In short: An active high-pass filter places a series capacitor and a shunt resistor (a CR network) in front of an op-amp so that signals above the cutoff frequency fc = 1/(2πRC) pass with gain while lower frequencies are attenuated at −20 dB/decade for a first-order stage and −40 dB/decade for a second-order Sallen–Key stage.
Blocking DC and low-frequency drift while letting the signal of interest through is one of the most common jobs in analog front-end design. A passive CR network does this, but it loads down the next stage and offers no gain. By pairing the CR network with an operational amplifier we get a buffered, gain-capable active high-pass filter. This article walks through the RC (strictly, CR) topology, derives the cutoff frequency, explains the roll-off slope, and compares first- and second-order designs with concrete component values an interviewer or a bench engineer would expect you to know.
The CR network and why order matters
A high-pass filter is simply a low-pass filter with the resistor and capacitor swapped. The capacitor sits in series with the signal path; the resistor shunts to ground. At DC the capacitor is an open circuit (XC = 1/(2πfC) → ∞), so no signal reaches the op-amp. As frequency climbs, XC shrinks until, well above fc, the capacitor looks like a short and the full signal appears across the resistor.
The order of the filter equals the number of reactive elements (capacitors, here) that set the response. A first-order stage uses one capacitor and rolls off at −20 dB per decade (−6 dB per octave). A second-order stage uses two and rolls off at −40 dB/decade, giving a far sharper transition between the stopband and the passband.
Cutoff frequency: the one equation to remember
The −3 dB cutoff is where the capacitor's reactance equals the resistance, XC = R. Solving 1/(2πfcC) = R gives:
fc = 1 / (2πRC) — at this frequency the output has dropped to 70.7% (−3 dB) of the passband value and the phase is +45°.
Worked example: for a microphone preamp that must reject rumble below roughly 160 Hz, pick C = 10 nF and R = 100 kΩ. Then fc = 1/(2π × 100,000 × 10×10−9) ≈ 159 Hz. Below 159 Hz the response falls at 20 dB/decade, so a 16 Hz power-line harmonic sits one decade down and is attenuated by about 20 dB. Choose the capacitor first from the E12 series, then solve for R — resistors come in finer increments and are cheaper to trim.
Gain in the passband
In the non-inverting configuration shown above, the passband (high-frequency) gain is set only by the feedback network, independent of the CR cutoff: Av = 1 + Rf/Rg. For unity gain, omit Rf and Rg and wire the output directly to the inverting input — a clean buffered high-pass. The full transfer function magnitude is:
|H(f)| = Av · (f/fc) / √(1 + (f/fc)²)
Watching the signal climb: animated response
Going second order: the Sallen–Key topology
When −20 dB/decade is not steep enough — say you need to reject a 50/60 Hz mains tone just below an audio band — cascade two reactive elements into a Sallen–Key stage. Two series capacitors and two resistors feed a single op-amp, with one resistor returned from the output to form a controlled positive feedback that shapes the response near fc.
The damping (and therefore the peaking) is set by the quality factor Q. Choosing component ratios for a Butterworth response (Q = 0.707) gives the flattest possible passband with no peaking; a Chebyshev ratio trades passband ripple for an even sharper knee; a Bessel ratio preserves phase linearity for pulse fidelity. For equal-component Sallen–Key (C1 = C2 = C, R1 = R2 = R), fc = 1/(2πRC) and Q is set by the gain: Q = 1/(3 − Av), so Av = 1.586 yields the Butterworth Q = 0.707.
Watch the stability margin: as Av → 3 in the equal-component design, Q → ∞ and the filter oscillates. Keep gain comfortably below 3, and in SPICE (Cadence Spectre or Synopsys HSPICE) run an AC sweep plus a transient to confirm there is no ringing on a step input.
| Property | First-order | Second-order (Sallen–Key) |
|---|---|---|
| Reactive elements | 1 capacitor | 2 capacitors |
| Roll-off slope | −20 dB/decade | −40 dB/decade |
| Phase shift at fc | +45° | +90° |
| Tunable Q / peaking | No (Q fixed) | Yes (set by gain) |
| Typical use | DC blocking, coupling | Band edge, anti-rumble |
Practical design and verification tips
- Pick C from the E12 series, solve for R. Capacitor values are coarse; let the resistor absorb the fine adjustment.
- Mind the op-amp GBW. At high frequencies the amplifier's own gain roll-off interacts with the filter — keep fc at least a decade below the gain-bandwidth product so the op-amp is not the limiting pole.
- Bias the input. For a single-supply design, add a resistive divider to set the non-inverting input at mid-rail, since the series capacitor blocks any DC bias path.
- Simulate, then lay out. Verify the AC response in a SPICE tool and check component tolerance with a Monte Carlo run before committing to a PCB. Siemens PSpice and Cadence Spectre both automate this.
The high-pass is the mirror image of the low-pass, so it pays to study them together — see our companion guide to the active low-pass filter, and the combined treatment in our active band-pass filter article where a high-pass and a low-pass stage are cascaded. If you want hands-on bench training on analog front-ends, our partner ChipXpert VLSI training runs lab-based analog design courses.
Ready to design real front-ends? The CourseTron Analog Circuit Design course takes you from CR networks to multi-stage Sallen–Key and multiple-feedback filters, with SPICE labs and interview-style problem sets.
Frequently asked questions
Why is it called a CR filter and not an RC filter?
In a high-pass stage the capacitor is in series (first) and the resistor shunts to ground (second), so the signal meets the C before the R — hence CR. A low-pass puts the R first and C to ground, giving the familiar RC order. The cutoff equation fc = 1/(2πRC) is identical either way.
What sets the gain of an active high-pass filter?
In the non-inverting topology the passband gain is Av = 1 + Rf/Rg, set entirely by the feedback resistors and independent of the cutoff-setting C and R. For a unity-gain buffer, omit both feedback resistors.
How much steeper is a second-order filter?
A first-order stage rolls off at −20 dB/decade; a second-order stage at −40 dB/decade. One decade below fc, a first-order filter attenuates by about 20 dB while a second-order filter attenuates by about 40 dB — a 100× difference in rejection.
Can the Sallen–Key high-pass oscillate?
Yes. In the equal-component design Q = 1/(3 − Av), so as the gain approaches 3 the Q rises without bound and the circuit becomes an oscillator. Keep the passband gain safely below 3 and verify with a transient step in SPICE.