Active low-pass filter

What it computes

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.

Sallen-Key (unity gain, R1 = R2 = R)
  ω0 = 1 / (R √(C1 C2))        Q = ½ √(C1 / C2)
  C1 = 4 Q² · C2               R  = 1 / (2π f0 √(C1 C2))

MFB (inverting, K = R2 / R1)
  ω0 = 1 / √(R2 R3 C1 C2)
  Q  = √(R2 R3 C1 C2) / (C2 (R2 + R3 + R2 R3 / R1))
  real resistors need C1 ≥ 4 Q² (1 + K) C2   (tool defaults to 1.1× that)

Any second-order section, u = f / f0
  |H| = 1 / √((1 − u²)² + (u / Q)²)
  peak (Q > 0.707) at f0 √(1 − 1/2Q²), height Q / √(1 − 1/4Q²)
  op amp GBW ≥ 100 · K · f0 · Q

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

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 C₁spreads the resistors apart, which is fine until R₃ gets small enough to load the source.

Where it stops 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 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.

Further reading

• TI SLOA049, Active Low-Pass Filter Design — the Sallen-Key and MFB equations used here, pole tables for higher orders, GBW discussion.
• TI SBOA093, Filter Design in Thirty Seconds — the cookbook version, good for sanity-checking values.
• TI SLOA024, Analysis of the Sallen-Key Architecture — the derivation, gain sensitivity, and why the stop-band comes back up.
• ADI Basic Linear Design, chapter 8: Analog Filters — topology comparison, sensitivity, full pole tables to tenth order.