RC low and high pass filters

A filter is a circuit that allows signals in a certain range of frequencies to pass and attenuates signals of other frequencies. Both RC filters below have cut-off frequency fc = 1/(2πRC) (ωc = 1/RC), where |H| = 0.707 (−3 dB).

RC low pass filter

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

The output is taken across C.

• Transfer function: H(jω) = 1/(1 + jωRC)
• Magnitude: |H| = 1/√(1 + (ω/ωc)²), phase φ = −tan⁻¹(ω/ωc)
• |H| = 1 at low frequency, 0.707 at fc, and falls at 20 dB/decade above fc.
• Passes low frequencies; blocks high frequencies.

RC high pass filter

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

The output is taken across R.

• Transfer function: H(jω) = jωRC/(1 + jωRC)
• Magnitude: |H| = (ω/ωc)/√(1 + (ω/ωc)²), phase φ = 90° − tan⁻¹(ω/ωc)
• |H| rises at 20 dB/decade below fc, equals 0.707 at fc, and tends to 1 at high frequency.
• Passes high frequencies; blocks low frequencies and dc.