Butterworth Filter Designer Calculator

Design Inputs

Use the form below to design lowpass, highpass, bandpass, or bandstop Butterworth filters from attenuation targets and frequency edges.

Filter Type

Frequency Unit

Gain (dB)

Optional output gain added to the plotted response.

Passband Attenuation Ap (dB)

Stopband Attenuation As (dB)

Passband Edge fp

Used by lowpass and highpass designs.

Stopband Edge fs

For lowpass use fs > fp. For highpass use fp > fs.

Lower Frequency 1

Bandpass: lower stopband edge. Bandstop: lower passband edge.

Lower Frequency 2

Bandpass: lower passband edge. Bandstop: lower stopband edge.

Upper Frequency 3

Bandpass: upper passband edge. Bandstop: upper stopband edge.

Upper Frequency 4

Bandpass: upper stopband edge. Bandstop: upper passband edge.

Example Data Table

Case	Type	Input Example	Typical Output Snapshot
1	Lowpass	fp = 1 kHz, fs = 2 kHz, Ap = 1 dB, As = 40 dB	Prototype order ≈ 8, cutoff ≈ 1.089 kHz, monotonic passband
2	Highpass	fp = 5 kHz, fs = 2.5 kHz, Ap = 1 dB, As = 30 dB	Prototype order ≈ 6, cutoff ≈ 4.465 kHz, steep low frequency rejection
3	Bandpass	fs1 = 0.5 kHz, fp1 = 1 kHz, fp2 = 2 kHz, fs2 = 4 kHz	Prototype order ≈ 4, realized order ≈ 8, center frequency ≈ 1.414 kHz
4	Bandstop	fp1 = 0.8 kHz, fs1 = 1 kHz, fs2 = 2 kHz, fp2 = 2.5 kHz	Prototype order depends on attenuation target, notch centered near 1.414 kHz

Formula Used

1) Butterworth ripple parameter

ε = √(10^(Ap/10) − 1)

2) Required order for lowpass or highpass

n = ceil{ log10[(10^(As/10) − 1) / (10^(Ap/10) − 1)] / [2 log10(r)] }

For lowpass, r = fs / fp. For highpass, r = fp / fs.

3) Lowpass and highpass cutoff

Lowpass: ωc = ωp / ε^(1/n)

Highpass: ωc = ωp × ε^(1/n)

4) Band transformations

Bandpass: Ω = |(ω² − ω0²) / (Bω)|

Bandstop: Ω = |Bω / (ω² − ω0²)|

Here, ω0 = √(ω1ω2) and B = ω2 − ω1.

5) Magnitude response

|H(jω)| = 1 / √(1 + (Ω/Ωc)^(2n))

For lowpass and highpass, Ω becomes ω or ωc/ω. For band filters, Ω is the transformed lowpass variable.

How to Use This Calculator

Choose the filter family that matches your design target.
Select the working frequency unit before entering any frequency values.
Enter passband attenuation Ap and stopband attenuation As in decibels.
For lowpass or highpass, supply one passband edge and one stopband edge.
For bandpass or bandstop, enter four ordered frequencies that match the on-screen guidance.
Optionally enter output gain if you want the chart shifted vertically in decibels.
Press Design Filter to generate order, cutoff, stage data, poles, and the response curve.
Use the export buttons to save summary data as CSV or capture the result area as a PDF report.

FAQs

1) What makes a Butterworth filter different from other filters?

A Butterworth response is maximally flat in the passband. It avoids ripple and keeps a smooth magnitude curve, which is useful when clean amplitude behavior matters more than the sharpest possible transition.

2) Why does the required order increase quickly?

Order rises when stopband attenuation becomes stricter or when passband and stopband edges move closer together. A narrow transition band forces a steeper rolloff, which needs more poles.

3) Why do bandpass and bandstop designs show doubled realized order?

A lowpass prototype pole maps into two actual poles after a band transformation. Because of that mapping, the implemented bandpass or bandstop filter has twice the prototype order.

4) What does the Q value mean in the stage table?

Q describes the damping of each second order section. Higher Q means lower damping and a narrower resonant behavior. It helps when you convert the design into practical active or passive stages.

5) Does this tool design analog or digital filters?

This page designs analog Butterworth filters from edge frequencies and attenuation targets. You can still use the prototype information as a starting point before applying a digital transformation elsewhere.

6) Why can a design fail input validation?

Validation fails when the frequency order is physically inconsistent, such as a lowpass stopband below the passband, or when band edges are too close to satisfy the attenuation constraints.

7) What is the difference between prototype cutoff and passband edge?

The passband edge is your specification point. The computed cutoff shifts slightly so the response at that edge exactly satisfies the allowed passband attenuation for the chosen order.

8) Can I use the exported CSV for documentation or audit records?

Yes. The CSV includes summary metrics, stage data, pole locations, and sampled frequency response values. That makes it useful for reports, calculations review, and design traceability.