Butterworth Response
|H(jω)|² = 1 / [1 + (ω/ω_c)^(2N)] N = filter order −3 dB exactly at ω_c for all N Roll-off: −20·N dB/decade beyond cutoff No passband ripple (maximally flat at ω=0) Monotonically decreasing magnitude
Butterworth Pole Locations
N poles lie on the left half of the unit circle in the s-plane, uniformly spaced by 180°/N:
sₖ = ω_c · exp(j·π·(2k+N−1)/(2N)) k = 1,2,…,N N=3 poles: s = ω_c·(−1), ω_c·(−½±j√3/2) Normalized: (s+1)(s²+s+1) [standard 3rd-order Butterworth]
Normalized Lowpass Prototype Element Values
| Order N | g₁ | g₂ | g₃ | g₄ | g₅ |
|---|---|---|---|---|---|
| 1 | 2.000 | — | — | — | — |
| 2 | 1.4142 | 1.4142 | — | — | — |
| 3 | 1.0000 | 2.0000 | 1.0000 | — | — |
| 4 | 0.7654 | 1.8478 | 1.8478 | 0.7654 | — |
| 5 | 0.6180 | 1.6180 | 2.0000 | 1.6180 | 0.6180 |
(g values are normalized inductances/capacitances for LC ladder prototype with Z₀=1Ω, ω_c=1 rad/s)
Frequency Transformation to Bandpass
LPF → BPF substitution: s → Q(s/ω₀ + ω₀/s) ω₀ = center freq = √(f_low·f_high) Q = ω₀ / (f_high − f_low) = loaded Q Each LPF element → resonant pair: LPF inductor L → series LC (with series resonance at ω₀) LPF capacitor C → parallel LC (with parallel resonance at ω₀)
RF View Filter Analysis: Load a measured Butterworth BPF .s2p file into RF View. BW Marker automatically reads the 3 dB bandwidth and center frequency. Compare measured response against theoretical Butterworth rolloff of 20N dB/decade.