RC Filter Formulas: Cutoff Frequency and Gain
RC low-pass and high-pass filter formulas: cutoff frequency, time constant, gain in dB and phase at a frequency, and the resistor for a chosen cutoff.
The formulas
| To find | Formula |
|---|---|
| Cutoff frequency | fc = 1 / (2π × R × C) |
| Time constant | τ = R × C, so fc = 1 / (2π × τ) |
| Low-pass gain | |H| = 1 / √(1 + (f / fc)²) |
| High-pass gain | |H| = (f / fc) / √(1 + (f / fc)²) |
| Gain in dB | 20 × log10(|H|) |
| Phase (low-pass) | φ = −arctan(f / fc) |
| Resistor for a chosen cutoff | R = 1 / (2π × fc × C) |
R is in ohms, C in farads and f in hertz. At f = fc the gain is 0.7071, which is −3.01 dB, and the phase of a first-order low-pass is −45°. Above the cutoff a first-order filter falls at 20 dB per decade. The formulas assume the source resistance is small and the load resistance is large compared with R.
Worked example: 10 kΩ and 100 nF
- fc = 1 / (2π × 10,000 × 100 × 10⁻⁹) = 159.2 Hz
- τ = 10,000 × 100 × 10⁻⁹ = 1 ms
- At 1 kHz, f / fc = 6.283, so the low-pass gain is 0.1572 = -16.07 dB
- The phase at 1 kHz is −arctan(6.283) = −80.96°
Worked example: choose the resistor
A cutoff of 1 kHz with a 10 nF capacitor.
- R = 1 / (2π × 1,000 × 10 × 10⁻⁹) = 15.92 kΩ, so use the nearest standard value and check the cutoff
Cutoff frequency with C = 100 nF
| Resistor | Cutoff frequency | Time constant |
|---|---|---|
| 1 kΩ | 1,592 Hz | 0.1 ms |
| 4.7 kΩ | 338.6 Hz | 0.47 ms |
| 10 kΩ | 159.2 Hz | 1 ms |
| 47 kΩ | 33.86 Hz | 4.7 ms |
Frequently asked questions
What is the difference between a low-pass and a high-pass RC filter?
In a low-pass filter the output is taken across the capacitor and low frequencies pass. In a high-pass filter the output is taken across the resistor and high frequencies pass. Both have the same cutoff frequency.
Why is the gain −3 dB at the cutoff?
At f = fc the resistance and the capacitor reactance are equal, so the output voltage is 1 / √2 of the input, which is 0.7071 or −3.01 dB.
How steep is a single RC filter?
It falls at 20 dB per decade, or about 6 dB per octave, above the cutoff. A steeper response needs more stages or an active filter.
Does the load change the cutoff?
Yes. A load resistance in parallel with the capacitor lowers the gain and moves the cutoff. Use a buffer or keep the load much larger than R.
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