This article describes the basic types of filter circuits. An understanding of calculus is helpful, but not necessary. A basic understanding of voltage and current is assumed. A basic knowledge of capacitors and inductors is also assumed. This article uses a higher level of mathematics than most of the tutorials.
Filters are circuits that allow certain signals to pass through the circuit, but prevent other types of signals from passing through the circuit. The frequencies that the circuit allows through is called the pass band. A high pass filter allows high frequency signals to pass through, but rejects or filters off lower frequency filters. A low pass filter does the opposite, allowing low frequency signals to pass through, but rejects high frequency signals. A band pass filter allows frequencies in a certain band to pass through, but rejects signals of a frequency above or below this band. A band reject filter allows frequencies above and below the band to pass through, but filters off any frequencies within the band.
The cutoff frequency is the frequency at which the filter begins to operate. In other words, for a high pass filter, the frequency above which signals are passed is called the cutoff frequency. Anything below the cutoff frequency in this type of filter is rejected. Filters generally do not cutoff immediately at a frequency, but instead have a more gradual response. The cutoff frequency is usually defined as the -3 db point. For those that are not familiar with the db scale, this is the half power point. If a signal is cut by 3 db, its strength is cut in half. If the signal is cut by an additional 3 db (or 6 db total), then it is cut in half again, or is now one fourth of its original value.
If a filter cuts off a signal very quickly past its cutoff frequency, it is said to have a fast rolloff.
Filters have many, many uses. If your stereo has a bass, midrange, and treble control, the bass is a low pass filter, the midrange is a bandpass filter, and the treble is a high pass filter. The radio receiver uses a bandpass filter as part of its receving circuit, so that it only receives the station you want to listen to, and rejects all other stations. Your television uses a bandpass filter to select the channel, and to seperate audio and video tracks from the signal. Power supplies use low pass filters with a very low cutoff frequency to pass DC (a frequency of zero) and reject any other noise that might be present. Your infrared remote control for your TV or stereo uses a coded or modulated signal. The receiver uses a filter so that it is very sensitive to this frequency, but rejects other frequencies. This helps to cut out interference from lights in the room and sunlight, allowing your remote control to function with a longer range and better efficiency.
One important note about these types of filters is that the voltage divider that you set up is disturbed by the load that you place upon it. The impedences of the load (usually the next stage in your circuit) must be significantly higher than the impedences of the filter, so that the load does not effect the filter characteristics. For audio circuits, a simple buffer stage with a reasonably high input impedence (like an op-amp circuit) will usually do the trick.
The cutoff frequency for these types of filters is given by the following formula (using a w for a greek omega because it's the closest thing I can type):
w = 1/(RC)
(where w = 2(pi)f, and f is the frequency)
If you want to rearrange that for the frequency, f=1/(2(pi)RC). The RC term is often called the time constant, and is sometimes abbreviated with a greek letter tau. Anyone who has gone through all of the math of a first year electrical engineering student is all too familiar with RC time constants. They show up in plenty of equations. Fortunately, hobbyists don't have to worry about them to much (but if you are a beginning EE, you have plenty of worse things yet to come...).
Note that inductors and capacitors do not act exactly like frequency dependent resistors. For example, in a capacitor, the output current is i=C(dv/dt), so for a sine wave input, the output is a cosine wave, or is shifted by 90 degrees from the input. This is called phase distortion. For many ciruits phase distortion is not important. For example, your ear cannot hear the phase of a signal unless it beats against a signal that is almost identical in frequency. Instead, your ears have cells in them that act as itty bitty bandpass filters. You have zillions of these cells (ok, I admit I don't really know how many, but it's a lot), which all respond to different frequencies. Your brain therefore only gets information about the frequency (which cells are responding) and how loud the signal is. The phase shift caused by a capacitor would be unnoticable.
Other types of RLC filters are also common.
I just had to give this one a brief mention. When I was in college, I purchased an old Collins short wave receiver at a yard sale (I think I paid a dollar or two for it). Not only did this thing contain a lot of tubes and things that I was not familiar with, but it had one unique thing in it that I haven't seen since: a mechanical filter. This device was a rather odd looking contraption of round metal disks that happened to resonate at just the right frequency. While it was certainly a novel idea, I doubt that the frequency characteristics of the filter were all that great. Otherwise we would have probably seen more of these filters over the years. Modern electronics have definately made this technique obsolete.