What is the low pass filter cutoff frequency formula?
For an RC low pass filter, fc = 1 / (2πRC), with R in ohms, C in farads and fc in hertz. At fc the output is 3.01 dB below the input (70.7 % of the amplitude) and lags it by 45°.
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RC filter calculator (low pass)
Put in R and C and this RC filter calculator returns the low-pass cutoff frequency, or give it the cutoff you want and get the resistor or capacitor, rounded to standard values. See the Bode plot, the attenuation at any frequency and what a load does to the result.
Solve for
Examples
Ω
F
Hz
Optional: gain and phase at this frequency.
Ω
Optional: input resistance of whatever the output drives.
Cutoff frequency (−3 dB)
159.2 Hz
- Time constant τ
- 1 ms
- Rise time (10–90%)
- 2.197 ms
- −20 dB at
- 1.584 kHz
- −40 dB at
- 15.91 kHz
Assumes a low-impedance source and a high-impedance load. Add a load resistance if the next stage draws current.
Gain (dB)Phase (°)
How do you calculate the cutoff frequency of an RC low pass filter?
Use fc = 1 ÷ (2πRC). A 10 kΩ resistor with a 100 nF capacitor has a time constant of 1 ms, so fc = 159 Hz. At that frequency the output is 3 dB down (70.7 % of the input) and lags by 45°; above it the output falls by 20 dB for every tenfold increase in frequency.
RC low pass filter formula
A resistor in series with the signal and a capacitor to ground form a voltage divider whose ratio depends on frequency. The capacitor’s reactance falls as frequency rises, so more of the signal is shunted to ground.
fc = 1 ÷ (2π × R × C)
- fc
- cutoff (−3 dB) frequency in Hz
- R
- series resistance in Ω
- C
- capacitance to ground in F
Rearranged for the part you are choosing:
R = 1 ÷ (2π × fc × C) C = 1 ÷ (2π × fc × R)
The gain and phase at any frequency f follow from the same ratio f / fc:
|H(f)| = 1 ÷ √(1 + (f / fc)²) φ(f) = −arctan(f / fc)
RC low pass filter examples
Each one can be rerun in the calculator above: a cutoff from known parts, a PWM smoother and an anti-aliasing filter.
Example 1: cutoff of 10 kΩ and 100 nF
τ = RC = 10,000 × 100 × 10⁻⁹ = 1 ms. fc = 1 ÷ (2π × 0.001) = 159 Hz. The 10–90 % rise time is 2.2 ms, and the filter is 20 dB down at 1.58 kHz.
Example 2: smoothing 1 kHz PWM into a DC level
For a 10 Hz cutoff with a 1 µF capacitor, R = 1 ÷ (2π × 10 × 1 µF) = 15.9 kΩ. The nearest E24 value is 16 kΩ, giving fc = 9.95 Hz. At the 1 kHz PWM frequency the ripple is attenuated by 40.0 dB, roughly a hundredfold.
Example 3: a 10 kHz anti-aliasing filter with a 1 kΩ resistor
C = 1 ÷ (2π × 10 kHz × 1 kΩ) = 15.9 nF. The nearest E12 capacitor, 15 nF, gives 10.6 kHz. Use C0G/NP0 ceramic here: its capacitance does not drift with voltage or temperature the way X7R does.
RC filter cutoff frequency for common R and C values
fc = 1 / (2πRC) for standard resistor and capacitor values. Find the cutoff you need and read off a pair.
| R \ C | 1 nF | 10 nF | 100 nF | 1 µF | 10 µF |
|---|---|---|---|---|---|
| 1 kΩ | 159 kHz | 15.9 kHz | 1.59 kHz | 159 Hz | 15.9 Hz |
| 4.7 kΩ | 33.9 kHz | 3.39 kHz | 339 Hz | 33.9 Hz | 3.39 Hz |
| 10 kΩ | 15.9 kHz | 1.59 kHz | 159 Hz | 15.9 Hz | 1.59 Hz |
| 47 kΩ | 3.39 kHz | 339 Hz | 33.9 Hz | 3.39 Hz | 339 mHz |
| 100 kΩ | 1.59 kHz | 159 Hz | 15.9 Hz | 1.59 Hz | 159 mHz |
Attenuation and phase versus frequency
Every first-order low-pass filter has the same shape; only fc moves it. Multiply your cutoff by the ratio in the first column.
| f / fc | Gain | Output amplitude | Phase |
|---|---|---|---|
| 0.1× | -0.04 dB | 99.5 % | -5.7° |
| 0.5× | -0.97 dB | 89.4 % | -26.6° |
| 1× | -3.01 dB | 70.7 % | -45.0° |
| 2× | -6.99 dB | 44.7 % | -63.4° |
| 5× | -14.15 dB | 19.6 % | -78.7° |
| 10× | -20.04 dB | 10.0 % | -84.3° |
| 100× | -40.00 dB | 1.0 % | -89.4° |
Choosing parts for an RC low pass filter design
Capacitor type matters as much as value. Class 2 ceramics (X7R, X5R) lose a large share of their capacitance with DC bias and temperature, which moves the cutoff; C0G/NP0 ceramics and film capacitors stay put. The source impedance adds to R, and the input resistance of the next stage appears in parallel with C: if either is within a factor of ten of R, include it (the calculator’s load field does this for the load). When you need a sharper edge than 20 dB per decade, cascade a second RC through a buffer, or use a second-order active filter.
Glossary
- Cutoff frequency (fc)
- The frequency where the output has fallen 3 dB from the passband, also called the −3 dB or corner frequency.
- Time constant (τ)
- R × C, the time for the step response to reach 63.2 % of its final value.
- Bode plot
- Gain in dB and phase in degrees against frequency on a log axis.
- First-order filter
- A filter with one pole, rolling off at 20 dB per decade.
- Passband
- The frequencies below the cutoff, which pass with little attenuation.
- Anti-aliasing filter
- A low-pass filter in front of an ADC that removes content above half the sample rate.
Related guides
- RC time constant calculator: the same RC in the time domain.
- Voltage divider calculator: the DC half of a loaded filter.
Frequently asked questions
How does an RC low pass filter work?
R and C form a voltage divider whose ratio depends on frequency. At low frequencies the capacitor’s reactance is high and the output follows the input; as frequency rises the reactance falls and more of the signal is shunted to ground.
What RC cutoff frequency do 10 kΩ and 100 nF give?
159 Hz. 10,000 Ω × 100 × 10⁻⁹ F = 1 ms, and 1 ÷ (2π × 1 ms) = 159 Hz.
How fast does an RC low pass filter roll off?
20 dB per decade (6 dB per octave) above the cutoff, because it has one pole. Ten times the cutoff is about −20 dB and a hundred times is about −40 dB. If you need steeper filtering, cascade stages through a buffer or use a second-order active filter such as a Sallen-Key.
How do I choose R and C for a given cutoff?
Pick the capacitor first, because capacitors come in fewer standard values (often E6 or E12) and C0G/NP0 ceramics are limited in size. Then compute R = 1 / (2π fc C) and round it to the nearest E24 or E96 value. Keep R between about 1 kΩ and 100 kΩ: much lower loads the source, much higher picks up noise and is shifted by the next stage’s input current.
Does the load change the cutoff frequency?
Yes. A resistive load RL across the capacitor makes a divider with R, so the passband gain drops to RL / (R + RL) and the cutoff rises to 1 / (2π (R ∥ RL) C). Enter the load in the calculator above to see both. A load at least ten times R changes the cutoff by 10 % or less.
What is the time constant of an RC filter?
τ = RC. The step response reaches 63.2 % after one τ and 99.3 % after five. The 10–90 % rise time is τ × ln 9 ≈ 2.2 τ, which is the same as 0.35 ÷ fc.
Can I use an RC low pass filter to turn PWM into an analog voltage?
Yes: set the cutoff well below the PWM frequency. With 1 kHz PWM, a 10 Hz cutoff (16 kΩ and 1 µF) attenuates the ripple by about 40 dB, at the cost of a slow response (τ ≈ 16 ms). A higher PWM frequency or a second filter stage gets less ripple with a faster response.
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