Active low-pass filter calculator

A 1 kHz Butterworth Sallen-Key with a 10 nF C₂ needs C₁ = 20 nF and R = 11.25 kΩ. The nearest E24 value, 11 kΩ, moves the corner to 1023 Hz. Q stays 0.707, because in TI SLOA049's equations only the capacitor ratio sets it. The ideal-op-amp model holds while the GBW exceeds 100·K·f₀·Q, 71 kHz here.

Second-order low-pass with an op amp: pick the corner, the shape and one capacitor, and get the resistors, what the E24 values actually give you, the peaking to expect, and how fast the op amp has to be.

Fig 1 — Sallen-Key: f0 1.0 kHz, Q 0.707, −40 dB/decade above the knee.

Topology

Response

Corner frequency (Hz)

C2, the capacitor to ground (nF)

Check attenuation at (0 to skip) (Hz)

C1 (junction to output): 20.0 nF
C2 (+ input to ground): 10.0 nF
R1 = R2, exact: 11.3 kΩ → E24 11.0 kΩ
With E24: f0 / Q: 1.02 kHz · 0.707
Designed f0 / Q: 1.00 kHz · 0.7071
Response at fc = 1.00 kHz: -2.82 dB
Op amp GBW, at least: 72.3 kHz

Flat passband, −3 dB at fc.

How this is calculated

Standard: TI SLOA049 (active low-pass filter design); TI SBOA093

$\omega_0 = \frac{1}{R \sqrt{C_1 \cdot C_2}}, \quad Q = \frac{\sqrt{C_1 / C_2}}{2}$: Sallen-Key at unity gain, with R1 = R2 = R.
$C_1 = 4 Q^2 C_2, \quad R = \frac{1}{2\pi f_0 \sqrt{C_1 C_2}}$: The same two relations rearranged into the parts to fit, given a corner frequency and the capacitor you already have.
$\omega_0 = \frac{1}{\sqrt{R_2 R_3 C_1 C_2}}$: Multiple-feedback, with inverting gain K = R_2 / R_1.
$Q = \frac{\sqrt{R_2 R_3 C_1 C_2}}{C_2 \left(R_2 + R_3 + \frac{R_2 R_3}{R_1}\right)}$
$C_1 \ge 4 Q^2 (1 + K) C_2$: The multiple-feedback capacitor ratio needed for the resistors to come out real.
$f_{peak} = f_0 \sqrt{1 - \frac{1}{2Q^2}}, \quad \frac{|H|_{peak}}{K} = \frac{Q}{\sqrt{1 - \frac{1}{4Q^2}}}$: Where the response peaks and how high, for Q above 0.7071. Below that it does not peak.
$\text{GBW} \ge 100 \cdot K \cdot f_0 \cdot Q$: Gain-bandwidth the op amp needs so that its own roll-off does not move the corner.
$|H(f)| = \frac{K}{\sqrt{\left(1 - \left(\frac{f}{f_0}\right)^2\right)^2 + \left(\frac{f}{f_0 Q}\right)^2}}$: Second-order low-pass magnitude. Q is 0.7071 for Butterworth, 0.5773 for Bessel and 0.9565 for Chebyshev with 1 dB of ripple.

Assumptions

Ideal op amp. Below about a hundredth of the GBW that is a fair model; the tool tells you the GBW it wants.
Capacitors are exact. Use C0G/NP0: X7R tolerance and DC-bias loss move f0 and Q more than any resistor rounding does.
Bessel and Chebyshev pole frequencies are scaled so that fc is the −3 dB point (Bessel) or the ripple-band edge (Chebyshev). The Bessel and 1 dB Chebyshev pairs are SLOA049's tables 9-2 and 9-3; SLOA049 has no 0.5 dB table, so that pair is derived from the standard Chebyshev pole formulas and re-derived in the unit tests.
Single section only. A fourth-order filter is two of these with different Q values, not two identical ones.
Sallen-Key stop-band comes back up at high frequency because the op amp output impedance rises; the plot does not show that.

What sets a Sallen-Key or MFB filter's corner and Q

A second-order low-pass around one op amp, from a corner frequency, a response shape and a capacitor you already have. Sallen-Key is the unity-gain, equal-resistor form: R₁ = R₂ = R, C₁ from the resistor junction to the op-amp output, C₂ from the + input to ground. MFB (multiple feedback) is the inverting form with gain K = R₂/R₁, C₁ to ground at the summing node and C₂ in the feedback.

Both topologies, the capacitor ratio the multiple-feedback form needs for its resistors to come out real, the magnitude response, the peak a high Q puts just below f₀, and the gain-bandwidth the op amp has to have are all under How this is calculated above. Where the ratio bound applies the tool fits 1.1 times it, so the arithmetic does not land on the boundary.

The shape sets Q and where the pole f₀ sits relative to the corner you asked for. Butterworth: Q 0.7071, f₀ = f_c. Bessel: Q 0.5773, f₀ = 1.2736 f_c. Chebyshev 0.5 dB: Q 0.8637, f₀ = 1.2313 f_c. Chebyshev 1 dB: Q 0.9565, f₀ = 1.05 f_c. For Butterworth and Bessel f_c is −3 dB; for Chebyshev it is the ripple-band edge. Equations from TI SLOA049. The tool rounds resistors to E24 and reports the f₀ and Q you actually get.

Worked example: a 1 kHz Butterworth from a 10 nF capacitor

Sallen-Key, 1 kHz Butterworth, starting from a 10 nF C0G on the reel.

Q = 0.7071, f0 = 1000 Hz
C1 = 4 × 0.7071² × 10 nF = 20.0 nF
R  = 1 / (2π × 1000 × √(20n × 10n)) = 1 / (2π × 1000 × 14.14n) = 11.25 kΩ
E24: 11 kΩ  →  f0 = 1 / (2π × 11k × 14.14n) = 1023 Hz,  Q = ½ √(20/10) = 0.707 (unchanged)

Q depends only on the capacitor ratio, so resistor rounding moves the corner 2 % and leaves the shape alone. Ask for Bessel at the same 1 kHz and the tool puts the pole at 1273.6 Hz, Q 0.5773: C₁ = 13.3 nF, R = 10.8 kΩ. The soft Bessel roll-off needs the pole up there for −3 dB to land on 1 kHz.

MFB, 1 kHz Butterworth, gain 2, from C₂ = 1 nF.

min C1 = 4 × 0.7071² × (1 + 2) × 1 nF = 6.0 nF      tool uses 1.1× = 6.6 nF

a  = 1 / (2π × 1000)      = 1.5915e-4
p  = a² / (C1 C2)         = 2.533e-8 / 6.6e-18 = 3.838e9      (= R2 R3)
b  = a / (Q C2)           = 1.5915e-4 / 0.7071e-9 = 2.2508e5  (= R2 + R3 (1+K))
disc = √(b² − 4 (1+K) p)  = √(5.066e10 − 4.606e10) = 6.786e4
R3 = (b − disc) / (2 (1+K)) = 1.5722e5 / 6 = 26.2 kΩ
R2 = p / R3 = 3.838e9 / 26 203 = 146.5 kΩ
R1 = R2 / K = 73.2 kΩ

E24: R1 75 kΩ, R2 150 kΩ, R3 27 kΩ
  f0 = 1 / (2π √(150k × 27k × 6.6n × 1n)) = 973 Hz
  Q  = 0.708,  gain = 150/75 = 2.00
GBW wanted: 100 × 2 × 1000 × 0.707 = 141 kHz

The 6 nF minimum is the discriminant reaching zero; the 10 % back-off keeps tolerance from making the design unsolvable. A bigger C₁ spreads the resistors apart, which is fine until R₃ gets small enough to load the source.

Where the Sallen-Key and MFB equations stop being valid

The op amp is part of the filter. Once loop gain at f₀ drops toward the section's own gain, the pole moves and Q rises. 100·K·f₀·Q keeps the error near 1 %; at a tenth of that the shape is visibly off. High-Q sections at tens of kHz want megahertz op amps.

Sallen-Key comes back up. C₁ dumps the stop-band signal into the op-amp output, and the output is not a short: its closed-loop impedance rises with frequency as loop gain falls. The response flattens out a decade or two above f_c and then rises again; floors of 40 to 60 dB are typical. MFB's feedback capacitor works into a virtual ground and keeps falling until parasitics take over, which is why it is the better anti-alias filter when the aliasing band is far above f_c.

Sensitivity. Unity-gain SK sets Q by the capacitor ratio and barely notices the resistors. Add gain (the tool does not) and the equal-component form has Q = 1/(3 − K): K of 2.9 is Q 10, K of 3 oscillates, and 1 % in the gain leg is 10 % in Q. MFB spreads the sensitivity over resistor ratios instead, so nothing blows up but every 1 % resistor contributes, and C₁/C₂ ratios of 10 or more pull the two capacitors from different decades.

Fourth order is not two of these. Two Butterworth sections in series give Linkwitz-Riley, −6 dB at f_c. A true fourth-order Butterworth is two sections at the same f₀ with Q 0.5412 and 1.3066; Bessel and Chebyshev spread f₀ as well. Take the pair from the pole tables in SLOA049, run the tool once per section, and put the higher-Q section last so it does not clip on the peaking.

Common active filter mistakes

X7R capacitors. ±15 % over temperature plus DC-bias loss goes straight into Q (the ratio) and f₀ (the product). C0G/NP0 for both, film above what C0G stocks.
Picking resistors first. Resistors come in E96; C0G capacitors in E6 or E12. Pick C₂ from stock, let the tool derive C₁, fit the resistors last. If C₁ lands off-series, parallel two parts or let f₀ move.
Forgetting the MFB inverts. Gain is −K; either the next stage inverts again or the firmware flips the sign. On a single supply the + input needs a mid-rail reference that can sink the signal current through R₁.
Driving an MFB from a high-impedance source. Its input impedance is R₁, 73 kΩ in the example and often much less; a few kΩ of source resistance changes gain and Q. Buffer first, or use Sallen-Key.
Trusting the SK stop-band for anti-aliasing. Measure the floor or use MFB. Either way the filter goes directly in front of the ADC, after the gain stages, with a small RC on its output as the converter's charge reservoir.
Reading the Chebyshev f_c as −3 dB. It is the ripple-band edge. The 1 dB design's Q of 0.9565 peaks 1.0 dB at 0.67 f₀; that is the ripple, not a fault.