Question 19.11: Obtain the y parameters of the op amp circuit in Fig. 19.37....
Obtain the y parameters of the op amp circuit in Fig. 19.37. Show that the circuit has no z parameters.
Because no current can enter the input terminals of the op amp, \pmb{I_1} = 0, which can be expressed in terms of \pmb{V_1} \text{ and } \pmb{V_2} as
\pmb{I _1}=0 \pmb{V _1}+0 \pmb{V _2} (19.11.1)
Comparing this with Eq. (19.8) gives
(19.8): \pmb{I _1}= \pmb{y _{11} V _1+ y _{12} V _2}\\\pmb{I _2}= \pmb{y _{21} V _1+ y _{22} V _2}
\pmb{y _{11}}=0= \pmb{y _{12}}
Also,
\pmb{V _2}=R_3 \pmb{I _2}+ \pmb{I _o}\left(R_1+R_2\right)
where \pmb{I_o} is the current through R_1 \text{ and } R_2. \text{ But } \pmb{I_o} = \pmb{V_1}/R_1. Hence,
\pmb{V _2}=R_3 \pmb{I _2}+\frac{ \pmb{V _1}\left(R_1+R_2\right)}{R_1}
which can be written as
\pmb{I _2}=-\frac{\left(R_1+R_2\right)}{R_1 R_3} \pmb{V _1}+\frac{ \pmb{V _2}}{R_3}
Comparing this with Eq. (19.8) shows that
\pmb{ y _{21}}=-\frac{\left(R_1+R_2\right)}{R_1 R_3}, \quad \pmb{y _{22}}=\frac{1}{R_3}
The determinant of the [y] matrix is
\Delta_y= \pmb{ y _{11} y _{22}- y _{12} y _{21}}=0
Since ∆_y = 0, the [y] matrix has no inverse; therefore, the [z] matrix does not exist according to Eq. (19.34). Note that the circuit is not reciprocal because of the active element.
(19.34): [ \pmb{y }]=[ \pmb{z} ]^{-1}