$($iii$)$ Figure $2($b$)$ shows an inverting Buck$-$Boost converter which can provide an output voltage with a magnitude which is greater or less than the input voltage depending on the value of the duty cycle.

The following assumptions can be applied to the circuit:

$-$ The circuit operates in steady state with a DC output voltage which is negative, i$.$e$.$ inverted with respect to the input.

$-$ The switch is operated with a duty cycle $\backslash ($ D $\backslash )$ with a switching period $\backslash ($ T $\backslash ).$

$-$ When the switch is on$,$ the diode is reversed biased.

$-$ All components are ideal with no losses.

$-$ The converter operates in continuous conduction mode.

$-$ Sketch the voltage waveform for the inductor.

$-$ By applying the volt$-$sec balance principle to the inductor in steady state, derive the relationship between $\backslash ($ V$\_\{\backslash$text $\{$out $\}\}\backslash )$ and $\backslash ($ V$\_\{\backslash$text $\{$in $\}\}\backslash )$ in terms of the $[15\backslash \%]$ duty cycle, $\backslash ($ D $\backslash ).$

$-$ For what values of duty cycle will the magnitude of the output

voltage be greater than the input voltage?