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ENT 163
FUNDAMENTALS OF ELECTRICAL
ENGINEERING
LECTURE #3
METHODS OF ANALYSIS
HASIMAH ALI
Programme of Mechatronics,
School of Mechatronics Engineering, UniMAP.
Email: [email protected]
Contents
• Introduction
• Nodal Analysis
• Nodal Analysis with Voltage Sources
• Mesh Analysis
• Mesh Currents 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 v1, v2, ......vn 1, 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
NODAL ANALYSIS
The first step is selecting a node as the reference or datum node. The
reference node is commonly called the ground since it is assumed to
have zero potential.
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:
Node1 : I1  I 2  i1  i2
Node2 : I 2  i2  i3
NODAL ANALYSIS
Apply Ohm’s Law, where current flows from a higher
potential to a lower potential in a resistor:
i
We obtain:
vhigher  vlower
R
v1  0
v1  v2
v2  0
i1 
, i2 
, i3 
R1
R2
R3
Substitute into equation:
I1  I 2 
I2 
v1 v1  v2

R1
R2
v1  v2
v
 2
R2
R3
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.
NODAL ANALYSIS
Solution:
Using the elimination technique
NODAL ANALYSIS
Practice Problem
Obtain the node voltages in the
circuit in Fig shown.
Answer: v1= -2V, v2 = -14V
NODAL ANALYSIS
Example 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.
Using elimination
technique
NODAL ANALYSIS WITH VOLTAGE SOURCES
Possibilities:
1. If a voltage source is connected between the reference node and a
nonrefence node, 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 connected
between two nonreference nodes and any elements
connected in parallel with it.
NODAL ANALYSIS WITH VOLTAGE SOURCES
From the figure:
• v1 = 10V
• Nodes 2 and 3 form a supernode
• KCL at supernode:
i1  i4  i2  i3
v1  v2 v4  v3 v2  0 v3  0



2
4
8
6
KVL at supernode:
 v2  5  v3  0  v2  v3  5
NODAL ANALYSIS WITH VOLTAGE SOURCES
Practice Problem:
Find v and i in the circuit in Fig 3.11:
Answer: -0.2V, 1.4A
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.
6. A circuit may have crossing branches and still be planar if it can be
redrawn such that it has no crossing branches.
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:
Mesh1 : V1  R1i1  R3 (i1  i2 )  0
Mesh 2 : R2i2  V2  R3 (i2  i1 )  0
3. Solve for mesh currents i1 and i2
MESH ANALYSIS
Example: For the circuit in Figure, find the branch currents I1, I2 and I3
using mesh analysis.
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 CURRENT SOURCES
Possibilities:
1. When a current source exists only in one mesh – set the current
as equal to the source.
• set i2 = -5A
• Mesh equation:
10  4i1  6(i1  i2 )  0, i1  2 A
MESH ANALYSIS CURRENT SOURCES
1. 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
MESH ANALYSIS CURRENT SOURCES
•
From the above figure:
1. Apply KVL to supermesh:
 20  6i1  10i2  4i2  0
2. Applying KCL to node 0:
i2  i1  6
3. Solving:
i1  3.2 A, i2  2.8 A
MESH ANALYSIS CURRENT SOURCES
Problem: For the circuit shown, find i1 to i4 using mesh analysis.
MESH ANALYSIS CURRENT SOURCES
Solution:
Applying KVL to the larger supermesh,
MESH ANALYSIS CURRENT SOURCES
Properties of a supermesh:
1. The current source in the supermesh provides the
constraint.
2. A supermesh has no current of its own.
3. A supermesh requires the application of both KVL
and KCL
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