Exercise 4.9 A degenerated NMOS amplifier

a)

An answer:

b)

An answer:
$$\begin{array}{rcll}& I_D& =\frac{1}{2}K{\left(V_{\mathit{GS}}-V_T\right)}^2& \\ & & V_{\mathit{GS}}=V_{\mathit{GG}}-I_D{R}_S& \\ & I_D& =\frac{1}{2}K{\left(V_{\mathit{GG}}-V_T-I_D{R}_S\right)}^2& \\ & & =\frac{1}{2}K{\left(V_x-I_D{R}_S\right)}^2,\mathrm{\text{with }}{V}_x=V_{\mathit{GG}}-V_T& \\ & & =\frac{1}{2}K\left(V_x^2-2V_x{I}_D{R}_S+I_D^2{R}_S^2\right)& \\ \Rightarrow \mathrm{\text{(solve quadratic equation)}}& & & \\ 0& & =I_D^2\cdot \frac{1}{2}K{R}_S^2-I_D\left(1+K{V}_x{R}_S\right)+\frac{1}{2}K{V}_x^2& \\ I_D& & =\frac{1+K{V}_x{R}_S\pm \sqrt{{\left(1+K{V}_x{R}_S\right)}^2-{\left(K{V}_x{R}_S\right)}^2}}{K{R}_S^2}& \end{array}$$

Of these 2 solutions, only one is valid. This valid solution corresponds to $V_{\mathit{GS}}>V_T$.

c)

An answer:

$$\begin{array}{lll}\hfill g_m=\sqrt{2K{I}_D},\mathrm{\text{substitute }}{I}_D\mathrm{\text{ from (b)}}& & \hfill \end{array}$$

d)

An answer:
$$\begin{array}{lll}\hfill v_{\mathit{out}}=-g_m{v}_{\mathit{in}}{R}_D\Rightarrow \frac{v_{\mathit{out}}}{v_{\mathit{in}}}=-g_m{R}_D& & \hfill \end{array}$$

e)

An answer:
The input resistance is infinite.