In short: An active low-pass filter uses an op-amp to pass signals below a cutoff frequency fc = 1/(2πRC) while rolling off higher frequencies — a first-order stage gives −20 dB/decade and a second-order Sallen-Key gives −40 dB/decade with a tunable Q, and the op-amp sets the passband gain independently of the RC that sets fc.
Passive RC filters are simple, but they load the source, cannot provide gain, and their response sags as the next stage draws current. An active filter wraps the frequency-shaping network around an op-amp so the output impedance is near zero, the passband can have gain, and stages cascade without interacting. For any analog front end — a sensor amplifier, an audio tone control, or the anti-aliasing filter ahead of an ADC — the active low-pass filter is the workhorse you reach for first.
What "active low-pass" actually means
A low-pass filter passes DC and low frequencies and attenuates everything above its cutoff. "Active" simply means an amplifying device (here, an op-amp) is inside the loop. The op-amp does three jobs at once: it buffers the RC network so nothing downstream detunes it, it supplies gain so a weak sensor signal can be lifted to a usable level, and it keeps the transfer function clean by enforcing a virtual short between its inputs.
The defining number is the −3 dB cutoff: the frequency where output power is half the passband value, i.e. gain has dropped to 0.707 of its flat-band figure. For a single RC pair this is always fc = 1/(2πRC).
The first-order stage: cutoff and gain
The most common first-order topology is a non-inverting amplifier with an RC to ground on the input. Take R = 16 kΩ and C = 10 nF: fc = 1/(2π · 16000 · 10×10−9) ≈ 995 Hz, essentially 1 kHz. Above that, the response falls at −20 dB/decade (a factor of ten in frequency costs a factor of ten in amplitude).
Crucially, fc and the gain are set by different components. The passband gain of a non-inverting stage is Av = 1 + Rf/Rg. So you can place the knee at exactly 1 kHz with the RC, then dial a gain of, say, 11 (Rf = 100 kΩ, Rg = 10 kΩ) without moving the cutoff at all. That orthogonality is the whole reason analog designers prefer active stages over a passive divider.
Watching the signal roll off (animated)
The figure below shows a Bode magnitude plot with a sweeping marker. As the test tone climbs past fc, the gain curve bends from flat to a steady downhill slope — that bend is the filter earning its name.
Why a single pole is often not enough
A first-order filter's −20 dB/decade is gentle. If you are feeding a 100 kHz-sampled ADC and want to suppress a 48 kHz tone to 60 dB, a single pole at 20 kHz gets nowhere near. You need a steeper skirt, and that means more poles. Cascading two first-order stages helps, but the smarter move is a second-order section that places two poles with one op-amp and lets you shape the transition.
The Sallen-Key second-order filter
The Sallen-Key topology is the classic single-op-amp, two-pole, unity-or-controlled-gain low-pass filter. It uses two resistors and two capacitors around the amplifier to realise a complex-conjugate pole pair. The result is a −40 dB/decade rolloff and a second knob the first-order stage never had: the quality factor Q, which sets how peaked or flat the response is right at fc.
Q decides the filter's character. A Butterworth alignment (Q = 0.707) gives the flattest possible passband — no ripple, no peaking — and is the default for anti-aliasing. A Bessel alignment (Q ≈ 0.577) sacrifices flatness for linear phase and minimal overshoot, ideal when pulse shape matters. A Chebyshev alignment uses higher Q to get a sharper corner at the cost of passband ripple.
| Property | First-order | Sallen-Key 2nd-order |
|---|---|---|
| Rolloff slope | −20 dB/decade | −40 dB/decade |
| Poles per op-amp | 1 | 2 |
| Q adjustable? | No (fixed 0.5) | Yes (set by R/C ratios & gain) |
| Response shaping | Fixed | Butterworth / Bessel / Chebyshev |
| Typical use | Simple bandwidth limit | Anti-aliasing, precision audio |
For an equal-component Sallen-Key (R1=R2=R, C1=C2=C), fc = 1/(2πRC) just like the first-order case, but Q is then fixed by the amplifier gain as Q = 1/(3 − Av). A gain of 1.586 lands you exactly on a Butterworth Q of 0.707.
A practical caution from the bench: Sallen-Key Q rises toward infinity as Av approaches 3, so the gain-setting resistors must be tight-tolerance (1% or better) or the filter can peak unexpectedly — or even oscillate. This is why engineers simulate the stage first. In Cadence Virtuoso or an LTspice/PSpice deck you run an AC analysis to confirm the −3 dB point and peaking, then a transient to check step overshoot before committing to silicon or PCB.
Design procedure in practice
- Pick fc from the system requirement (e.g. half the ADC sample rate for anti-aliasing).
- Choose C first (good capacitor values are scarcer than resistor values), then solve R = 1/(2πfcC).
- Select the alignment (Butterworth for flatness, Bessel for clean pulses) and read off the required Q and gain.
- Verify the op-amp has enough gain-bandwidth product: a rule of thumb is GBW ≥ 100 × fc × gain so the amplifier's own rolloff does not corrupt the filter shape.
- Simulate, then build and measure — real capacitor tolerance and op-amp input capacitance shift fc.
Low-pass and active high-pass filters are mirror images — swap the R and C positions and the same math produces a high-pass response. Combine a low-pass and a high-pass and you get a band-pass filter. If you want the full derivation and hands-on labs, our analog design courses walk through each topology on real hardware, and for team training or a custom front-end design you can also talk to Avecas analog design services.
Ready to design filters that actually meet spec? Join the CourseTron Analog Circuit Design track and build op-amp filters from first principles to tape-out-ready schematics.
Frequently asked questions
Why use an op-amp instead of a plain RC filter?
The op-amp buffers the RC network so the following stage cannot detune it, provides a near-zero output impedance, and lets you add gain. A passive RC divider can only attenuate and is loaded by whatever it drives.
How do I change the cutoff without changing the gain?
In a non-inverting active low-pass, fc is set by the input R and C, while gain is set by the feedback pair Rf and Rg. They are independent, so you can retune either one alone.
When should I use a Sallen-Key instead of a first-order stage?
Use Sallen-Key whenever a −20 dB/decade slope is too gentle — most anti-aliasing and precision audio needs the −40 dB/decade and the controllable Q that a second-order section provides.
What Q gives a flat passband?
Q = 0.707 is the Butterworth value: the maximally flat response with no peaking. Lower Q (Bessel, ~0.577) trades flatness for better pulse fidelity; higher Q sharpens the corner but introduces ripple or peaking.