Problem $2(20$pts$)$:

A pair of thyristors connected in inverse$-$parallel, as shown in Fig. $1$ below, is used to supply adjustable voltage to a resistive load of $R=10.$ If the $50$ Hz supply voltage is $v_i=311\mathrm{.}13sint$ and the thyristors are each triggered at an angle $$ after their respective anode voltage zeros. Th$1$ turns on during the source positive half$-$cycle, while Th$2$ turns on during the source negative half$-$cycle

a$)$ Sketch the load $($output$)$ voltage waveform for $=45.$

b$)$ Find the r$.$m$.$s$.$ value $V_o$ of the load $($output$)$ voltage.

c$)$ Find the r$.$m$.$s$.$ value $Io$ of the load $($output$)$ current.

d$)$ If thyristor Th$2$ fails to conduct, how much would be the value of the average dc output voltage of the converter at $=45.$

Fig. $1$