Exercise 6.4 A circuit with an opamp

The opamp circuit below uses an ideal opamp ($r_{\mathit{in}}\to \infty ,r_{\mathit{out}}=0\mathrm{\Omega },a_V\to \infty )$.

a)

Derive an equation for (the large signal) $v_{\mathit{OUT}}(v_{\mathit{IN}})$

b)

In this circuit schematic, the output signal is fed back to the non-inverting input of the opamp. Show or derive that the system is stable despite feedback to the positive input.

c)

Derive an equation for the large signal $v_X$ as a function of $v_{\mathit{IN}}$ and draw a graph of it. For the MOS transistor, you can assume that it is operated in saturation with $i_D=\frac{1}{2}k{(v_{\mathit{GS}}-V_T)}^2$.

d)

It was assumed that the transistor works in saturation, for which the square law relation is valid. Identify the range of $v_{\mathit{IN}}$ where this assumption is satisfied.

e)

What happens to $v_X$ when $v_{\mathit{IN}}$ approaches $V_{\mathit{DD}}$? Does it asymptotically approach a value? Does it go to 0, $V_{\mathit{DD}}$, $\infty$, -$\infty$, or some other value?