The figure below can be used to represent a single$-$phase switch$-$mode AC$-$DC converter. Assume that the grid voltage is $220\frac{V}{60}Hz,$ the coupling inductor is $8$ mH $,$ and the load resistance is $25.$

A$)$ If the AC$-$DC converter, including the coupling inductor, operates with $S=5$ kVA and displacement power factor $($DPF$)$ of $0\mathrm{.}85$ leading, Compute the average value of the voltage across the load. Hint: Recall that the active power in the AC side of the converter is reflected to the DC side, but not the reactive power.

B$)$ Draw the phasor diagram showing the following waveforms: $V_s,I_{s1},V_{\text{c}\text{o}\text{n}\text{v}1},V_{xL1}.$

C$)$ Compute the DC side capacitor so that the $2^{nd}$ order voltage ripple across the load is limited to $15$ V peak. Consider that the DC side current harmonic flows entirely through the capacitor. Note: For the calculation of the DC side current ripple, consider the voltage and current waveforms in the AC side of the converter.

D$)$ Assuming that the converter operates with unipolar SPWM and $m_a=1,$ compute the frequency of the triangular carrier so that the magnitude of the dominant harmonic in the current drawn from the grid is $5\%$ of the fundamental component.

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