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ECE 1311

Chapter 3 Methods of Analysis

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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

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Nodal Analysis without Voltage Sources

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Nodal analysis is based on KCL.

It makes use of node voltages as circuit variables.

Steps to determine node voltages - assume a circuit with nnodes:

1. Select a node as the reference node.

Reference nodea node with 0 potential.

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

The voltages are referenced with respect to the reference node.

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

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

4. Solve the resulting simultaneous equations to obtain the unknownnode voltages.

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Example 1

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Obtain node voltages for the circuit shown.

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Answers:

v1=-2V

v2=-14V

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Example 2

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Find the voltages at the three non-reference nodes.

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Answers:

v1=80V

v2=-64V

v3=156V

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Nodal Analysis with Voltage Sources

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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.

Supernodeit is formed by enclosing a (dependent/independent) voltagesource connected between two non-reference nodes and any elementsconnected in parallel with it.

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Example 3

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Find vand iin the circuit below.

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Answers:

v=-600mV

i=4.2A

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Example 4

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Find v1, v2andv3in the circuit shown.

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Answers:

v1=3.043V

v2=-6.956V

v3=652.2mV

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Mesh Analysis

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A mesh is a loop that does not contain any other loopswithin 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 circuita circuit that can be drawn in a plane with no

branches crossing one another; otherwise, it is non-planar.

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A Planar Circuit

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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

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Mesh analysis is based on KVL.

It makes use of mesh currents as circuit variables.

Steps to determine node voltages - assume a circuit with

nmeshes:1. Assign mesh currents i1, i2, , into the nmeshes.

Assume each mesh current flows clockwise.

2. Apply KVL to each of the nmeshes.

Use Ohms law to express the voltages in terms of the meshcurrents.

3. Solve the resulting simultaneous equations to get the mesh

currents.

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Example 5

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Calculate the mesh currents, i1and i2.

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Answers:

i1=2A

i2=0A

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Example 6

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Find I0in the circuit shown using mesh analysis.

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Answers:

I0=-5A

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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 anyelements connected in series with it.

A supermesh results when two meshes have a (dependent or

independent) current source in common.

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Example 7

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Use mesh analysis to determine i1, i2andi3in the circuitshown.

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Answers:

i1=3.474A

i2=473.7mA

i3=1.1052A

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Nodal Analysis by Inspection

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Implemented when allsources in the circuit are

independent current

sources.

Convert resistors to

conductors.

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Example 8

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By inspection, obtain the node-voltage equations for thecircuit below.

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Mesh Analysis by Inspection

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Implemented when all sources are independentvoltage sources.

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Example 9

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By inspection, obtain the mesh-current equations for thecircuit below.

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