Feedback OTA $[20$ pts$]$

Shown below is an equivalent ac model of an OTA in a feedback loop. Assume that all devices operate in saturation. Neglect ch$$nnel length modulation and capacitances other than the ones explicitly shown. The triangular $"-1"$ gain element has infinite input impedance and zero output impedance. Use $g_m=2mS,R=5k,$ and $C_1=5$ pF $.$

a$)$ Write an expression for the loop gain, $T(s),$ in terms of $g_m,R,C_1,$ and $C_L.$

b$)$ In this circuit, a good phase margin $($PM$)$ is possible for either very small or large values of $C_L.$ First, consider the case of a very small $C_L.$ Calculate the loop crossover frequency $f_{C1}$ and the corresponding value of $C_L$ for a PM of $70.$ Make reasonable approximations to simplify your algebra.

c$)$ Consider now the case for large $C_L.$ Calculate the crossover frequency $f_{c2}$ and the corresponding value of $C_L$ for a PM of $70.$

d$)$ Sketch each case's bode plot $($magnitude and phase$)$ of the loop gain on one diagram. Label pertinent breakpoints.

e$)$ Now assume $C_1=C_L.$ Determine the phase margin of the circuit. To improve your phase margin, you can add a compensation capacitor. For this purpose, where would be the best place to add the capacitor $($drain of Drain of $M2/$ across gate and drain of $M_1/$across gate and drain of $M_2)?$ Briefly explain how it would help your phase margin.