For the amplifier circuit given below,

$($a$)$ Draw DC equivalent circuit.

$($b$)$ Knowing $I_D=4mA,$ determine the $V_{GS}$ and $V_{DS}.$ Also state the operation region.

$($c$)$ Determine the value of $R_1$

$($d$)$ Calculate the small signal gain $g_m.$

$($e$)$ Draw the complete AC small signal equivalent circuit. Show where is the input and output resistances of the amplifier.

$($f$)$ Calculate $R_{in}$ and $R_{\text{o}\text{u}\text{t}}.$

$($g$)$ Calculate overall gain $G_v=\frac{v_{\text{o}\text{u}\text{t}}}{v_{\text{s}\text{i}\text{g}}}.$

Circuit Parameters

$R_{\text{s}\text{i}\text{g}}=50k,R_2=400k$

$R_S=250,R_L=250$

$V_{DD}=12V,V_A=$

$V_t=1V,K_n=0\mathrm{.}5m\frac{A}{V^2}$