The purpose of this problem is to guide you through the analysis necessary to establish a design procedure for determining the circuit components in a filter circuit. The circuit to be analyzed is shown in Fig. P15.50.
a) Analyze the circuit qualitatively and convince yourself that the circuit is a low-pass filter with a passband gain of $R_2 / R_1$.
b) Support your qualitative analysis by deriving the transfer function $V_o / V_i$. (Hint: In deriving the transfer function, represent the resistors with their equivalent conductances, that is, $G_1=1 / R_1$, and so forth.) To make the transfer function useful in terms of the entries in Table 15.1, put it in the form
$$
H(s)=\frac{-K b_o}{s^2+b_1 s+b_o}
$$
c) Now observe that we have five circuit components- $R_1, R_2, R_3, C_1$, and $C_2$ - and three transfer function constraints $-K, b_1$, and $b_0$. At first glance, it appears we have two free choices among the five components. However, when we investigate the relationships between the circuit components and the transfer function constraints, we see that if $C_2$ is chosen, there is an upper limit on $C_1$ in order for $R_2\left(G_2\right)$ to be realizable. With this in mind, show that if $C_2=1 \mathrm{~F}$, the three conductances are given by the expressions
$$
\begin{aligned}
G_1 & =K G_2 \\
G_3 & =\left(\frac{b_o}{G_2}\right) C_1 \\
G_2 & =\frac{b_1 \pm \sqrt{b_1^2-4 b_d(1+K) C_1}}{2(1+K)}
\end{aligned}
$$
For $G_2$ to be realizable,
$$
C_1 \leq \frac{b_1^2}{4 b_o(1+K)}
$$
d) Based on the results obtained in (c), outline the design procedure for selecting the circuit components once $K, b_o$ and $b_1$ are known.
(Figure Cant Copy)
Figure P15.50