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Butterworth Filter Lowpass Prototype Element Values

Simulations of Normalized and Denormalized LP, HP, BP, and BS Filters

Lowpass Filters

(above)

Highpass Filters

(above)

Bandpass and Bandstop Filters

(above)

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, vol. 7, 1930, pp. 536-541.

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 denormalization equations.  Complex poles are here.

 Capacitor Input Inductor Input
 Capacitor Input, RS=RL=1 Ω, f=1 rad/sec 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 Inductor Input, RS=RL=1 Ω, f=1 rad/sec