A low-pass filter passes signals below its cutoff frequency f_c and progressively attenuates higher frequencies. The cutoff is conventionally the −3 dB point — the frequency at which the output amplitude has fallen to 1/√2 (≈ 0.707) of the input, i.e. half the power. Below f_c the filter is "passband"; above it is "stopband"; the transition between them depends on filter order.
RC and RL — 1st-order filters
A single resistor + capacitor (RC) or resistor + inductor (RL) makes a 1st-order low-pass with f_c = 1 / (2π·τ), where τ is the RC product (R·C) or L/R time constant. Above f_c the magnitude rolls off at −20 dB/decade (-6 dB/octave) — a relatively gentle slope. Phase shifts from 0° at DC to −90° at high frequency, passing through −45° exactly at f_c. RC and RL have identical mathematical responses; the choice between them depends on impedance and component-cost trade-offs.
Sallen-Key — 2nd-order active filters
The Sallen-Key topology uses an op-amp plus two R-C pairs to make a 2nd-order low-pass with −40 dB/decade rolloff (twice as steep) and a tuneable Q factor. Q controls the shape near f_c:
- Q = 0.5 (critically damped) — slowest, no overshoot in step response, slight passband sag.
- Q = 0.577 (Bessel) — maximally-flat group delay. Best for audio / pulse fidelity. Soft transition.
- Q = 0.707 (Butterworth) — maximally-flat passband magnitude. The standard general-purpose choice.
- Q ≥ 1 (Chebyshev) — sharper rolloff at the cost of passband ripple AND step-response ringing.
Component selection in practice
Resistor values typically range from 1 kΩ to 100 kΩ — too low loads the source; too high adds Johnson noise and is sensitive to op-amp bias currents. Capacitor values are then determined by f_c. Standard E12 or E24 capacitor values (1.0, 1.2, 1.5, 1.8, 2.2, 2.7, ..., 8.2, 10) × decade are easiest to source. Inductors are bulkier and less ideal than capacitors at audio frequencies, so RC is far more common than RL for audio work.
Step response — what it tells you
The step response shows how the filter reacts to an instantaneous voltage change at its input. A 1st-order RC filter's step response is a simple exponential rise to the final value, reaching 63.2% at one time constant. A 2nd-order Q=0.707 (Butterworth) filter shows a modest 4% overshoot before settling. As Q increases past 1, ringing becomes pronounced — useful when "snap" is desired (e.g., gunshot-edge anti-aliasing) but undesirable for most audio.
Should I use RC or RL?
RC for almost everything. Capacitors are smaller, cheaper, closer to ideal, and don't pick up magnetic noise the way inductors do. RL low-pass shows up in (a) power-supply filtering where you want low DC resistance and high HF impedance, (b) crossover networks for loudspeakers where the inductor is also doing useful current handling, (c) RF circuits where parasitic capacitances of resistors at high frequency make RC harder to control. For everything else, use RC.
Why is the cutoff at −3 dB and not −6 dB or 0 dB?
By convention. The −3 dB point is the "half-power" frequency: power is amplitude squared, and 10·log₁₀(0.5) = −3.01 dB. It's a natural boundary because it's exactly where the RC pole's magnitude is 1/√2 — that is, where the reactive impedance equals the resistive impedance. Older AM radio and telephone-grade systems sometimes used −6 dB (half-amplitude) instead, but modern signal-processing literature universally uses −3 dB.
Why does the Sallen-Key need an op-amp?
The 2nd-order response comes from two cascaded RC stages, but if you just connect two RC sections directly the second loads the first and you don't get the sharp 2nd-order rolloff. The op-amp provides buffering between the two stages (it's wired as a non-inverting amplifier with gain K = 1 + Rb/Ra) and also provides the positive feedback path that creates the resonant Q. Without the op-amp you'd need a much more complex passive LC ladder to get the same response.
What's the difference between Butterworth, Bessel and Chebyshev?
All three are 2nd-order low-pass filters with the same −40 dB/decade rolloff — they differ in how they transition. Butterworth (Q=0.707) has the flattest possible passband and a smooth knee. Bessel (Q=0.577) sacrifices passband flatness for the most-linear phase response → least step-response overshoot, best for audio waveforms. Chebyshev (Q≥1) gives a sharper transition at the cost of ripple in the passband and a more pronounced step-response overshoot. For analog audio: Butterworth or Bessel. For sharp anti-aliasing where some passband ripple is acceptable: Chebyshev.
Can I cascade two 1st-order filters to make a 2nd-order?
Yes, but only with buffering between them (e.g., a unity-gain op-amp follower). Without buffering, the second stage loads the first and the response isn't a true product of two 1st-order responses — it's worse. A cascaded buffered pair of RC stages with same f_c has Q = 0.5 (critically damped, no overshoot). To get higher Q you need positive feedback like Sallen-Key uses. Cascaded 1st-orders are "Linkwitz-Riley" when used in audio crossovers — Q = 0.5, no peaking, allows 2 sections to sum flat.
What if I need a steeper rolloff than 40 dB/decade?
Cascade more sections. Two Sallen-Key stages give 80 dB/decade (4th order), three give 120 dB/decade (6th order). For Butterworth 4th order, the two stages need different Q values (~0.541 and 1.307) to combine flatly — the values for these stage Qs are tabulated in any filter design handbook. For the steepest possible rolloff in a small package, consider switched-capacitor filter ICs (e.g., MAX291, LTC1064) which offer 8th-order Butterworth or Chebyshev in a single chip.
What about digital filters?
For audio in the digital domain, IIR (biquad) and FIR filters can implement any response shape with arbitrary precision and no component tolerances. The math here transfers directly: a digital biquad implementing an analog low-pass uses the bilinear transform of the s-domain transfer function, with the same Q and f_c parameters. The Frequency Detector and other audio analysis tools on this site use digital filtering internally. For analog hardware design (the focus of this calculator), the RC/RL/Sallen-Key topologies remain the dominant approach.