Introduction to Electric Circuits

7.3 Nodal voltage analysis 153

recourse to Kirchhoff's law by using the pattern noted in the bullet points
leading to Equations (7.4) and (7.7) above. Thus:

9 the coefficient of V2 is the sum of all the conductances connected to node 2,

i.e. (G~ + G 2 + G3);

9 the coefficient of V3 is (-1) times the conductance connected between

nodes 2 and 3, i.e. (-G3);

9 the right-hand side of the equation is the current fed directly into the node
from source Vs1 (G1Vs1) and source Is (Is), i.e. a total of (Is + G~Vs~).

Putting in the numbers we have

(1/4 + 1/20 + 1/2)V2 - (1/2)I/3 - [2 + (1/4)200]

0.8 V 2 - 0.5 V 3 - 52 (7.8)

For node 3"

9 the coefficient of ~ is the sum of all the conductances connected to node 3

(G3 + G4 + Gs);

9 the coefficient of V2 is (-1) times the conductance connected between
nodes 3 and 2 (-G3);

9 the right-hand side of the equation is the total current fed directly into the
node from the voltage source Vs2 (GsVs2) and the current source Is (-Is)
negative because the current is flowing away from the node.
Thus we have
(G3 + G4 -+- G5)V3- G3V2- (G5Vs2- Is)
Putting in the values we have
(1/2 + 1/25 + 1//5)V3- (1/2)V2- (1/5)220- 2
0.74 V3- 0.5 V2- 42 (7.9)
In matrix form Equations (7.8) and (7.9) become

[0.8 --0.5 ][ g2] = 152] (7.10)

-0.5 0.74 1/'3 42

Using Cramer's rule to solve for V2,

V2- AI/A

0.8 -0.5

A=

-0.5 0.74

= (0.8 X 0.74) - (-0.5 x -0.5) -0.592- 0.25 = 0.342