Introduction to Electric Circuits

7.3 Nodal voltage analysis 147

Solution

In matrix form the equations may be written [A][I] = [B] where

A=I521 _3102 _11 and B= Ii!l

Using Cramer's rule we have I~ - A~/A.

A is the determinant of A and

A= 5 2 1

1 10 -2

2 -3 4

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

(5 x 10 x 4) + (2 • -2 x 2) + (1 x 1 • -3) = 200- 8 - 3 = 189

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

(1x10x2)+(2xl x4)+(5x-2x-3)=20+8+30=58

Therefore

A = 189- 58 = 131

To find A1 we replace the first column of matrix A with the column matrix B

and calculate the determinant of the resulting matrix. Thus

AI = 5 2 1

10 10 -2

0 -3 4

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

(5•215215215 •215

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

(1 x 10 x 0) + (2 x 10 x 4) + (5 x -2 x -3) = 80 + 30 = 110

Therefore

A1 = 170- 110 = 60

Finally, I 1 -- Al/A = 60/131 - 0.458 A

7.3 NODAL VOLTAGE ANALYSIS

This method of circuit analysis involves identifying the nodes in a circuit,

selecting one of them as the reference node and then referring the voltages at