You can use Matlab for generating the Bode and Nyquist plot for this homework.

$(1)$ Consider the buck converter circuit below. $.$ The component values are: $R1=9k,R2=1k,RL=10,L=0\mathrm{.}1mH,$ and $C=100uF.$ The peak sawtooth wave in the PWM generator is $V_{\text{r}\text{a}\text{m}\text{p}}=5v.$ The input voltage $V_{in}$ is $12$ v $.$ The objective of the circuit is to produce an output that is as close to $5$ v as possible.

$(1.$a$)$ Sketch a block diagram for the circuit $($assume $R3=C3=0)$ and determine the value of the feedforward term. $(5$ points$)$

$(1.$b$)$ Determine the value of Kp so that the circuit's phase margin is $30($assume $R3=C3=0)(5$points$)$

$(1.$c$)$ Using the kp value found in $(4.$b$),$ what is the value of Vref so that the output voltage is $5$ v $.(5$ points$)$

$(4.$d$)$ Using the kp value found in $(4.$b$),$ find the value of R$3$ and C$3$ so that the phase margin is about $60$ degrees. $(10$ points$).$

$11/24/2024$

$1$

$(2)$ Design a compensator to stabilize the following system. The compensator must be $1^{st}$ order and you should try to get at least $30$ degree phase margin. The DC gain from input to output should be within $5\%$ from unity. What is the DC disturbance rejection ratio based on your design? $(25$ points$)$

$H(s)=\frac{1}{s^2-1}.$