$2.$ Consider the dc$-$to$-$dc conversion system shown in Figure P$2\mathrm{.}4.$ The system is configured with two buck converters powered from a common voltage source. Referring to the switch drive signals for Q$1$ and Q$2,$ answer the questions.

$($a$)$ First assume that all the circuit components are ideal and the power conversion is performed losslessly. Evaluate the average current flowing through each semiconductor switch, $\backslash ($ i$\_\{$Q $1\},$ i$\_\{$D $1\},$ i$\_\{$Q $2\}\backslash ),$ and $\backslash ($ i$\_\{$D $2\}\backslash ).$

$($b$)$ To calculate the conduction losses of a power switch, should the current be the effective $($RMS$)$ value or the average value of the switch current?

$($c$)$ To calculate the conduction losses of a power diode, should the current be the effective $($RMS$)$ value or the average value of the diode switch current?

$($d$)$ Now, assume the following practical devices for the semiconductor switches: Q$1$ and Q$2$: MOSFET with the channel resistance of RDS $=\backslash (0\mathrm{.}5\backslash$Omega $\backslash$mathrm$\{$D$\}1\backslash )$ and D$2$: Schottky diode with the turn$-$on voltage of VD $=0\mathrm{.}5$ V $,$ Calculate the average power dissipated at each practical switch.

$($e$)$ Lastly, calculate the overall efficiency of the system using the following formula $\backslash (\backslash$eta$=\backslash$frac$\{$P$\_\{\backslash$text $\{$out $\}\}\}\{$P$\_\{\backslash$text $\{$out $\}\}+$P$\_\{\backslash$text $\{$loss $\}\}\}\backslash )$ where Pout is the power delivered to the load resistors and Ploss is the sum of the power losses, each evaluated in $($b$)$ for the respective practical switch.