$4.($a$)$ A separately excited DC motor with motor constant of $\backslash (0\mathrm{.}2\backslash$mathrm$\{$~V$\}/\backslash$mathrm$\{$rpm$\}\backslash )$ and resistance of $\backslash (0\mathrm{.}5\backslash$Omega $\backslash )$ is operated by a single$-$phase full$-$bridge converter with single$-$phase squarewave voltage supply. When the motor is operated at firing angle $\backslash (\backslash$alpha$=0^\{\backslash$circ$\}\backslash ),$ its speed is $1200$ rpm with an armature current of $30$ A $.$

$($i$)$ Find the voltage of the single$-$phase square$-$wave voltage supply.

$($ii$)$ Based on the voltage found in part $($a$)($i$),$ find the armature current at an operating speed of $800$ rpm and firing angle $\backslash (\backslash$alpha$=30^\{\backslash$circ$\}\backslash ).$

$($iii$)$ If the armature current is adjusted to $10$ A $,$ find the firing angle $($in degree$)$ if the motor is operated at operating speed of $1000$ rpm $.$

$($iv$)$ If the armature current is adjustable, based on the firing angle found in part $($a$)($iii$),$ find the maximum operating speed $($in rpm$)$ that the motor can achieve.

$($v$)$ If the voltage supply is replaced with a sinusoidal$-$wave voltage input, based on the same conditions, i$.$e$.,$ the motor can be operated at firing angle $\backslash (\backslash$alpha$=0^\{\backslash$circ$\}\backslash )$ to provide speed of $1200$ rpm with armature current of $30$ A $,$ find the voltage of the sinusoidal$-$wave voltage supply.

$(13$ Marks$)$

$($b$)$ A separately excited DC motor with resistance of $\backslash (0\mathrm{.}2\backslash$Omega $\backslash )$ is operated by a buck converter with $\backslash (120-\backslash$mathrm$\{$V$\}\backslash )$ dc input supply. The converter is operated with duty ratio between $0\mathrm{.}3$ and $0\mathrm{.}8.$ The motor is kept driving a constant$-$torque load of $18$ Nm $.$ It can reach a minimum speed of $300$ rpm $.$

$($i$)$ Find the range of output voltage of the buck converter.

$($ii$)$ Find the value of the armature current. $($Hint: the value of armature current should be less than $20$ A $)$

$($iii$)$ Find the motor constant $($in V$/$rpm$).$

$($iv$)$ Based on the motor constant found in part $($b$)($iii$),$ find the maximum speed $($in rpm$).$