Design a multi$-$purpose power supply with an internal $24$V DC voltage source $($battery$).$ The output specifications are: one $5-$V DC voltage source and one $110-$V$($rms$),60-$Hz single$-$phase AC voltage source $($RL load, R $=100,$ L $=100$ mH$).$

$($a$)$ Design a complete power conversion circuit with the above functions and explain the operating principle of the designed circuit $($including the on$/$off mechanism of all power switches$).$

$($b$)$ Draw the voltage and current waveforms of the designed circuit, and describe with mathematical formulas how the designed circuit can correctly output the required AC$/$DC voltage values $($please specify the required parameter values, such as inductance, capacitance, resistance, switching frequency, etc.$).$

Hint: Consider using a Buck converter for the DC$-$DC conversion and a Resonant inverter for the DC$-$AC conversion.