Pick a Butterworth, Bessel or Chebyshev response and this Sallen-Key filter calculator gives the resistors for a unity-gain low-pass or high-pass stage, rounded to standard values. It also runs in reverse: enter the four parts you have and read off f0, Q, the −3 dB frequency and the Bode plot.
−3 dB point for Butterworth and Bessel; edge of the ripple band for Chebyshev.
Optional. Blank picks the smallest E12 value that reaches the Q (C2 ≥ 4Q² × C1).
Optional: gain and phase at this frequency.
R1 and R2 (nearest E96)
7.87 kΩ, 14.7 kΩ
exact 7.86 kΩ and 14.6 kΩ
Butterworth cutoff is the −3 dB point. Gain is exactly 1 (0 dB) in the passband because the op amp is a follower.
C2 choices for C1 = 10 nF (E12 capacitors, nearest E96 resistors)
| C2 | R1 | R2 | Cutoff | Q |
|---|---|---|---|---|
| 22 nF | 7.87 kΩ | 14.7 kΩ | 997.6 Hz (−0.24%) | 0.707 |
| 27 nF | 5.49 kΩ | 16.9 kΩ | 1.006 kHz (+0.56%) | 0.707 |
| 33 nF | 4.22 kΩ | 18.2 kΩ | 999.7 Hz (−0.03%) | 0.710 |
| 39 nF | 3.4 kΩ | 19.1 kΩ | 1 kHz (+0.01%) | 0.707 |
| 47 nF | 2.74 kΩ | 19.6 kΩ | 1.002 kHz (+0.18%) | 0.711 |
For a unity-gain low-pass, f0 = 1 ÷ (2π√(R1R2C1C2)) and Q = √(R1R2C1C2) ÷ ((R1 + R2)C1); f0 is the −3 dB cutoff only for a Butterworth response. For a 1 kHz Butterworth (Q = 0.7071) with C1 = 10 nF and C2 = 33 nF, R1 = 4.19 kΩ and R2 = 18.3 kΩ; the nearest 1 % values 4.22 kΩ and 18.2 kΩ give 999.7 Hz. The feedback capacitor must be at least 4Q² times C1.
A Sallen-Key stage is two RC sections with positive feedback from the op amp output into the middle node. The feedback raises Q above the 0.5 limit of a passive RC-RC network. With the op amp as a follower (K = 1), the passband gain is exactly 1.
The high-pass swaps the parts: C1 and C2 in series with the signal, R1 from the + input to ground and R2 from the capacitor junction to the output. Its Q has the same form with the roles exchanged:
For a Sallen-Key filter design, look up Q and the frequency scaling factor FSF for the response, then set the pole frequency to f0 = FSF × fc for a low-pass or fc ÷ FSF for a high-pass. With R1 = mR, R2 = R, C1 = C and C2 = nC (TI SLOA049, Appendix A.1.2), Q = √(mn) ÷ (m + 1), which needs n ≥ 4Q²:
For the equal-capacitor high-pass, R1 = Q ÷ (π f0 C) and R2 = 1 ÷ (4π Q f0 C).
Q = 0.7071 and FSF = 1, so f0 = 1 kHz. With C1 = 10 nF and C2 = 33 nF, n = 3.3 and m = 0.2288. R = 1 ÷ (2π × 1 kHz × 10 nF × √(mn)) = 18.3 kΩ, so R2 = 18.3 kΩ and R1 = m × R = 4.19 kΩ. Rounded to E96, 4.22 kΩ and 18.2 kΩ give f0 = 999.7 Hz and Q = 0.710. TI lists 4.22 kΩ and 18.4 kΩ for the same circuit.
For a Butterworth high-pass, f0 = fc = 20 Hz. R1 = 0.7071 ÷ (π × 20 × 220 nF) = 51.2 kΩ and R2 = 1 ÷ (4π × 0.7071 × 20 × 220 nF) = 25.6 kΩ, a 2:1 ratio. The E96 values 51.1 kΩ and 25.5 kΩ give 20.04 Hz. Film capacitors suit this: an X7R ceramic in the signal path loses capacitance with bias and adds distortion.
R1 = R2 = 1 kΩ and C1 = C2 = 1 nF: f0 = 1 ÷ (2π × 1 µs) = 159.2 kHz and Q = √(10⁻¹²) ÷ (2 × 10⁻⁶) = 0.50, matching the 159 kHz and Q = 1/2 TI quotes. Because Q is below 0.707, the −3 dB point is lower than f0, at 102.4 kHz.
Second-order values computed from the pole positions of each response. They match TI SLOA049 Tables 9-1 to 9-4 within 0.13 % (TI rounds the Bessel FSF to 1.2736 from rounded pole values) and Op Amps for Everyone Table 16-6 for the 0.5 dB Chebyshev. Gains are for a low-pass, relative to the DC gain.
| Response | Q | FSF (f0/fc) | Min C2/C1 | Peak | Gain at 2fc | Gain at 10fc |
|---|---|---|---|---|---|---|
| Butterworth | 0.7071 | 1.0000 | 2.000 | none | −12.3 dB | −40.0 dB |
| Bessel | 0.5774 | 1.2720 | 1.333 | none | −9.8 dB | −35.9 dB |
| Chebyshev 0.5 dB | 0.8637 | 1.2313 | 2.984 | +0.50 dB | −7.9 dB | −36.3 dB |
| Chebyshev 1 dB | 0.9565 | 1.0500 | 3.660 | +1.00 dB | −10.4 dB | −39.1 dB |
| Chebyshev 3 dB | 1.3047 | 0.8414 | 6.809 | +3.00 dB | −14.0 dB | −43.0 dB |
Unity gain, C1 the E6 value nearest 1/(2π fc × 10 kΩ), C2 the smallest E12 value of at least 2 × C1, resistors rounded to E96. The last column is f0 with those rounded parts.
| fc | C1 | C2 | R1 | R2 | Actual f0 |
|---|---|---|---|---|---|
| 100 Hz | 150 nF | 330 nF | 5.23 kΩ | 9.76 kΩ | 100.1 Hz |
| 1 kHz | 15 nF | 33 nF | 5.23 kΩ | 9.76 kΩ | 1.001 kHz |
| 3.4 kHz | 4.7 nF | 10 nF | 5.36 kΩ | 8.87 kΩ | 3.367 kHz |
| 10 kHz | 1.5 nF | 3.3 nF | 5.23 kΩ | 9.76 kΩ | 10.01 kHz |
| 20 kHz | 680 pF | 1.5 nF | 5.76 kΩ | 10.7 kΩ | 20.07 kHz |
| 50 kHz | 330 pF | 680 pF | 5.62 kΩ | 8.06 kΩ | 49.92 kHz |
Use C0G/NP0 ceramic or film capacitors and 1 % resistors: Q depends on part ratios, and X7R capacitance moves with DC bias and temperature. TI SLOA049 suggests resistors from a few hundred ohms to a few kilohms and capacitors of at least 10 pF, so keep the capacitors well above the op amp’s input capacitance. A low-pass Sallen-Key does not keep attenuating forever: well above the cutoff the signal couples through C2 to the op amp output, whose output impedance rises with frequency, so the stopband floor is set by the amplifier (SLOA024B, section 3.5). A small RC after the output fixes it if you need deep high-frequency rejection. A high-pass is really a band-pass whose upper edge is the op amp’s bandwidth. The source impedance adds to R1 of a low-pass, so drive the filter from a low-impedance source or fold the source resistance into R1. Bessel tables differ in how they normalize: the FSF of 1.2720 used here puts the −3 dB point at fc, while tables normalized for unit group delay put the poles elsewhere, so check which one a reference uses before mixing values (a 1 kHz Bessel with C1 = 10 nF takes C2 = 15 nF here).
Two RC sections around an op amp, with positive feedback from the output into the middle node. The feedback raises Q above the 0.5 limit of a passive RC-RC network, and with the op amp as a follower the passband gain is exactly 1.
For the unity-gain low-pass, f0 = 1 / (2π √(R1 R2 C1 C2)) and Q = √(R1 R2 C1 C2) / ((R1 + R2) C1), where C1 goes from the op-amp + input to ground and C2 is the feedback capacitor. For a Butterworth response f0 equals the −3 dB cutoff and Q = 0.7071.
Q = 0.5 at unity gain, below a Bessel (0.577). The −3 dB point then sits at 0.644 × f0: with 1 kΩ and 1 nF, f0 = 159.2 kHz but the −3 dB point is 102.4 kHz. To get a sharper corner at unity gain, make the feedback capacitor C2 larger than C1.
At unity gain the highest Q a low-pass can reach is √(C2/C1) / 2, which happens with equal resistors. A Butterworth (Q = 0.7071) needs C2/C1 ≥ 2, a Bessel ≥ 1.33 and a 3 dB Chebyshev ≥ 6.81. Leave C2 blank in the calculator and it picks the smallest E12 value that works.
Swap the resistors and capacitors. With two equal capacitors C, R1 (to ground) = Q / (π f0 C) and R2 (feedback) = 1 / (4π Q f0 C). For a 20 Hz Butterworth with 220 nF that is 51.2 kΩ and 25.6 kΩ. For Bessel and Chebyshev the pole frequency is fc divided by the same FSF the low-pass multiplies by.
TI SLOA049 sums it up: the unity-gain Sallen-Key filter has the best gain accuracy because its gain does not depend on component values, while the MFB is less sensitive to component variation and keeps attenuating better at high frequency. Sallen-Key is non-inverting; MFB inverts.
TI’s rule of thumb (Op Amps for Everyone, section 16.8.4) for about 1 % gain error is a unity-gain bandwidth of 100 × the −3 dB frequency for Q below 1, and 100 × Q × f0 × √((Q² − 0.5)/(Q² − 0.25)) above it. A 1 kHz Butterworth needs 100 kHz; a 1 kHz, 3 dB Chebyshev about 99.9 kHz.
Cascade second-order stages, each with its own f0 and Q from a filter table (TI SLOA049 section 9), plus one RC section for odd orders. A fourth-order Butterworth uses stages with Q = 0.5412 and 1.3065, both at fc. Put the lowest-Q stage first to reduce clipping.
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