$1.$ Derive the transfer function of the circuit given above, assuming $\backslash (\backslash$mathrm$\{$I$\}(\backslash$mathrm$\{$s$\})\backslash )$ and $\backslash (\backslash$mathrm$\{$V$\}(\backslash$mathrm$\{$s$\})\backslash )$ are the excitation and the response, respectively.

This system can be categorized as follows:

i$.$ if the transfer function has two real poles, the system is said to be overdamped

ii$.$ if it has a double pole, the system is said to be critically damped

iii. and for the third case, the system is said to be underdamped.

$2.$ Derive conditions in terms of $\backslash ($ R$,$ L $\backslash ),$ and $\backslash ($ C $\backslash )$ for the poles of the transfer function to be two real poles, a double pole, or complex poles.

$3.$ Characterize the poles when $\backslash ($ R $\backslash$rightarrow $\backslash$infty $\backslash ).$

$4.$ Assume that $\backslash ($ L$=0\mathrm{.}1\backslash$mathrm$\{$H$\}\backslash )$ and $\backslash ($ C$=0\mathrm{.}1\backslash$mathrm$\{$~F$\}\backslash ).$ Choose three different values for $\backslash ($ R $\backslash )$ so that the system is overdamped, critically damped, or underdamped.

$5.$ Write the transfer function for each $\backslash ($ R $\backslash )$ value that you chose in part $4$ and for $\backslash ($ R $\backslash$rightarrow $\backslash$infty $\backslash ).$ Then calculate the impulse response and the step response of the svstem for each $\backslash ($ R $\backslash )$ value.