Introduction to Electric Circuits

156 Nodal and mesh analysis

V2 = 105/--101.5~ ~ 194.8/--22.2 ~

The potential difference across the admittance Yz is

1/2 - V4 = V2 = 194.8/_ - 22.2 ~ V.

Supernodes

If the voltage source Vs is connected between a node N and the reference node,
the voltage of node N becomes Vs as we have seen in the previous two
examples. If, however, the voltage source is connected between two nodes
neither of which is the reference, we introduce the notion of a supernode.

Figure 7.6

Vsl

~1 _ I1 2 R3 I3 3

I

rh

!

R~ Vs2

4

In the circuit of Fig. 7.6 the nodes 1 and 2 and the voltage source Vsl together

form a supernode. Two other nodes are identified as 3 and 4. We choose node 4

as the reference and its voltage is V4 = 0. Applying KCL to the supernode we

have

11+12+[3=0

(V1- V4)/R1 + (V2- V4)/R2 + (V2- V3)/R3 = 0

Now (V3 - V4) = V3 -- Vs2 so that, using conductances, we have

G1V~ + G2V2 + G3(V2- Y~2)= 0

But V1 = Vsl + V2, therefore

G,(V~I + Y2) + G~V2 + G3V~- G3V~2 = 0

G~Ys~ + G~V2 + GzV2 + G3V2- G3Vs2 = 0

(G1 + G2 + G3)V2 = G3Vs2- G.,Vs,

V2- (G3Vs2- OlVs,)/(G, + G2 + G3) (7.11)

Example 7.12

Determine the voltage across the resistors R 2 and R4 in the circuit of Fig. 7.7.