Introduction to Electric Circuits

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158 Nodal and mesh analysis

7.4 MESH CURRENT ANALYSIS


Whereas in the nodal voltage method of analysis we used Kirchhoff's current
law to set up equations from which we could determine the voltages at the
various nodes, in the mesh current method of analysis we use Kirchhoff's
voltage law to set up equations from which the currents in the various meshes
can be calculated. To illustrate the method we will consider the two-mesh
circuit of Fig. 7.8. Remember from Chapter 3 that meshes cannot have loops
inside them so the loop containing Vs~, R1, R3 and Vs2 is not a mesh. We assign
the mesh currents 11 and I2 to the meshes i and 2. Note that the branch currents
/4 and 15 are, respectively, equal to the mesh currents 11 and I2, while the branch
current 13 is 11 -- 12.


R1 R3


Vsl ~ Vs2


Figure 7.8


Applying KVL to mesh 1 and taking the clockwise direction to be positive,
we have


Vsl- RII 4 -- R2I 3 = 0
Vsl - R~Ia - R2(I1- I2) = 0
(R1 + R2)I1- R212 = Vsl (7.12)
Applying KVL to mesh 2 and taking the clockwise direction to be positive,
we have


R213 - R315 - Vs2 = 0
R2(I1- I2) - R312 - Vs2 -- 0
RzI 1 - RzI 2 - R3I 2 -- Vs2--0
-R211 + (R2 + R3)I2 = -Vs2
In matrix form Equations (7.12) and (7.13) may be written

(7.13)

-R 2 (R2 + R3)JLI2J -Vs2_]

Equations (7.12) and (7.13) can be solved simultaneously to determine 11 and 12.
Alternatively, Cramer's rule can be applied to Equation (7.14).
Let us now consider the circuit of Fig. 7.9. This is a three-mesh circuit, the
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