# Transfer Function of Active Notch Filter

I have rigged up a notch filter from here. See below: -

Its the first circuit. Now I am trying to calculate its transfer function. I have calculated two equations, but I need the third one:

1. (Vin-V2)/(1/SC1) = V2/R3
2. (Vin-V1)/R1 = (V1-Vo)/(R2+(1/SC2))

V1 and V2 are the potentials at the input of the op-Amp.

• As long as the opamp works in its linear region you can set V1=V2 (because the open-loop gain is very large).
– LvW
Jun 10 '16 at 7:20
• I didn`t check your two equations - however, did you forget the R4 path?
– LvW
Jun 10 '16 at 10:01
• This filter doesn't work. So now I need to design a new filter. Jun 11 '16 at 10:05
• Interesting circuit, without R4 and C2 this circuit look just like a ordinary all pass filter.
– G36
Jun 12 '16 at 12:01

Assuming an ideal op-amp, you have $v_1=v_2$ (as pointed out in a comment by LvW). Also note that your second equation is wrong because there is a current into or out of the output of the op-amp, so you can't just add up the impedances of $R_2$ and $C_2$.

Introducing another unknown voltage $v_x$ at the output of the op-amp, you can write down three equations:

$$(v_i-v_1)sC_1-v_1/R_3=0\\ (v_i-v_1)/R_1+(v_x-v_1)/R_2=0\\ (v_i-v_o)/R_4-(v_o-v_x)sC_2=0$$

where $v_i$ and $v_o$ are the input and output voltages, respectively, and $v_1$ is the voltage at both inputs of the op-amp. These equations can be solved for the transfer function $H(s)$, i.e., for the ratio $v_o/v_i$. With $C=C_1=C_2$ and $R=R_3=R_4$ and $R_1=R_2$ you get

$$H(s)=R\frac{s^2+\frac{1}{R^2C^2}}{s^2+s\frac{2}{RC}+\frac{1}{R^2C^2}}\tag{1}$$

From the numerator of $(1)$ you immediately get for the notch frequency

$$\omega_0=\frac{1}{RC}\tag{2}$$

Note that as long as you choose $R_1=R_2$, the actual value of these resistors doesn't show up in the transfer function.