Please derive the control$-$to$-$output transfer function and the input$-$to$-$output transfer function for the Buck$-$Boost converter on paper. Please show details of work.

Use the following parameters:

$($a$)$ Vin $=5$ V $($input voltage$)$

$($b$)$ Vout $=12$ V $($output voltage$)$

$($c$)$ f $=300$ kHz $($frequency$)$

$($d$)$ L $=30$ H $($inductance$)$

$($e$)$ RL $=1$ m$($series resistance with inductor$)$

$($f$)$ RCin $=1$ m$($series resistance with input capacitor$)$

$($g$)$ RCout $=1$ m$($series resistance with output capacitor$)$

$($h$)$ Rload $=0\mathrm{.}5($converter load$)$

$($i$)$ Cin $=100$ F $($input capacitance$)$

$($j$)$ Cout $=100$ F $($output capacitance$)$