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• • • www.Hobby-Electronics.info Twin T Notch Filter calculations

The diagram below shows a simple Twin T notch filter. 1. I1 = (Vin-V1)/2R
2. I2 = (V1-AV2)2jpf2C = (V1-AV2)4jpfC     (AV2 is the output voltage of opamp U1B, and can be adjusted by R5)
3. I3 = (V1-V2)/2R = (V2-V3)2jpfC    (I3 flows through R2 and C4, since the input resistance of the opamp is infinite)
4. I4 = (Vin-V3)2jpfC
5. I5 = (V3-AV2)/R

I2+I3=I1 => (V1-AV2)4jpfC + (V1-V2)/2R = (Vin-V1)/2R
Multiplying everything with 2R gives: (V1-AV2)4jpfC2R + V1-V2 = Vin-V1 =>

1. (V1-AV2)8jpfRC = Vin-2V1+V2

Equation 3. gives: V1-V2=(V2-V3)4jpfRC =>

1. V1=(V2-V3)4jpfRC+V2

Substituting 7. in 6. gives: [(V2-V3)4jpfRC + (1-A)V2]8jpfRC = Vin-V2-(V2-V3)8jpfRC+V2 =>

1. (1+4jpfRC)(V2-V3)8jpfRC + (1-A)8jpfRCV2 = V1-V2

I3+I4=I5 => (V2-V3)2jpfC + (Vin-V3)2jpfC = (V3-AV2)/R => (V2-Vin-2V3)2jpfC = (V3-AV2)/R =>
(V3-AV2)/2jpfRC = V2-Vin-2V3 => (2 + 1/2jpfRC)V3 = V2+Vin+(AV2)/2jpfRC =>
([4jpfRC+1]/2jpfRC)V3 = V2+Vin+(AV2)/2jpfRC => V3 = (V22jpfRC + Vin2jpfRC + AV2)/(1+4jpfRC) =>

1. V3 = (Vin2jpfRC + V2[2jpfRC+A])/(1+4jpfRC)

=> V3-V2 = (Vin2jpfRC + V2[2jpfRC+A - 1 - 4jpfRC])/(1+4jpfRC) = (Vin2jpfRC + V2[A - 1 - 2jpfRC])/(1+4jpfRC) =>

1. V2-V3 = -(Vin2jpfRC + V2[A - 1 - 2jpfRC])/(1+4jpfRC)

Substituting 10. in 8. gives: -[Vin2jpfRC + V2(A-1-2jpfRC)]8jpfRC = -(1-A)8jpfRCV2 + Vin - V2 =>
[Vin2jpfRC + V2(A-1-2jpfRC)]8jpfRC = (1-A)8jpfRCV2 - Vin + V2
Deviding everything by 8jpfRC gives: Vin2jpfRC + V2(A-1-2jpfRC) = (1-A)V2 + (Vin-V2)/8jpfRC =>
V2(A-1-2jpfRC) + V2(A-1) - V2/8jpfRC = -Vin2jpfRC - Vin/8jpfRC => And since opamp U1A is a unity gain amplifier, Vout = V2. So the equation above also describes the output response.
Vout/Vin will be 0 if the nominator becomes 0. So 16(jpfRC)2 + 1 = 0 => 16(jpfRC)2 = -1 => j2(4pfRC)2 = -1 =>
-(4pfRC)2 = -1 => f=1/(4pRC)

The Vout/Vin equation also shows that the width of the notch is determined by A. If A=1, the notch becomes so narrow that Vout/Vin=1. The notch becomes wider as A decreases.

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