# Twin T Notch Filter calculations

The diagram below shows a simple Twin T notch filter.

(1) I1 = (Vin-V1)/2R

(2) I2 = (V1-AV2)2jπf2C = (V1-AV2)4jπfC     (AV2 is the output voltage of opamp U1B, and can be adjusted by R5)

(3) I3 = (V1-V2)/2R = (V2-V3)2jπfC    (I3 flows through R2 and C4, since the input resistance of the opamp is infinite)

(4) I4 = (Vin-V3)2jπfC

(5) I5 = (V3-AV2)/R

I2+I3=I1 => (V1-AV2)4jπfC + (V1-V2)/2R = (Vin-V1)/2R
Multiplying everything with 2R gives: (V1-AV2)4jπfC2R + V1-V2 = Vin-V1 =>

(6) (V1-AV2)8jπfRC = Vin-2V1+V2

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

(7) V1=(V2-V3)4jπfRC+V2

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

(8) (1+4jπfRC)(V2-V3)8jπfRC + (1-A)8jπfRCV2 = V1-V2

I3+I4=I5 => (V2-V3)2jπfC + (Vin-V3)2jπfC = (V3-AV2)/R => (V2-Vin-2V3)2jπfC = (V3-AV2)/R =>
(V3-AV2)/2jπfRC = V2-Vin-2V3 => (2 + 1/2jπfRC)V3 = V2+Vin+(AV2)/2jπfRC =>
([4jπfRC+1]/2jπfRC)V3 = V2+Vin+(AV2)/2jπfRC => V3 = (V22jπfRC + Vin2jπfRC + AV2)/(1+4jπfRC) =>

(9) V3 = (Vin2jπfRC + V2[2jπfRC+A])/(1+4jπfRC)

=> V3-V2 = (Vin2jπfRC + V2[2jπfRC+A - 1 - 4jπfRC])/(1+4jπfRC) = (Vin2jπfRC + V2[A - 1 - 2jπfRC])/(1+4jπfRC) =>

(10) V2-V3 = -(Vin2jπfRC + V2[A - 1 - 2jπfRC])/(1+4jπfRC)

Substituting (10) in (8) gives: -[Vin2jπfRC + V2(A-1-2jπfRC)]8jπfRC = -(1-A)8jπfRCV2 + Vin - V2 =>
[Vin2jπfRC + V2(A-1-2jπfRC)]8jπfRC = (1-A)8jπfRCV2 - Vin + V2
Deviding everything by 8jπfRC gives: Vin2jπfRC + V2(A-1-2jπfRC) = (1-A)V2 + (Vin-V2)/8jπfRC =>
V2(A-1-2jπfRC) + V2(A-1) - V2/8jπfRC = -Vin2jπfRC - Vin/8jπfRC =>

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(jπfRC)2 + 1 = 0 => 16(jπfRC)2 = -1 => j2(4πfRC)2 = -1 =>
-(4πfRC)2 = -1 => f=1/(4πRC)

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.