The first-order RC filter
A resistor and a capacitor make the simplest filter. Wired with the capacitor to ground after the resistor, it is a low-pass filter: low frequencies pass and high frequencies are attenuated. Swap the two parts and it becomes a high-pass filter, which blocks DC and low frequencies.
Cutoff frequency
Both versions share the same cutoff, where the output is 3 dB down (about 70.7% of the input amplitude):
fc = 1 ÷ (2π × R × C)
Gain and phase at any frequency
With x = f ÷ fc:
- low-pass: |H| = 1 ÷ √(1 + x²), phase = −arctan(x);
- high-pass: |H| = x ÷ √(1 + x²), phase = 90° − arctan(x).
Far beyond the cutoff, a first-order filter rolls off at 20 dB per decade (6 dB per octave).
Example
R = 10 kΩ, C = 10 nF:
- fc = 1 ÷ (2π × 10,000 × 0.00000001) ≈ 1.59 kHz;
- time constant τ = RC = 0.1 ms;
- a 10 kHz signal through the low-pass filter: x ≈ 6.28, gain ≈ 0.157, or −16.1 dB, with a phase shift of about −81°.
Uses
- Smoothing a PWM signal into an analog voltage.
- Anti-aliasing before an ADC.
- Removing hiss or hum, or blocking DC between audio stages (high-pass).
- Debouncing switches and filtering sensor noise.
Design tips
- The source impedance adds to R, and the load impedance appears in parallel with the output. Keep the load at least ten times larger than R, or buffer the filter with an op-amp.
- For steeper roll-off, cascade stages or use an active filter (Sallen-Key, for example). Two passive stages in a row interact unless the second has a much higher impedance.
- Capacitor tolerance (often ±10%) shifts the cutoff by the same amount; choose C0G/NP0 or film capacitors for accuracy.