chapter_3_-methods_of_analysis- (1).pdf

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CHAPTER 3

METHODS OFANALYSIS

Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.


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7/6/2012 Chapter 3 : Method ofAnalysis

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Methods of Analysis - Chapter 3

3.1 Motivation

3.2 Nodal analysis.

3.3 Nodal analysis with voltage sources.

3.4 Mesh analysis.

3.5 Mesh analysis with current sources.

3.6 Nodal versus mesh analysis.


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What are the things which we need to

know in order to determine the answers?

3.1 Motivation (1)

If you are given the following circuit, how canwe determine (1) the voltage across eachresistor, (2) current through each resistor. (3)power generated by each current source, etc.


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Things we need to know in solving anyresistive circuit with current and voltagesources only:

How should we apply these laws todetermine the answers?

Kirchhoffs Current Laws (KCL)Kirchhoffs Voltage Laws (KVL)

Ohms Law

3.1 Motivation (2)


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3.2 Nodal Analysis (1)

It provides a general procedure for analyzing circuitsusing node voltages as the circuit variables.

Practice Problem 3.1 (pg85) circuit independentcurre

nt source only

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Steps to determine the node voltages:

1. Select a node as the reference node.

2. Assign voltages v1,v2,,vn-1 to the remainingn-1 nodes. The voltages are referenced withrespect to the reference node.

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

nodes. Use Ohms law to express the branchcurrents in terms of node voltages.

4. Solve the resulting simultaneous equationsto obtain the unknown node voltages.


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v1 v2

Practice Problem 3.1 (pg85) circuit independentcurrent source only

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Apply KCL atnode 1 and 2


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8*Answer v1 = -2V, v2 = -14V

i1

i2

i3


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Example 3.2 (pg86)

current with dependant currentsource


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Practice Problem 3.2 (Pg 88)

Find the voltages at the three nonreference nodes in the following circuit:


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Solution:


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3.3 Nodal Analysis withVoltage Source (1)

Example 3.3 (pg90)circuit with independent voltagesource

How to handle the 2V voltage source?


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A super-node is formed by enclosing a (dependentor independent) voltage source connected betweentwo non-reference nodes and any elementsconnected in parallel with it.

*Note: We analyze a circuit with super-nodes usingthe same three steps mentioned above except that

the super-nodes are treated differently.


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Basic steps:

1. Take off all voltage sources in super-nodes and apply KCL to super-nodes(Combine the nodes at supernode).

2. Put voltage sources back to the nodesand apply KVL to relative loops.


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Example 3.3 (pg90) circuit with independent voltagesource

Super-node => 2 =i1+i2+7

Apply KVL => v1+2-v2 = 0


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Example 3.4 circuit with two voltage sources(Dependant & Independent)


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Example 3.4 circuit with two independent voltagesources

i1+i2 = i3 +10 i1 = i3+i5+i4 v1-20-v2 = 0 v3-3vx-v4 = 0


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Practice Problem 3.3 (Pg 90) THIS IS SUPERNODE!

Find vand iin the circuit of the following figure:

V V1

Solution:


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Solution:


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Practice Problem 3.4 (Pg 93) ANOTHER SUPERNODE!

Find v1, v2, and v3 in the following circuit using nodal analysis.

Solution:


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Solution:

i=V1/2


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3.4 Mesh Analysis (1)

1. Mesh analysis provides another generalprocedure for analyzing circuits using meshcurrents as the circuit variables.

2. Nodal analysis applies KCL to find unknownvoltages in a given circuit, while mesh analysisapplies KVL to find unknown currents.

3. A mesh is a loop which does not contain anyother loops within it.


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Steps to determine the mesh currents:

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

n meshes.

2. Apply KVL to each of the n meshes. Use Ohmslaw to express the voltages in terms of the

mesh currents.

3. Solve the resulting n simultaneous equationsto get the mesh currents.


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circuit with independent voltage sources

Note:

i1 and i2 are mesh current (imaginative, not measurable directly)

I1, I2 and I3 are branch current (real, measurable directly)

I1 = i1; I2 = i2; I3 = i1 - i2


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3.4 Mesh Analysis (4)Example 3.5 (pg95) circuit with dependent voltagesource

Find I1 , I2 , I3


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Practice Problem 3.5 (pg96) circuit with dependentvoltage source

Calculate the mesh currents i1 and i2 in the circuit below :


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Example 3.6 (pg96) circuit with dependent voltagesource

Find I0


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Chapter 3 : Method ofAnalysis 34

*Answer Io = I1-I2 =1.5A

I1 = 2.25A, I2= 0.75A, I3 = 1.5A

3


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Practice Problem 3.6 (pg98) circuit with dependent& Independent voltage sources

Using mesh analysis, find ioin the circuit below:


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Chapter 3 : Method ofAnalysis 36

*Answer I1 = -3.214A, I2 = -9.643A, I3 = -5A,

I0 = I3


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Example (pg98) circuit with current source

Using mesh analysis, find current through 6 resistor in the circuitbelow:


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*Answer I1 I2 = -2 + 5 = 3A (Down)


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3.5 Mesh Analysis withCurrent Source (1)

Circuit with current source in between SUPERMESH!(pg98)

A super-mesh results when two meshes have a (dependent orindependent) current source in common as shown in (a). We createa super-mesh by excluding the current source and any elementsconnected in series with it as shown in (b).


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Steps to solve the supermesh circuit:

1. Assign mesh currents i1, i2, , in to then meshes.

2. Bring out the supermesh branch and get theKVL.

3. Bring in back the supermesh branch and the

KCL in the supermesh circuit.


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Example 3.7 (pg99) circuit with dependent&independent voltage source

Combined Supermesh, must try!

For the circuit below, find i1 to i4 using mesh analysis.


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Practice Problem 3.7 (pg100)

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


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3.5 Mesh Analysis withCurrent Source (2)

The properties of a super-mesh:

1. The current source in the super-mesh is not

completely ignored; it provides the constraintequation necessary to solve for the meshcurrents.

2. A super-mesh requires the application of both

KVL and KCL.

3 6 N d l M h


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3.6 Nodal versus MeshAnalysis (1)

To select the method that results in the smaller number ofequations. For example:

1. Choose nodal analysis for circuit with fewer nodes thanmeshes.

*Choose mesh analysis for circuit with fewer meshesthan nodes.

*Networks that contain many series connectedelements, voltage sources, or supermeshes aremore suitable for mesh analysis.

*Networks with parallel-connected elements, currentsources, or supernodes are more suitable for nodalanalysis.

2. If node voltages are required, it may be expedient toapply nodal analysis. If branch or mesh currents arerequired, it may be better to use mesh analysis.

chapter_3_-methods_of_analysis- (1).pdf

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