RC low pass filter operation

A filter is a circuit that passes signals of some frequencies and blocks (attenuates) signals of other frequencies. It is built from resistors, capacitors, inductors or active devices.

RC low pass filter

A low pass filter (LPF) passes low frequencies and attenuates high frequencies. A resistor R is in series with the signal and a capacitor C is connected across the output.

 Vin o----[ R ]----+----o Vout
                   |
                  === C
                   |
 GND o-------------+----o GND

Operation

• Capacitive reactance is Xc = 1/(2πfC). At low frequency Xc is very large, so almost no current flows and the capacitor takes nearly the whole input voltage: Vout ≈ Vin.
• At high frequency Xc is very small, so the capacitor acts almost like a short circuit and Vout ≈ 0.
• So low frequencies pass and high frequencies are blocked.

Transfer function

Using the voltage divider with phasors (s = jω):

H(jω) = Vout/Vin = (1/jωC) / (R + 1/jωC) = 1 / (1 + jωRC)

Let the cut-off frequency be ωc = 1/RC, that is fc = 1/(2πRC). Then

H(jω) = 1 / (1 + j(ω/ωc))

• Magnitude: |H| = 1 / √(1 + (ω/ωc)²)
• Phase: φ = −tan⁻¹(ω/ωc)

Frequency response

 |H|
 1 |------------.
   |             `.
0.707|- - - - - - - -`.
   |                  `--.
   +-----------|---------------- f
              fc

• The gain is flat (1) in the pass band, 0 to fc.
• At fc the output power is half of the maximum, so fc is called the half-power or −3 dB frequency.
• Beyond fc the gain falls at 20 dB per decade (6 dB per octave).
• The output lags the input; the lag increases from 0° to 90°.

Example: R = 1.59 kΩ, C = 0.1 µF gives fc = 1/(2π × 1590 × 0.1×10⁻⁶) ≈ 1 kHz.

Uses: removing noise from signals, power supply ripple filtering, anti-aliasing before an ADC.