Question $4(10$ Marks$)$

A controllable AC$-$DC converter is shown in Fig. $4.$ Its AC$-$side is connected to an ideal transformer T with turns ratio $1$:$N.$ Its DC $-$side is connected to a pure resistive load with resistance $R.$ The primary and secondary voltages of the ideal transformer T are denoted by $u_1$ and $u_2,$ respectively. The secondary current of the ideal transformer T is denoted by $i_2.$ The voltage and current of the load resistor are denoted by $u_d$ and $i_d,$ respectively. $N=2$;$u_1=155sin(250t)V.$ The firing angles $$ of the thyristors are all $30.$

$($a$)$ Sketch the waveforms of $u_2,u_d,i_2,$ and $i_d.$

$(6$ marks$)$

$($b$)$ Calculate the average value of the output voltage $u_d.$

$(4$ marks$)$