Question 19.1: Determine the z parameters for the circuit in Fig. 19.7...

■ METHOD 1 To determine z_{11}   and  z_{21}   , we apply a voltage source V_{1}   to the input port and leave the output port open as in Fig. 19.8(a). Then , 

z_{11} = \frac{V_{1}}{I_{1}} = \frac{(20  +  40) I_{1}}{I_{1}} = 60  Ω

that is, z_{11}   is the input impedance at port 1.

z_{21} = \frac{V_{2}}{I_{1}} = \frac{  40 I_{1}}{I_{1}} = 40  Ω

To find  z_{12}  and  z_{22}   , we apply a voltage source V_{2}   to the output port and leave the input port open as in Fig. 19.8(b). Then,

z_{12} = \frac{V_{1}}{I_{2}} = \frac{  40 I_{2}}{I_{2}} = 40  Ω ,              z_{22} = \frac{V_{2}}{I_{2}} = \frac{ (30  +  40) I_{2}}{I_{2}} = 70  Ω

Thus,

[z] = \begin{bmatrix} 60\Omega & 40\Omega \\ 40\Omega & 70\Omega \end{bmatrix}

■ METHOD 2 Alternatively, since there is no dependent source in the given circuit, z_{12} = z_{21}   and we can use Fig. 19.5(a). Comparing Fig. 19.7 with Fig. 19.5(a), we get

z_{12}  =  40 Ω  = z_{21}

z_{11}  –  z_{12}  = 20        ⇒             z_{11}  =  20  +  z_{12}  =  60  Ω

z_{22}  –  z_{12}  = 30        ⇒             z_{22}  =  30  +  z_{12}  =  70  Ω