ECE 1131

Chapter 3 Methods of Analysis

1

Outlines

Nodal analysis without voltage sources

Nodal analysis with voltage sources

Mesh analysis without current sources

Mesh analysis with current sources

Nodal analysis by inspection

Mesh analysis by inspection

2

Nodal Analysis without Voltage Sources

3

Nodal analysis is based on KCL.

It makes use of node voltages as circuit variables.

Steps to determine node voltages - assume a circuit with n nodes:

1. Select a node as the reference node.

Reference node a node with 0 potential.

2. Assign voltages v1, v2, , vn-1 to the remaining n-1 nodes.

The voltages are referenced with respect to the reference node.

3. Apply KCL to each of the n-1 non-reference nodes.

Use Ohms law to express the branch currents in terms of node voltages.

4. Solve the resulting simultaneous equations to obtain the unknown node voltages.

Example 1

4

Obtain node voltages for the circuit shown.

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Answers:

v1=-2V

v2=-14V

Example 2

5

Find the voltages at the three non-reference nodes.

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Answers:

v1=80V

v2=-64V

v3=156V

Nodal Analysis with Voltage Sources

6

Two cases to consider:

Case 1: A voltage source is connected between a reference node and a non-reference node.

Set the voltage at the non-reference node equal to the voltage of the voltage source.

Case 2: A voltage source is connected between two non-reference nodes.

The two non-reference nodes form a supernode.

Apply both KCL and KVL to determine the node voltages.

NOTE:

Supernode it is formed by enclosing a (dependent/independent) voltage source connected between two non-reference nodes and any elements connected in parallel with it.

Example 3 (Fig. 3.7)

7

Find the node voltages in the circuit below.

Example 4 (approximately same as p.p 3.3

page 90)

8

Find v and i in the circuit below.

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Answers:

v=-600mV

i=4.2A

Example 5 (about same as p.p 3.4 page 93)

9

Find v1, v2 and v3 in the circuit shown.

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Answers:

v1=3.043V

v2=-6.956V

v3=652.2mV

Mesh Analysis

10

A mesh is a loop that does not contain any other loops

within it.

Recap: a loop is any closed path with no node pass more than

once.

Mesh analysis is only applicable to a circuit that is planar.

Planar circuit a circuit that can be drawn in a plane with no branches crossing one another; otherwise, it is non-planar.

A Planar Circuit

11

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A Non-Planar Circuit

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No way to redraw it and avoid the branches crossing.

A non-planar circuit can be handled using nodal analysis.

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Mesh Analysis without Current Sources

13

Mesh analysis is based on KVL.

It makes use of mesh currents as circuit variables.

Steps to determine node voltages - assume a circuit with

n meshes:

1. Assign mesh currents i1, i2, , in to the n meshes.

Assume each mesh current flows clockwise.

2. Apply KVL to each of the n meshes.

Use Ohms law to express the voltages in terms of the mesh

currents.

3. Solve the resulting simultaneous equations to get the mesh

currents.

Example 6

14

Calculate the mesh currents, i1 and i2.

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Answers:

i1=2A

i2=0A

Example 7

15

Find I0 in the circuit shown using mesh analysis.

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Answers:

I0=-5A

Mesh Analysis with Current Sources

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Two cases to consider:

Case 1: A current source exists in only one mesh.

Set the mesh current = current source.

Case 2: A current source exists between two meshes.

Create a supermesh by excluding the current source and any

elements connected in series with it.

NOTE:

A supermesh results when two meshes have a (dependent or

independent) current source in common.

Example 8

17

Use mesh analysis to determine i1, i2 and i3 in the circuit

shown.

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Answers:

i1=3.474A

i2=473.7mA

i3=1.1052A

Nodal Analysis by Inspection

18

Implemented when all

sources in the circuit are

independent current

sources.

Convert resistors to

conductors.

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Example 9

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By inspection, obtain the node-voltage equations for the

circuit below.

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Mesh Analysis by Inspection

20

Implemented when all sources are independent

voltage sources.

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Example 10

21

By inspection, obtain the mesh-current equations for the

circuit below.

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