a$)$ A $20$ resistive load requires an average current that varies from $4\mathrm{.}5$A to $8\mathrm{.}0$A$.$ An isolation transformer is placed between a $120$V rms $60$Hz AC source and a controlled single$-$phase full$-$wave rectifier. Design a circuit to meet the current requirements. Specify the transformer turns$-$ratio and the range of delay angle. b$)$ Show that the power factor for the controlled rectifier with a resistive load is given by: