v $.$ If a $220$ V single$-$phase AC source is used to supply power for the shunt motor, please properly design the single$-$phase full$-$wave diode rectifier and the DC$/$DC power converter for connecting the $220$ V single$-$phase AC source to the DC motor, the detailed requirements as below:

$-$ Design the single$-$phase full$-$wave diode rectifier and determine the average output current and voltage of the rectifier $($show theoretical calculations$).$

$-$ Select a C filter capacitor to get the smoother output current and voltage for the full$-$wave diode rectifier.

$-$ Sketch a clearly labeled equivalent circuit of a DC$-$DC converter and briefly describe the operation of the converter.

$-$ Design the DC$/$DC power converter having a switching frequency of $25$ kHz $,$ a ripple in the output voltage is limited to $\backslash (5\backslash \%\backslash ),$ and a ripple in the continuous inductor current is limited to $\backslash (10\backslash \%\backslash )($please present how to appropriately select the inductor, capacitor in the DC$-$DC power converter designed$).$

$-$ Calculate the input current of the DC$-$DC converter, the average inductor current, and the maximum and minimum currents of the inductor.

vi$.$ Assume that a load whose torque is proportional to the square of the speed. When running at $900$ rpm $,$ please calculate the armature current and the terminal voltage needed to control the motor in this case.

vii. Assume that a load whose torque is proportional to the square of the speed. When running at $1200$ rpm $,$ please calculate the armature current and the terminal voltage needed to control the motor in this case.

viii. In case the load torque is constant, please determine the speed and output power of the shunt DC motor to lift the object of $1500$ kg at a velocity $($ v $)$ of $\backslash (1\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$; and how to adjust the armature voltage$/$current to get the bove$-$calculated speed$/$output power of the motor.