Electrical and Computing Fundamentals

GENG 2340

Dr. A. Emadi
Department of Electrical and Computer Engineering

1
Methods of Analysis
• Nodal Analysis
• Nodal Analysis with Voltage Sources
• Mesh Analysis
• Mesh Analysis with Current Sources
• Nodal Versus Mesh Analysis

Chapter #3
Images and Texts are from the course textbook @ McGraw-Hill and Wiley, Refer to Syllabus
Fundamentals of Electric Circuits by C. Alexander and M. Sadiku, 7th Edition
Fundamentals of Electric Circuit Analysis by C. R. Paul
2
Introduction
Use three laws (Ohm’s, KCL, KVL) to develop two analysis
techniques:
1) Nodal analysis
• Systematic application of KCL
• Write nodal equations
• Solve for unknown node voltages
2) Mesh analysis
• Systematic application of KVL
• Write mesh equations
• Solve for unknown mesh/loop currents
Obtain a set of simultaneous equations that can be solved to
obtain the values of current or voltage. 3
Nodal Analysis
• Node – Voltage method
• General procedure for analyzing circuits using node voltages:
• Use node voltages (referenced to ground or another node) instead of
element voltages as it is convenient and reduces the number of
simultaneous equations
• Interested in finding the node voltages

4
Nodal Analysis – Steps
1) Setup Circuit
• Select a node as the reference node
• Assign voltages v1,v2,…,vn-1 to the remaining n-1 nodes
• Voltages are referenced with respect to the reference node
2) Setup Equations
• Apply KCL to each of the non-reference nodes
• Use Ohm’s law to express the branch currents in terms of node
voltages
3) Solve the resulting simultaneous equations
• Solve to obtain the unknown node voltages

5
Nodal Analysis – Step 1
• Select a reference node (e.g. the
ground, zero potential, node 0).

• All voltages will be referenced from

this node.

• Few symbols:
• a – common ground
• b – ground
• c – chassis ground

6
Nodal Analysis – Step 2
• Nodes 1 and 2 are assigned voltages v1 and
v2 respectively.
• Redraw circuit to include only node voltages
and currents at these nodes
• Apply KCL to each non-reference node.
• Add currents through resistors.

I1 = I 2 + i1 + i2 at 1
I 2 + i2 = i3 at 2
7
Nodal Analysis – Step 2
• Apply Ohm’s Law to unknown currents: i1, i2 and i3
• Current flows from a higher potential to a lower
potential
• Choose a direction for current
• Label the current
• Voltage at the beginning of arrow minus voltage at the
end of the arrow divided by R

8
Nodal Analysis – Step 2
• Substituting back to the original nodal
equations:
v1 v1 - v2
I1 = I 2 + +
R1 R2
v1 - v2 v2
I2 + =
R2 R3
or
I1 = I 2 + v1G1 + (v1 - v2 )G2
I 2 + (v1 - v2 )G2 = v2G3
9
Nodal Analysis – Step 2
• Place equations in matrix form:

éG1 + G2 -G2 ù é v1 ù é I1 - I 2 ù
ê -G ú ê ú =ê ú
ë 2 G2 + G3 û ëv2 û ë I 2 û
from
I1 = I 2 + v1G1 + (v1 - v2 )G2
I 2 + (v1 - v2 )G2 = v2G3

10
Nodal Analysis – Step 3
• Use Cramer’s Rule

éa b ù é e ù é g ù
êc d ú ê f ú = ê h ú
ë ûë û ë û
D = ad - cb
D1 = gd - hb D 2 = ah - cg
D1 D2
e= f =
D D
• For 2x2 only!
11
Nodal Analysis
(a) a general circuit
(b) definition of the node voltages
(c) labeling of the original circuit with the node voltages.

12
Example #1 – Nodal Analysis
Find V2:

13
Example #2 – Nodal Analysis
Applying KCL at node 1 gives:

14
Example #2 Cont. – Nodal Analysis
Applying KCL at node 2 gives:

15
Example #2 Cont. – Nodal Analysis
Calculate v1 and v2.

16
Example #3 – Nodal Analysis
Calculate I.
Solution: assume d as ref (ground) point.

17
Example #4 – Nodal Analysis
Find node voltages.

18
Example #5 – Nodal Analysis
Find node voltages.

19
Example #6 – Homework
Find node voltages.

20
Nodal Analysis with Voltage Sources
• Consider how voltage sources affect nodal analysis; two
possibilities:
• A voltage source is connected between the reference node and a
non-reference node:
• set the voltage at the non-reference node equal to the voltage of the
voltage source.
• A voltage source (dependent or independent) is connected
between two non-reference nodes:
• the two non-reference nodes (with the voltage source and any
elements connected in parallel with it) form a generalized node or
supernode
• apply both KCL and KVL to determine the node voltages.

21
Nodal Analysis with Voltage Sources

• First Case:
• v1 = 10V
• We know the voltage at this particular node.

• Second Case:
• A supernode is formed by enclosing a
(dependent or independent) voltage source
connected between two non-reference
nodes and any elements connected in
parallel with it.

22
Nodal Analysis with Voltage Sources

• KCL on supernode:
i1 + i4 = i2 + i3
v1 - v2 v1 - v3 v2 - 0 v3 - 0
+ = +
2 4 8 6
• KVL on supernode loop:

-v2 + 5 + v3 = 0
v2 - v3 = 5

23
Nodal Analysis with Voltage Sources

• The voltage source inside the supernode

provides a constraint equation needed to
solve for the node voltages.
• A supernode requires the application of both
KCL and KVL.

24
Example #7 – Nodal Analysis
Perform nodal analysis at v0

25
Example #8 – Nodal Analysis
Perform nodal analysis for v0

26
Example #9 – Nodal Analysis
Determine I. b

ref

27
Example #10 – Nodal Analysis
Find node voltages.

28
Example #10 – Nodal Analysis
Find node voltages.

29
Mesh Analysis
• Mesh analysis is another general procedure for analyzing circuits
• Uses mesh currents as the circuit variables
• Can reduce the number of simultaneous equations

• Loop and Mesh

• A loop is a closed path with no node passed more than once
• A mesh is a loop that does not contain any other loop within it

• Analysis types (so far):

• Nodal analysis applies KCL to find unknown voltages
• Mesh analysis applies KVL to find unknown currents

30
Mesh Analysis
• Concept:

31
Mesh Analysis
• Mesh analysis is not as general as nodal analysis.
• Only applies to a planar circuit
• A circuit that can be drawn in a plane with no
branches crossing one another.
• Otherwise it is nonplanar.
• A circuit may have crossing branches and still be
planar if it can be redrawn such that it has no
crossing branches.
• Circuit on right has two crossing branches, but it
can be redrawn so that is planar.

32
Mesh Analysis

• This circuit is nonplanar as there is no

way to redraw it and avoid the
branches crossing.
• Nonplanar circuits can be handled
using nodal analysis.

33
Mesh Analysis

1) Setup Circuit
• Assign mesh currents i1,i2,…,in to the n meshes
2) Setup Equations
• Apply KVL to each of the n meshes
• Use Ohm’s law to express voltages in terms of mesh currents
3) Solve the resulting simultaneous equations
• Solve to obtain the unknown mesh currents

34
Mesh Analysis – Step 1

• Mesh currents i1 and i2 are assigned

to meshes 1 and 2.
• The direction of mech current is
arbitrary (CW or CCW). Convention
to assume current is clockwise.
However, the direction does not
affect the validity of solution.
• Although path “abcdefa” is a loop and not a mesh, KVL still holds.
• This is the reason for loosely using the terms loop analysis and mesh
analysis to mean the same thing.
35
Mesh Analysis – Step 2

• Apply KVL at each mesh.

Mesh 1:
-V1+R1I1+R3I3=0
-V1+R1I1+R3(I1-I2)=0
(R1+R3)I1-R3I2=V1

Mesh 2:
𝐼! = 𝑖!, 𝐼" = 𝑖"
R2I2+V2-R3I3=0
R2I2+V2-R3(I1-I2)=0 𝐼# = 𝑖! − 𝑖"= 𝐼! − 𝐼"
-R3I1+(R2+R3)I2=-V2 36
Mesh Analysis – Step 3

• Solve the equations.

(R1+R3)I1-R3I2=V1
-R3I1+(R2+R3)I2=-V2

𝑅! + 𝑅" −𝑅" 𝐼! 𝑉!
=
−𝑅" 𝑅# + 𝑅" 𝐼# −𝑉#

I! = i! , I# = i# , I" = i! − i#

Notice that the branch currents can be different from the mesh currents
unless the mesh is isolated.
37
Example #11 – Mesh Analysis
How are v1 and v2 related?

38
Example #12 – Mesh Analysis
Find v1 and v2?

39
Example #13 – Mesh Analysis
Which circuit is planar?

40
Example #14 – Mesh Analysis
Which circuit is planar?

41
Example #15 – Mesh Analysis
Perform mesh analysis on the left loop.

42
Example #16 – Mesh Analysis
Perform mesh analysis on the right loop.

43
Example #17 – Homework
Find the currents through all branches.

Hint:

We know: I1+I2+I3=0

44
Example #18 – Mesh Analysis
Use mesh analysis to find Io.

45
Example #19 – Homework
Use nodal analysis to find Io and validate your
answer with Example #18.

Hint:
label all nodes
use branches current (and NOT mesh
currents)
Write KCL

46
Example #20 – Homework
Use mesh analysis to find Io.

Hint: write 3 KVL in 3 meshes and 1 KCL at

Node A.
Answer:
i1=2.25A
i2=0.75A
i3=1.5A
Io=1.5A

47
Mesh Analysis with Current Sources

• Applying mesh analysis to circuits

containing current sources is actually
much easier than previous methods.
• The current sources reduce the
number of equations.

• Example 21, First case:

• Current source is in only one i2 = -5A
mesh.
-10 + 4i1 + 6 ( i1 - i2 ) = 0
i1 = -2A
48
Mesh Analysis with Current Sources
• Second Case:
• A current source exists between two meshes
• Create a supermesh by excluding the current
source and any elements connected in series
with it.
• Example 22, KVL at the supermesh:
-20 + 6i1 + 10i2 + 4i2 = 0
6i1 + 14i2 = 20
• KCL to a node in the branch where the two
meshes interconnect, and solve:

i2 = i1 + 6
i1 = -3.2A i2 = 2.8A 49
Mesh Analysis with Current Sources

• The current source in the supermesh provides

the constraint equation necessary to solve for
the mesh currents.
• A supermesh requires the application of both
KVL and KCL.

50
Example #23 – Mesh Analysis
What is the equation for the supermesh?

51
Example #24 – Mesh Analysis
What is the constraining equation?

52
Example #23 & 24 – Mesh Analysis
Find currents of all elements.

53
Example #23 & 24 – Mesh Analysis
Find voltages of all elements.

54
Example #25 – Mesh Analysis
Find I0.

55
Example #26 – Mesh Analysis
Find v.

56
Example #27 – Mesh Analysis
Find V.

57
Example #28 – Mesh Analysis
Find V.

58
Example #29 – Homework
Use mesh analysis to find i1, i2, and i3.

59
Nodal Versus Mesh Analysis

Nodal Analysis Mesh Analysis

• Parallel-connected elements, • Series-connected elements,
current sources, or supernodes voltage sources, or supermeshes
• Fewer nodes than meshes • Fewer meshes than nodes
• Node voltages are required • Branch or mesh currents are
• Non-planar network required
• Computer solution • Transistor circuits

60
Nodal Versus Mesh Analysis

• Select the method that results in the smaller number of

equations.

• One method can be used to check the results from the other
method, if possible.

• Since each method has its limitations, only one method may
be suitable for a particular problem.

61