chapter 3 methods of analysis

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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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• 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:


R1

1

V1

i V11

V22

1

12211

RVVVi

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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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supermesh

010)(823506)(842

344

4003212

24331

iiiiiiiiiiand

iiiii

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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

chapter 3 methods of analysis

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