chapter 3 mesh and nodal analysis

ELECTRICAL TECHNOLOGY ENT 188/3

Methods of Analysis

• Introduction

• Nodal Analysis

• Nodal Analysis with Voltage Sources

• Mesh Analysis

• Mesh Analysis with Current Sources

INTRODUCTION

Two powerful techniques for circuit analysis:

1. Nodal analysis ( application of KCL)

2. Mesh analysis ( application of KVL)

NODAL ANALYSIS

1. In nodal analysis, we are interested in finding the node voltages.

2. Steps to determine node voltage:

1. Select reference node.

2. Assign voltage to the remaining n-1 nodes (with respect to reference node)

3. Apply KCL to each nonreference nodes. Use Ohm’s Law to express the branch currents.

4. Solve the resulting simultaneously equations to solve for node voltage

,1,2,1 ...... −nvvv

The first step is selecting a node as the reference node. The reference node is commonly called the ground since it is assumed to have zero potential.

NODAL ANALYSIS

NODAL ANALYSIS

Consider the above figure as an example:

1. Ground has been chosen as the reference node

2. Assign v1 and v2 as node 1 and 2 respectively. (node voltage = voltage of node with respect to the reference node).

3. Apply KCL:

322

2121

:2:1

iiINodeiiIINode

=+++=

NODAL ANALYSISApply Ohm’s Law, where current flows from a higher potential to a lower potential in a resistor:

3

23

2

212

1

11

0,,0R

viR

vviR

vi −=

−=

−=

Rvv

i lowerhigher −=

3

2

2

212 R

vR

vvI =−

+

2

21

1

121 R

vvRvII −

++=

We obtain:

Substitute into equation:

Solve for v1 and v2 using elimination technique/ Cramer’s rule.

NODAL ANALYSIS

Example 3.1:

Calculate the node voltages in the circuit shown in Figure 1.

1. References node

2. Nodes Voltage,

v1 & v2

3.Applying KCL

(refer next slide)

NODAL ANALYSISSolution:

Using the elimination technique

Eq. (3.1.1)

NODAL ANALYSIS

NODAL ANALYSISExample 3.2:

Determine the voltages at the nodes in Figure shown.

Solution:

NODAL ANALYSIS

We have three simultaneous equation to solve to get the node voltages v1,v2, v3.

We can get the answer by Using elimination technique

Answer :

Vvv

vvinSubstitute

vvvv

Fromvv

vv

vvvvvv

vvv

4.2125

12914)4()6(

)6......(..........242

)5()5..(..........042

)3()2()4...(..........1297

)2()1()3....(..........032)2(..........074)1..(..........1223

2

2

22

21

21

21

21

321

321

321

==

=−

==

=+−+

=−−

=+−=−+−

=−−

Continue with v1 & v3..!!!!!!!

Answer : v1 = 4.8V

v2 = 2.4V

v3 = -2.4V

NODAL ANALYSIS WITH VOLTAGE SOURCES

Possibilities:

1. If a voltage source is connected between the reference node and a nonrefencenode, simply set the voltage at the nonreference node equal to the voltage of the voltage source

2. If the voltage is connected between two nonreference nodes, the two nonreference nodes from a generalized node or supernode; apply both KCL and KVL to determine the node voltages.

Supernode: formed by enclosing voltage source connectedbetween two nonreference nodes and any elements

connected in parallel with it.

From the figure:

• v1 = 10V

• Nodes 2 and 3 form a supernode

• KCL at supernode:

60

80

42323121 −

+−

=−

+− vvvvvv

505 3232 =−⇒=++− vvvvKVL at supernode:

3241 iiii +=+

NODAL ANALYSIS WITH VOLTAGE SOURCES

NODAL ANALYSIS WITH VOLTAGE SOURCES

Practice Problem:

Find v and i in the circuit in Fig 3.11:

Answer: -0.2V, 1.4A

At node 1

)4......(....................6

02

03

04

33221 −+

−+

−=

− vvvvv

i1

i2

i3

i41 2 3 )1.......(....................71 Vv =

At node 2

)2.....(..........4321 iiii ++=

Node 2 & node 3 form super node

)3.....(....................303

23

32

+==+−−

vvvv

From (2)

To be continued……….

NODAL ANALYSIS WITH VOLTAGE SOURCES

Practice problem:

For the circuit shown in Figure 3.9, find the node voltages.

Answer: v1= -7.333V, v2= -5.333V

MESH ANALYSIS

1. Mesh analysis is also known as loop analysis or the mesh-current method.

2. Mesh is a loop which does not contain any other loops within it.

3. Application: to find unknown currents

4. Only capable to a planar circuit

5. Planar circuit: can be drawn in a plane with no branches crossing one another.

MESH ANALYSIS

Fig. 3.15 a) a Planar circuit with crossing branches.

b)The same circuit redrawn with no crossing branches b)

MESH ANALYSIS

Steps in determining node voltage:

1. Assign mesh currents i1,i2,…in to the n meshes

2. Apply KVL to each of the n meshes. Use Ohm’s Law to express the voltages in terms of the mesh currents.

3. Solve the resulting n simultaneously equations to get the mesh currents.

MESH ANALYSIS

Consider the figure below:1. Assign i1 and i2 as meshes 1 and 2.

2. Apply KVL to each mesh:

3. Solve for mesh currents i1 and i2

0)(:20)(:1

123222

213111

=−++=−++−

iiRViRMeshiiRiRVMesh

i3

i1i2

I3= i1 - i2

MESH ANALYSIS

Example: For the circuit in Figure, find the branch currents I1, I2 and I3 using mesh analysis.

i3

i1i2

I3=i1-i2 or I3 = -(i2-i1)

MESH ANALYSIS

Practice Problem:

Calculate the mesh currents i1 and i2 in the circuit of Figure shown.

Answer: i1= 2/3 A, i2=0A

MESH ANALYSIS WITH CURRENT SOURCESPossibilities:

1. When a current source exists only in one mesh – set the current as equal to the source.

• set i2 = -5A

• Mesh equation:

Aiiii 2,0)(6410 1211 −==−++−

MESH ANALYSIS CURRENT SOURCES

2. When the current source exists between two meshes – create a supermesh (by excluding the current source and any elements connected in series with it).

A supermesh results when two meshes have a (dependent or independent) current source in common.

Fig: a) Two meshes having a current source in common,

b) a supermesh, created by excluding the current source

• From the above figure:

1. Apply KVL to supermesh:

2. Applying KCL to node 0:

3. Solving:

AiAi 8.2,2.3 21 =−=

0410620 221 =+++− iii

612 += ii

MESH ANALYSIS CURRENT SOURCES

FURTHER READING

1. Fundamentals of Electric Circuits, 2nd Edition,McGrawhill Alexander, C. K. and Sadiku, M. N. O.

2. Electric Circuit, 8th Edition, Pearson, Nillson and Riedel