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
(highpass or lowpass), integrators and differentiators.
For an introduction to AC circuits, resistors and
capacitors, see AC
circuits.

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 lowcost 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.
The voltage across the capacitor is IX
_{C} = I/
wC.
The voltage across the series combination is IZ
_{RC} = I(R
^{2} + (1/
wC)
^{2})
^{1/2},
so the gain is
From the phasor diagram for this filter, we see that
the output lags the input in phase.
 
At the angular frequency
w =
w_{o}
= 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 k
W
and C = 0.47
mF, then 1/RC
=
w_{o} = 2.1 10
^{3} rad.s
^{1},
so f
_{o} =
w_{o}/2
p
= 340 Hz.
At this frequency, the gain = 1/2^{0.5} = 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 f_{o}.
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 = f_{o}
= 1/2pRC is attenuated by 3 dB,
lower frequencies are less attenuated and high frequencies
more attenuated.
At w = w_{o},
the phase difference is p/4 radians
or 45¡, as shown in the plot of f(w).
High pass filter
The voltage across the resistor is IR. The voltage across
the series combination is IZ
_{RC} = I(R
^{2} + (1/
wC)
^{2})
^{1/2},
so the gain is
From the phasor diagram for this filter, we see that
the output leads the input in phase.
 
Once again, when w =
w_{o}
= 1/RC, the gain = 1/2
^{0.5} = 0.71, as shown on the
plot of g(
w). A signal with frequency
f = f
_{o} = 1/2
pRC is attenuated
by 3 dB, higher frequencies are less attenuated and lower
frequencies more attenuated. Again, when
w
=
w_{o}, 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. 

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.)


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.


Filger gains are usually written in decibels. see
the
decibel scale.
Integrator
Here we have an AC source with voltage
v_{in}(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.
 
Differentiator
Again we have an AC source with voltage
v_{in}(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.
 
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