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Textbook Question
100% Chapter 3, Problem 3.1P

Formulate node-voltage equations for the circuit in Figure P3-1. Arrange the results in matrix form Ax = b .  To determine

The node voltage equations in matrix form Ax=b .

The required node voltage equation in the form of matrix is [ 1 R 5 + 1 R 1 1 R 1 1 R 5 1 R 1 1 R 1 + 1 R 2 + 1 R 3 1 R 3 1 R 5 1 R 3 1 R 3 + 1 R 4 + 1 R 5 ]+[ v A v B v C]=[ i S00] .

### Explanation of Solution

Calculation:

The given circuit diagram is shown in Figure 1 Mark the branch current in the given diagram.

The required diagram is shown in Figure 2 Apply the KCL at the node vA .

iSvAvCR5vAvBR1=0vAvCR5+vAvBR1=iSvA(1 R 5 +1 R 1 )vB(1 R 1 )vC(1 R 5 )=iS ....... (1)

Apply the KCL at the node vB .

vAvBR1vBvCR3vBR2=0vB(1 R 1 +1 R 2 +1 R 3 )vA(1 R 1 )vC(1 R 3 )=0vA(1 R 1 )+vB(1 R 1 +1 R 2 +1 R 3 )vC(1 R 3 )=0 ....... (2)

Apply the KCL at the node vC .

vBvCR3+vAvCR5vCR4=0vC(1 R 3 +1 R 4 +1 R 5 )+vA(1 R 5 )+vB(1 R 3 )=0vA(1 R 5 )vB(1 R 3 )+vC(1 R 3 +1 R 4 +1 R 5 )=0 ....... (3)

Arrange the Equation (1), (2) and (3) in the matrix form.

[ 1 R 5 + 1 R 1 1 R 1 1 R 5 1 R 1 1 R 1 + 1 R 2 + 1 R 3 1 R 3 1 R 5 1 R 3 1 R 3 + 1 R 4 + 1 R 5 ]+[ v A v B v C]=[ i S00]

Conclusion:

Therefore, the required node voltage equation in the form of matrix is [ 1 R 5 + 1 R 1 1 R 1 1 R 5 1 R 1 1 R 1 + 1 R 2 + 1 R 3 1 R 3 1 R 5 1 R 3 1 R 3 + 1 R 4 + 1 R 5 ]+[ v A v B v C]=[ i S00] .

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