
Butterworth Lowpass Filter Poles


Butterworth
poles lie along a circle and are spaced at equal angular distances around a circle. It is designed
to have a frequency response which is as flat as mathematically possible in the passband, and
is often referred to as a 'maximally flat magnitude' filter. Prototype value real and imaginary
pole locations (ω=1 at the 3 dB cutoff point) for Butterworth filters are presented in the table
below.
The Butterworth type filter was first described by the British engineer Stephen
Butterworth in his paper "On the Theory of Filter Amplifiers", Wireless Engineer (also called
Experimental Wireless and the Wireless Engineer), vol. 7, 1930, pp. 536541.
See my
online filter calculators and plotters
here.
Butterworth filter
prototype element values are here.
Pole locations are calculated as follows, where K=1,2,...,n. n is the filter
order.
Order (n) 
Re Part (σ) 
Im Part (±jω) 
1 
1.0000 

2 
0.7071 
0.7071 
3 
0.5000 1.0000 
0.8660 
4 
0.9239 0.3827 
0.3827 0.9239 
5 
0.8090 0.3090 1.0000 
0.5878 0.9511 
6 
0.9659 0.7071 0.2588 
0.2588 0.7071 0.9659 
7 
0.9010 0.6235 0.2225 1.0000 
0.4339 0.7818 0.9749 
8 
0.9808 0.8315 0.5556 0.1951 
0.1951 0.5556 0.8315 0.9808 
9 
0.9397 0.7660 0.5000 0.1737 1.0000 
0.3420 0.6428 0.8660 0.9848 
10 
0.9877 0.8910 0.7071 0.4540 0.1564 
0.1564 0.4540 0.7071 0.8910 0.9877 
The Butterworth polynomials may be written in complex form as above, but are usually
written with real coefficients by multiplying pole pairs which are complex conjugates, such
as s_{1} and s_{n}. The polynomials are normalized by setting ω_{c}
= 1.
The normalized Butterworth polynomial equations have the general form:
n 
Factors of Polynomial B_{n}(s) 
1 
(s + 1) 
2 
(s^{2} + 1.4142s + 1) 
3 
(s + 1)(s^{2} + s + 1) 
4 
(s^{2} + 0.7654s + 1)(s^{2} + 1.8478s + 1) 
5 
(s + 1)(s^{2} + 0.6180s + 1)(s^{2} + 1.6180s + 1) 
6 
(s^{2} + 0.5176s + 1)(s^{2} + 1.4142s + 1)(s^{2} + 1.9319s
+ 1) 
7 
(s + 1)(s^{2} + 0.4450s + 1)(s^{2} + 1.2470s + 1)(s^{2}
+ 1.8019s + 1) 
8 
(s^{2} + 0.3902s + 1)(s^{2} + 1.1111s + 1)(s^{2} + 1.6629s
+ 1)(s^{2} + 1.9616s + 1) 
Data taken from "Filter Design," by Steve Winder, Newnes
Press, 1998. This is a great filter design book, and I recommend you purchase a copy of it.



