RC filters, integrators and differentiators

Joe Wolfe
School of Physics, The University of New South Wales.

This page explains how RC circuits work as filters (high-pass or low-pass), integrators and differentiators. For an introduction to AC circuits, resistors and capacitors, see AC circuits.

filter pic

A triangle wave (upper trace) is integrated to give a rounded, parabolic wave.

Overview. A low pass filter passes low frequencies and rejects high frequencies from the input signal. And vice versa for a high pass filter. The simplest of these filters may be constructed from just two low-cost electrical components. Over appropriate frequency ranges, these circuits also integrate and differentiate (respectively) the input signal.

Low pass filter

A first order, low pass RC filter is simply an RC series circuit across the input, with the output taken across the capacitor. We assume that the output of the circuit is not connected, or connected only to high impedance, so that the current is the same in both R and C.
    low pass filter
The voltage across the capacitor is IXC = I/wC. The voltage across the series combination is IZRC = I(R2 + (1/wC)2)1/2, so the gain is

    filter algebra
From the phasor diagram for this filter, we see that the output lags the input in phase.
    filter algebra

gain and phase shift of low pass filter

At the angular frequency w = wo = 1/RC, the capacitive reactance 1/wC equals the resistance R. We show this characteristic frequency on all graphs on this page. For instance, if R = 1 kW and C = 0.47 mF, then 1/RC = wo = 2.1 103 rad.s-1, so fo = wo/2p = 340 Hz.

At this frequency, the gain = 1/20.5 = 0.71, as shown on the plot of g(w). The power transmitted usually goes as the gain squared, so the filter transmits 50% of maximum power at fo. Now a reduction in power of a factor of two means a reduction by 3 dB (see What is a decibel?). A signal with frequency f = fo = 1/2pRC is attenuated by 3 dB, lower frequencies are less attenuated and high frequencies more attenuated.

At w = wo, the phase difference is p/4 radians or 45, as shown in the plot of f(w).


High pass filter

    high pass filter
The voltage across the resistor is IR. The voltage across the series combination is IZRC = I(R2 + (1/wC)2)1/2, so the gain is

    filter algebra
From the phasor diagram for this filter, we see that the output leads the input in phase.
    filter algebra

gain and phase shift of high pass filter

Once again, when w = wo = 1/RC, the gain = 1/20.5 = 0.71, as shown on the plot of g(w). A signal with frequency f = fo = 1/2pRC is attenuated by 3 dB, higher frequencies are less attenuated and lower frequencies more attenuated. Again, when w = wo, the phase difference is p/4 radians or 45.


Filter applications and demonstrations

RC and other filters are very widely used in selecting signals (which are voltage components one wants) and rejecting noise (those one doesn't want). A low pass filter can 'smooth' a DC power supply: allow the DC but attenuate the AC components. Conversely, a high pass filter can pass the signal into and out of a transistor amplifier stage without passing or affecting the DC bias of the transistor. They can also be used to sort high frequency from low frequency components in a purely AC signal. Capacitors are often used in 'cross over' networks for loudspeakers, to apply the high frequencies to the 'tweeter' (a small, light speaker) and the low frequencies to the 'woofer' (a large, massive speaker). We include some sound examples here as demonstrations.

No filter.
    First we recorded the sound from a microphone into the computer sound card, without any (extra*) filtering.

    * The sound coming out of your computer speakers will nevertheless have been filtered by the time you hear it. A speaker already filters the sound, because its impedance is partly inductive, due to the speaker coil. Further, its acoustic efficiency is a strong function of frequency. Nevertheless, you should notice the differences among these sound files, particularly if you switch from one to another in succession.

mic, amplifier and speaker
gain of unity

Low pass filter.
    At high frequencies, the capacitor 'shorts out' the input to the sound card, but hardly affects low frequencies. So this sound is less 'bright' than the example above.

    This sound is quieter than the previous sample. We have cut out the frequencies above 1 kHz, including those to which your ear is most sensitive. (For more about the frequency response of the ear, see Hearing Curves.)

mic, amplifier and speaker in low pass filter
gain of low pass filter

High pass filter.
    At low frequencies, the reactance of the capacitor is high, so little current goes to the speaker. This sound is less 'bassy' than those above.

    Losing the low frequencies makes the sound rather thin, but it doesn't reduce the loudness as much as removing the high frequencies.

mic, amplifier and speaker in high pass filter
gain of high pass filter

Filger gains are usually written in decibels. see the decibel scale.

Integrator

Here we have an AC source with voltage vin(t), input to an RC series circuit. The output is the voltage across the capacitor. We consider only high frequencies w >> 1/RC, so that the capacitor has insufficient time to charge up, its voltage is small, so the input voltage approximately equals the voltage across the resistor.

integrator pic

    integrator algebra

Differentiator

Again we have an AC source with voltage vin(t), input to an RC series circuit. This time the output is the voltage across the resistor. This time, we consider only low frequencies w << 1/RC, so that the capacitor has time to charge up until its voltage almost equals that of the source.

integrator pic

    integrator algebra
More accurate integration and differentiation is possible using resistors and capacitors on the input and feedback loops of operational amplifiers. Such amplifiers can also be used to add, to subtract and to multiply voltages. An analogue computer is a combination of such circuits, and may be used to solve simultaneous, differential and integral equations.

Go back to AC circuits


Joe Wolfe / J.Wolfe@unsw.edu.au, phone 61- 2-9385 4954 (UT + 10, +11 Oct-Mar).

School of Physics, University of New South Wales, Sydney, Australia.

 Joe's scientific home page
 A list of educational links
 Music Acoustics site

Pic of the author

 

 

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