What order do I need?

Each order adds -20 dB/decade of roll-off. 2nd order gives -40 dB/dec, 4th gives -80 dB/dec. Choose the minimum order that provides enough attenuation at your stopband frequency.

Normalised prototype values from filter tables. They are denormalised (scaled) to your specific frequency and impedance. The calculator handles this automatically.

Butterworth Filter Calculator

Maximally Flat Passband

−20n dB/decade roll-off

|H(jω)|² = 1/(1+(ω/ω_c)²ⁿ)

Butterworth Filter Design

f_c Cutoff Frequency

Z Load Impedance

n Filter Order (1–8)

Higher order = sharper roll-off but more components

Enter fc, Z and order

Butterworth Design

Cutoff fc
—
Hz

Roll-off

—

Gain at 10×f_c

—

Figure 1: Butterworth filters have the flattest possible passband response. Higher orders provide steeper roll-off but require more components. All orders cross −3 dB at exactly f_c.

Table of Contents

Fundamentals

What Is a Butterworth Filter?
How It Works

Worked Examples

2nd Order Audio Low Pass
4th Order Anti-Aliasing

Deep Dive

Butterworth vs Chebyshev vs Bessel
Prototype and Denormalisation

Reference

Frequently Asked Questions
Related Filter Calculators

What Is a Butterworth Filter?

The Butterworth filter is designed for the flattest possible magnitude response in the passband — no ripples, no peaking. It is named after Stephen Butterworth who described it in 1930. The trade-off is a less steep transition band compared to Chebyshev or elliptic designs, but for many applications the smooth response is more important than the transition sharpness. The Filter Cutoff Frequency Calculator covers the fundamental frequency calculation that underpins filter design.

How It Works

Butterworth filters use a ladder network of alternating inductors and capacitors (for passive) or cascaded second-order active stages. The normalised prototype g-values are scaled to the desired frequency and impedance. The RC Filter Calculator provides simpler single-stage designs.

Worked Example — 2nd Order Audio Low Pass

Given: fc = 1 kHz, Z = 50 Ω, n = 2

C₁ = g₁/(ωc×Z) = 1/(2π×1000×50) = 3.183 µF
L₂ = g₂×Z/ωc = 1.4142×50/(2π×1000) = 11.25 mH

The RL Filter Calculator can compare this with other filter types.

Worked Example — 4th Order Anti-Aliasing

Given: fc = 10 kHz, Z = 600 Ω, n = 4 (anti-aliasing)

A 4th-order filter gives −80 dB/decade, attenuating 100 kHz signals by 80 dB — sufficient for 16-bit ADC anti-aliasing.

The LC Filter Calculator simplifies single-stage design.

Butterworth vs Chebyshev vs Bessel

Butterworth has zero passband ripple. Chebyshev allows ripple for steeper roll-off. Bessel has linear phase (best step response). Choose based on priority: amplitude flatness (Butterworth), selectivity (Chebyshev), or time-domain performance (Bessel). The Low Pass Filter Calculator applies Butterworth principles to active designs.

Prototype and Denormalisation

Filter design starts with a normalised prototype (fc = 1 rad/s, Z = 1 Ω). The g-values from Butterworth tables are then denormalised: C = g/(ωc×Z) and L = g×Z/ωc. This calculator handles denormalisation automatically. The High Pass Filter Calculator covers complementary filter designs.

Resonant Frequency Calculator: Each LC stage in a Butterworth ladder resonates near fc — the Resonant Frequency Calculator analyses individual stages.

Frequently Asked Questions

How many poles do I need?

Each pole adds −20 dB/decade. 2 poles (−40 dB/dec) is adequate for most audio. 4 poles (−80 dB/dec) for anti-aliasing. 6-8 poles for sharp RF selectivity.

Can I cascade Butterworth stages?

Yes. Two cascaded 2nd-order Butterworth stages make a 4th-order Butterworth — but only if each stage has Q = 0.541 and Q = 1.307 (not both 0.707). The calculator provides correct per-stage Q values.

Is Butterworth the best filter?

It depends on your priority. Butterworth is best for flat amplitude. Bessel is best for pulse response. Chebyshev is best for sharp selectivity. There is no single best — only best for a specific application.

What about component tolerances?

Use 1% resistors and 5% or better capacitors. For filters above 4th order, tighter tolerances (1-2% capacitors) are recommended as the response becomes more sensitive to component errors.

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Last updated: March 2026