An active low pass filter with an op-amp, on a live Bode plot
Advanced
Or see every lesson.
- 505parts on the bench
- 45boards running now
- 1.00xreal time, on every board
A passive low pass filter is a resistor and a capacitor. An active one puts an op-amp around them, which buys two things: gain, and an output that does not sag when you connect something to it. This page runs the most common kind, an inverting op-amp low pass.
In the circuit above, a signal generator feeds Rin = 1 k into the op-amp’s inverting input. Rf = 10 k and Cf = 8 nF sit in parallel from the output back to that input. A 10 k and 10 k divider holds the other input at half of the 5 V supply. Press Run and open the Scope tab: the 100 Hz input comes out ten times bigger and upside down. Open it in the editor and sweep it to see the corner.
The two formulas
In the pass band the capacitor is nearly an open circuit, and the stage is a plain inverting amplifier. Its gain is set by the two resistors: |Av| = Rf / Rin = 10,000 / 1,000 = 10, which is 20 dB. The minus sign of an inverting stage shows up as 180 degrees of phase.
As the frequency rises, Cf starts to short out Rf and the gain falls. The corner, where the gain is 3 dB down, is fc = 1 / (2 pi Rf Cf) = 1 / (2 pi x 10,000 x 8e-9), about 1989 Hz. Above it the gain falls by 20 dB per decade, the slope of one capacitor.
Note that Rin is not in the corner formula. You can set the gain with Rin and the corner with Cf, more or less independently.
What the sweep shows
Open the circuit in the editor, choose Analysis, then Frequency response, drive gen1 and watch the op-amp’s output from 10 Hz to 100 kHz. The pass band is flat at 20.0 dB with the phase at 180 degrees. The gain is 3 dB down at about 2.0 kHz, within a fraction of a percent of the formula. A decade higher, at 20 kHz, the gain is close to 0 dB, another 20 dB down.
The corner lands so close to the formula because Rf and Cf are inside the feedback loop. Whatever the scope’s input does to the output node, the op-amp corrects for it. A passive RC filter has nothing correcting its output, and the scope or the next stage pulls its corner around.
Why it needs a bias on one supply
The op-amp here runs from 5 V and ground, with no negative supply, so its output cannot go below 0 V. An inverting stage referenced to ground would try to swing negative for every positive half of the input. The fix is to move the reference: the 10 k and 10 k divider sets the non-inverting input to 2.5 V, and the output then sits at 2.5 V with no signal and swings both ways around it.
The generator’s offset is 2.5 V too, so the input is centered on the same point. If you build this on a bench with a signal centered on 0 V, put a capacitor in series with Rin so the DC difference does not get amplified ten times.
Try these in the editor
Change the op-amp model to 741. A 741’s gain-bandwidth product is 1 MHz, a tenth of the default model’s 10 MHz, so it runs out of loop gain sooner. The corner comes down to about 1.95 kHz, about 2% under the formula. Real op-amps are not ideal, and at higher gains and frequencies the gain-bandwidth product is the number that decides.
Make Rin 10 k. The gain drops to 1, 0 dB, and the corner stays where it was, because it depends only on Rf and Cf.
Make Cf 2 nF, a quarter of the value. The corner moves up four times, to about 8 kHz.
Turn the input up. At an amplitude of 0.3 V the output wants 6 V peak to peak, more than a 5 V supply can give, and the tops and bottoms flatten on the scope. A filter is an amplifier too, and it clips like one.
Other active low pass filters
A non-inverting version puts the RC filter in front of a non-inverting amplifier, so the signal stays the right way up and the input impedance is high. The corner is still 1 / (2 pi R C) of that front RC.
A Sallen-Key filter uses two resistors and two capacitors around one op-amp for a second-order response: 40 dB per decade instead of 20, the slope the RLC circuit gives without an op-amp. Audio crossovers and anti-aliasing filters in front of an ADC are usually second order or higher. You can build both in the editor from the same parts.
Common mistakes
Putting Cf in series with Rf instead of in parallel. In series it turns the stage into something else entirely. Rf and Cf sit side by side, both from the output to the inverting input.
No bias on a single supply. The output pins at 0 V for half of every wave. Bias the non-inverting input to the middle of the supply.
Expecting a gain of 10 at any frequency. Above the corner the gain falls, and with a slow op-amp it falls sooner than the formula says.
Mixing up the corner formula. It is Rf with Cf, not Rin.
Open this circuit in the editor. Op-amp low pass: gain 10, corner near 2 kHz, wired as it runs at the top of this page. Change a part or a value and run it again.
Open the circuitQuestions
How do I calculate the cutoff frequency of an active low pass filter?
For the inverting design, fc = 1 / (2 pi Rf Cf). With Rf = 10 k and Cf = 8 nF it is about 1.99 kHz. The pass band gain is Rf / Rin.
What is the difference between an active and a passive low pass filter?
A passive filter is only resistors and capacitors, so it cannot add gain, and its output changes with the load. An active filter adds an op-amp, which can add gain and holds its output steady whatever is connected.
Can I use a 741 for an active filter?
Yes, for low frequencies. Its 1 MHz gain-bandwidth product limits it: with a gain of 10, the loop starts to run out near 100 kHz, and even at 2 kHz the corner lands about 2% low. A faster op-amp tracks the formula more closely.
Can I simulate an op-amp filter online for free?
Yes. The circuit above runs in your browser with no account, with a scope and a frequency response sweep. The op-amp simulator page has more op-amp circuits.
Keep going
Build this for real
Open the editor, change a value and watch the number move with it. Nothing to install, and no account needed.