D $**7\mathrm{.}124$ The MOSFET in the amplifier circuit of Fig. P$7.124$ has $V_t=0\mathrm{.}6V$ and $k_n=5m\frac{A}{V^2}.$ We shall assume that $V_A$ is sufficiently large so that we can ignore the Early effect. The input signal $v_{\text{s}\text{i}\text{g}}$ has a zero average.

$($a$)$ It is required to bias the transistor to operate at an overdrive voltage $V_{OV}=0\mathrm{.}2V.$ What must the dc voltage at the drain be$?$ Calculate the dc drain current $I_D.$ What value must $R_D$ have?

$($b$)$ Calculate the value of $g_m$ at the bias point.

$($c$)$ Use the small$-$signal equivalent circuit of the amplifier to show that

$\frac{v_o}{v_{\text{s}\text{i}\text{g}}}=\frac{1+(\frac{R_2}{R_1})}{1+\frac{(1+\frac{R_2}{R_1})}{g_m{R}_D^{\text{'}}}}$

and

$R_{in}=\frac{1}{g_m}(1+g_m{R}_D^{\text{'}}\frac{R_1}{R_1+R_2})$

where

$R_D^{\text{'}}=R_D||(R_1+R_2)$

$($d$)$ Evaluate $\frac{v_o}{v_{\text{s}\text{i}\text{g}}}$ and $R_{in}.$