$($b$)$ The specifications of a switch$-$mode buck$-$converter with a common input and output ground are shown below:

Supply dc voltage: $200$ V

Output de voltage: $48$ V

Output power: $300$ W

Switching frequency: $40$ kHz

Output current ripples: peak to peak $1$ A

Operating mode: continuous conduction mode $($CCM$)$

$($i$)$ Calculate the average input current of the converter

$(2$ marks$)$

$($ii$)$ Calculate the duty cycle of the switching device.

$(2$ marks$)$

$($iii$)$ The minimum value of the inductor to meet the output current ripple requirement.

$(6$ marks$)$