RC high pass filter operation

Circuit diagram

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

A capacitor C is in series with the signal path and the resistor R is connected across the output. A high pass filter (HPF) passes frequencies above a cut-off frequency and attenuates lower frequencies.

Operation

• The reactance of the capacitor is Xc = 1/(2πfC).
• At low frequency Xc is very large. Most of the input voltage drops across C and very little appears across R, so Vout ≈ 0. For dc (f = 0) the capacitor is an open circuit and Vout = 0.
• At high frequency Xc is very small. The capacitor behaves like a short circuit, the whole input appears across R and Vout ≈ Vin.

Transfer function

Using the voltage divider (the output is taken across R):

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

With cut-off frequency fc = 1/(2πRC) (ωc = 1/RC):

• |H| = (ω/ωc) / √(1 + (ω/ωc)²)
• Phase φ = 90° − tan⁻¹(ω/ωc)

Frequency response

 |H|
 1 |                 ____________
   |             .-'
0.707|- - - - - .'
   |        .-'
   |  __.-'
   +-----------|---------------- f
              fc

• Below fc the output rises at 20 dB per decade.
• At fc the gain is 0.707 of the maximum (−3 dB point).
• Above fc the gain is nearly flat at 1. The output leads the input in phase.

Uses: coupling and dc-blocking capacitors in amplifiers, removing low-frequency hum or drift, and as a differentiator circuit when RC is small.