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🔌 Sallen-Key Filter Calculator

Calculate Sallen-Key active filter component values (capacitors and resistors) for unity-gain 2nd-order low-pass and high-pass configurations. Enter cutoff frequency, resistor value, and Q factor to get exact component values.

Filter Type

Cutoff Frequency (fc)

Frequency Unit

R1 = R2 value (Ohms) for LP, or C1 = C2 value (Farads) for HP

Q Factor (0.707 = Butterworth)

What is this tool?

The Sallen-Key topology is one of the most popular active filter configurations, introduced by R.P. Sallen and E.L. Key in 1955 at MIT's Lincoln Laboratory. Unlike passive LC filters such as the Butterworth or Chebyshev designs that require inductors, the Sallen-Key filter uses only resistors, capacitors, and an operational amplifier (op-amp). This makes it especially attractive for audio and instrumentation applications where inductors are bulky, expensive, or impractical — particularly at low frequencies where inductor values would be enormous. The most common variant is the **unity-gain** Sallen-Key filter, where the op-amp is wired as a voltage follower (output connected directly to the inverting input). This simplifies the design, eliminates the need for gain-setting resistors, and provides excellent buffering with high input impedance and low output impedance. A key advantage of active filters is that the Q factor (quality factor) can be set independently of the cutoff frequency by choosing the ratio of component values. For a **Butterworth response**, Q = 0.707 (damping ratio ζ = 1/(2Q) ≈ 0.707). For a Bessel response, Q = 0.577. Higher Q values create peaking near the cutoff frequency, while lower Q values produce a more gradual transition. This calculator handles both **low-pass** and **high-pass** unity-gain Sallen-Key configurations. For low-pass design, you specify R1 = R2 = R and the desired Q, and the calculator computes C1 and C2. For high-pass design, you specify C1 = C2 = C and the calculator computes R3 and R4. In both cases, the cutoff frequency fc is determined by the RC time constant, while Q is set by the ratio of the two capacitors (or resistors in the HP case). For passive LC alternatives, see our Butterworth filter calculator and Chebyshev filter calculator. For op-amp power requirements, the Ohm's law calculator can help determine supply voltage needs.

How it works

The Sallen-Key filter is a 2nd-order active RC filter whose transfer function has two poles set by the RC network. The unity-gain low-pass configuration uses two resistors (R1, R2) and two capacitors (C1, C2) with an op-amp voltage follower. **Key formulas (unity-gain low-pass):** - Cutoff frequency: fc = 1 / (2π × √(R1 × R2 × C1 × C2)) - Quality factor: Q = √(R1 × R2 × C1 × C2) / (C2 × (R1 + R2)) - Damping ratio: ζ = 1 / (2Q) When we set R1 = R2 = R, the formulas simplify to: - fc = 1 / (2π × R × √(C1 × C2)) - Q = √(C1 × C2) / (2 × C2) = ½ × √(C1 / C2) Solving for C1 and C2 given fc, R, and Q: - C2 = 1 / (2 × Q × 2π × fc × R) - C1 = 2 × Q² × C2 = Q / (π × fc × R) The table below shows the relationship between Q factor, damping ratio, and filter response type: | Q Factor | Damping Ratio ζ | Filter Response | Characteristics | |---|---|---|---| | 0.500 | 1.000 | Overdamped (Bessel) | Slowest roll-off, no overshoot, best step response | | 0.577 | 0.866 | Bessel | Maximally linear phase, excellent transient response | | 0.707 | 0.707 | Butterworth | Maximally flat passband, most commonly used | | 1.000 | 0.500 | Chebyshev (mild) | Slight peaking (~1.25 dB), steeper roll-off | | 1.500 | 0.333 | Peaking | Significant peaking (~6 dB), ringing in step response | | 2.000 | 0.250 | High peaking | Strong resonance near fc, poor transient response | For high-pass Sallen-Key (unity gain, C1 = C2 = C), the resistor values are: - R3 = 1 / (2 × Q × 2π × fc × C) - R4 = Q / (π × fc × C) The following table shows pre-computed capacitor values for common Butterworth (Q = 0.707) low-pass filters with R = 10 kΩ: | fc | C2 (F) | C1 (F) | C1/C2 Ratio | |---|---|---|---| | 1 kHz | 11.25 nF | 22.50 nF | 2.000 | | 10 kHz | 1.125 nF | 2.250 nF | 2.000 | | 100 kHz | 112.5 pF | 225.0 pF | 2.000 | | 1 MHz | 11.25 pF | 22.50 pF | 2.000 | Note: The C1/C2 ratio of exactly 2.0 is characteristic of the Butterworth Q = 0.707 design when R1 = R2.

How to use

Select the filter type (Low-Pass or High-Pass) using the dropdown.
Enter the desired cutoff frequency (fc) in Hz, kHz, or MHz.
For low-pass: enter the resistor value (R1 = R2 = R) in ohms. For high-pass: enter the capacitor value (C1 = C2 = C) in farads.
Enter the Q factor (0.707 for Butterworth, 0.577 for Bessel, or your custom value).
Click Calculate to get C1 and C2 (low-pass) or R3 and R4 (high-pass), plus the damping ratio.

Frequently Asked Questions

What is a Sallen-Key filter and how does it differ from passive LC filters?

A Sallen-Key filter is an active filter that uses an operational amplifier (op-amp) combined with resistors and capacitors — no inductors required. This is a major advantage over passive LC filters like Butterworth or Chebyshev designs, especially at low frequencies (audio range) where inductors would be physically large and expensive. The op-amp also provides impedance buffering, meaning the filter does not load the source and can drive subsequent stages directly.

Why does the Sallen-Key filter with equal components (R1=R2, C1=C2) only give Q = 0.5?

When R1 = R2 = R and C1 = C2 = C, the quality factor simplifies to Q = 0.5, which corresponds to an overdamped (Bessel-like) response with a very gradual transition. This is not a Butterworth response. To achieve Butterworth Q = 0.707 with R1 = R2, you need C1/C2 = 2 (C1 is twice C2). This calculator automatically computes the correct C1 and C2 values for any Q you specify.

What Q factor should I use for my Sallen-Key filter?

The most common choice is Q = 0.707 for a Butterworth (maximally flat) response. For better transient response with less ringing, use Q = 0.577 (Bessel). For slightly steeper initial roll-off at the cost of ~1.25 dB passband peaking, Q = 1.0 works well. Avoid Q above 1.5 unless you specifically want peaking near resonance. The damping ratio \u03B6 = 1/(2Q) gives you an intuitive feel: \u03B6 = 1 is critically damped, \u03B6 < 1 is underdamped (some overshoot), and \u03B6 > 1 is overdamped.

Can I use this calculator for higher-order filters?

This calculator designs a single 2nd-order Sallen-Key stage. For a 4th-order filter, cascade two stages; for 6th-order, cascade three. Each stage has its own Q factor — for a Butterworth response, the stage Q values follow a specific sequence (e.g., for 4th-order Butterworth: Q1 = 0.541, Q2 = 1.306). You would use this calculator twice with different Q values and connect the stages in series.

What op-amp gain-bandwidth product do I need for my Sallen-Key filter?

For unity-gain Sallen-Key filters, the op-amp GBW should be at least 100 times the cutoff frequency to maintain accurate filter response. For example, a 10 kHz filter needs an op-amp with GBW \u2265 1 MHz. Popular choices: TL072 (3 MHz GBW) for audio, OPA2134 (8 MHz) for high-quality audio, and AD8055 (>300 MHz) for high-speed applications. Insufficient GBW causes the actual cutoff frequency to be lower than designed and can introduce distortion.

Why does my built Sallen-Key filter not match the calculated response?

Several factors can cause deviation: component tolerance (use 1% resistors and 5% or better capacitors), op-amp non-idealities (finite GBW, slew rate, input capacitance), PCB parasitic capacitance, and power supply limitations. At frequencies above 100 kHz, op-amp limitations become the dominant source of error. Always prototype and measure with a network analyzer or frequency sweep to verify the actual response. If the cutoff is shifted, adjust component values empirically.

Tips & Advice

When building a Sallen-Key filter, op-amp selection is critical. The op-amp must have sufficient gain-bandwidth product (GBW) — as a rule of thumb, GBW should be at least 100× the filter cutoff frequency for accurate response. For audio filters (up to 20 kHz), general-purpose op-amps like the TL072 or NE5532 work well. For RF or high-speed applications, consider current-feedback or high-speed op-amps. Use 1% or better tolerance components for predictable Q and fc. Avoid electrolytic capacitors due to their poor tolerance and high dielectric absorption; film or C0G/NP0 ceramic capacitors are preferred. Remember that real op-amps have finite input capacitance and output impedance that can shift the filter response at high frequencies. For steeper roll-off, cascade two or more 2nd-order Sallen-Key stages — each stage contributes 40 dB/decade (12 dB/octave) of attenuation.

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