Sallen Key Lowpass⚠ unverified
Electrical / Filters · Compute the cutoff frequency of a Sallen-Key low-pass filter
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| R1 | R1 | — | 1.0 | First resistance |
| R2 | R2 | — | 1.0 | Second resistance |
| C1 | C1 | F | 1.0 | First capacitance |
| C2 | C2 | F | 1.0 | Second capacitance |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | fc | Hz | Cutoff frequency, in hertz (Hz) |
The science & history
Understanding the Parameters
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$R_1 R_2 C_1 C_2$ — the cutoff depends on the geometric mean of the two RC products; it is the two‑pole generalisation of the single‑section $1/(2\pi RC)$.
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Component ratios (not captured by $f_c$ alone) set the quality factor $Q$, which controls passband peaking and how sharply the response turns over (see Q Factor).
Derivation (Approaching a Proof)
The unity‑gain Sallen–Key places the op‑amp as a buffer around an RC ladder with positive feedback through $C_1$. Writing node equations and using the ideal‑op‑amp buffer ($V_{out} = V_+$) yields a standard second‑order low‑pass transfer function
$$H(s) = \frac{\omega_0^2}{s^2 + \dfrac{\omega_0}{Q}s + \omega_0^2}, \qquad \omega_0 = \frac{1}{\sqrt{R_1 R_2 C_1 C_2}}.$$
The natural frequency $\omega_0$ (equivalently $f_c = \omega_0/2\pi$) sets where the response turns down, while $Q$ (a separate function of the component ratios) sets the peaking. Converting $\omega_0$ to hertz,
$$f_c = \frac{1}{2\pi\sqrt{R_1 R_2 C_1 C_2}}.$$
The op‑amp buffer is what lets the two RC sections interact to make a true second‑order response without loading each other — something two cascaded passive RC stages cannot do cleanly.
History
R. P. Sallen and E. L. Key published this topology in 1955 at MIT Lincoln Laboratory ("A practical method of designing RC active filters"). It became the workhorse second‑order building block for active‑RC filters — Butterworth, Chebyshev, and Bessel responses are realised by cascading Sallen–Key stages with the right $f_c$ and $Q$.
Related Concepts: Active Filter Gain, Butterworth Cutoff, Filter Order Rolloff, Q Factor, RC Low-Pass Cutoff
Notes: Registry calculator sallen-key-lowpass (unverified; resistors mislabelled dimensionless —
should be ohms). Unity‑gain form; the same $f_c$ formula holds, with $Q$ set separately by ratios.