Exercise 4.5 AN NMOS amplifier circuit

a)

An answer:
Non-inverting. This can be shown in multiple ways. You could derive a proper equation for voltage gain and have a look at its sign (note that by setting reactances to shorts or opens there is no phase shift). Another way is to do an if-this-than-that analysis:

$$\begin{array}{rcll}& & \mathrm{\text{an increase in }}{v}_{\mathit{in}}\Rightarrow & \\ & & \mathrm{\text{a decrease in }}{v}_{\mathit{GS}}\Rightarrow & \\ & & \mathrm{\text{a decrease in }}{i}_{D_1}\Rightarrow & \\ & & \mathrm{\text{a lower voltage drop across }}{R}_d\Rightarrow & \\ & & \mathrm{\text{an increase of }}{v}_{\mathit{OUT}}& \end{array}$$

b)

An answer:

c)

An answer:
$$\begin{array}{rcll}& a_v& =\frac{v_{\mathit{out}}}{v_{\mathit{in}}}& \\ & & v_{\mathit{out}}=-g_m{v}_{\mathit{gs}}{R}_d& \\ & & v_{\mathit{gs}}=-v_{\mathit{in}}& \\ & & \Rightarrow & \\ & a_v& =g_m{R}_d& \end{array}$$

d)

An answer:
$$\begin{array}{rcll}& r_{\mathit{in}}& =\frac{v_{\mathit{in}}}{i_{\mathit{in}}}& \\ & & i_{\mathit{in}}=-g_m{v}_{\mathit{gs}}& \\ & & v_{\mathit{gs}}=-v_{\mathit{in}}& \\ & & \Rightarrow & \\ & r_{\mathit{in}}& =\frac{v_{\mathit{in}}}{g_m{v}_{\mathit{in}}}=\frac{1}{g_m}& \end{array}$$