Comprehensive Problem

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Comprehensive Problem

19.99 Assume that the two circuits in Fig. 19.135 are equivalent. The parameters of the two circuits must be equal. Using this factor and the z parameters, derive Eqs. (9.67) and (9.68).

Figure 19.135 For Prob. 19.99.

Appendix A

Simultaneous Equations and Matrix Inversion

In circuit analysis, we often encounter a set of simultaneous equations having the form

$$a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = b_1$$

\n

$$a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = b_2$$

\n

$$\vdots \qquad \vdots$$

\n

$$a_{n1}x_1 + a_{n2}x_2 + \dots + a_{nn}x_n = b_n$$

\n(A.1)

where there are n unknown x1, x2, … , xn to be determined. Equation (A.1) can be written in matrix form as

$$\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = \begin{bmatrix} b_2 \\ b_2 \\ \vdots \\ b_n \end{bmatrix}$$

(A.2)

This matrix equation can be put in a compact form as

$$AX = B \tag{A.3}$$

where

$$\mathbf{A} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}, \quad \mathbf{X} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix}$$

(A.3)

A is a square (n × n) matrix while X and B are column matrices.

There are several methods for solving Eq. (A.1) or (A.3). These in clude substitution, Gaussian elimination, Cramer’s rule, matrix inver sion, and numerical analysis.

A.1 Cramer’s Rule

In many cases, Cramer’s rule can be used to solve the simultaneous equa tions we encounter in circuit analysis. Cramer’s rule states that the solution to Eq. (A.1) or (A.3) is