Introduction to Electric Circuits

9.2 The impedance or z-parameters 207

Example 9.1

Determine the z-parameters of the T-circuit shown in Fig. 9.3.

Figure 9.3

11 Za

+o .~ !_ F~-

(10+j0)D

Vl

-----t

(5+j0)D.

Z2 I2

l-- ~ O+

(5+j0)D ['~ Z3

t

l~ + 12

--O __ O--

V2

Solution

From Equation (9.4) z~ - V~/I~ with I2 = 0 and applying KVL with I2 - 0 we

have

V1 = IlZl + Ill3 = I1(Z1 + Z3)

SO

zal = V1/I1- Zl Jr- Z 3

From Equation (9.6) z2~ - V2/I1 with 12 - 0 and with 12 = 0, 1/2 - 11Z3, so

Z21-'- V2/I1 = Z3

From Equation (9.7) z22 - V2/I2 with 11 = 0 and applying KVL with I1 = 0 we

have

V 2 - 12z 2 -F 12Z 3 -- 12(z 2 -F Z3).

Therefore

z:~ = Vii2 = Z: + Z3

From Equation (9.5) z12 - V1/I2 with I~ = 0 and with I~ = 0, V1 = 12 Z3, so

z12 = Viii2 = Z3

Note that Za2 = Z21.

Putting in the values for Z~, Z2 and Z3 from the diagram we have

Zll = Z1 + Z3- 10 + 5 = 15 f~

z22 = Z2 + Z3 = 5 + 5 = 101q

Z12- Z21 "- Z3 - 5 ~-~

In matrix form we have

= [1: 1:]