In this problem, we design a two$-$stage op amp based on the topology shown in the figure. Assume a power budget of $6$ mW$,$ a required output swing of $2\mathrm{.}5$ V$,$ and Leff $=0\mathrm{.}5$ m for all devices.

$($a$)$ Allocating a current of $1$ mA to the output stage and roughly equal overdrive voltages to M$5$ and M$6,$

determine $($W$/$L$)5$ and $($W$/$L$)6.$ Note that the gate$-$source capacitance of M$5$ is in the signal path, whereas that of M$6$ is not. Thus, M$6$ can be quite a lot larger than M$5.$

$($b$)$ Calculate the small$-$signal gain of the output stage.

$($c$)$ With the remaining $1$ mA flowing through M$7,$ determine the aspect ratio of M$3($and M$4)$ such that

VGS$3=$ VGS$5.$ This is to guarantee that if Vin $=0$ and hence Vx $=$ Vy $,$ then M$5$ carries the expected current.

$($d$)$ Calculate the aspect ratios of M$1$ and M$2$ such that the overall voltage gain of the op amp is equal to $500.$