Free Twin-T Active Notch Filter Tool

1. Physics of the Twin-T Notch Filter

The classic Twin-T (parallel-T) network consists of two parallel RC tee branches connected between input and output:

• Low-Pass Tee (R-R-C): Two series resistors $R$ with a shunt capacitor of value $2C$ tied to AC ground. At low frequencies, signals pass freely through the resistors; at high frequencies, the shunt capacitor bypasses signal to ground.
• High-Pass Tee (C-C-R): Two series capacitors $C$ with a shunt resistor of value $R/2$ tied to AC ground. At high frequencies, signals pass cleanly through the capacitors; at low frequencies, the shunt resistor shorts signals to ground.

At the exact resonant center frequency $f_0 = rac{1}{2pi R C}$, the phase shift through the low-pass tee reaches $+90^circ$, while the phase shift through the high-pass tee reaches $-90^circ$. The two equal-amplitude outputs are exactly $180^circ$ out of phase, canceling each other out completely and creating a deep null (infinite theoretical notch depth).

2. Passive $Q = 0.25$ vs Active Bootstrapping

In a purely passive Twin-T with shunt elements grounded, the quality factor is mathematically fixed at: $$Q_{passive} = rac{1}{4} = 0.25$$ A $Q$ of $0.25$ produces an extremely wide notch: for a $60 ext{ Hz}$ filter, the $-3 ext{dB}$ bandwidth spans $BW = 60 / 0.25 = 240 ext{ Hz}$! This devastates legitimate signals between $20 ext{ Hz}$ and $200 ext{ Hz}$ (guitar bass, kick drums, ECG P-Q-R-S waves).

By connecting the node between $2C$ and $R/2$ not to ground, but to a fraction $K$ of the output via a low-impedance op-amp voltage follower, positive feedback sharpens the notch dramatically: $$Q_{active} = rac{1}{4(1 - K)}$$ Setting $K = 0.95$ boosts $Q$ to $5.0$, narrowing the $-3 ext{dB}$ notch bandwidth to just $12 ext{ Hz}$ ($54 ext{ Hz}$ to $66 ext{ Hz}$), rejecting hum while keeping adjacent frequencies intact.

3. Component Matching & Practical Notch Depth

The depth of the notch in a physical circuit is strictly limited by component tolerances. A $1%$ resistor and capacitor mismatch shifts the cancellation vector, degrading theoretical infinite attenuation to roughly $-40 ext{ dB}$ ($99%$ hum reduction). With $0.1%$ matched parts or trimming potentiometers, notch depths exceeding $-60 ext{ dB}$ ($99.9%$ rejection) are attainable.