Exercise 3.5 Biasing an amplifier with an NMOS transistor

a)

An answer:
A MOSFET works in saturation if $V_{\mathit{DS}}>V_{\mathit{GS}}-V_T$.

In the circuit, there is zero current through $R_g$, meaning $V_D=V_G$ from which it follows that $V_{\mathit{GS}}=V_{\mathit{DS}}$. Consequently, the saturation condition is always satisfied.

b)

An answer:
$$\begin{array}{rcll}& R_D& =\frac{V_{R_D}}{I_D}& \\ & & =\frac{V_{\mathit{DD}}-V_{\mathit{DS}}}{I_D}& \\ & & =\frac{V_{\mathit{DD}}-V_{\mathit{GS}}}{I_D}& \\ & & V_{\mathit{GS}}=V_T+\sqrt{\frac{2I_D}{k}}& \\ & R_D& =\frac{V_{\mathit{DD}}-V_T+\sqrt{\frac{2I_D}{k}}}{I_D}& \end{array}$$

where the equation for $V_{\mathit{GS}}$ is a reworked version of the element equation (the square law relation).