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4.7.1 
Principle of mesh Analysis
A circuit with n nodes, b elements and l loops will have n KCL equations, l KVL equations and 
b element equations involving 2b element variables. Only (n
- 
1) KCL equations out of n will 

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4.28
Nodal Analysis and Mesh Analysis of Memoryless Circuits
be linearly independent. Only (b
- 
n
+ 
1) KVL equations out of l such equations will be linearly 
independent. Any set of KCL equations written at (n
- 
1) nodes of the circuit will form a linearly 
independent set. However, any set of (b
- 
n
+ 
1) equations drawn from the set of l voltage equations 
need not form a linearly independent set.
In Node Analysis, KVL equations are used to show that all element voltages can be expressed in 
terms of (n
- 
1) node voltages. KCL equations along with element relations are used subsequently to 
set up (n
- 
1) node equations needed for determining the node voltages.
Analogously, we try to use the KCL equations 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, 
we set up (b
- 
n
+ 
1) KVL equations involving these currents to determine them.
Which (b
- 
n
+ 
1) loops do we choose for writing these KVL equations? The loops have to be chosen 
in such a way that the KVL equations will form an independent set. Two equations are necessarily 
independent if both equations contain terms that belong to only one of them. Consider the following 
equations. 
v
v
v
v
v
v
1
2
3
1
2
4
2 0
7 0
− + − =
− + + − =
and
Obviously, no combinations like 
a v
v
v
b v
v
v
1
2
3
1
2
4
2
7
− + −
(
)
+ − + + −
(
)
can become equal to zero 
for all time for any combination of values for a and b for the simple reason that v
3
and v
4
cannot be got 
rid of. v
3
is present in only one equation and v
4
too is present only in one equation.
Thus, a sufficient but not necessary condition for a set of linear equations to form an independent 
set is that each equation should have at least one variable that does not appear in any other equation 
in the set.
Refer to the circuit in Fig. 4.7-2 .
M1
+
+
–
–
+
–
+
–
+
–
+
–
–
+
+
–
+
–
M2
M3
R
3
R
3
i
R
4
R
4
i
R
5
R
5
i
R
1
R
1
i
R
2
R
2
i
v
2
i
v
1
i
v
3
i
v
4
i
R
1
v
R
3
v
R
5
v
R
4
v
R
2
v
V
1
V
4
V
3