Sallen-Key High-Pass Filter Calculator
This calculator designs and analyzes ideal second-order Sallen-Key high-pass filters using a documented VCVS topology convention. It calculates natural frequency, Q, damping ratio, high-frequency passband gain, passband-relative -3 dB frequency, phase, op-amp gain resistors, and sweep tables.
It is not a duplicate of the first-order active high-pass calculator. The first-order page sizes a simple active RC stage; this page models a second-order Sallen-Key topology where Q depends on component ratios and non-inverting op-amp gain.
Engineering tool
Sallen-Key High-Pass Filter Calculator
Design and analyze ideal second-order Sallen-Key high-pass filters, including natural frequency, Q, damping, Butterworth gain, passband-relative cutoff, op-amp gain resistors, and frequency response.
First resistor in the adopted Sallen-Key denominator convention.
Second resistor in the adopted Sallen-Key denominator convention.
Capacitor C1 in the adopted Sallen-Key topology convention.
Capacitor C2 in the adopted Sallen-Key topology convention.
Frequency where the transfer function is evaluated.
Result console
- Natural frequency
- 1.591549kHz
- Q factor
- 0.70710656
- Damping ratio
- 0.707107
- High-frequency passband gain
- 1.585786
- High-frequency passband gain
- 4.0049dB
- Gain at frequency
- 1.12132009
- Gain at frequency
- 0.9946dB
- Passband-normalized gain
- 0.70710682
- Passband-normalized gain
- -3.0103dB
- Phase
- 90°
- Passband-relative -3 dB frequency
- 1.59155kHz
- Peak status
- No peaking
- Classification
- Butterworth high-pass response
This V1 model assumes an ideal op-amp. Real high-frequency response is limited by GBW, slew rate, output swing, parasitics, source impedance, and load impedance.
The high-pass passband gain is the high-frequency gain K; -3 dB is referenced to that passband gain.
Formula reference
Sallen-Key High-Pass Formulas
Adopted topology convention: standard VCVS Sallen-Key high-pass with high-frequency passband gain K = 1 + Rf/Rg. The denominator uses the same documented R1, R2, C1, C2, and K convention as the matching Sallen-Key low-pass tool; the high-pass numerator is the K a2s² term.
H(s) = K a2s² / (a2s² + a1s + 1)a2 = R1R2C1C2a1 = C2(R1 + R2) + C1R1(1 - K)ω0 = 1 / √a2f0 = 1 / (2π√(R1R2C1C2))Q = √a2 / a1ζ = 1 / (2Q)Equal components: Q = 1 / (3 - K)Butterworth equal components: K = 3 - √2 ≈ 1.585786High-pass normalized response: Hn(s) = (s/ω0)² / ((s/ω0)² + s/(Qω0) + 1)Passband-relative -3 dB: |H| / K = 1 / √2Gain dB = 20log10(|H|)Variable definitions
- R1
- First Sallen-Key resistor in the adopted denominator convention
- R2
- Second Sallen-Key resistor in the adopted denominator convention
- C1
- Capacitor C1 in the adopted topology convention
- C2
- Capacitor C2 in the adopted topology convention
- K
- Non-inverting high-frequency passband gain
- f0
- Natural or pole frequency
- Q
- Quality factor
- ζ
- Damping ratio
- H(jω)
- High-pass voltage transfer function
Worked Examples
Equal components f0
Known: R1 = R2 = 10 kΩ, C1 = C2 = 10 nF
f0 = 1/(2πRC) ≈ 1591.55 Hz.
Unity-gain equal components
Known: K = 1
Q = 1/(3 - K) = 0.5, not Butterworth.
Butterworth gain
Known: Q = 1/√2
K = 3 - 1/Q ≈ 1.58578644.
Butterworth response at f0
Known: R = 10 kΩ, C = 10 nF, K ≈ 1.585786
Passband-normalized magnitude at f0 is approximately 0.70710678.
Butterworth -3 dB point
Known: Q = 1/√2
Passband-relative f3dB equals f0.
Low-frequency rejection
Known: f = f0 / 100
The high-pass gain is near zero because the numerator contains s².
High-frequency passband
Known: f = 100 × f0
Absolute gain approaches K and normalized gain approaches 1.
K = 2 equal components
Known: R1 = R2, C1 = C2, K = 2
Q = 1, so response peaking and ringing risk should be reviewed.
Invalid gain boundary
Known: K approaches 3 with equal components
The damping coefficient approaches zero and the ideal model becomes impractical.
Solve R
Known: f0 = 1 kHz, C = 10 nF
R = 1/(2πf0C) ≈ 15.9155 kΩ.
Solve C
Known: f0 = 10 kHz, R = 10 kΩ
C = 1/(2πf0R) ≈ 1.59155 nF.
Solve Rf
Known: K ≈ 1.585786, Rg = 10 kΩ
Rf = (K - 1)Rg ≈ 5.85786 kΩ.
Solve Rg
Known: K ≈ 1.585786, Rf ≈ 5.85786 kΩ
Rg = Rf/(K - 1) ≈ 10 kΩ.
Non-Butterworth cutoff
Known: Q = 1 instead of 0.707
The passband-relative -3 dB frequency is not forced to equal f0.
Phase at low frequency
Known: f << f0
The ideal transfer phase approaches approximately 180 degrees.
Phase at high frequency
Known: f >> f0
The ideal transfer phase approaches approximately 0 degrees.
Frequency unit equivalence
Known: 1000 Hz and 1 kHz
Both evaluate the same transfer function frequency.
Unequal components
Known: R1, R2, C1, C2 not equal
f0, Q, damping, and response are calculated from denominator coefficients.
Sallen-Key Filter
A Sallen-Key stage is an active second-order filter using an op-amp as a VCVS gain element.
High-Pass Behavior
The ideal high-pass response rejects DC and approaches K at high frequency.
Natural Frequency
f0 depends on R1, R2, C1, and C2 through the product R1R2C1C2.
Q Factor
Q depends on component ratios and K. It is not controlled by f0 alone.
Butterworth Response
For the equal-component convention used here, Butterworth response requires K ≈ 1.585786.
Unity Gain
Equal R and C with unity gain gives Q = 0.5, so it is not Butterworth.
Passband Reference
High-pass -3 dB frequency is solved relative to the high-frequency passband gain K.
Op-Amp Limits
GBW, slew rate, output swing, common-mode range, noise, and bias current affect practical response.
Tolerance
Capacitor and resistor tolerance shift f0 and Q; high-Q designs are especially sensitive.
Phase
The ideal high-pass phase moves from about 180 degrees at very low frequency toward 0 degrees in the passband.
Common Mistakes
Support reference
FAQ
What is a Sallen-Key high-pass filter?
A Sallen-Key high-pass filter is an active second-order filter topology that uses two resistors, two capacitors, and a non-inverting op-amp gain stage to attenuate low-frequency content and pass higher-frequency content.
How is this different from a first-order active high-pass filter?
A first-order active high-pass filter has one pole and usually depends on one RC cutoff. This calculator models a second-order Sallen-Key stage where natural frequency, Q, damping, gain, phase, and passband-relative cutoff behavior are all important.
What transfer function convention does this calculator use?
The calculator uses H(s) = K a2s² / (a2s² + a1s + 1), where a2 = R1R2C1C2 and a1 = C2(R1+R2)+C1R1(1-K). The numerator is the high-pass term, while the denominator convention matches the documented Sallen-Key VCVS model.
How do I calculate the natural frequency?
Natural frequency is f0 = 1/(2π√(R1R2C1C2)). It is the pole frequency of the second-order stage, not always the same as the passband-relative -3 dB frequency.
How is Q calculated?
The calculator uses Q = √a2/a1 from the adopted denominator coefficients. For equal R and equal C, the simplified relationship is Q = 1/(3-K).
What gain gives a Butterworth high-pass response?
For the equal-component convention used here, Butterworth Q = 1/√2 requires K = 3 - √2, approximately 1.585786.
Why is the -3 dB frequency relative to passband gain?
A high-pass Sallen-Key stage approaches K at high frequency. The -3 dB point is therefore solved relative to that high-frequency passband gain, not as a fixed absolute magnitude of 0.707 unless K = 1.
What happens at very low frequency?
The ideal high-pass numerator contains s², so gain approaches zero as frequency approaches DC. Phase approaches approximately 180 degrees at very low frequency and approaches 0 degrees in the high-frequency passband.
Can equal components with unity gain produce Butterworth response?
No. Equal R and equal C with K = 1 gives Q = 0.5, which is more damped than Butterworth Q = 0.70710678.
Why does op-amp gain affect high-pass Q?
In this Sallen-Key convention, non-inverting gain K appears in the damping coefficient. Increasing K raises Q and can make the response peak or become sensitive to component tolerance.
How much op-amp bandwidth is required?
The op-amp must have enough gain-bandwidth, slew-rate, input common-mode range, and output swing margin for the intended frequency and amplitude. Always verify the ideal result with the op-amp datasheet and simulation for critical designs.
Can this calculator analyze unequal components?
Yes. Analyze, natural-frequency, Q/gain, and sweep modes support unequal R1, R2, C1, and C2 values using the coefficient-based model.
Planned Engineering Guides
Planned guide
Sallen-Key Filters Explained
Planned guide
Second-Order High-Pass Filter Design
Planned guide
Butterworth Active Filter Design
Planned guide
Understanding Active Filter Q
Planned guide
Op-Amp Bandwidth in Active Filters
Planned guide
Component Tolerance in Sallen-Key Filters
Planned Filter Calculators
Multiple-Feedback Band-Pass Filter Calculator
Butterworth Filter Calculator
Bessel Filter Calculator
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Engineering Disclaimer
This calculator uses an ideal Sallen-Key high-pass model. It assumes ideal op-amp behavior, sufficient GBW and slew rate, no output saturation, no common-mode violation, ideal components, and no PCB parasitics. Validate critical designs with op-amp datasheets, SPICE, tolerance analysis, and bench measurement.