Introduction to Electric Circuits

7.3 Nodal voltage analysis 151

9 the coefficient of V1 is the sum of all the conductances connected to node 1,

9 the coefficient of V2 is (-1) times the conductance connected between node

2 and node 1,

9 the right-hand side of the equation is the current being fed directly into the

node from source Vs~;

and for node 2,

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

9 the coefficient of V~ is (-1) times the conductance connected between node

1 and node 2,

* the right-hand side of the equation is the current being fed directly into the

node from source Vs2.

In matrix form Equations (7.5) and (7.6) may be written

-G 3 (G3 + G 4 + Gs) Vs2GsJ

Using Cramer's rule to solve for V~ we have V1 - A1/~k. Now

A

(G 1 + G 2 + G3) -G 3

-G3 (G3 + G4 + Gs)

The sum of the products of the elements along the diagonals to the right is

(G, + G2 + G3)(G3 + G4 + Gs)

The sum of the products of the elements along the diagonals to the left is
(-- G3)(- G3) = G3 2, so
A=(G, +G2+G3)(G3+G4+Gs)-G3 2

Now A 1 is A with the first column replaced by the column vector on the right-
hand side of the Equation (7.7), so that

A1

VslG1

Vs2G5

-G3

(G 3 4- G 4 4- Gs)

The sum of the products of the elements along the diagonals to the right is

(Vs~G~)(G3 + G4 + Gs). The sum of the products of the elements along the

diagonals to the left is (-Ga)(VsEG3). Therefore

A1 = (Vs, G,)(G3 + G4 + Gs) - (-G3)(Vs2Gs)

= (VsIGI)(G3 + G4 + Gs) + Vs2G3G5