How do you calculate the cutoff frequency of an RC filter?
fc = 1 / (2πRC). With 10 kΩ and 100 nF that is about 159 Hz.
CIRCUIT DIAGRAM EXAMPLE
RC Low-Pass Filter Circuit Diagram and Cutoff
A series resistor and a capacitor to ground make the simplest filter there is. It passes DC and slow signals and attenuates fast ones, which is why one sits on so many sensor inputs, PWM outputs and noisy signal lines.
How this circuit works
R1 and C1 form a voltage divider whose bottom half depends on frequency. A capacitor's reactance is XC = 1 / (2πfC): very large at low frequencies and small at high ones. At DC, C1 is an open circuit and Vout equals Vin.
As the frequency rises, C1's reactance falls and it shunts more of the signal to ground. The cutoff frequency, where the reactance equals R1, is fc = 1 / (2πRC) ≈ 159 Hz here. At that point the output is 1/√2 of the input, about −3 dB, and lags it by 45°.
Above the cutoff the output falls by 20 dB per decade: about a tenth of the input at 1.59 kHz and a hundredth at 15.9 kHz. One RC section is a first-order filter. For a steeper roll-off, use a second-order filter such as the Sallen-Key.
In the time domain the same parts give a time constant τ = RC = 1 ms. After a step at the input, the output reaches about 63 % of the step after 1 ms and over 99 % after 5 ms.
Formulas
- Cutoff frequency
fc = 1 / (2π × R × C) = 1 / (2π × 10 kΩ × 100 nF) ≈ 159 Hz- Time constant
τ = R × C = 10 kΩ × 100 nF = 1 ms- Gain at a frequency f
|Vout / Vin| = 1 / √(1 + (f / fc)²)At 1.59 kHz, ten times the cutoff, this is about 0.1, or −20 dB.
Parts list
| Ref | Part | Value | What it does |
|---|---|---|---|
| C1 | Capacitor | 100 nF | 100 nF capacitor to ground. A C0G or X7R ceramic is fine. |
| R1 | Resistor | 10 kΩ | Series resistor. With C1 it sets the cutoff. |
Netlist
Recovered from the drawing itself: every pin on each net below is joined by a wire, a junction or a shared supply or ground symbol.
| Net | Connects |
|---|---|
| Vin | R1.1 |
| Vout | R1.2, C1.1 |
| GND | C1.2 |
Common mistakes
- Loading the output. A resistive load in parallel with C1 lowers the passband gain and shifts the cutoff; buffer the output with an op-amp if the load is heavy.
- Ignoring the source resistance. Whatever drives Vin adds its own resistance to R1 and moves the cutoff down.
- Swapping R and C by accident. A series capacitor with a resistor to ground is a high-pass filter with the same cutoff.
- Expecting a sharp cutoff. A first-order filter only falls 20 dB per decade, so a signal at twice the cutoff still comes through at about 45 % of its input.
Frequently asked questions
What is the difference between an RC low-pass and a high-pass filter?
The same two parts in swapped positions. Low-pass: resistor in series, capacitor to ground. High-pass: capacitor in series, resistor to ground. The cutoff formula is the same.
Why is the cutoff called the −3 dB point?
At fc the output amplitude is 1/√2 ≈ 0.707 of the input, which is −3 dB; the output power has halved.
Can an RC filter smooth a PWM signal into a DC voltage?
Yes. Put the cutoff well below the PWM frequency. The lower it is, the less ripple is left, at the cost of a slower response to changes.
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