4.30
Nodal Analysis and Mesh Analysis
of Memoryless Circuits
However, a separate symbolic representation of mesh current is used in circuit analysis in order
to emphasize the clockwise direction of flow in the definition of mesh current and to highlight the
point that mesh current is a current that is common to all elements in the mesh. This is shown in
Fig. 4.7-4.
2
Ω
3
Ω
1
Ω
1
Ω
4
Ω
+
–
+
–
+
+
+
–
–
–
+
+
–
–
+
–
+
–
6 V
5 V
2 V
–11 V
R
3
i
3
i
2
i
1
R
1
R
5
R
4
R
2
V
1
V
2
V
3
V
4
Fig. 4.7-4
Circuit
for mesh analysis
The clockwise arrow and the symbol nearby in every mesh stand for the mesh current variable.
Mesh current magnitude itself gives the magnitude of current in all the elements owned by that mesh.
If the assumed current direction in such an element coincides
with that of mesh current,
element
current is same as mesh current. If the assumed current direction in such an element is opposite to that
of mesh current, element current is same as negative of mesh current. If an element is shared by two
meshes, its current is given by the difference between the two mesh currents with due attention to be
placed on current directions.
The procedure of Mesh Analysis is now illustrated using the circuit in Fig. 4.7-4 as an example.
Three mesh currents
i
1
,
i
2
and
i
3
are assigned in the three meshes in clockwise direction as shown. The
KVL equations for the three meshes are written now with element equations
employed to convert the
voltage variables into mesh current variables. We follow a convention in writing these KVL equations.
We start at the leftmost corner of the mesh and traverse the mesh in clockwise direction. We enter
the voltages we meet with in a sum. A voltage is entered in the sum with the same polarity as its first
polarity marking that we meet during our traversal- if we meet its positive polarity first we enter it with
positive sign and if we meet its negative polarity first we enter it with negative sign.
The mesh equation for first mesh is derived below.
The first voltage that we meet is that of
V
1
. We meet its negative polarity first. Therefore, –
V
1
gets
into the equation. Then, we see the voltage across
R
1
with positive polarity first. The current through
it is same as the mesh current
i
1
and hence
R
1
i
1
enters the equation. The next voltage we meet with is
that of
R
2
with its negative polarity first. The current through
R
2
is
i
2
-
i
1
in the direction assumed for
it in the diagram. Hence, –
R
2
(
i
2
-
i
1
) enters the equation. The last voltage we meet with in first mesh
is that of
V
2
, positive polarity coming first. Hence,
+
V
2
enters the equation. Thus, the mesh equation
for
the first mesh is
− +
−
− +
=
+
−
= −
V
R i
R i
i
V
i e
R
R i
R i
V
V
1
1 1
2
2
1
2
1
2
1
2 2
1
2
0
(
)
. ., (
)
With our convention of traversing a mesh in clockwise direction, the mesh current variable in the
mesh where KVL is being applied will appear with positive sign and other mesh current variables
will appear with negative sign in the equation. And the
net rise in voltage contributed by all the
independent voltage sources in that mesh will appear with
positive sign on the right side of equation.
The remaining two mesh equations
for this circuit are
Mesh Analysis of Circuits with Resistors and Independent Voltage Sources
4.31
R i
i
R i
R i
i
V
V
i e
R i
R
R
R i
R i
2
2
1
3 2
4
3
2
2
3
2 1
2
3
4
2
4 3
(
)
(
)
. .,
(
)
− +
−
−
=
−
−
+
+
+
−
==
−
−
+
=
−
−
+
+
=
−
V
V
R i
i
R i
V
V
i e
R i
R
R i
V
V
2
3
4
3
2
5 3
3
4
4 2
4
5
3
3
4
(
)
. .,
(
)
We
can solve for i
1
,
i
2
and
i
3
using the three equations listed below.
(
)
R
R i
R i
i
V
V
1
2
1
2 2
3
1
2
0
+
−
+
= −
−
+
+
+
−
=
−
−
+
+
=
−
R i
R
R
R i
R i
V
V
i
R i
R
R i
V
V
2 1
2
3
4
2
4 3
2
3
1
4 2
4
5
3
3
4
0
(
)
(
)
We followed a certain convention in writing these mesh equations in Eqn. 4.2-2. Adhering to such
a convention has resulted in certain symmetry in these equations. Let us express these equations in
matrix notation to see the symmetry clearly.
(
)
(
)
(
)
R
R
R
R
R
R
R
R
R
R
R
i
i
i
1
2
2
2
2
3
4
4
4
4
5
1
2
3
0
0
+
−
−
+
+
−
−
+
=
−
−
−
1
1 0
0
1
1
0
0
1
0
0
1
1
2
3
4
V
V
V
V
i e
. .,
Do'stlaringiz bilan baham: 
