A voltage waveform vi $=5$sin$(200$t$)$ V is fed into a Full$-$wave rectifier with a load resistor, R $=5$ k$.$ Silicon diodes are used in this circuit where, VD$0=0\mathrm{.}6$ V$.$

$($a$)$Draw the rectifier circuit. Label the input and output voltages properly. Briefly explain the application of the circuit.

$($b$)$Calculate the DC value of the output voltage, Vdc and the output frequency fo$.$

$($c$)$Draw the Voltage Transfer Characteristics $($VTC$)$ of the Full$-$wave rectifier and label it properly.

$($d$)$ Now, you have to connect a capacitor in parallel with the load resistor. You have two capacitors of $5$F and $1$F at your disposal. Which capacitor will you use? Explain briefly with necessary calculations.

$($e$)$ A different input waveform is fed into the Full$-$wave rectifier. The new peak$-$to$-$peak ripple voltage is $50\%$ of the previous one calculated from $\text{'}($d$)\text{'}$ with the $5$F capacitor. The new output frequency is $300$ Hz$.$ Determine the equation of the input waveform.