Butterworth Filter Order
Minimum Butterworth order to hit a stopband spec given passband ripple, attenuation, and the two edge frequencies.
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The engineering
The Butterworth response is maximally flat in the passband with no ripple, trading that smoothness for a gentler rolloff than Chebyshev or elliptic designs. This card returns the minimum integer order that meets both edges: at most Aₚ loss at fₚ and at least A_s loss at f_s. Rolloff far out is 20·n dB/decade, so each added order buys another 6 dB/octave.
Because the order rounds up, the finished filter overshoots your stopband spec — the 'actual stopband loss' row shows by how much. The two cutoff values bracket where you can place the 3-dB natural frequency ω_c: anchor to the passband edge to guarantee ripple margin, or to the stopband edge to maximize rejection. For a real analog build, expect the achievable order to top out near 8–10 before component tolerances dominate.
Where this math comes from
Stephen Butterworth was a British physicist working at the Admiralty Research Laboratory when he published 'On the Theory of Filter Amplifiers' in the October 1930 issue of Experimental Wireless. He was chasing a practical problem — cascading valve amplifier stages without the ragged passband that plagued the multi-section designs of the day — and showed that a specific pole placement gives a gain curve as flat as mathematically possible near DC.
Butterworth could only build a few stages by hand and never named the filter after himself; the label came later as engineers formalized the approximation-theory zoo alongside Cauer's elliptic and Chebyshev's equiripple forms. The order formula used here is the standard closed-form spec-to-order map that shows up in every filter-design text from the 1950s onward.
- 1930Stephen ButterworthPublishes the maximally-flat amplifier filter response in Experimental Wireless.
- 1931Wilhelm CauerDevelops the elliptic filter, framing Butterworth's design within general approximation theory.
- 1958Louis WeinbergTabulates normalized Butterworth pole locations and element values for practical synthesis.
- 1975IEEE / analog-design textsStandardize the log-ratio order formula this card evaluates.
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