Sallen-Key Active Filter Designer
Sallen-Key Active Filter Designer
Second-order Sallen-Key low-pass or high-pass, unity-gain or equal-component, to a Butterworth, Bessel or 0.5 dB Chebyshev response: the exact R and C values, the nearest E-series parts, the response those real parts give, the op-amp speed it needs, and the Bode plot.
Sallen-Key second-order section
a 1 kHz Butterworth low-pass, unity gain, seeded with a 10 nF capacitor and rounded to E24
The Sallen-Key second-order section
- Q
- 0.7071 for Butterworth, 0.5774 for Bessel, 0.8637 for 0.5 dB Chebyshev — the response IS the Q
- ω₀
- the pole frequency, which is not the −3 dB frequency unless Q happens to be 0.7071
- K
- the passband gain, 1 + Rb/Ra; 1 in the unity-gain form
- 1/Q (low-pass)
- √(R₂C₂/R₁C₁) + √(R₁C₂/R₂C₁) + (1−K)√(R₁C₁/R₂C₂)
Worked example
a 1 kHz Butterworth low-pass, unity gain, seeded with a 10 nF capacitor and rounded to E24
Butterworth second order is Q = 0.7071 and f₀ = fᴄ, so ω₀ = 2π × 1,000 = 6,283.2 rad/s
Unity gain with R₁ = R₂: C₁ = 4Q²C₂ = 20 nF and R = 1 ÷ (2Qω₀C₂) = 11.254 kΩ
Nearest E24 parts: R₁ = R₂ = 11 kΩ, C₁ = 20 nF, C₂ = 10 nF
Those parts give f₀ = 1.023 kHz and Q = 0.7071, so the real −3 dB point is 1.023 kHz — 2.31% high, and the Q is untouched because both capacitors moved together
The op-amp needs GBW ≥ 100 × gain × fᴄ = 100 kHz; 10 MHz clears it
The three responses, second order
| Response | Q | f₀ ÷ f(−3 dB) | What it is for |
|---|---|---|---|
| Butterworth | 0.7071 | 1.0000 | The flattest passband of any all-pole filter. The default when nothing else argues, and the only one whose pole frequency IS its −3 dB frequency. |
| Bessel | 0.5774 | 1.2720 | Maximally flat group delay, so a step through it barely overshoots and a pulse keeps its shape. Pays for it with the gentlest roll-off of the three. |
| Chebyshev 0.5 dB | 0.8637 | 0.8860 | The steepest skirt, bought with 0.5 dB of ripple in the passband and the worst step response. Its ripple band edge is at 0.7196 × the −3 dB frequency. |
Designing a Sallen-Key section
The Sallen-Key section is the standard way to build a second-order filter round one op-amp. Two resistors and two capacitors set the pole frequency; the feedback path round the amplifier sets Q, the peaking. Its transfer function is the textbook biquad, Kω₀² ÷ (s² + (ω₀/Q)s + ω₀²), and the whole of filter design is choosing Q. The three responses on this page are three values of Q and nothing more: 0.7071 gives Butterworth, 0.5774 Bessel, 0.8637 a 0.5 dB Chebyshev.
Which response. Butterworth has the flattest passband and is the right default for an anti-aliasing or noise filter where what matters is not colouring the band you keep. Bessel has maximally flat group delay, so it passes a step or a pulse with almost no overshoot — use it wherever the waveform’s shape in time matters, which is most of instrumentation. Chebyshev gets the steepest transition for a given order by allowing a ripple in the passband, so it is the one to reach for when a stop-band spec is tight and a little passband error is acceptable; its step response rings. Because the three are normally tabulated against different reference frequencies, this page defines the cut-off as the −3 dB point for all of them, and reports the Chebyshev ripple band edge separately.
Which topology. The unity-gain form uses no gain resistors at all and keeps the passband at 0 dB. Q comes from the ratio of the two capacitors in a low-pass (Q = ½√(C₁/C₂)) or the two resistors in a high-pass, so a high Q needs a wide component ratio — fine at Q below about 3. The equal-component form uses one resistor value and one capacitor value and sets Q with the amplifier’s gain, Q = 1 ÷ (3 − K), which is tidy to build but means you cannot choose the gain and the response independently, and it is sensitive: near K = 3 the section oscillates. Start from a capacitor value you have, because capacitors come in far fewer values than resistors, and let the resistors fall out.
What the op-amp has to do. The design equations assume an ideal amplifier. A real one has finite gain–bandwidth, and Texas Instruments’ design guidance for this circuit puts the requirement at GBW = 100 × gain × fᴄ. Below that the pole frequency sags, Q rises, and a high-pass section quietly becomes a band-pass because the amplifier runs out of gain above the passband. The non-inverting amplifier calculator does the closed-loop bandwidth and slew-rate checks on the amplifier itself. For a passive first-order RC or a series RLC instead, the RC and RLC filter calculator covers those, and the decibel calculator converts the gains here to and from ratios. Finally, real parts: the values shown are the nearest E-series neighbours, rounded to the closest value by ratio rather than up or down, and the response reported is the one those parts give, not the one you asked for.
Frequently asked questions
What Q do I need for a Butterworth filter?
0.7071 for a second-order section, and the pole frequency is the −3 dB frequency. Bessel needs Q = 0.5774 with f₀ 1.272 times the cut-off, and a 0.5 dB Chebyshev needs Q = 0.8637 with f₀ at 0.886 times it — both for a low-pass; a high-pass uses the reciprocal of each ratio.
How do I choose R and C for a Sallen-Key low-pass?
Pick the capacitor first. In the unity-gain form with equal resistors, C₁ = 4Q²C₂ and R = 1 ÷ (2Qω₀C₂). For 1 kHz Butterworth seeded with 10 nF that is C₁ = 20 nF and R = 11.25 kΩ, so 11 kΩ from E24.
Why is my filter’s cut-off not where I designed it?
Usually the standard values: rounding 11.25 kΩ to 11 kΩ moves the corner 2.3% up. Then component tolerance, then the op-amp — if its gain–bandwidth product is less than about 100 × gain × fᴄ the corner sags and Q rises.
What op-amp gain-bandwidth product does a Sallen-Key filter need?
At least 100 × passband gain × cut-off frequency, which is what TI’s Sallen-Key design procedure specifies. A 1 kHz unity-gain section needs 100 kHz; a 100 kHz section at a gain of 2 needs 20 MHz.
Unity gain or equal component?
Unity gain if you want the passband flat at 0 dB and no gain resistors. Equal component if using one R value and one C value is worth having the gain fixed by the response — and keep K well below 3, where the circuit oscillates.
Related calculators
References
- Sallen RP, Key EL. A Practical Method of Designing RC Active Filters. IRE Transactions on Circuit Theory, vol. 2 no. 1, March 1955, pp. 74–85. The original topology.
- Karki J. Analysis of the Sallen-Key Architecture. Texas Instruments application report SLOA024B. The transfer functions used here, including the equal-component simplification Q = 1/(3−K) and the unity-gain form.
- Texas Instruments. Analog Engineer’s Circuit: single-supply, second-order Sallen-Key low-pass filter, SBOA226. States the op-amp requirement as GBW = 100 × gain × fᴄ.
- Zumbahlen H (ed.). Basic Linear Design, chapter 8: Analog Filters. Analog Devices, 2007. Filter design tables; for a 2-pole 0.5 dB Chebyshev the ratio of the 3 dB bandwidth to the ripple bandwidth is 1.38974, which this page reproduces from the pole construction.
- Horowitz P, Hill W. The Art of Electronics, 3rd ed. Cambridge University Press, 2015. Chapter 6: active filters, the VCVS (Sallen-Key) realisation and the Butterworth, Bessel and Chebyshev families.