A $25$ kW three$-$phase modular multi$-$level converter $($MMC$)$ includes half$-$bridge in each sub$-$module. The inverter has four sub$-$modules per phase. It is fed by $600$ VDC and is expected to achieve an RMS output line$-$to$-$line voltage that varies between $300$ and $420$ V RMS$.$

a$.$ From the literature, choose a modulation or switching strategy that allows this MMC to achieve the desired output voltage. Derive the mathematical representation of that voltage, or reiterate the derivation from the literature for this specific number and type of sub$-$modules.

b$.$ Simulate this inverter with a resistive load at full power, and verify that the switching or modulation strategy you selected achieves the correct RMS values across the desired output voltage range. Note that load power is at $25$ kW for the $420$ V RMS output case, and the power will drop for the $300$ V RMS case.