Sallen Key Calculator: Low-Pass Formulas & Worked Examples

The Unity-Gain Sallen-Key Low-Pass Formula & Symbol Definitions

When designing active filters, the unity-gain Sallen-Key low-pass topology is the bench standard. It requires only one op-amp, two resistors, and two capacitors, while avoiding the stability issues introduced by positive feedback in high-gain variants. To use a Sallen Key calculator effectively, you must start with the foundational transfer function parameters: the cutoff frequency (-3dB point) and the Quality factor (Q), which dictates the filter alignment (Butterworth, Bessel, Chebyshev).

The governing equations for the standard unity-gain topology are:

• Cutoff Frequency: fc = 1 / (2π × √(R1 × R2 × C1 × C2))
• Quality Factor: Q = √(R1 × R2 × C1 × C2) / [C2 × (R1 + R2)]

Rearranged Forms & The Component Selection Decision Tree

Plugging random values into a Sallen Key calculator rarely yields standard component values. In practice, designers force a ratio to simplify the math. The most robust method for physical builds is the Equal-Resistor Design, where we set R1 = R2 = R. This minimizes the variety of resistor values you need to stock and keeps impedance balanced.

When R1 = R2 = R, the formulas collapse into these highly practical rearranged forms:

• Solve for C₁: C1 = 4 × Q2 × C2
• Solve for R: R = 1 / (4 × π × fc × Q × C2)
• Solve for C₂ (if R is fixed): C2 = 1 / (4 × π × fc × Q × R)
• Solve for f_c (verification): fc = 1 / (2 × π × R × √(C1 × C2))

The Decision Path: From Spec to BOM

Follow this exact sequence to terminate your design with concrete, purchasable part numbers:

1. Choose Q: Select based on application. Butterworth (Q = 0.707) for flat passband audio; Bessel (Q = 0.577) for linear phase/ADC anti-aliasing; Chebyshev (Q > 0.707) for steep roll-off where passband ripple is acceptable.
2. Pick C₂: Choose a standard E12/E24 capacitor value that keeps impedance in the 1kΩ–100kΩ sweet spot. (Usually 1nF to 100nF).
3. Calculate C₁: Use C1 = 4 × Q2 × C2. Round to the nearest standard E24 value.
4. Calculate R: Use the rearranged R formula. Round to the nearest E96 (1%) resistor value for precision filters.
5. Verify: Recalculate actual f_c using your rounded, standard values to ensure the error is within your system tolerance.

Worked Example 1: 2 kHz Audio Crossover (Butterworth)

Target: f_c = 2,000 Hz, Butterworth alignment (Q = 0.7071).
Application: Midrange driver protection in a hi-fi speaker crossover.

Step 1: Pick C₂
For audio frequencies, we want to avoid electrolytic capacitors due to ESR and distortion. We target film or C0G ceramics in the nanofarad range. Let's choose C2 = 10 nF (which is 10 × 10-9 F in base SI units).

Step 2: Calculate C₁
C1 = 4 × (0.7071)2 × 10 nF
C1 = 4 × 0.5 × 10 nF = 20 nF
20 nF is a standard E24 value. We will use C1 = 20 nF (20 × 10-9 F).

Step 3: Calculate R
R = 1 / (4 × π × 2000 × 0.7071 × 10 × 10-9)
R = 1 / (0.00017771)
R = 5,627 Ω
The nearest standard E24 1% resistor is 5.60 kΩ (5600 Ω).

Step 4: Verify Actual Cutoff
Using R = 5600 Ω, C₁ = 20nF, C₂ = 10nF:
fc = 1 / (2 × π × 5600 × √(20 × 10-9 × 10 × 10-9))
fc = 1 / (35185 × 1.414 × 10-7) = 2,010 Hz
Result: A 0.5% deviation from target. Perfectly acceptable for audio crossovers.

Worked Example 2: 50 kHz ADC Anti-Aliasing Filter (Bessel)

Target: f_c = 50,000 Hz, Bessel alignment (Q = 0.577) for linear phase response to prevent step-response ringing in digitized sensor data.
Application: Pre-filtering a piezoelectric vibration sensor feeding a 16-bit SAR ADC.

Step 1: Pick C₂
At 50 kHz, we need smaller capacitors to keep resistor noise manageable. Choose C2 = 1 nF (1 × 10-9 F).

Step 2: Calculate C₁
C1 = 4 × (0.577)2 × 1 nF
C1 = 4 × 0.3329 × 1 nF = 1.331 nF
The nearest standard E24 value is 1.3 nF (1.3 × 10-9 F).

Step 3: Calculate R
R = 1 / (4 × π × 50000 × 0.577 × 1 × 10-9)
R = 1 / (0.00036254)
R = 2,758 Ω
For a 16-bit ADC, we need tight tolerance. We select the nearest E96 (1%) value: 2.74 kΩ (2740 Ω).

Step 4: Verify Actual Cutoff
fc = 1 / (2 × π × 2740 × √(1.3 × 10-9 × 1 × 10-9))
fc = 1 / (17216 × 1.140 × 10-9) = 50,968 Hz
Result: The corner shifted up by ~960 Hz (1.9%). If the ADC samples at 200 kSPS, this 51 kHz corner still provides adequate attenuation above the 100 kHz Nyquist limit.

Unit Traps, Parasitics, and Op-Amp GBW Limits

A Sallen Key calculator is only as good as the numbers you feed it and the physical reality of your workbench. Here is where designs fail in practice:

The Unit Scaling Trap

The most common error is plugging "10" into the calculator for a 10nF capacitor, or mixing kΩ with μF without tracking the powers of ten. Always convert to base SI units (Ohms, Farads, Hertz) before calculating. If you use 10nF, type 0.00000001 or 10e-9. A single missed micro- prefix will result in a calculated resistor value that is off by a factor of one million.

Parasitic Reality: Capacitor Dielectrics

Do not use X7R or Y5V ceramic capacitors for the C₁ and C₂ positions in precision or audio filters. X7R dielectrics exhibit severe voltage coefficient (capacitance drops as applied voltage increases) and microphonic piezoelectric effects.

• For f_c < 100 kHz: Use C0G/NP0 ceramics, polypropylene film, or polyester film. C0G holds its tolerance to ±5% regardless of voltage or temperature.
• For f_c > 100 kHz: C0G/NP0 is mandatory. Stray PCB capacitance (typically 2pF to 5pF per pad) will alter your Q if your calculated C values drop below 100pF. Keep C > 470pF to swamp out PCB parasitics.

The Op-Amp GBW Bottleneck

The mathematical derivation assumes the op-amp has infinite bandwidth. In reality, the op-amp's internal pole introduces phase shift that degrades the filter's Q. According to Texas Instruments SLOA088 Active Filter Design Techniques, the op-amp's Gain-Bandwidth Product (GBW) must satisfy:

GBW > 100 × fc × Q

If you are building the 50 kHz Bessel filter (Example 2), your required GBW is 100 × 50,000 × 0.577 = 2.88 MHz. A standard LM358 (GBW ~1 MHz) will fail here, causing severe passband droop. You must step up to an OPA2134 (8 MHz) or TLV2171 (3 MHz).

Final Component Recommendation & BOM

Stop guessing and standardizing on sub-par components. For a universal, high-performance Sallen-Key low-pass stage that covers DC to 100 kHz with minimal distortion and low noise, procure the following specific BOM:

By anchoring your design to the equal-resistor rearranged formulas and strictly enforcing C0G dielectrics and E96 resistor tolerances, your physical breadboard or PCB will match your Sallen-Key active filter simulations within 2%, eliminating the need for iterative bench tuning.