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IN OUT K {{ topologyLabel }}
The circuit map, equation path, result labels, and response curve change together.
Enter the schematic value and its printed unit.
Use the actual schematic value rather than a rounded target.
Required only for the Sallen-Key band-pass topology.
Changing the unit preserves the physical capacitance.
Enter the value used in the selected topology.
Type an exact gain or tune the slider from 1.00 to 2.90 V/V.
{{ formatNumber(Number(gain_k), 3) }} V/V
V/V
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Leave zero when output swing is not part of this screen.
Vpp
Enter the datasheet gain-bandwidth product, or leave zero.
MHz
Zero keeps the nominal frequency result unchanged.
%
Zero disables capacitor-driven frequency spread.
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Op-Amp Gain Calculator Calculate closed-loop op-amp gain or solve a resistor value and check output swing against speed limits and resistor tolerance under the entered load.
Resonant Frequency Calculator Calculate LC resonant frequency from inductance and capacitance, then check tolerance limits, loaded Q, bandwidth, and reactance.
Attenuator Pad Calculator Design a matched Pi or T attenuator from target loss and impedance, then audit preferred resistor values, input match and power rating.
Impedance Calculator Calculate ideal series RLC impedance from reactances or component values with phase, power factor, current and resonance results.

A second-order active filter shapes more than a cutoff. Its pole frequency sets where the response turns, its quality factor Q sets damping or peaking, and its amplifier gain can change both the passband level and the stability of the chosen component ratios. Those relationships make a Sallen–Key stage useful for audio, sensor conditioning, and analog front ends, but also sensitive to seemingly small part changes.

Low-pass stages retain lower frequencies and attenuate higher ones. High-pass stages do the reverse. A band-pass stage favors a region around a center frequency. In each case, the same resistor and capacitor labels have meaning only in the matching topology; copying values from one schematic arrangement into another can produce the wrong frequency or non-positive damping.

QuantityPractical meaningWhat changes it
Natural frequency f0The pole-pair reference frequencyR/C products and, for band-pass, the R3 damping path
Quality factor QDamping and resonance around f0Component ratios and closed-loop gain K
Bandwidth f0/QA second-order width referenceBoth f0 and Q
Gain-bandwidth headroomA first-pass check on amplifier speedOp-amp GBW, f0, Q, and K

Nominal equations assume an ideal small-signal op amp and exact components. Real response also depends on gain-bandwidth product, slew rate, output swing, input and output loading, noise, parasitics, and resistor and capacitor tolerances. High-Q designs amplify those departures, so calculation is a starting point for simulation and bench measurement rather than a substitute for either.

How to Use This Tool:

Start from the exact topology and the component values printed on its schematic.

  1. Choose Low-pass, High-pass, or Band-pass. Enter R1, R2, C1, and C2 with their units; band-pass also requires the R3 damping path.
  2. Enter the non-inverting Closed-loop gain K from 1.00 to 2.90 V/V. If the result reports non-positive damping, reduce K or change the component ratios.
  3. Use Advanced to add input amplitude, op-amp GBW, and resistor and capacitor tolerances. Zero leaves the corresponding swing, headroom, or frequency-spread check inactive.
  4. Review Stage results and Design audit, then inspect the Response curve. Verify promising values with the intended op-amp model and real component tolerances.

Interpreting Results:

Natural frequency f0 is the pole-pair reference, not automatically the exact −3 dB point for every Q and gain. Bandwidth is reported as f0/Q. Use the gain and phase response to see what the selected topology does at 0.1, 0.5, 1, 2, and 10 times f0.

A Q near 0.707 is close to the familiar second-order Butterworth target. Q at or above 1.2 is flagged as sensitive because amplifier and component errors can produce large response changes. The displayed peak is the largest of the five sampled response points, so it may not equal the true continuous-frequency peak.

A GBW headroom ratio of 20× or more receives a comfortable first-pass note, but this is a screening rule, not a guarantee. Check the op amp's open-loop response, slew rate, noise, output swing, load drive, and stability in the final circuit.

Technical Details:

The component values are first converted to ohms and farads. Each topology then produces two positive denominator coefficients, A and B, from which natural frequency and Q follow. A non-positive A or B is rejected because the selected gain and ratios do not describe a stable damped pole pair in this model.

Formula Core

Once A and B are known, the principal stage results are:

f0=12πB Q=BA Bandwidth=f0Q GdB=20log10(G)

Rule Core

Low-pass and high-pass share the same B term but use different damping terms. Band-pass introduces R3 and an intermediate angular-frequency expression.

TopologyAB or angular-frequency rule
Low-passR1C1 + R2C1 + R1C2(1 − K)R1R2C1C2
High-passR2C2 + R2C1 + R1C2(1 − K)R1R2C1C2
Band-passThe normalized s coefficient divided by ω02ω02 = (1/R1 + 1/R2) / (R3C1C2), then B = 1/ω02

For band-pass, the s coefficient is [C1/R3 + (C1 + C2)/R1 + C2(1 − K)/R2] / (C1C2). Center gain is KB/(R1C2A). Low-pass and high-pass use K as passband gain.

The normalized frequency ratio r = f/f0 gives a denominator with real part 1 − r2 and imaginary part r/Q. The low-pass numerator is K, the high-pass numerator is −Kr2, and the band-pass numerator is imaginary and proportional to r/Q. Magnitude becomes 20 log10|H| in decibels, while phase comes from the complex numerator angle minus the denominator angle.

Tolerance screening assumes every resistor and capacitor moves to the same worst-case limit. For resistor tolerance tR and capacitor tolerance tC, the nominal frequency is divided by (1 + tR)(1 + tC) for the low bound and by (1 − tR)(1 − tC) for the high bound. This does not model Q sensitivity or statistical tolerance distributions.

The GBW ratio is GBW/[f0 × max(1,Q) × max(1,K)]. If input amplitude is supplied, output peak-to-peak voltage uses the largest sampled gain. Display precision changes formatting only; the calculations retain full numeric precision.

Limitations:

The response is an ideal second-order small-signal estimate. It does not replace a SPICE model using the chosen amplifier or hardware verification.

  • The response contains five normalized sample frequencies, not a dense analytic sweep.
  • The tolerance window moves f0 uniformly and does not calculate independent part combinations, Monte Carlo spread, or Q variation.
  • Input bias currents, source and load impedance, thermal and op-amp noise, slew limiting, clipping, parasitic capacitance, and PCB layout are outside the model.

Worked Examples:

Equal-part low-pass near Butterworth Q

With R1 = R2 = 10 kΩ, C1 = C2 = 10 nF, and K = 1.586, the nominal natural frequency is about 1.592 kHz and Q is about 0.7072. The near-Butterworth note is useful, but the five-point response and ideal op-amp assumption still require simulation with the intended amplifier and component tolerances.

References: