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.
The table below lists prototype element values for the normalized lowpass function, which assumes a cutoff frequency
of 1 rad/sec and source and load impedances of 1 Ω. Either an input capacitor
(top title line in table) or an input inductor (bottom title line in table) can
be used.
Convert Butterworth prototype values to other cutoff frequencies, impedances, and to highpass, bandpass or bandstop
using the equations here.
See my online Butterworth filter calculators and plotters
here.
Complex poles are here.
Capacitor Input
Inductor Input
Order 
C1 
L2 
C3 
L4 
C5 
L6 
C7 
L8 
C9 
L10 
1 
2.000 









2 
1.41421 
1.41421 








3 
1.00000 
2.00000 
1.00000 







4 
0.76537 
1.84776 
1.84776 
0.76537 






5 
0.61803 
1.61803 
2.00000 
1.61803 
0.61803 





6 
0.51764 
1.41421 
1.93185 
1.93185 
1.41421 
0.51764 




7 
0.44504 
1.24698 
1.80194 
2.00000 
1.80194 
1.24698 
0.44504 



8 
0.39018 
1.11114 
1.66294 
1.96157 
1.96157 
1.66294 
1.11114 
0.39018 


9 
0.34730 
1.00000 
1.53209 
1.87938 
2.00000 
1.87938 
1.53209 
1.00000 
0.34730 

10 
0.31287 
0.90798 
1.41421 
1.78201 
1.97538 
1.97538 
1.78201 
1.41421 
0.90798 
0.31287 

L1 
C2 
L3 
C4 
L5 
C6 
L7 
C8 
L9 
C10 
