Problem $6$: Active filters

For the circuit topology $($a multi$-$feedback bandpass filter with voltage gain$)$ shown below, we claim that the transfer function is given by:

$H=\frac{V_{\text{o}\text{u}\text{t}}}{V_{in}}=-[\frac{\frac{s}{C_2{R}_1}(1+(\frac{R_5}{R_4}))}{s^2+(\frac{C_1+C_2}{C_1{C}_2{R}_3}-(\frac{R_1+R_2}{R_1{R}_2{C}_2})(\frac{R_5}{R_4}))s+\frac{R_1+R_2}{C_1{C}_2{R}_1{R}_2{R}_3}}]$

a$)$ Derive the transfer function for this circuit. Is the claimed transfer function correct?

We build a $3$ stage filter using this circuit with the following component values:

All Stages: $,C_1=C_2=160pF$

Stage $1$:

$R_1=46\mathrm{.}8k,R_2=1\mathrm{.}02k,R_3=99\mathrm{.}5k,R_4=9\mathrm{.}8k,R_5=147$

Stage $2$:

$R_1=91\mathrm{.}9k,R_2=984,R_3=97\mathrm{.}3k,R_4=9\mathrm{.}66k,R_5=169$

Stage $3$:

$R_1=95\mathrm{.}9k,R_2=1\mathrm{.}03k,R_3=102k,R_4=9\mathrm{.}66k,R_5=169$

b$)$ Do a bode plot of the $3$ stage filter over an appropriate frequency range. What is the approximate center frequency, bandwidth, and gain?

Problem $6$: Active filters

For the circuit topology $($a multi$-$feedback bandpass filter with voltage gain$)$ shown below, we claim that the transfer function is given by:

$H=\frac{V_{\text{o}\text{u}\text{t}}}{V_{in}}=-[\frac{\frac{s}{C_2{R}_1}(1+(\frac{R_5}{R_4}))}{s^2+(\frac{C_1+C_2}{C_1{C}_2{R}_3}-(\frac{R_1+R_2}{R_1{R}_2{C}_2})(\frac{R_5}{R_4}))s+\frac{R_1+R_2}{C_1{C}_2{R}_1{R}_2{R}_3}}]$

a$)$ Derive the transfer function for this circuit. Is the claimed transfer function correct?

We build a $3$ stage filter using this circuit with the following component values:

All Stages: $,C_1=C_2=160pF$

Stage $1$:

$R_1=46\mathrm{.}8k,R_2=1\mathrm{.}02k,R_3=99\mathrm{.}5k,R_4=9\mathrm{.}8k,R_5=147$

Stage $2$:

$R_1=91\mathrm{.}9k,R_2=984,R_3=97\mathrm{.}3k,R_4=9\mathrm{.}66k,R_5=169$

Stage $3$:

$R_1=95\mathrm{.}9k,R_2=1\mathrm{.}03k,R_3=102k,R_4=9\mathrm{.}66k,R_5=169$

b$)$ Do a bode plot of the $3$ stage filter over an appropriate frequency range. What is the approximate center frequency, bandwidth, and gain?