Electronics Fundamentals/Electronic Filter
A filter is an electronic network where the output voltage, current, or power varies in a predictable way with the input frequency. Filters can be designed to pass high frequencies and block low frequencies, to pass only low frequencies, to pass only a specific band of frequencies, to block only a specific band of frequencies, or any combination thereof. Generally a filter will exhibit the same output for all input frequencies within its "pass band", that range of frequencies which it is designed to allow to pass.
The calculations around filters will, of course, involve frequency. To make the math easier, we have to use a concept that may be hard to understand, the value which is defined as the square root of -1. (In mathematics, this value is called , but in electronics we already use the symbol i for current, which would cause confusion.) The square root of minus 1 is, of course, not possible to calculate, so any number that is a multiple of is called "imaginary". A number which has both imaginary and real parts is called "complex". We assume that the student has been introduced to complex numbers elsewhere.
It is a useful abstraction to think of the voltage going through the circuit as a complex value. It always has the same value of voltage, current, or power, however that value varies consistently between real and imaginary. If you picture a plane, with real numbers on the horizontal axis, and imaginary numbers on the vertical, the voltage (e.g.) can be represented as a point at a constant distance from the origin, making a perfect circle around the origin once every cycle. The real value of the voltage then appears to be a sine wave, the imaginary part is a cosine. Continuing this abstraction, we can see that the point is actually moving a fixed number of radians per second; for instance a 1Hz signal can be represented by a point moving about the origin in the complex plane at a rate of 6.28 () radians per second. The angle in radians that the point makes with the positive real axis at any time is represented in our calculations by the symbol , and the frequency is often expressed as radians per second.
This abstraction is particularly useful in determining the "phase angle" change of a signal as it goes through a filter. A filter does not only cut frequencies, it also changes the phase of the signal by slowing or advancing the wave as it passes through the electronic network. This change of phase, whether it is wanted or not, is a necessary side effect of the filter action, and no filter network analysis is complete without analysis of the associated phase changes.