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.
Documentation
Related Engineering Resources
Guides and lookup references linked to this tool.
Engineering Guide
Active and Application-Specific Filter Topologies
Compare active, notch, PWM smoothing, anti-alias, and pi-filter implementations with loading, tolerance, and device-limit checks.
22 min · Intermediate
Engineering Guide
Butterworth Filter Design: Order, Poles, Stage Q, and Cascaded Verification
Design Butterworth filters from attenuation requirements through order, cutoff, normalized poles, stage Q, cascade implementation, tolerance, and verification.
18 min · Intermediate
Engineering Guide
Practical Filter Design: Cutoff Frequency, Q Factor, Bandwidth, and Damping
Learn cutoff frequency, -3 dB points, center frequency, bandwidth, Q factor, damping, filter order, passive and active tradeoffs, and real hardware limits.
18 min · Intermediate
Engineering References
Engineering Reference
Filter Response, Q, Bandwidth and Damping Reference
Lookup tables for filter response types, cutoff conventions, center frequency, bandwidth, Q, damping ratio, order, asymptotic slope, and approximation terminology.
14 min · Reference
Engineering Reference
Active Filter Topologies and Approximation Reference
Lookup for Sallen-Key, multiple-feedback, twin-T notch, Butterworth, Bessel, Chebyshev, elliptic, cascaded-stage, and ADC anti-alias filter terminology.
12 min · Reference
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.