chapter 3 circuit analysis -...

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

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.

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)

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−

=

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

Example 1

Find the matrix equation for the following circuit.

Solution

1. Label the nodes (including the reference node) and draw the direction of currents

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

−=+

−

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 −

+=

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

Example 2

Find V1 and I for the following circuit.

2A 3A

Ω2

Ω4 Ω8+

−1V

I

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

=+−

=

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

+

+

=

+=

+

+=+

=++

−=−−

=+−

Example 3

Calculate the node voltages for the following circuit.

5A

10AΩ2

Ω4

Ω6

Example 4

Determine the voltages at the node 1, 2 and 3 of the following figure.

Ω4

Ω2 Ω8

Ω43A 2ix

1 2 3ix

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.

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

Supernode (cont.)

• Potential difference across the voltage source, i.e. supernode

sBAvvv =−

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

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.

Example 5

Write the support and supernode equations for the following circuit.

supernode

1IB 2IB1R3R

a b

c

1I

2I 3I

4IVB

Solution

• Support equation:

• Supernode equation:

supernode

1IB 2IBR1 3R

a b

c

1I

2I 3I

4IVB

Example 6

Find V1 for the following circuit.

2A

c

7V

Ω4 Ω2 Ω2V1

Solution

• Support equation:

• Supernode equation:

a b1I

2I 3I

4I

2A

c

7V

Ω4 Ω2 Ω2V1

Homework [1]

Find Ia for the following circuit.

+

−16V9V

Ω2 Ω6

Ω3

6Ia

Ia

[Answer : Ia = 3.67A]

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

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

Example 1

Write the mesh equation for the following circuit.

1VB

3R1R

4R2VB2R

5R

Example 2

Find Vo for the following circuit.

4A

Ω2 Ω341V

Ω4Ω62A Vo

Example 3

For the following circuit, find i1, i2 and i3 using mesh analysis.

41V

41V

6Ω5Ω

10Ω4Ω

1i 2i

3i

Example 4

Use mesh analysis to find the current Io in the following circuit.

+

−

24V

10Ω 24Ω

4Ω

12Ω 4Io

Io

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

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.

Example 5

Write the supermesh and support equation for the following circuit.

1VB 2VB

1R

2R

3R

1I 2I

Ib

Example 6

Find V3 for the following circuit.

2A

4Ω 1Ω

3Ω

5A

38V

+ −3V

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

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

Nodal versus Mesh Analysis (cont.)

• Information required

–Mesh analysis: branch or mesh currents are required

–Nodal analysis: node voltages are required

Nodal versus Mesh Analysis (cont.)

Most important:

Be familiar with both methods

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