0
\$\begingroup\$

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

enter image description here

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.

\$\endgroup\$
4
  • 1
    \$\begingroup\$ As long as the opamp works in its linear region you can set V1=V2 (because the open-loop gain is very large). \$\endgroup\$
    – LvW
    Jun 10 '16 at 7:20
  • \$\begingroup\$ I didn`t check your two equations - however, did you forget the R4 path? \$\endgroup\$
    – LvW
    Jun 10 '16 at 10:01
  • \$\begingroup\$ This filter doesn't work. So now I need to design a new filter. \$\endgroup\$ Jun 11 '16 at 10:05
  • \$\begingroup\$ Interesting circuit, without R4 and C2 this circuit look just like a ordinary all pass filter. \$\endgroup\$
    – G36
    Jun 12 '16 at 12:01
1
\$\begingroup\$

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.

\$\endgroup\$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.