For circuit shown in Fig. E$2$t$-1,$ where the input voltage is $\backslash ($ v$\_\{$i$\}($t$)\backslash ).$ The output of the system is the voltage across the capacitor $\backslash ($ v$\_\{$C$\}($t$)\backslash ).$

a$)[5$ pts$]$ Define the input$-$output relationship for this circuit using a first order differential equation in the time domain. $[$ECE$201$ knowledge!$]$

b$)[5$ pts$]$ Find the transfer function, $\backslash (\backslash$mathrm$\{$H$\}(\backslash$mathrm$\{$s$\})\backslash )$ for this system.

c$)[10$ pts$]$ Use the following MATLAB commands for generating the magnitude and phase response of the system $(\backslash ($ s$=$j $\backslash$omega $\backslash )).$ Plot the magnitude and the phase plots and determine what type of filter this system corresponds to$.$

$```$

$>>$w $=0$:$1$:$100($you do not need to adopt the $[0,100]$ range here$)$; $\%$

choose the range of the frequency properly!

$>>$H$=\%$ plug in the expression for your H$($j$\backslash$omega$)$;

$>>$magH$=$abs$($H$)$;

$```$

angH$=$angle $($H$)*180/$pi;

d$)$ Apply the sawtooth waveform shown in Textbook Example $17\mathrm{.}1($Fig$.17\mathrm{.}2)$ as the input

voltage v$\_($i$)($t$),$ also shown here as Fig. E$2$t$-2.$ Calculate by hand the output voltage across the capacitor

first $[5$ pts$]$ and plot the v$\_($C$)($t$)$ using MATLAB $[5$ pts$].$ You can refer to Example $17\mathrm{.}1$ on the calculation of

F$.$S$.$

Figure. E$2$t$-2($Fig$.17\mathrm{.}2$ in Textbook$)$

e$)[5$ pts bonus$]$ Use PSPICE or LTspice to find the output voltage by given the input voltage in

Problem d$).$

Please answer all problems with all steps! Will upvote!