A boost converter shown in Figure $3$ has an input voltage of $V_g=24V$ and peak value of the $[40]$ sawtooth waveform in pulse width modulator is $V_m=3V.$ The reference voltage is set to $V_{\text{r}\text{e}\text{f}}=4V$ and sensor gain is $H(s)=\frac{1}{10}.$ Assume inductor current is continuous.

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Figure $3$ A boost converter for a regulated output voltage.

$($a$)$ The compensator is constructed around an operational amplifier as illustrated in Figure $2$ where $R_1=3\mathrm{.}2k,C_1=0\mathrm{.}1F,C_2=2\mathrm{.}5nF,$ and $R_{s1}=32k.$ Determine the value of $R_{s2}.$ Derive the transfer function $G_c(s)$ of this compensator and draw the Bode plot of magnitude and phase response.

$($b$)$ Calculate the steady$-$state values of control voltage $v_c(t)$ and output voltage $V.$