The Butterworth filter is a type of signal processing filter designed to have a frequency response that is as flat as possible in the passband. It is also referred to as a maximally flat magnitude filter. It was first described in 1930 by the British engineer and physicist Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers".

Butterworth had a reputation for solving very complex mathematical problems thought to be 'impossible'. At the time, filter design required a considerable amount of designer experience due to limitations of the theory then in use. The filter was not in common use for over 30 years after its publication. Butterworth stated that:

"An ideal electrical filter should not only completely reject the unwanted frequencies but should also have uniform sensitivity for the wanted frequencies".

Such an ideal filter cannot be achieved, but Butterworth showed that successively closer approximations were obtained with increasing numbers of filter elements of the right values. At the time, filters generated substantial ripple in the passband, and the choice of component values was highly interactive. Butterworth showed that a low-pass filter could be designed whose gain as a function of frequency (i.e., the magnitude of its frequency response) is:

where ${\displaystyle \omega }$ is the angular frequency in radians per second and ${\displaystyle n}$ is the number of poles in the filter—equal to the number of reactive elements in a passive filter. Its cutoff frequency (the half-power point of approximately −3 dB or a voltage gain of 1/√2 ≈ 0.7071) is normalized to 𝜔 = 1 radian per second. Butterworth only dealt with filters with an even number of poles in his paper, though odd-order filters can be created with the addition of a single-pole filter applied to the output of the even-order filter. He built his higher-order filters from 2-pole filters separated by vacuum tube amplifiers. His plot of the frequency response of 2-, 4-, 6-, 8-, and 10-pole filters is shown as A, B, C, D, and E in his original graph.

Butterworth solved the equations for two-pole and four-pole filters, showing how the latter could be cascaded when separated by vacuum tube amplifiers and so enabling the construction of higher-order filters despite inductor losses. In 1930, low-loss core materials such as molypermalloy had not been discovered and air-cored audio inductors were rather lossy. Butterworth discovered that it was possible to adjust the component values of the filter to compensate for the winding resistance of the inductors.

He used coil forms of 1.25″ diameter and 3″ length with plug-in terminals. Associated capacitors and resistors were contained inside the wound coil form. The coil formed part of the plate load resistor. Two poles were used per vacuum tube and RC coupling was used to the grid of the following tube.

Butterworth also showed that the basic low-pass filter could be modified to give low-pass, high-pass, band-pass and band-stop functionality.