# Twin T Notch Filter calculations

The diagram below shows a simple Twin T notch filter.

(1) I_{1} = (V_{in}-V_{1})/2R

(2) I_{2} = (V_{1}-AV_{2})2jπf2C = (V_{1}-AV_{2})4jπfC (AV_{2} is the output voltage of opamp U1B, and can be adjusted by R5)

(3) I_{3} = (V_{1}-V_{2})/2R = (V_{2}-V_{3})2jπfC (I_{3} flows through R2 and C4, since the input resistance of the opamp is infinite)

(4) I_{4} = (V_{in}-V_{3})2jπfC

(5) I_{5} = (V_{3}-AV_{2})/R

I_{2}+I_{3}=I_{1} => (V_{1}-AV_{2})4jπfC + (V_{1}-V_{2})/2R = (V_{in}-V_{1})/2R

Multiplying everything with 2R gives: (V_{1}-AV_{2})4jπfC2R + V_{1}-V_{2} = V_{in}-V_{1} =>

(6) (V_{1}-AV_{2})8jπfRC = V_{in}-2V_{1}+V_{2}

Equation (3) gives: V_{1}-V_{2}=(V_{2}-V_{3})4jπfRC =>

(7) V_{1}=(V_{2}-V_{3})4jπfRC+V_{2}

Substituting (7) in (6) gives: [(V_{2}-V_{3})4jπfRC + (1-A)V_{2}]8jπfRC = V_{in}-V_{2}-(V_{2}-V_{3})8jπfRC+V_{2} =>

(8) (1+4jπfRC)(V_{2}-V_{3})8jπfRC + (1-A)8jπfRCV_{2} = V_{1}-V_{2}

I_{3}+I_{4}=I_{5} => (V_{2}-V_{3})2jπfC + (V_{in}-V_{3})2jπfC = (V_{3}-AV_{2})/R => (V_{2}-V_{in}-2V_{3})2jπfC = (V_{3}-AV_{2})/R =>

(V_{3}-AV_{2})/2jπfRC = V_{2}-V_{in}-2V_{3} => (2 + 1/2jπfRC)V_{3} = V_{2}+V_{in}+(AV_{2})/2jπfRC =>

([4jπfRC+1]/2jπfRC)V_{3} = V_{2}+V_{in}+(AV_{2})/2jπfRC => V_{3} = (V_{2}2jπfRC + V_{in}2jπfRC + AV_{2})/(1+4jπfRC) =>

(9) V_{3} = (V_{in}2jπfRC + V_{2}[2jπfRC+A])/(1+4jπfRC)

=> V_{3}-V_{2} = (V_{in}2jπfRC + V_{2}[2jπfRC+A - 1 - 4jπfRC])/(1+4jπfRC) = (V_{in}2jπfRC + V_{2}[A - 1 - 2jπfRC])/(1+4jπfRC) =>

(10) V_{2}-V_{3} = -(V_{in}2jπfRC + V_{2}[A - 1 - 2jπfRC])/(1+4jπfRC)

Substituting (10) in (8) gives: -[V_{in}2jπfRC + V_{2}(A-1-2jπfRC)]8jπfRC = -(1-A)8jπfRCV_{2} + V_{in} - V_{2} =>

[V_{in}2jπfRC + V_{2}(A-1-2jπfRC)]8jπfRC = (1-A)8jπfRCV_{2} - V_{in} + V_{2}

Deviding everything by 8jπfRC gives: V_{in}2jπfRC + V_{2}(A-1-2jπfRC) = (1-A)V_{2} + (V_{in}-V_{2})/8jπfRC =>

V_{2}(A-1-2jπfRC) + V_{2}(A-1) - V_{2}/8jπfRC = -V_{in}2jπfRC - V_{in}/8jπfRC =>

And since opamp U1A is a unity gain amplifier, V_{out} = V_{2}. So the equation above also describes the output response.

V_{out}/V_{in} will be 0 if the nominator becomes 0. So 16(jπfRC)^{2} + 1 = 0 => 16(jπfRC)^{2} = -1 => j^{2}(4πfRC)^{2} = -1 =>

-(4πfRC)^{2} = -1 => f=1/(4πRC)

The V_{out}/V_{in} equation also shows that the width of the notch is determined by A. If A=1, the notch becomes so narrow that V_{out}/V_{in}=1. The notch becomes wider as A decreases.