Obtain Z-parameters in terms of ABCD parameters.

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written 2.5 years ago by

prajapatijaimin • 3.3k

• modified 2.5 years ago

Solution:

standard equations of $A B C D$ parameters are given by

$\begin{gathered} V_1(S)=A V_2(S)-B I_2(S)......(1) \\\\ I_1(S)=C V_2(S)-D I_2(S)....(2) \\\\ \Delta T=\left|\begin{array}{ll} \\\\ A & -B \\\\ C & -D \\ \end{array}\right|=-(A D)+(B C) \\\\ =-[A D-B C] \end{gathered} \\$

Rearranging equation (2)

$\begin{aligned} \\ &C V_2(s)=I_1(s)+D I_2(s) \\\\ &V_2(s)=\frac{1}{C} I_1(s)+\frac{D}{C} I_2(s) \\ \end{aligned} \\$ put equation (3) in (1)

$\begin{aligned} \\ V_1(s) &=A\left[\frac{1}{C} I_1(s)+\frac{D}{C} I_2(s)\right]-B I_2(s) \\\\ &=\frac{A}{C} I_1(s)+\frac{A D}{C} I_2(s)-B I_2(s) \\\\ &=\frac{A}{C} I_1(s)+\left[\frac{A D-B C}{C}\right] I_2(s) \\\\ V_1(s) &=\frac{A}{C} I_1(s)+\left[\frac{-\Delta T}{C}\right] I_2(s) \\ \end{aligned}$

from equations (3) and (4) $z$ -parameters are,

$\left[\begin{array}{ll} \ Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \ \end{array}\right]=\left[\begin{array}{cc} \ \frac{A}{C} & \frac{-\Delta T}{C} \\ \frac{1}{C} & \frac{D}{C} \ \end{array}\right] \$

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