Low-Pass, High-Pass, Band-Pass & Active Filter Topology
Master active filter design with the Sallen-Key topology — learn low-pass and high-pass transfer functions, component selection, cutoff frequency tuning, and real-world design trade-offs for precision analog signal conditioning.
Design low-pass and high-pass active filters instantly — choose cutoff frequency, filter order, and get component values.
The Sallen-Key topology (also known as a voltage-controlled voltage-source or VCVS filter) is an active filter design using a single op-amp per pole in a non-inverting configuration. It is the most widely used active filter topology for low-pass and high-pass applications because of its simplicity, low component count, and excellent stability. The cutoff frequency fc = 1 / (2π√R1R2C1C2), and the quality factor Q determines peaking and damping behavior. For a Butterworth response (maximally flat passband), Q = 1/√2 ≈ 0.707; for Bessel (linear phase), Q ≈ 0.577; for Chebyshev (steep roll-off with ripple), Q > 0.707.
The Sallen-Key filter was first described by R. P. Sallen and E. L. Key of MIT Lincoln Laboratory in 1955. It remains the most popular active filter architecture for low-frequency applications (up to approximately 100 kHz) due to its elegant design: a single op-amp, four passive components (two resistors and two capacitors), and no requirement for precision inductors that are bulky, expensive, and non-ideal at low frequencies.
The topology uses positive feedback through the capacitor network to achieve frequency-dependent gain shaping. The op-amp is configured as a unity-gain amplifier (voltage follower) or a non-inverting amplifier with gain K = 1 + Rf/Rg. When K = 1 (voltage follower), the filter is called the “unity-gain Sallen-Key” and offers the simplest design equations. Higher gain values allow independent control of the quality factor Q but introduce additional design complexity.
Key advantages of the Sallen-Key topology include: low component sensitivity (the filter’s response is relatively insensitive to component tolerances), easy cascading for higher-order filters (a 2n-order filter requires n stages), good high-frequency performance with modern wideband op-amps, and the ability to realize all standard filter responses — Butterworth, Bessel, Chebyshev, and Linkwitz-Riley.
The second-order low-pass Sallen-Key filter has a transfer function characterized by a natural frequency ω0 and quality factor Q. The standard form is:
Cutoff Frequency: fc = 1 / (2π × √(R1R2C1C2))
Quality Factor: Q = √(R1R2C1C2) / (R1C1 + R2C1 + R1C2(1-K))
DC Gain: K = 1 + Rf / Rg
Roll-Off: -40 dB/decade (second order)
The high-pass version is obtained by swapping the positions of resistors and capacitors in the low-pass topology. The transfer function becomes:
Cutoff Frequency: fc = 1 / (2π × √(R1R2C1C2))
Quality Factor: Q = √(R1R2C1C2) / (C1R1 + C2R1 + C2R2(1-K))
When using a unity-gain configuration (K = 1) with R1 = R2 = R and C1 = C2 = C, the equations simplify dramatically:
Q = 1/3 (for unity-gain, equal components)
Butterworth (Q = 0.707) requires K = 1.586 (Rf/Rg = 0.586)
Bessel (Q = 0.577) requires K = 1.268 (Rf/Rg = 0.268)
Chebyshev 0.5 dB ripple (Q = 0.864) requires K = 1.842 (Rf/Rg = 0.842)
| Filter Response | Quality Factor (Q) | Gain K | Rf/Rg Ratio | Key Characteristic |
|---|---|---|---|---|
| Butterworth (Maximally Flat) | 0.707 | 1.586 | 0.586 | Flattest passband response; no ripple; good compromise for general use |
| Bessel (Linear Phase) | 0.577 | 1.268 | 0.268 | Best step-response with no overshoot; constant group delay in passband |
| Chebyshev 0.5 dB | 0.864 | 1.842 | 0.842 | Steeper roll-off near cutoff; 0.5 dB ripple in passband |
| Chebyshev 1.0 dB | 0.956 | 2.114 | 1.114 | Sharper roll-off at expense of 1 dB ripple |
| Chebyshev 2.0 dB | 1.128 | 2.546 | 1.546 | Maximum roll-off steepness; 2 dB passband ripple |
| Linkwitz-Riley (Crossover) | 0.500 | 1.000 | 0 | Unity-gain; zero phase difference at crossover for audio systems |
| Gaussian to 6 dB | 0.544 | 1.172 | 0.172 | Near-Gaussian pulse response; minimal overshoot |
| Transitional Gaussian 12 dB | 0.653 | 1.459 | 0.459 | Compromise between Gaussian and Butterworth |
| Component | Selection Criteria | Recommended Range | Impact on Performance |
|---|---|---|---|
| Resistors (R1, R2) | Standard 1% metal film; low temperature coefficient (±50 ppm/°C or better) | 1 kΩ – 100 kΩ | Low values improve noise but increase power; high values increase noise and parasitic effects |
| Capacitors (C1, C2) | Film (polypropylene, polyester) for stability; NP0/C0G ceramic for small values; avoid X7R for precision | 1 nF – 1 μF | C1/C2 ratio affects Q sensitivity; mismatched caps change filter shape |
| Op-Amp | GBW ≥ 100 × fc; rail-to-rail IO; low noise for precision | — | Insufficient GBW causes Q-enhancement and high-frequency peaking |
| Feedback (Rf, Rg) | Same type as filter resistors; ratio sets gain K | 1 kΩ – 100 kΩ | Higher gain K increases Q sensitivity to component tolerances |
Design a second-order low-pass Sallen-Key filter with a cutoff frequency fc = 1 kHz and a Butterworth (maximally flat) response. The filter will be used as an anti-aliasing stage for a 16-bit ADC sampling at 10 kS/s.
Butterworth response: Q = 0.707, K = 1.586, Rf/Rg = 0.586
Select C = 10 nF (standard NP0 ceramic, good temperature stability).
Set C1 = C2 = C = 10 nF for simplicity.
fc = 1 / (2πRC)
R = 1 / (2π × 1 kHz × 10 nF) = 1 / (2π × 103 × 10-8)
R = 1 / (6.283 × 10-5) = 15,915 Ω ≈ 15.8 kΩ (nearest 1% E96 value)
Rf/Rg = 0.586. Choose Rg = 10 kΩ.
Rf = 0.586 × 10 kΩ = 5.86 kΩ → nearest 1% value: 5.90 kΩ
Using R = 15.8 kΩ, C = 10 nF:
fc = 1/(2π × 15800 × 10-8) = 1/(9.925 × 10-4) = 1007 Hz — within 0.7% of target.
Gain K = 1 + 5.90/10 = 1.590 — within 0.25% of Butterworth ideal.
Required GBW ≳ 100 × fc = 100 kHz → use TL081 (GBW = 3 MHz) or MCP602 (GBW = 10 MHz) for ample margin and low noise.
Design a second-order high-pass Sallen-Key filter with fc = 80 Hz for a subwoofer crossover. Use a Linkwitz-Riley response (Q = 0.5, K = 1) for zero phase difference at the crossover frequency.
Linkwitz-Riley (LR2): Unity-gain K = 1, Q = 0.5. This is the standard for active audio crossovers.
Select C = 100 nF (polyester film capacitor for audio quality).
R = 1 / (2π × 80 Hz × 100 nF) = 1 / (5.027 × 10-5) = 19,894 Ω
Nearest 1% value: 19.6 kΩ (E96 series).
Actual fc = 1/(2π × 19600 × 10-7) = 81.2 Hz — acceptable for crossover application.
Since K = 1 (voltage follower), connect the op-amp output directly to the inverting input. Rf is open and Rg is shorted — no gain-setting resistors needed.
For audio applications, choose a low-noise op-amp: NE5532 (noise = 5 nV/√Hz) or OPA2134 (FET input, excellent audio quality).
Sallen-Key filters are notoriously sensitive to op-amp GBW. If the op-amp’s open-loop gain drops near the cutoff frequency, the filter exhibits Q-enhancement — the actual Q becomes larger than designed, causing peaking and potential oscillation. Rule of thumb: select an op-amp with GBW ≥ 100 × fc for Butterworth and Bessel responses, and ≥ 1000 × fc for high-Q Chebyshev designs.
Using X7R or Z5U ceramic capacitors for the filter network is a common error. These dielectrics have voltage and temperature coefficients exceeding ±15%, causing the cutoff frequency to drift significantly. Always use NP0/C0G ceramics or film capacitors (polypropylene preferred) for the frequency-determining components. A 10% capacitor tolerance can shift fc by 10% and alter Q by 5–10%.
Setting R1 = R2 and C1 = C2 with K = 1 (unity gain) fixes Q at exactly 1/3, which is highly overdamped. The step response will be sluggish and the roll-off will be shallower than expected. For a Butterworth response (Q = 0.707), you must either use unequal components or set K = 1.586 with gain-setting resistors.
When building fourth-order or higher filters by cascading second-order stages, each stage must be designed with a different Q value for the overall response to be correct. For example, a fourth-order Butterworth requires two stages with Q1 = 0.541 and Q2 = 1.306 — not two identical Q = 0.707 stages. Using identical stages will produce incorrect roll-off shape and excessive passband ripple.
Sallen-Key filters are sensitive to parasitic capacitance between the non-inverting input and ground, and between the feedback path and output. Ensure the feedback resistor Rf is placed close to the op-amp, keep traces short, and use a ground plane. A stray capacitance of just 1 pF at the inverting node can cause measurable Q-enhancement above 10 kHz.
Use LTspice, PSpice, or Micro-Cap to simulate your Sallen-Key filter design before prototyping. Pay attention to: (1) AC analysis showing the actual −3 dB point vs. design target; (2) group delay to check for phase linearity; (3) Monte Carlo analysis to verify robustness with realistic component tolerances; and (4) transient response to confirm acceptable overshoot and settling time.
A: The Sallen-Key topology uses a non-inverting op-amp configuration with positive feedback through capacitors, offering higher input impedance and simpler design equations. The MFB (multiple-feedback) topology uses an inverting configuration with both positive and negative feedback, providing better high-frequency performance and lower sensitivity to op-amp GBW. MFB is preferred for high-Q designs and applications above 100 kHz, while Sallen-Key is the simpler choice for general-purpose filters below 100 kHz.
A: Cascade two second-order Sallen-Key stages. For a fourth-order Butterworth low-pass: Stage 1 with Q1 = 0.541 and fc1 = fc; Stage 2 with Q2 = 1.306 and fc2 = fc. Use filter design tables or polynomial factorization to find the pole locations for your target response (Bessel, Chebyshev, etc.). Each stage is designed independently using the same formulas but with different Q values.
A: While the classic Sallen-Key topology is optimized for low-pass and high-pass, band-pass and band-reject (notch) versions do exist. The Sallen-Key band-pass uses additional components, while notch filters often use the twin-T or Bainter topology. For band-pass applications, consider the multiple-feedback (MFB) or state-variable filter topologies instead — they offer more straightforward design equations and better performance.
A: Q determines the shape of the filter’s frequency response near the cutoff frequency. Low Q (<0.5) produces an overdamped response with gradual roll-off and no overshoot in the step response. Q = 0.707 (Butterworth) gives the flattest passband with minimal overshoot (~4.3%). Higher Q (>0.707) creates peaking at the cutoff frequency, sharper roll-off, and increased overshoot in the time domain. Q > 2 risks instability and excessive ringing.
A: Key selection criteria: (1) GBW at least 100× the cutoff frequency (1000× for high-Q designs); (2) Rail-to-rail output if operating from low supply voltages; (3) Low noise for precision applications (choose < 10 nV/√Hz); (4) FET/CMOS input for high-impedance designs (avoid bias current errors). Recommended devices: TL081/TL084 (general purpose), OPA2134 (audio), OPA1642 (high-performance audio), MCP602 (low-power), ADA4625 (precision, wide bandwidth).
A: Practically, Sallen-Key filters work well up to about 100 kHz. Beyond this, the op-amp’s finite GBW and phase shift cause significant Q-enhancement and response distortion. With a very high-speed op-amp (GBW > 100 MHz), cutoff frequencies up to 1 MHz are achievable but require careful PCB layout and compensation. For filters above 1 MHz, consider LC passive filters, active filters using current-feedback op-amps, or MFB topologies.
A: If you’re stuck with a marginally-fast op-amp, you can pre-distort the design: (1) Use the actual GBW value to calculate the expected Q-enhancement; (2) Reduce the design Q so the actual Q meets your target; (3) Add a small capacitor (1–10 pF) in parallel with Rf to roll off gain at high frequencies; or (4) Switch to the multiple-feedback (MFB) topology which is less sensitive to GBW effects.
A: Passive filters use only resistors, capacitors, and inductors — no power supply required, but they suffer from insertion loss, loading effects, and require bulky inductors at low frequencies. Active filters (like Sallen-Key) use op-amps to provide gain, high input impedance, low output impedance, and no inductors. Active filters are preferred below 100 kHz; passive LC filters dominate at RF frequencies where op-amp GBW becomes insufficient.
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This design guide is part of the Analog Electronics category, dedicated to continuous-signal circuit design and analysis. Topics span active and passive filters, operational amplifier circuits, sensor interfaces, signal conditioning, and precision analog system design. Each resource combines theoretical foundations with practical design examples to help engineers build reliable, production-ready analog circuits.