Sallen-Key Low-Pass Filter

What is modelled

An ideal Sallen-Key stage: two resistors in series, a capacitor to ground at the second node, a feedback capacitor from the first node to the amplifier output, and a non-inverting amplifier of gain K. It is a second-order low-pass filter: flat at low frequency, falling at 40 dB per decade above the corner, with a resonance whose height depends on K.

Equations

• Transfer function H(s) = K / [R1 R2 C1 C2 s² + (C2(R1+R2) + R1 C1 (1-K)) s + 1].
• Natural frequency ω0 = 1/√(R1 R2 C1 C2), quality factor Q = √(R1 R2 C1 C2) / [C2(R1+R2) + R1 C1(1-K)], damping ζ = 1/(2Q).
• Circuit equations used in the checks: C2 dVB/dt = (VA - VB)/R2 and C1(dVA/dt - K dVB/dt) = (Vin - VA)/R1 - (VA - VB)/R2, V_out = K V_B.

Checks run in the page

• RK4 integration of the circuit against the closed-form step response.
• Steady sine response (gain and phase) against the transfer function.
• The -3 dB corner against bisection; the resonant peak against a frequency scan.

Honest simplifications

The op amp is ideal: no finite bandwidth, slew rate, output limits or noise, and component tolerances are ignored. Above a few hundred kilohertz a real stage would behave differently. K above 2.85 would be unstable and is excluded by the slider.

• Choose Resonant peak and look at the Bode plot: the peak height matches the formula and the step response rings.
• Choose Low Q: the response is smooth, the poles move onto the real axis.
• Switch to the square wave at 3 kHz and watch the filter remove the higher harmonics.