To maximize the time$-$averaged power delivered from a generator tilde$(Z{)}_g=(60+j80)$ to a load tilde$(Z{)}_L=(75-j25),$ a lossless impedance matching network is employed as shown below.

$($a$)$ Determine the value of tilde$(Z{)}_{in}$ and tilde$(Z{)}_{\text{o}\text{u}\text{t}}$ to maximize the time$-$averaged power delivered from the generator to the load. $(50$ pts $)$

$($b$)$ Calculate a value for the maximum time$-$averaged power delivered to the load if the generator voltage is given by $v_g(t)=600cos(t-60),(mV).(50pts)$