Given the following circuit,

Determine the transfer function between Vout and Vs when Vs$($t$)=11$ u$($t$)($u$($t$)$ is unit step response$)$ V$,$ R$1=2$ ohms, R$2=4$ ohms, R$3=6$ ohms, L$=1$H$,$ and C$=0\mathrm{.}5$F$.(20$ pts$)$

Determine poles, zeros, $2^("$nd $")$ order system type $($which case$)$ and the stability of the system. $(15$ pts$)$

If case $3,$ determine the damping ratio and natural frequency of the system. $(5$ pts$)$

Derive $($set up the equations for magnitude and phase response$)$ and plot the frequency response using the asymptotic approach. $(20$ pts$)$

How would you characterize the system behavior $($in terms of frequency response magnitude plot, i$.$e$.$ lowpass, $.....)(10$ pts$)$

The system needs to be redesigned and speeded up$.$ The speed$-$up has to be $10$ times faster than the current system. How would you change the system to meet this new design requirement? $(15$ pts$)$

What is the effect of the redesign on the frequency response? Explain. $(15$ pts$)$