Consider the circuit below:

$($a$)$ Show that the transfer function is:

H$($s$)=$

$1$

L$1$L$2$C

s$3+$

$(1$

L$1$

$+1$

L$2$

$)$

s$2+$

$($L$1+$ L$2+$ C

L$1$L$2$C

$)$

s $+2$

L$1$L$2$C

$($b$)$ Determine component values so that the circuit realizes a $3$rd order Butterworth

gain characteristic. Note: In this design, k $=1$

$2$ and the cutoff frequency is

normalized for the final transfer function.

$($c$)$ Find parameter values of a $3$rd order low pass Butterworth filter having a cutoff

frequency of $20$ krad$/$s and a resistor value of $1$ k$$