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example: 4.9-2
Apply mesh analysis on the circuit in Example 4.9-2.
V
1
R
1
R
2
R
3
R
4
R
5
i
x
2
i
x
–11 V
I
2
i
1
I
1
4 V
+
+
+
+
+
–
–
+
–
+
+
+
+
–
–
–
–
–
–
–
5 V
2 
Ω
3 
Ω
1 
Ω
1 
Ω
4 
Ω
V
2
V
3
v
x
v
x
12
Fig. 4.9-3 
Circuit for Example 4.9-2 
Solution
Step-1: Carry out mesh reduction by employing source transformation, if relevant.
There are no current sources appearing in parallel with any resistor. Hence, no mesh reduction is 
possible.
Step-2: Identify meshes and assign mesh current variables.
The second mesh current is directly constrained by dependent current source I
1
and the third mesh 
current is indirectly constrained by i
3
=
I
2
+
I
1
. Hence, there is only one mesh current variable and that 
is i
1
in the first mesh.
Step-3: Identify the controlling variable of dependent sources in terms of mesh current variables 
and express dependent source functions in terms of mesh current variables.
i
x
is the controlling variable of I
1
. i
x 
is the current flowing in R
1
from left to right and hence it is 
equal to i
1
. Therefore, the source function of I
1
is 2i
1
A.
v
x
is the controlling variable for I
2
. v
x
is the voltage across R
5
. Therefore, v
x
=
R
5
(I
2
+
I
1
) 
=
R
5
(I
2
+ 
2 i
1
) 
=
R
5
(2 i
1
+
v
x
÷
12) 
=
(8 i
1
+
v
x
/3).
\
v
x
=
12 i
1
and the source function of I
2 
=
i
1
A.
Step-4: Prepare the mesh equations and solve them.
The mesh equation for the first mesh is written with the dependent source functions expressed in 
terms of mesh current variables.
− +
−
− + =
=
4 2
3 2
5 0
1
1
1
1
1
i
i
i
i e
i
(
)
. .,

---

4.42
Nodal Analysis and Mesh Analysis of Memoryless Circuits
Therefore, the mesh currents are i
1
=
1A, i
2
=
2A and i
3
=
3A.
Step-5: Apply KCL at various nodes of the circuit to find all the element currents and resistor 
voltages.
Consider R
4
. Applying KCL at the node formed by R
3
, R
4 
and R
5
, we get the current flowing from 
top to bottom in R
4 
as i
2
-
i
3
. However, the reference direction that was chosen for current in R
4
is from 
bottom to top. Hence, current in R
4
=
-
(i
2
-
i
3
) in the direction marked in The value is 1 A. 
Currents through other resistors and voltage sources may be obtained in a similar manner.
Step-6: Apply KVL in various meshes to obtain the voltage across the current sources.
Apply KVL to the third mesh first.
v
v
I
I
2
2
1
3 2
4 3
11
0
2
+ × − + × + −
= ⇒
= −
(
)
(
)
V
Now, apply KVL to the second mesh.
− + × − +
+ × −
− −
= ⇒
= −
5 3
2 1
1 2 1 3 2
0
1
1
2
1
(
)
(
)
v
v
v
I
I
I
V
The complete circuit solution is shown in Fig. 4.9-4.
V
1
4 V
–1 A
–1 V
–1 A
–2 V
1 A
2 A
2 A
1 A
1 A
1 A
3 A
3 A
2 V
2 V
+
+
+
+
+
–
–
–
+
–
+
+
+
+
–
–
–
–
–
–
5 V
3 V
1 V
–11 V
12 V
1A
2A
3A
V
2
V
3
Fig. 4.9-4 
Complete mesh analysis solution for the circuit in Example 4.9-2 
4.10 
summAry
• This chapter dealt with two systematic procedures for solving the circuit analysis problem in 
the case of memoryless circuits. Memoryless circuits contain linear resistors, linear dependent 
sources and independent sources.
• Circuit analysis problem for an n-node, b-element circuit involves the determination of b element 
voltage variables and b element current variables, given the source functions of all independent 
sources present in the circuit.
• Node voltage is the voltage of a node in a circuit with respect to a chosen reference node in the 
circuit. In Nodal Analysis, KVL equations are used to show that all element voltages can be 
expressed in terms of (n
- 
1) node voltages and KCL equations along with element relations are 
used subsequently to set up the (n
- 
1) node equations needed for determining the node voltages. 
Nodal Analysis is applicable to any circuit that has a unique solution.
• Mesh current is a fictitious current that flows in clockwise direction in the internal periphery of 
a mesh. In Mesh Analysis, the KCL equations are used to show that all element currents can be 
expressed in terms of a reduced set of (b
- 
n
+ 
1) specially defined currents called mesh currents. 
Subsequently, (b
- 
n
+ 
1) KVL equations involving these currents are set up to determine them. 
Mesh Analysis is applicable only to circuits that are planar.

---

Summary 
4.43
• Source Transformation Theorem is a valuable aid in both analysis procedures. It states that a 
voltage source v
S
(t) in series with a resistance R
S
can be replaced by a current source i
S
(t) 
=
v
S
(t) / R
S
in parallel with R
S 
without affecting any voltage/current/power variable external to the 
source. The direction of current source is such that current flows out of the terminal at which the 
positive of the voltage source is presently connected. 
• The procedure for Nodal Analysis of a circuit containing linear resistors, linear dependent sources 
and independent sources is summarized below.