Butterworth Filter Formula, Equations, & Calculations

- the basic calculations formulae and calculations for the Butterworth filter together with a table of some values.

The Butterworth filter provides a maximally flat response. However this also has the advantage that the calculations are somewhat simpler than those for other forms of filter.

This simplicity combined with a level of performance that is more than adequate for many applications means that the Butterworth filter is widely used in many areas of electronics from RF to audio active filters.

Using the equations for the Butterworth filter, it is relatively easy to calculate and plot the frequency response as well as working out the values needed.


Butterworth filter calculation for frequency response

As the Butterworth filter is maximally flat, this means that it is designed so that at zero frequency, the first 2n-1 derivatives for the power function with respect to frequency are zero.

Thus it is possible to derive the formula for the Butterworth filter frequency response:


Butterworth filter frequency response calculation


Where:
   f = frequency at which calculation is made
   fo = the cut-off frequency, i.e. half power or -3dB frequency
   Vin = input voltage
   Vout = output voltage
   n = number of elements in the filter

The equation can be re-written to give its more usual format. Here H(jω) is the transfer function and it is assumed the filter has no gain, i.e. it is not an active filter.


Butterworth filter frequency response calculation


Where:
   H(jω) = transfer function at angular frequency ω
   ω = angular frequency and is equal to 2πf    ωo = cutoff frequency expressed as an angular value and is equal to 2πfo

Note: It does not matter whether ω/ωo or f/fo is used as it is purely a ratio of the two figures. If ω which is 2πf is used, then the factor 2π cancels out as it is on both the top and bottom of the fraction.

When wanting to express the loss of the Butterworth filter at any point, the Butterworth formula below can be used. This gives the attenuation in decibels at any point.

Butterworth filter equation for attenuation


Butterworth filter calculation example

To provide an example of the response of the Butterworth filter calculation, take an example of the circuit given below. As is normal with these calculations normalised values are used where the cut-off frequency is 1 radian, i.e. 1/2Π Hz, the impedance is 1 Ω and values are given in Farads and Henries.

The example below uses some of the simplest values, with an impedance of 1Ω, and values for the capacitor of 2 Farads and the series inductors each 1 Henry.

Butterworth filter circuit
Butterworth filter circuit

Using the formula above and a knowledge of the cut-off point being 0.159Hz, it is possible to calculate values of response at various frequencies:


Frequency (Hz) Relative Power Output
0.00 1.00
0.07 0.99
0.095 0.95
0.159 0.50
0.223 0.117
0.254 0.056
0.318 0.015

Butterworth filter poles

The poles of a Butterworth low-pass filter with cut-off frequency ωc are evenly-spaced around the circumference of a half-circle of radius ωc centred upon the origin of the s-plane.

The poles of a two-pole filter are at ±45°. Those of a four-pole filter are at ±22.5° and ±67.5°. Other cases can also be deduced in a similar fashion.

However the table below provides the poles of the low-pass Butterworth filters with one to eight poles and cut-off frequency 1 rad/s, i.e. for a normalised filter.


Poles of the Normalized Butterworth Polynomials
Order Poles
1 −1 ± j 0
2 −0.707 ± j 0.707
3 −1 ± j 0, −0.5 ± j 0.866
4 −0.924 ± j 0.383, −0.383 ± j 0.924
5 −1 ± j 0, −0.809 ± j 0.588, −0.309 ± j 0.951
6 −0.966 ± j 0.259, −0.707 ± j 0.707, −0.259 ± j 0.966
7 −1 ± j 0, −0.901 ± j 0.434, −0.624 ± j 0.782, −0.222 ± j 0.975
8 −0.981 ± j 0.195, −0.832 ± j 0.556, −0.556 ± j 0.832, −0.195 ± j 0.981

By Ian Poole


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