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