$3.($a$)$ A boost converter can be operated by varying its duty ratio. Assume the converter is ideal.

$($i$)$ In continuous conduction mode, sketch a plot of inductor voltage against operating time, and then derive a relationship between input voltage $(\backslash ($ V$\_\{$d$\}\backslash ))$ and output voltage $(\backslash ($ V$\_\{$o$\}\backslash )).$

$($ii$)$ Derive a relationship between output boundary current $\backslash (\backslash$left$($I$\_\{$o B$\}\backslash$right$)\backslash )$ against duty ratio. Assume the converter consists of continuous output with constant output voltage and constant switching frequency. Then, sketch a plot of output boundary current against duty ratio.

$($iii$)$ Derive a relationship between output boundary current against duty ratio with constant input voltage. Then, sketch a plot of output boundary current against duty ratio.

$($iv$)$ In terms of the inductor value, find the expression of the maximum output boundary current with constant input voltage and constant switching frequency.

$(15$ Marks$)$

$($b$)$ To provide a variable$-$speed range, a fan needs an input dc voltage between $4$ V and $6$ V $.$ To provide this voltage range for the fan, a boost converter is installed with two AAA batteries connected in series. Each AAA battery has a DC voltage of $1\mathrm{.}2$ V$.$ Consider the boost converter with specifications as follows:

$\backslash [$

$\backslash$begin$\{$array$\}\{$l$\}$

$\backslash$text $\{$ Switching frequency $\}=50\backslash$mathrm$\{$kHz$\}\backslash \backslash$

$\backslash$text $\{$ Output current $\}=50\backslash$mathrm$\{$~mA$\}\backslash$text $\{$ to $\}150\backslash$mathrm$\{$~mA$\}$

$\backslash$end$\{$array$\}$

$\backslash ]$

Assume the converter needs to be operated in continuous conduction mode $($CCM$)$ for all time.

$($i$)$ Design the inductor with critical value for this converter.

$($ii$)$ Verify the inductor value in part $($b$)($i$)$ that can fulfill the CCM requirement for all duty ratios.