Sallen-Key Filter Calculator

Filter response:

Choose the topology you want to analyze from entered component values.

R1:

Enter the first resistor in the frequency-setting network.

R2:

Use the same unit as R1 when you are checking an equal-value prototype.

R3 damping path:

Only used for the band-pass topology and ignored by low-pass/high-pass modes.

C1:

Enter the first capacitor and select its unit.

C2:

Match C1 for a quick equal-component check, or enter the actual schematic value.

Passband gain K: {{ gainValueLabel }}

Slider and exact value both update the Q, chart, checks, and export payload.

V/V

{{ error }}

Input amplitude:

Optional signal level for output swing rows.

Vpp

Op-amp GBW:

Leave the default unless you already know the op-amp gain-bandwidth product.

MHz

Resistor tolerance:

Enter the expected resistor tolerance as a percent.

Capacitor tolerance:

Use the datasheet or measured capacitor tolerance.

Display precision:

Adjust output precision without changing the calculation.

Part	Entered	SI value	Nearest E24	Nearest E96	Copy
{{ row.part }}	{{ row.entered }}	{{ row.si }}	{{ row.e24 }}	{{ row.e96 }}

Check	Current	Reference	Action	Copy
{{ row.check }}	{{ row.current }}	{{ row.reference }}	{{ row.action }}

Frequency	f / f0	Gain	Phase	Copy
{{ row.frequency }}	{{ row.normalized }}	{{ row.gain }}	{{ row.phase }}

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When a sensor front end, audio circuit, or analog-to-digital converter input needs more roll-off than one RC pole can provide, a Sallen-Key stage is a common compact answer. It uses an op amp with a two-pole resistor-capacitor network so useful frequencies pass while unwanted frequencies fall faster than they would through a first-order filter.

The circuit belongs to the voltage-controlled voltage-source family of active filters. Two resistors and two capacitors set the time constants, while the non-inverting amplifier path supplies gain and controlled positive feedback. That feedback is useful because it can raise Q above what a passive two-stage RC network can reach, but it also makes high-Q designs more sensitive to component values, op amp bandwidth, and output swing.

Natural frequency f0: The pole-pair frequency set mainly by the R and C product. In a band-pass stage it is read as the center frequency.
Quality factor Q: The damping measure. Lower Q gives a gentler response, while higher Q creates peaking and a narrower transition.
Passband gain K: The non-inverting gain around the op amp. Raising K can also raise Q, so gain is not just an output-level choice.
Component spread: The ratio between large and small same-type parts. Wide spreads can make leakage, noise, and parasitic effects harder to ignore.

Sallen-Key RC network feeding an op amp, with feedback shaping Q and the RC path setting f0. — A Sallen-Key second-order stage uses an RC network and amplifier feedback to set natural frequency and Q.

Low-pass, high-pass, and band-pass Sallen-Key stages share the same second-order idea, but the parts shape different parts of the response. In low-pass and high-pass work, f0 is the natural frequency of the pole pair and is often close to the familiar cutoff only for a Butterworth-like Q near 0.707. In band-pass work, f0 is the center frequency and the bandwidth is tied directly to Q.

The common mistake is treating the nominal frequency as the finished circuit. Real resistors and capacitors come from preferred-value series and tolerances, capacitors change with temperature and voltage, and the op amp has finite open-loop gain, gain-bandwidth product, slew rate, common-mode range, and output swing. A design that looks tidy on paper can peak, shift, clip, or miss its target after parts and layout are included.

Sallen-Key math is most useful as an early screening step. It narrows component choices, exposes damping problems, and shows whether the selected op amp has enough room to behave like the ideal amplifier assumed by the equations. Simulation and bench measurement still matter for high-Q, high-frequency, high-output-swing, or production-critical filters.

How to Use This Tool:

Start with the topology and actual part values, then use the checks and curve to decide what needs verification.

Choose Filter response: Low-pass, High-pass, or Band-pass. Band-pass mode adds R3 damping path.
Enter R1, R2, C1, and C2 with their units. Use the schematic values you intend to analyze, not only rounded target values.
Adjust Passband gain K. The exact value field and slider update the summary, Q, chart, and tables together. Keep K between 1.00 and 2.90 V/V.
Raising K can sharpen Q and peaking, so treat gain as part of the filter shape rather than only an output-level control.
Open Advanced when you want output swing, op-amp headroom, tolerance spread, or display precision. Input amplitude affects only the output swing estimate, and Display precision changes formatting only.
Read the summary for f0, Q, peak gain, and the response label. If validation appears, fix zero or negative R/C values, out-of-range K, negative input amplitude, invalid GBW, or a non-positive damping result before trusting any output.
The non-positive damping warning means the entered gain and component ratios do not produce a stable ideal second-order denominator.
Use Component Ledger to compare entered values with nearest E24 and E96 values, then use Design Checks for natural or center frequency, Q profile, feedback ratio, GBW headroom, tolerance span, part ratio spread, and output swing.
Open Gain Response Curve and Sweep Points to inspect gain and phase around f0. Use JSON when you need the same result data for documentation or another calculation.

Interpreting Results:

The headline frequency is the natural frequency for low-pass and high-pass stages, and the center frequency for band-pass mode. For a Butterworth-like second-order section, f0 is close to the familiar -3 dB corner. When Q is much lower or higher, the response curve is the safer place to inspect the usable passband and peaking.

How to read Sallen-Key filter calculator outputs
Output	What to check	Common misread
f0	Natural or center frequency from the entered R and C values.	Assuming it is always the final measured cutoff frequency.
Q profile	Damping and peaking. Near 0.707 is a common flat-response target.	Reading high Q as better filtering without checking tolerance and op-amp limits.
GBW headroom	Screening ratio between entered op-amp GBW and the gain, Q, and frequency demand.	Treating a pass as proof of stability, slew rate, or noise performance.
Tolerance span	Worst-case f0 movement from resistor and capacitor tolerances.	Assuming nominal E-series parts land exactly on the calculated response.
Gain Response Curve	Log-frequency gain curve with f0 and -3 dB reference markers.	Ignoring peak gain when checking supply headroom.

A green-looking headroom number should still lead to a model check for demanding circuits. Verify the chosen op amp's gain-bandwidth product, slew rate, input common-mode range, output swing, load drive, and noise against the real operating point.

When the summary says Damped response, the stage may be intentionally gentle. When it says Peaking response, compare the Output swing estimate with supply rails and rerun the calculation with tighter component tolerances before moving on.

Technical Details:

A Sallen-Key stage is normally reduced to a second-order transfer function. The denominator can be written as 1 + a s + b s^2, where b sets the time scale and a sets damping. A smaller positive a raises Q; a non-positive a means the ideal pole pair is not safely damped.

Low-pass and high-pass forms use the same R1, R2, C1, C2, and K pattern with the resistor and capacitor roles arranged for the desired passband. The band-pass form adds a damping resistor and uses bandwidth as f0 / Q. Its response numerator also depends on K, C2, R2, and the R1/R2 ratio, so the plotted gain can change even when the denominator frequency scale is unchanged. These equations describe the ideal small-signal filter, so op amp non-ideal behavior is a separate check rather than part of the denominator formula.

Formula Core:

Values are first converted to ohms and farads. The main frequency and Q equations are:

\begin{array}{lcl} f_{0} & = & \frac{1}{2 π \sqrt{b}} \\ Q & = & \frac{\sqrt{b}}{a} \\ BW & = & \frac{f_{0}}{Q} \end{array}

The topology determines a and b:

\begin{array}{lcl} Low-pass b & = & R_{1} R_{2} C_{1} C_{2} \\ Low-pass a & = & C_{2} (R_{1} + R_{2}) + C_{1} R_{1} (1 - K) \\ High-pass b & = & R_{1} R_{2} C_{1} C_{2} \\ High-pass a & = & (1 - K) R_{2} C_{2} + R_{1} C_{2} + R_{1} C_{1} \\ Band-pass b & = & \frac{C_{1} C_{2} R_{1} R_{2}}{1 + \frac{R_{1}}{R_{2}}} \\ Band-pass a & = & C_{2} R_{2} (1 + \frac{R_{1}}{R_{3}} - (K - 1) \frac{R_{1}}{R_{2}}) \end{array}

With equal low-pass or high-pass parts, R1 = R2 = R and C1 = C2 = C, the equations simplify to f0 = 1 / (2*pi*R*C) and Q = 1 / (3 - K). That is why K = 1.586 gives a Butterworth-like Q near 0.707, while values close to 3 become very sensitive. For the band-pass form with equal R and C values, the denominator product is halved by the 1 + R1/R2 term, which moves the center frequency upward compared with the equal-part low-pass case.

Variables used in the Sallen-Key formulas
Symbol	Meaning	Units or note
`R1`, `R2`, `R3`	Frequency-setting resistors, with R3 used only by the band-pass damping path.	Converted to ohms before calculation.
`C1`, `C2`	Frequency-setting capacitors in the Sallen-Key network.	Converted to farads before calculation.
`K`	Non-inverting passband gain of the amplifier stage.	Dimensionless V/V gain; accepted range is 1.00 to 2.90.
`a`, `b`	Second-order denominator coefficients.	`a` controls damping; `b` controls frequency scale.
`Q`	Quality factor of the pole pair.	Higher values mean more peaking and tighter bandwidth.

A common equal-part check shows the scale. With R = 10 kohm and C = 10 nF, 2*pi*R*C is about 0.000628, so f0 is about 1.592 kHz. With K = 1.586, Q = 1 / (3 - K), which lands near 0.707.

Sallen-Key design checks and validation boundaries
Check	Rule used	Interpretation
Valid damping	`a > 0` and `b > 0`	Required before frequency, Q, and curve outputs are shown.
Q labels	`Q < 0.55`, `0.64 to 0.78`, and `Q >= 1.2` drive the summary labels.	Use the label as a warning cue, not as a formal filter-family classification.
GBW headroom	`op-amp GBW / (f0 * K * max(1, Q))`	The table treats 20x or more as comfortable for an initial screen.
Tolerance span	Worst-case R and C products are recomputed using the entered tolerances.	Shows how far f0 can move before op amp and layout errors are considered.
Part ratio spread	Largest same-type resistor ratio and capacitor ratio reported separately.	Very large spreads increase leakage, noise, and parasitic sensitivity.
Output swing	Input amplitude times the larger of passband gain or peak curve gain.	Compare against supply rails and load drive limits.

The nearest E24 and E96 values are preferred-value comparisons for part selection. They do not retune the circuit automatically. If you substitute rounded values, rerun the calculation with the values you will actually buy or measure.

Limitations and Privacy:

The calculation is an ideal small-signal estimate. It does not include finite op amp open-loop response, slew rate, output current, rail limits, temperature drift, noise, capacitor voltage coefficient, PCB parasitics, or measured part spread.

Use SPICE or bench measurement for high-Q, high-frequency, high-output-swing, or production-critical stages.
Enter realistic GBW and tolerances before using the checks as a design review aid.
The entered component values are calculated in the browser and are not uploaded for a remote calculation.

Worked Examples:

Equal-part low-pass stage

With Low-pass, R1 = 10 kohm, R2 = 10 kohm, C1 = 10 nF, C2 = 10 nF, and K = 1.586 V/V, the summary reports f0 near 1.592 kHz and Q near 0.707. The Q profile sits near the Butterworth target, so the Gain Response Curve should look flat before the roll-off.

Band-pass damping check

Switching to Band-pass with R1 = R2 = R3 = 10 kohm, C1 = C2 = 10 nF, and K = 1.586 V/V moves the reported center frequency to about 2.251 kHz and gives Q near 0.500. That is a broad, damped passband rather than a narrow resonator.

Non-positive damping recovery

A low-pass entry such as R1 = R2 = 10 kohm, C1 = 100 nF, C2 = 10 nF, and K = 2.90 V/V can trigger the warning that gain and component ratios produce non-positive damping. Lower K or rebalance the capacitor ratio before reading f0, Q, or sweep rows.

FAQ:

Is f0 the same as cutoff frequency?

Not always. For a Butterworth-like second-order low-pass or high-pass stage, f0 is close to the -3 dB point. With other Q values, inspect the Gain Response Curve and -3 dB reference marker.

Why does increasing K raise the risk of peaking?

In Sallen-Key stages, amplifier gain participates in the damping term. Raising K can increase Q, and excessive Q can make the ideal response peak or become invalid for the entered ratios.

Why is R3 shown only for band-pass mode?

The low-pass and high-pass equations use R1, R2, C1, C2, and K. Band-pass mode also uses R3 damping path, so that extra resistor is requested only for that topology.

Do the nearest E24 and E96 columns change the result?

No. They compare entered resistor and capacitor values with common preferred-value series. To see the real effect of a substituted part, enter that chosen value and rerun the calculation.

Can this replace circuit simulation?

No. It gives an ideal Sallen-Key screening result with helpful checks. Use op amp simulation and measurement when GBW, slew rate, supply swing, tolerance, noise, or layout parasitics matter.

Glossary:

Natural frequency: The second-order frequency set by the denominator coefficient b; shown as f0.
Q: Quality factor, a damping measure that controls peaking and bandwidth.
Passband gain K: The non-inverting gain value used by the Sallen-Key stage.
GBW: Gain-bandwidth product, an op amp limit that affects whether the ideal response is realistic.
E24 and E96: Preferred resistor and capacitor value series used for nearest-value comparisons.
Bandwidth: For band-pass mode, the center frequency divided by Q.

References:

Analysis of the Sallen-Key Architecture, Texas Instruments, September 2002.
Low-pass Sallen-Key filter circuit, Texas Instruments, June 28, 2021.
AN-649: Using the Analog Devices Active Filter Design Tool, Analog Devices.
ADALM2000 Activity: Active Filtering - Part 2, Analog Devices, July 2024.
IEC 60063:2015 Preferred number series for resistors and capacitors, International Electrotechnical Commission, March 27, 2015.