Sallen-Key Active Filter Calculator
Cutoff frequency and Q of a unity-gain Sallen-Key 2nd-order low-pass filter, with a live peaking-aware response chart.
R1,R2,C1,C2 → fc & Q
Design (Equal-R Butterworth)
fc = 1/(2π√(R1R2C1C2)) • Q = √(R1R2C1C2) / (C2(R1+R2))
10k/20k, 10nF/5nF (general)
10k/10k, 20nF/10nF (Butterworth)
10k/10k, 100nF/10nF (peaking)
R1=R2=R (chosen) • C2 = 1/(2πRfc√2) • C1 = 2C2 (gives Q=0.7071, maximally flat)
1kHz cutoff, R=10kΩ
3kHz cutoff, R=4.7kΩ
What a Sallen-Key Filter Is, and Why Q Matters So Much
A Sallen-Key filter is a widely-used active (op-amp based) second-order filter topology: two resistors and two capacitors arranged with a single op-amp (wired as a unity-gain buffer in the most common version) to realize a clean, two-pole low-pass response without needing an inductor — useful because inductors are bulky, lossy, and hard to make precisely at audio frequencies. The two defining numbers of any second-order filter are its cutoff frequency fc (where the response is down roughly 3 dB, for the Butterworth case) and its Q factor, which controls the shape of the response right around fc.
Why Q changes the whole character of the filter
- Q = 0.5: the two poles are real (not complex), giving an over-damped, sluggish roll-off with no peaking at all — rarely the design target on its own.
- Q = 0.5774 (1/√3): a Bessel response — slightly softer roll-off than Butterworth, but the best possible step response (least overshoot/ringing) — preferred for pulse and square-wave signals.
- Q = 0.7071 (1/√2): a Butterworth response — "maximally flat" in the passband with no peaking, exactly −3.01 dB at fc, and the most common general-purpose target.
- Q > 0.7071: the response starts to peak above 0 dB near fc before rolling off — useful in some equalizer/resonance designs, but usually undesirable in a general anti-aliasing or smoothing filter since it emphasizes frequencies near fc instead of passing them flat.
| Quantity | Formula |
|---|
| Cutoff frequency | fc = 1/(2π√(R1R2C1C2)) |
| Q factor (unity-gain) | Q = √(R1R2C1C2) / (C2(R1+R2)) |
| Equal-R Butterworth design (R1=R2=R) | C1=2C2, giving Q=1/√2 automatically |
| Roll-off rate | −40 dB/decade (2nd-order), same steepness as the passive LC filter but without an inductor |
The classic "equal component" simplification — setting R1=R2=R and choosing C1=2C2 — is popular precisely because it always lands exactly on the Butterworth Q=0.7071 target regardless of the actual R and C2 values chosen, which is why the "Design" tab above uses it.
Real-World Applications & Fully-Explained Examples
- Anti-aliasing filters: a clean, flat-passband filter ahead of an ADC.
- Audio crossover networks: active alternatives to passive LC crossovers.
- Signal conditioning: removing high-frequency noise from sensor signals without an inductor.
- Multi-stage filter design: cascading Sallen-Key sections for higher-order (4th, 6th order) filters.
- Audio equalizers: deliberately peaking (high-Q) sections for resonance/boost effects.
- Function generator output filtering: smoothing a stepped waveform into a cleaner sine.
Worked examples — explained in full
1. General case: R1=10 kΩ, R2=20 kΩ, C1=10 nF, C2=5 nF. fc=1/(2π√(10000×20000×10−8×5×10−9))≈1591.6 Hz. Q=√(10000×20000×10−8×5×10−9)/(5×10−9×30000)≈0.667 — slightly below Butterworth, so a touch of extra roll-off softness but still no peaking.
2. Equal R, equal C: R1=R2=10 kΩ, C1=C2=10 nF. fc≈1591.6 Hz (same as example 1, since R1R2C1C2 is identical), but Q=0.500 exactly — the over-damped case, since equal capacitors (rather than the 2:1 ratio) don't hit the Butterworth target.
3. Equal-R Butterworth: R1=R2=10 kΩ, C2=10 nF, C1=20 nF (the 2:1 ratio). fc=1/(2π√(10000²×20×10−9×10×10−9))≈1125.4 Hz, and Q comes out to exactly 0.7071 — confirming the classic C1=2C2 equal-R recipe always gives the maximally-flat Butterworth response.
4. Designing directly for fc=1 kHz Butterworth with R=10 kΩ. Using the Design tab: C2=1/(2π×10000×√2×1000)≈11.25 nF, C1=2×11.25≈22.51 nF. Plugging these back into the general formula confirms fc=1000.0 Hz and Q=0.7071 exactly, as designed.
5. High-Q peaking filter: R1=R2=10 kΩ, C1=100 nF (10×C2), C2=10 nF. fc≈503.3 Hz, Q≈1.58 — well above the Butterworth threshold. The response actually peaks by about +4.4 dB near 450 Hz (slightly below fc) before rolling off, rather than flattening out — a clearly audible resonance if used in an audio path.
6. Comparing the response right at fc across Q values. At exactly f=fc: Q=0.5 gives −6.02 dB (over-damped, already rolling off), Q=0.7071 gives the Butterworth −3.01 dB, and Q=1.58 gives +3.98 dB (above unity — the signal is actually boosted at fc, not attenuated) — a concrete illustration of just how much Q reshapes the filter around its cutoff.
Frequently Asked Questions
What is a Sallen-Key filter?
It is a widely-used active (op-amp based) second-order filter topology built from two resistors, two capacitors, and one op-amp (in the common unity-gain configuration), used to realize a low-pass (or high-pass) response without needing a bulky, imprecise inductor.
What is the cutoff frequency formula for a Sallen-Key low-pass filter?
fc = 1/(2π√(R1R2C1C2)), the same general 2nd-order form as an LC or RLC filter, but built with resistors and capacitors around an op-amp instead of an inductor.
What is Q factor and why does it matter in a Sallen-Key filter?
Q describes how the filter behaves right around its cutoff frequency: low Q (below 0.7071) rolls off gently with no peaking, Q=0.7071 gives the maximally-flat Butterworth response, and Q above 0.7071 causes the response to peak above 0 dB near fc before rolling off, which can be desirable (equalizers) or a problem (unwanted resonance) depending on the application.
What is the easiest way to design a Butterworth Sallen-Key filter?
Set R1=R2=R (any convenient resistor value) and choose C1=2×C2. This combination automatically gives Q=0.7071 (Butterworth) regardless of the specific R and C2 chosen, as long as the ratio is exactly 2:1.
What Q gives a Bessel response instead of Butterworth?
Q=1/√3≈0.5774 gives a 2nd-order Bessel response, which has a slightly gentler roll-off than Butterworth but the best possible step response (least overshoot and ringing) — often preferred for filtering pulse or square-wave signals where waveform shape matters more than a maximally flat passband.
Why does my filter show a peak in its response instead of a flat passband?
Your Q factor is above 0.7071 (Butterworth). Any Q above that threshold causes the response to rise above 0 dB somewhere near the cutoff frequency before rolling off — check your R1/R2/C1/C2 ratios against the equal-R Butterworth recipe (C1=2×C2) if a flat passband was intended.
How steep is a Sallen-Key filter's roll-off compared to a simple RC filter?
A Sallen-Key filter is second-order, rolling off at −40 dB/decade — twice as steep as a single RC stage's −20 dB/decade — while, unlike a passive LC filter with the same roll-off, needing no inductor.
Can a Sallen-Key filter provide gain as well as filtering?
Yes — the classic version shown here uses the op-amp as a unity-gain buffer (gain=1), but a modified version can add passband gain by using the op-amp in a non-inverting gain configuration instead of a simple buffer, at the cost of a slightly different Q formula.
How do I build a higher-order (4th order or steeper) filter from Sallen-Key stages?
Cascade two or more Sallen-Key sections in series, each designed with a specific Q value from a standard filter-design table (Butterworth, Chebyshev, or Bessel tables list the required Q for each stage of a given order) to build up the desired overall response shape.
Does the op-amp's bandwidth limit how high a cutoff frequency I can design for?
Yes — the op-amp's own gain-bandwidth product must comfortably exceed the filter's cutoff frequency (with margin for the filter's own Q-dependent peaking), or the op-amp's limited bandwidth will distort the intended response, especially for high-Q or high-frequency designs.
What is the difference between this and the passive LC filter calculator?
The LC filter calculator sizes a passive inductor-capacitor section (no active devices, needs an inductor, but truly lossless in the ideal case). This Sallen-Key calculator sizes an active resistor-capacitor-op-amp section (no inductor needed, but requires a powered op-amp and has its own bandwidth/noise limitations). Both achieve a similar 2nd-order, −40 dB/decade roll-off.