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Fundamental of Electric Circuit : Chapter 3 ; Method of Analysis : Mesh &Nodal
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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
12
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
16
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
19
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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