Introduction to Electric Circuits

7.4 Mesh current analysis 159

mesh currents being 11, 12 and 13. None of the other loops are meshes because

they have other loops inside them. We note that the branch currents 16, 17 and 18

are the mesh currents I~, 12 and 13, respectively. Also the branch current 14 is the

difference of two mesh currents (I~ - I2). Similarly the branch current 15 is the

difference of two mesh currents (12 - 13).

Figure 7.9

I6 R1 R3 17 R5 I8

' I5

",01

Applying KVL to mesh 1:

Vs] - RII1 - R214 = 0

Vs1 -- R~I1 - n2(I1 -/2) = 0

Vsl- Rail - RzIl + R212 = 0

(R 1 + R2)11- R212 = Vs1

Applying KVL to mesh 2:

R214- R312- R4~ = 0

R2(I] - 12) - R312 -R4(I2 - 13) = 0

RzI1- R212- R312- R412 + R413-0

-R2I] + (R2 + R3 + R4)I2- R413 = 0

Applying KVL to mesh 3:

R415 - R fl3 - Vs2 = 0

R4(I 2 - 13) - Rs/3 - Vs2 = 0

R412- (R4 + Rs)~ = Vs2

Multiplying throughout by -1 we have

-R4I 2 + (R 4 + Rs)I 3 = _Vs2

In matrix form Equations (7.15), (7.1.6) and (7.17) may be written

-R 2 (R 2 + R 3 + R4) -R4 12 -

0 -R 4 (R 4 + L-* 2A

(7.15)

(7.16)

(7.17)

Equations (7.15), (7.16) and (7.17) can be solved simultaneously to determine

(7.18)