Butterworth poles lie along a circle and are spaced at equal angular distances around a circle. Prototype value real and imaginary pole locations (ω=1 at the 3dB cutoff point) for Butterworth filters are presented in the table below.
Pole locations are calculated as follows, where K=1,2,...,n. n is the filter order.
Order (N)
Re Part (-σ)
Im Part (ąjω)
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
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.
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