Chapter 3 Circuit Analysis -...
Transcript of Chapter 3 Circuit Analysis -...
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Mazita – Sem 1 1112BEL10103
Chapter 3 Circuit Analysis
1. Nodal Analysis, Nodal analysis with voltage sources
2. Mesh Analysis, Mesh analysis with current sources
3. Nodal versus Mesh analysis
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Learning Outcome...
At the end of this topic, students should be able to:
• Solve the circuits’ problems by using the appropriate methods, i.e. nodal and mesh analysis.
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Circuit AnalysisCircuit Analysis
• We can analyze any linear circuit by:
– obtaining a set of simultaneous equations
– solving the simultaneous equations either using:1. Elimination technique
2. Cramer’s Rule, or
3. any other software such as MATLAB or MathCAD.
– obtaining the required values (voltage or current)
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Nodal Analysis
• Technique based on systematic application of KCL.
• Node voltages are used as the circuit variables.
• Important point when writing the expression for current flow resistance is a passive element, by the passive sign convention, current must always flow from a higher potential to a lower potential
R
vv lowerhigheri−
=
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Nodal Analysis (cont.)
Steps in analysing the circuit
1. Define nodes available and select one of the nodes as the ground node
2. Write KCL equations with the currents must be expressed in terms of the node potentials
3. Solve the equations
4. Compute the element currents and voltages of interest from the node potentials
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Example 1
Find the matrix equation for the following circuit.
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Solution
1. Label the nodes (including the reference node) and draw the direction of currents
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Solution (cont.)
2. Write the KCL equation for node a and b.
Apply KCL at node a.
………….Eq.(1)
Apply KCL at node b.
………….Eq.(2)
2
ba
1
a B1
R
vv
R
0v I
−+
−=
3
b
B2
2
ba
R
0vI
R
vv
−=+
−
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Cont…
3. Simplify Eq.(1) and (2).
……..Eq.(3)
……..Eq.(4)
221R
b
R
1
R
1 aB1
vvI −
+=
232R
a
R
1
R
1 bB2
vvI −
+=
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Solution (cont.)
4. Rewrite the equation in matrix form.
=
+−
+
B2
B1
b
a
322
221
I
I
V
V
R
1
R
1
R
1
R
1-
R
1
R
1
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Example 2
Find V1 and I for the following circuit.
2A 3A
Ω2
Ω4 Ω8+
−1V
I
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Consider the following circuit with current and voltage source.
VB IB
2R
1R3R
a b
c
[2].................R
VbIB
R
VbVa
b, Node
[1]................VBVa
a, Node
32
=+−
=
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Cont…
32
2
232
232
322
322
32
R
1
R
1
R
VBIB
Vb
R
VBIB
R
1
R
1Vb
R
VBIB
R
Vb
R
Vb
IBR
Vb
R
Vb
R
VB-
IBR
Vb
R
Vb
R
VB
R
VbIB
R
VbVB
[2] into [1] substitute
+
+
=
+=
+
+=+
=++
−=−−
=+−
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Example 3
Calculate the node voltages for the following circuit.
5A
10AΩ2
Ω4
Ω6
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Example 4
Determine the voltages at the node 1, 2 and 3 of the following figure.
Ω4
Ω2 Ω8
Ω43A 2ix
1 2 3ix
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Nodal Analysis with Voltage Source (Supernode)
• Consider a section of a network containing a voltage source that connects two nodes below
A surface that encircles a voltage source and its two attached nodes is called a supernode.
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Supernode (cont.)
KCL equations at node A and B are:
node A :
node B :
Adding equations (1) and (2)
(1) 21 iii =+
(2) 043 =++ iii
04321 =+++ iiii
The equation does not involve the source variable, i, even though it encloses the voltage source and the
two nodes
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Supernode (cont.)
• Potential difference across the voltage source, i.e. supernode
sBAvvv =−
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Properties of a Supernode
1. The voltage source inside the supernodeprovides a constraint equation needed to solve the node voltages
2. A supernode has no voltage of its own.
3. A supernode requires the application of both KCL and KVL
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How to deal with Supernode
1. Supernode equationcombination of KCL equation for the respective nodes.
2. Support equationequation for the voltage drop in between the combined nodes compared to the voltage source.
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Example 5
Write the support and supernode equations for the following circuit.
supernode
1IB 2IB1R3R
a b
c
1I
2I 3I
4IVB
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Solution
• Support equation:
• Supernode equation:
supernode
1IB 2IBR1 3R
a b
c
1I
2I 3I
4IVB
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Example 6
Find V1 for the following circuit.
2A
c
7V
Ω4 Ω2 Ω2V1
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Solution
• Support equation:
• Supernode equation:
a b1I
2I 3I
4I
2A
c
7V
Ω4 Ω2 Ω2V1
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Homework [1]
Find Ia for the following circuit.
+
−16V9V
Ω2 Ω6
Ω3
6Ia
Ia
[Answer : Ia = 3.67A]
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Mesh Analysis
• Technique based on systematic application of KVL.
• Mesh currents are used as the circuit variables.
• Mesh analysis is only applicable to a circuit that is planar
A planar circuit is one that can be drawn in a plane with no branches crossing one another
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Mesh Analysis (cont.)
Steps in analysing the circuit
1. Define a mesh current for each mesh
2. Write KVL equations with the voltages must be expressed in terms of the mesh currents
3. Solve the equations
4. Compute the element currents and voltages of interest from the mesh currents
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Example 1
Write the mesh equation for the following circuit.
1VB
3R1R
4R2VB2R
5R
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Example 2
Find Vo for the following circuit.
4A
Ω2 Ω341V
Ω4Ω62A Vo
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Example 3
For the following circuit, find i1, i2 and i3 using mesh analysis.
41V
41V
6Ω5Ω
10Ω4Ω
1i 2i
3i
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Example 4
Use mesh analysis to find the current Io in the following circuit.
+
−
24V
10Ω 24Ω
4Ω
12Ω 4Io
Io
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Mesh Analysis with Current Mesh Analysis with Current SourceSource
Case 1:
When a current source exists only in one mesh
24V
10Ω 4Ω
12ΩI1 I2 3A
I2=-3A
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Mesh Analysis with Current SourceMesh Analysis with Current Source(cont.)
Case 2:
When a current source exists in between of two meshes create supermesh.
Set the following equation:1. Supermesh equation
2. Support equation.
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Example 5
Write the supermesh and support equation for the following circuit.
1VB 2VB
1R
2R
3R
1I 2I
Ib
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Example 6
Find V3 for the following circuit.
2A
4Ω 1Ω
3Ω
5A
38V
+ −3V
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Nodal versus Mesh Analysis
Both methods use systematic approach in solving circuits’ problems.
So how to choose the most suitable method in analysing a given network?
∴Based on 2 factors →the nature of the network
→information required
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Nodal versus Mesh Analysis (cont.)
• Type of Network
– Mesh analysis : contains many series-connected elements, voltage sources or supermeshes
OR a circuit with fewer meshes compared to nodes
– Nodal analysis: contains many parallel-connected elements, currents sources or supernodes
OR a circuit with fewer nodes compared to meshes
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Nodal versus Mesh Analysis (cont.)
• Information required
–Mesh analysis: branch or mesh currents are required
–Nodal analysis: node voltages are required
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Nodal versus Mesh Analysis (cont.)
Most important:
Be familiar with both methods
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References
• Alexander Sadiku, Fundamentals of Electric Circuits, 4th
edition, McGraw-Hill, 2009
• Russell M. Mersereau and Joel R. Jackson, Circuit Analysis: A System Approach, Pearson-Prentice Hall, 2006
• Richard C. Dorf & James A. Svoboda, Introduction to Electric Circuits, 3rd edition, John-Wiley