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SJTU 1
Chapter 3
Methods of Analysis
SJTU 2
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.
SJTU 3
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.
SJTU 4
Fig. 3.2 Typical circuit for nodal analysis
SJTU 5
R
vvi lowerhigher
So at node 1 and node 2, we can get the following equations.
3
2
2
212
2
21
1
121
R
v
R
vvI
R
vv
R
vII
322
2121
iiI
iiII
SJTU 6
In terms of the conductance, equations become
232122
2121121
)(
)(
vGvvGI
vvGvGII
Can also be cast in matrix form as
2
21
2
1
32
2
2
21
I
II
v
v
GG
G
G
GG
Some examples
SJTU 7
Fig. 3.5 For Example 3.2: (a) original circuit, (b) circuit for analysis
SJTU 8
Nodal Analysis with Voltage Sources(1)
• Case 1
If a voltage source is connected between 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
SJTU 9
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
0
8
0
42
32
323121
3241
vvand
vvvvvvor
iiii
SJTU 10
• 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
R
VVVi
SJTU 11
Nodal Analysis with Voltage Sources(3)
Example P113
SJTU 12
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.
SJTU 13
Fig. 3.17 A circuit with two meshes
SJTU 14
223213
123131
123222
213111
)(
)(
0)(
0)(
ViRRiR
ViRiRRor
iiRViR
iiRiRV
In matrix form:
2
1
2
1
32
3
3
31
V
V
i
i
RR
R
R
RR
SJTU 15
Fig. 3.18 For Example 3.5
SJTU 16
Mesh Analysis with Current Sources(1)
• Case 1
When a current source exists only in one mesh: Consider the figure right.
Ai
iii
Ai
2
0)(6410
2
1
211
2
SJTU 17
• Case 2
When a current source exists between two meshes:Consider the figure right.
2 solutions:
1. Set v as the voltage across the current 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
SJTU 18
R2
V2V1 +
R3
R4 R5
R1
v-
Isai
bi
ci
Solution 1:
Isiiand
ViiRviR
iiRiiRiR
ViRviiR
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.
SJTU 19
Fig. 3.24 For Example 3.7
supermesh
010)(82
35
06)(842
344
4003212
24331
iii
iiiiiiiand
iiiii
SJTU 20
Fig. 3.31 For Example 3.10
SJTU 21
Fig. 3.32 For Example 3.10; the schematic of the circuit in Fig. 3.31.
SJTU 22
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
