RC filters, integrators and differentiators
From Physclips:
Mechanics with animations and film.
RC circuits work as filters (highpass or lowpass filters), integrators and differentiators. Here we explain how, and give sound files examples of RC filters in action. 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/ωC. The voltage across the series combination is
IZ_{RC} = I(R^{2} + (1/ωC)^{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 ω = ω_{o} = 1/RC, the capacitive
reactance 1/ωC equals the resistance R. We show this
characteristic frequency* on all graphs on this page. For instance, if
R = 1 kω and C = 0.47 mF, then 1/RC = ω_{o} =
2.1 10^{3} rad.s^{1}, so f_{o} = ω_{o}/2π = 340 Hz.
* Let's check the units: it should be quick. First, a radian is the ratio of an arc to a radius, so it is a pure number. R is in ohms, and we can use Ohm's law (V = IR) and the definition of current (I = dQ/dt) to write ohm = volt/ampere = volt/(coulomb/second) = volt*second/coulomb. C (charge per unit voltage) is in farads (coulomb/volt), so RC has the units of (volt*second/coulomb)(coulomb/volt) = second. See, it was quick: it only took us 30 ohm.farads!
At this frequency, the gain = 1/2^{0.5} = 0.71, as shown on the plot
of g(ω). 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} = 1/2πRC
is attenuated by 3 dB, lower frequencies are less attenuated and high
frequencies more attenuated.
At ω = ω_{o}, the
phase difference is π/4 radians or 45°, as shown in
the plot of φ(ω).
High pass filter
The voltage across the resistor is IR. The voltage across the series combination
is
IZ_{RC} = I(R^{2} + (1/ωC)^{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 = ω_{o} = 1/RC, the gain
= 1/2^{0.5} =
0.71, as shown on the plot of g(ω). A signal with frequency
f = f_{o} = 1/2πRC is attenuated by
3 dB, higher frequencies are less attenuated and lower frequencies more
attenuated. Again, when ω = ω_{o}, the phase difference is π/4 radians or 45f.
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.
(If you do not notice much difference with the high pass filter,
it may be because you are using tiny computer speakers that do
not radiate low frequencies well. Try it with headphones or with
hifi speakers.)


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 ω >> 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.


The photograph at the top of this page shows a triangle wave input to an RC
integrator, and the resulting output.
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 ω << 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 very
rapidly.
Go back to AC circuits
Happy
birthday, theory of relativity!
As of June 2005,
relativity is 100 years old. Our contribution is Einstein Light:
relativity in brief... or in detail. It explains the key
ideas in a short multimedia presentation, which is supported
by links to broader and deeper explanations. 


