Problem statement:

A single$-$phase half$-$wave circuit is shown with a purely resistive load R$.$ Assume the diode is ideal and the source voltage, $\backslash (\backslash$mathrm$\{$V$\}\_\{\backslash$mathrm$\{$s$\}\}\backslash ),$ is $\backslash (240-\backslash$mathrm$\{$V$\}\backslash$mathrm$\{$rms$\}\backslash )$ and the frequency is $50$ Hz $.$ Let $\backslash (\backslash$mathrm$\{$R$\}=20\backslash$Omega $\backslash )$

$($A$)$ As a function of time t $,$ neatly plot $\backslash (\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$s$\}\},\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$L$\}\}\backslash ),$ and $\backslash (\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$D$\}\}\backslash )$ for two cycles.

$($B$)$ Determine the following values: $($i$)$ efficiency; $($ii$)$ form factor; $($iii$)$ ripple factor; $($iv$)$ transformer utilization factor; and $($v$)$ peak inverse voltage of the diode.

$($C$)$ Find the Fourier expansion of the load voltage, $\backslash (\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$L$\}\}\backslash )$ in the cosine form

Show the expanded form of $\backslash ($ f$($t$)\backslash )$ for values of n from $1$ to $5$ only. Plot the Fourier series expansion choosing a higher value for n $.$

$($D$)$ If the diode is replaced by a silicon$-$controlled rectifier $($SCR$),$ find $($a$)$ the form factor, and $($b$)$ the Fourier expansion of the load waveform as in part $($C$)$ for a gate triggering angle of $\backslash (30^\{\backslash$circ$\}\backslash ).$

$($E$)$ Can the circuit in part $($D$)$ be used in the speed control of a dc motor if the load is the field winding of the dc motor? If yes, explain how.