Electronics > Beginners
Notch active filter transfer function
alex.martinez:
Good evening eefolks,
I have been looking for a one-opamp notch filter and I stumped across a topology which seemed interesting to me. I found it here. They provide the equations for the natural frequency for a simplified version. I simulated it on LTspice and sure enough it was a notch filter.
But I was wondering, what was the transfer function of the filter (to check what was the expression for its Q and its natural frequency for the non-simplified version). So I took my notebook and started to use KCL and KVL...
Thing is I don't really know what is going on in node "V". I know that V+ = V-, and that the output would be the difference between the two...But again if I apply V+ = V-, I'm missing C2 for the expression. I attach you what I wrote below.
Could someone enlighten me to what is going on at node "V"?
Thanks in advance!
kulky64:
Hi, I tried to derive the transfer function and got this:
F(s)=Vo(s)/Vi(s)=(R3*R4*C1*C2*s^2+(R3*C1-R2/R1*R4*C2)*s+1)/(R3*R4*C1*C2*s^2+(R4*C2+R3*C1)*s+1)
You forgot to include two currents into your equations:
1) current flowing out of signal generator to the input of filter,
2) current flowing from opamp output into "V" node.
alex.martinez:
--- Quote from: kulky64 on April 04, 2019, 08:28:09 pm ---Hi, I tried to derive the transfer function and got this:
F(s)=Vo(s)/Vi(s)=(R3*R4*C1*C2*s^2+(R3*C1-R2/R1*R4*C2)*s+1)/(R3*R4*C1*C2*s^2+(R4*C2+R3*C1)*s+1)
You forgot to include two currents into your equations:
1) current flowing out of signal generator to the input of filter,
2) current flowing from opamp output into "V" node.
--- End quote ---
Yup, I just arrived to the TF you posted... Since I could not actually get around what was going on on "V" I just went ahead and included it in the feedback loop as I show below.
Then I just set the equation Vo = Vp - ic2·Z2 (like a voltage divider), and found ic2 = i4 and solved for Vo/Vi ... arrived to your TF! Also, found the Q factor
Q = wo/(R4·C2 + R3·C1), which matches to 1/2 for R4 = R3 and C1 = C2 ! THank you a lot!
The Electrician:
--- Quote from: alex.martinez on April 04, 2019, 10:45:58 pm ---Q = wo/(R4·C2 + R3·C1), which matches to 1/2 for R4 = R3 and C1 = C2 ! THank you a lot!
--- End quote ---
The Q you have here is the pole Q. For a notch filter, the sharpness of the notch is related to the Q of the zeroes of the transfer function.
I made all the resistors 10k except R4 which was 11k, and the capacitors .01uF. This gave me a 60 dB notch and the Q of the zeroes was about 10
alex.martinez:
--- Quote from: The Electrician on April 05, 2019, 05:43:27 pm ---
--- Quote from: alex.martinez on April 04, 2019, 10:45:58 pm ---Q = wo/(R4·C2 + R3·C1), which matches to 1/2 for R4 = R3 and C1 = C2 ! THank you a lot!
--- End quote ---
The Q you have here is the pole Q. For a notch filter, the sharpness of the notch is related to the Q of the zeroes of the transfer function.
I made all the resistors 10k except R4 which was 11k, and the capacitors .01uF. This gave me a 60 dB notch and the Q of the zeroes was about 10
--- End quote ---
What do you mean by pole and zero Q? I've never heard of it :'(
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