1. Consider the circuit shown below.
(a) Using circuit analysis, derive an expression for the low-frequency gain, $A_{v(LF)}$. (25 pts)
(b) Using circuit analysis, derive an expression for the high-frequency gain, $A_{v(HF)}$. (25 pts)
(c) Derive an expression for the transfer function, $\tilde{A}_v(s) = \tilde{V}_o(s) / \tilde{V}_i(s)$. (200 pts)
(d) Using the transfer function found in part (c), derive an expression for the low-frequency gain, $A_{v(LF)}$. (25 pts)
(e) Using the transfer function found in part (c), derive an expression for the high-frequency gain, $A_{v(HF)}$. (25 pts)
(f) Express $\tilde{A}_v(s)$ in the following form. Find an expression for $a_1, b_1$, etc. (25 pts)
$$\tilde{A}_v(s) = + \frac{a_2s^2 + a_1s + a_0}{b_2s^2 + b_1s + b_0}$$
(g) Using the transfer function expression given in part (f), express $\tilde{A}_v(s)$ in the following form. Derive an expression for $A_{mid}$, $\omega_0$, and $Q$ in terms of, $a_i$, $b_i$, etc. (50 pts)
$$\tilde{A}_v(s) = + \frac{(A_{mid}/Q)(s/\omega_0)}{(s/\omega_0)^2 + (1/Q)(s/\omega_0) + 1}$$
(h) Using the transfer function expression given in part (g), derive an expression for the gain at resonance, $\tilde{A}_v(s = j\omega_0)$. (25 pts)
