Exercise 8.3 A current source

The figure below shows a current source circuit using a MOS transistor. For transistor $M_1$ $i_D=\frac{1}{2}k\cdot {(v_{\mathit{GS}}-V_T)}^2\cdot (1+\lambda \cdot v_{\mathit{DS}})$ with $k=1\frac{\mathit{mA}}{V^2}$ , $V_T=1V$ and $\lambda =0.01V^{-1}$.

a)

Derive an expression for the value of $R_S$ needed to generate a current $I_D$. As simplification you may ignore the term with $\lambda$.

b)

Calculate the numerical value of $R_S$.
For this you can use $R_{G1}=R_{G2}$ and $V_{\mathit{GG}}=V_{\mathit{DD}}=6V$.

c)

Draw a small-signal equivalent circuit for the current source and determine the value of the small-signal parameters $g_m$ and $r_{\mathit{ds}}$ for the transistor.

d)

Derive an expression for the small-signal output resistance $r_{\mathit{out}}=\frac{v_x}{i_d}$ of the current source.