Butterworth Cutoff⚠ unverified
Electrical / Filters · Compute the scaled cutoff frequency for a Butterworth filter
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| n | n | — | 1.0 | Filter order (number of poles) |
| fc | fc | Hz | 1.0 | Nominal cutoff frequency |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | f | Hz | Scaled cutoff frequency, in hertz (Hz) |
The science & history
Understanding the Parameters
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$n$ — how many identical stages are in series; more stages steepen the roll‑off ($-20n\,$dB/decade, see Filter Order Rolloff) but pull the combined corner further in.
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$f_c$ — the corner of a single section (e.g. an RC Low-Pass Cutoff).
Derivation (Approaching a Proof)
Each identical first‑order low‑pass section has magnitude‑squared $\dfrac{1}{1 + (f/f_c)^2}$. Cascading $n$ of them (with buffering, so they don't load each other) multiplies the responses:
$$|H|^2 = \left(\frac{1}{1 + (f/f_c)^2}\right)^{n}.$$
The overall $-3\,$dB point is where $|H|^2 = \tfrac12$:
$$\big(1 + (f/f_c)^2\big)^{n} = 2 \;\Rightarrow\; 1 + (f/f_c)^2 = 2^{1/n} \;\Rightarrow\; f = f_c\sqrt{2^{1/n} - 1}.$$
Relation to a true Butterworth filter. A genuine $n$‑th‑order Butterworth is maximally flat: its poles lie equally spaced on a circle so that $|H|^2 = 1/\big(1 + (f/f_c)^{2n}\big)$ — flatter in the passband than a cascade of identical RC stages. The formula here is the cascaded‑identical‑section approximation, useful for quickly de‑rating a stack of simple stages.
History
Stephen Butterworth introduced the maximally‑flat response in his 1930 paper "On the theory of filter amplifiers," seeking a passband as flat as possible at the cost of a gentler transition than Chebyshev designs. His name now labels the flattest‑passband standard response used throughout filter design.
Related Concepts: Filter Order Rolloff, RC Low-Pass Cutoff, Sallen Key Lowpass, Lowpass Passive Cutoff
Notes: Registry calculator butterworth-cutoff (unverified). Computes the cascaded‑identical‑section
$-3\,$dB point, not the poles of a true maximally‑flat Butterworth.