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Kirt Blattenberger,

BSEE
- KB3UON

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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. 536-541.

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.

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) |

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 |

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.