With reference to the circuit in Fig. $2,$ denote the impedances of the branches in sinusoidal steady state as $Z1(),Z2(),Z3()$ and $Z4().$

$($a$)$ calculate an analytical expression $($a formula with symbols only$)$ for the voltage $20\%$ transfer function $H()=$Vout$()Nin()$ in terms of $Z1,Z2,23$ and $Z4$;

$($b$)$ calculate an analytical expression $($a formula with symbols only$)$ for the $20\%$ transfer function $H()$ in terms of R$1,$ R$2,$ C$3,$ C$4$ and $$;

$($c$)$ if $R1=R2=10k$ and $C3=C4=1nF,$ calculate the frequency $f_0($in Hz$)$ at $30\%$ which the phase of $H()$ is $-90$; and

$($d$)$ with the same parameter values as in part $($c$),$ write the steady state output $30\%$ voltage Vout$($t$)$ as a function of time when Vin$(t)=$Acos$(f{l}_{t}t),$ where $A=1V.$

Fig. $2$