Exercise 3.7 Biasing a circuit with an NMOS transistor

In the circuit schematic below, transistor $M_1$ is to be biased at a drain current $I_{\mathit{BIAS}}$.

a)

Derive an expression for the value of $R_{G1}$, required to set a drain current $I_D=I_{\mathit{BIAS}}$, as a function of properties of the components in the circuit (resistor values, transistor parameters, supply voltage, ...).

b)

Calculate the numerical value of $R_{G1}$ for $I_{\mathit{BIAS}}=1\mathit{mA}$, $R_S=1k\mathrm{\Omega }$, $K=0.5\mathit{mA}∕V^2$, $V_T=2V$, $R_{G2}=10k\mathrm{\Omega }$, $V_{\mathit{DD}}=10V$. Note: you can also answer this question if you didn’t answer a) (yet). In that case, explain each step in your calculation.

c)

We use the circuit with the values above as an amplifier, where $V_D$ will be the output voltage. Therefore we want to maximize the voltage swing at the $V_D$ node. Derive and equation AND derive the numerical value of $R_D$ for maximum voltage swing such that the output signal will not clip to the supply or push $M_1$ out of saturation.