent nodes. Use Ohm’s law to express the branch currents in terms of node voltages.
3. Solve the resulting simultaneous equations to obtain the unknown node voltages.
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Fig. 3.2 Typical circuit for nodal analysis
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Rvv
i lowerhigher
So at node 1 and node 2, we can get the following equations.
3
2
2
212
2
21
1
121
Rv
RvvI
Rvv
RvII
322
2121
iiIiiII
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In terms of the conductance, equations become
232122
2121121
)()(
vGvvGIvvGvGII
Can also be cast in matrix form as
2
21
2
1
32
2
2
21
III
vv
GGG
GGG
Some examples
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Fig. 3.5 For Example 3.2: (a) original circuit, (b) circuit for analysis
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Nodal Analysis with Voltage Sources(1)
• Case 1 If a voltage source is connected bet
ween the reference node and a nonreference node, we simply set the voltage at the nonreference node equal to the voltage of the voltage source. As in the figure right:
Vv 101
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Nodal Analysis with Voltage Sources(2)
• Case 2 If the voltage source (dependent or
independent) is connected between two nonreference nodes, the two nonreference nodes form a supernode; we apply both KVL and KCL to determine the node voltages. As in the figure right:
56
08
042
32
323121
3241
vvand
vvvvvvor
iiii
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• Case 3 If a voltage source
(dependent or independent) is connected with a resistor in series, we treat them as one branch. As in the figure right:
Nodal Analysis with Voltage Sources(3)
R1
1
V1
i V11
V22
1
12211
RVVVi
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Nodal Analysis with Voltage Sources(3)
Example P113
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Mesh Analysis
• Steps to Determine Mesh Currents:1. Assign mesh currents i1, i2,…in to the n meshes.
2. Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of the mesh currents.
3. Solve the resulting n simultaneous equations to get the mesh currents.
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Fig. 3.17 A circuit with two meshes
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223213
123131
123222
213111
)()(
0)(0)(
ViRRiRViRiRRor
iiRViRiiRiRV
In matrix form:
2
1
2
1
32
3
3
31
VV
ii
RRR
RRR
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Fig. 3.18 For Example 3.5
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Mesh Analysis with Current Sources(1)
• Case 1 When a current source exists only i
n one mesh: Consider the figure right.
Aiiii
Ai
20)(6410
2
1
211
2
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• Case 2 When a current source exists betwe
en two meshes:Consider the figure right.
2 solutions:1. Set v as the voltage across the curre
nt source, then add a constraint equation.
2. Use supermesh to solve the problem.
Mesh Analysis with Current Sources(2)
R2
V2V1 +
R3
R4 R5
R1
v-
Isai
bi
ci
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R2
V2V1 +
R3
R4 R5
R1
v-
Isai
bi
ci
Solution 1:
IsiiandViiRviR
iiRiiRiRViRviiR
ca
bcc
abcbb
aba
2)(0)()(
1)(
35
231
42
Solution 2:
supermesh
01452)(3)(2 ViaRicRVibicRibiaRNote:1, The current source in the supermesh is not completely ignored; it provides the constraint equation necessary to solve for the mesh current.2, A supermesh has no current of its own.3, A supermesh requires the application of both KVL and KCL.
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Fig. 3.24 For Example 3.7
supermesh
010)(823506)(842
344
4003212
24331
iiiiiiiiiiand
iiiii
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Fig. 3.31 For Example 3.10
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Fig. 3.32 For Example 3.10; the schematic of the circuit in Fig. 3.31.
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Nodal Versus Mesh Analysis• Both provide a systematic way of analyzing a complex network.• When is the nodal method preferred to the mesh
method?1. A circuit with fewer nodes than meshes is better analyzed using nodal
analysis, while a circuit with fewer meshes than nodes is better analyzed using mesh analysis.
2. Based on the information required. Node voltages required--------nodal analysis Branch or mesh currents required-------mesh analysis