What Is a Butterworth Filter?

A Butterworth filter is a signal processing filter designed to keep the passband as flat as possible. No ripple, no gain variation before cutoff. It’s also called a maximally flat magnitude filter, which is just a formal way of saying it doesn’t distort the frequencies it’s letting through.

Butterworth Filter Equations

Transfer Function

The transfer function H(s) describes how the filter behaves mathematically in the s-domain. For an nth-order Butterworth low-pass filter, the gain magnitude drops by exactly -3 dB at the cutoff frequency, with poles distributed evenly along the left half of the complex s-plane.

Cutoff Frequency Formula

For a basic analog design, the -3 dB cutoff frequency comes directly from the RC time constant: fc = 1 / (2πRC). Higher-order responses stack additional sections around this same relationship, adjusting component ratios per stage while the core formula stays unchanged.

Magnitude in Decibels

Attenuation in decibels follows a logarithmic scale. Past the cutoff, magnitude drops at -20n dB per decade (where n is the filter order), so a 2nd-order design rolls off at -40 dB/decade, a 4th-order at -80 dB/decade, and so on.

Filter Order Determination Formula

The minimum order n comes from your passband and stopband attenuation requirements plugged into a log-ratio formula. The result is almost never a whole number, so always round up – never down.

Phase Response

The amplitude may be flat, but phase is a different story. Butterworth filters have a non-linear phase response, which causes group delay variation across frequencies. For pulse or time-sensitive signals, that matters.

Quality Factor for Second-Order Sections

For a second-order Butterworth section, Q is exactly 0.7071. That specific value is what produces the maximally flat response. Push Q higher and you get peaking near cutoff. Lower and the rolloff softens noticeably.

Normalized Butterworth Pole Coefficients Table

These are the normalized Butterworth polynomials for orders 1 through 8, all referenced to ωc = 1 rad/s. Use these to verify calculator outputs or manually derive component values before scaling to your actual cutoff frequency.

Order (n)	Factored Polynomial	Pole Locations
1	(s + 1)	-1.0000
2	(s² + 1.4142s + 1)	-0.7071 ± j0.7071
3	(s + 1)(s² + 1.0000s + 1)	-1.0000, -0.5000 ± j0.8660
4	(s² + 0.7654s + 1)(s² + 1.8478s + 1)	-0.3827 ± j0.9238, -0.9238 ± j0.3827
5	(s + 1)(s² + 0.6180s + 1)(s² + 1.6180s + 1)	-1.0000, -0.3090 ± j0.9511, -0.8090 ± j0.5878
6	(s² + 0.5176s + 1)(s² + 1.4142s + 1)(s² + 1.9319s + 1)	-0.2588 ± j0.9659, -0.7071 ± j0.7071, -0.9659 ± j0.2588
7	(s + 1)(s² + 0.4450s + 1)(s² + 1.2470s + 1)(s² + 1.8019s + 1)	-1.0000, -0.2225 ± j0.9749, -0.6235 ± j0.7818, -0.9010 ± j0.4339
8	(s² + 0.3902s + 1)(s² + 1.1111s + 1)(s² + 1.6629s + 1)(s² + 1.9616s + 1)	-0.1951 ± j0.9808, -0.5556 ± j0.8315, -0.8315 ± j0.5556, -0.9808 ± j0.1951

Butterworth Filter Topology Selection

Active vs Passive – Which to Choose

For most audio, instrumentation, and sensor-level work below 1 MHz, active filters are the practical default. No inductors, easier to tune, and straightforward to cascade into higher-order stages.

Passive designs earn their place when there’s no power supply available, or when the application involves RF frequencies or high-current power lines where op-amps simply don’t belong.

	Active Filter	Passive Filter
Power required	Yes	No
Frequency range	Below ~1 MHz	RF/microwave capable
Inductors needed	No	Often yes
Best for	Audio, sensors, instrumentation	RF, high-power, no-supply designs

Sallen-Key vs Multiple Feedback (MFB) Topology

Both are second-order active topologies, but they suit different situations. Sallen-Key is the more common starting point, with high input impedance, easy to design for unity gain, and well-documented everywhere. MFB handles component drift better and works well in inverting configurations, which matters more in precision or higher-frequency designs.

	Sallen-Key	MFB
Input impedance	High	Moderate
Gain configuration	Non-inverting	Inverting
Component drift sensitivity	Higher	Lower
Best for	General use, audio	Precision, higher-frequency stages

Op-Amp Selection for Active Butterworth Filters

The op-amp choice can quietly break an otherwise well-designed filter. A few things actually matter here:

• Gain-Bandwidth Product (GBW): Target at least 10–20× your cutoff frequency. Higher for precision work.
• Slew rate: Needs to keep up with your signal’s voltage swings without distorting the output.
• Input bias current: Matters most for low-frequency or DC-coupled sensor applications — JFET and CMOS inputs stay lowest here.
• Noise: The op-amp adds its own thermal and flicker noise into the signal path, so match the noise profile to your application’s sensitivity requirements.

Circuit Diagrams

Butterworth vs Other Filter Types – Which Should You Use?

Butterworth vs Chebyshev

The core tradeoff: passband flatness versus rolloff steepness. Butterworth gives you a clean, undistorted passband with no amplitude ripple. Chebyshev trades that flatness for a sharper transition into the stopband, which means better frequency separation with fewer stages, but your passband isn’t clean anymore.

Butterworth vs Bessel

Different problem entirely. Bessel filters prioritize constant group delay, which preserves the time-domain shape of a signal. If you’re filtering pulses or square waves and waveform distortion matters, Bessel is the right call. Butterworth wins on frequency-domain flatness; Bessel wins on phase linearity.

Butterworth vs Elliptic

Elliptic filters achieve the sharpest possible cutoff and deepest stopband rejection of any standard filter type. The cost is a ripple in both the passband and stopband, plus a nonlinear phase. If signal cleanliness matters, Butterworth. If aggressive frequency separation matters more, Elliptic.

When Butterworth Is the Right Default

When no specific requirement pushes you toward another design, Butterworth is the sensible starting point. No ripple, predictable rolloff, less phase distortion than the alternatives. Most audio, sensor, and instrumentation applications don’t need Chebyshev’s aggression or Elliptic’s complexity. Butterworth just works cleanly.

Common Design Errors and How to Fix Them

Calculated Cutoff Frequency Doesn’t Match Measured Response

The calculator gave you the right answer. The real world didn’t cooperate. Most commonly caused by:

• Component tolerances: Standard capacitors run 10–20% off. A theoretical 1kHz cutoff can land anywhere between 800Hz and 1.2kHz with off-the-shelf parts. Measure actual values with an LCR meter, then recalculate.
• Load impedance: Passive filters are sensitive to what’s connected after them. A low-impedance load forms a voltage divider that shifts the cutoff. Add an op-amp buffer between stages if needed.
• Op-amp bandwidth: If the signal frequency approaches the op-amp’s gain-bandwidth product, the amplifier loses gain, and the cutoff drifts downward. Check the datasheet. GBW should be at least 10–20× your target frequency.

Higher-Order Active Filter Is Unstable

Higher-order designs going unstable almost always comes down to one of three things:

• Wrong stage ordering: Each second-order section in a cascaded Butterworth has a different Q value. Sequence them, lowest Q first. Running a high-Q stage early causes it to saturate before the signal reaches the later stages.
• Insufficient op-amp GBW: Phase shift from inadequate bandwidth eats into phase margin. Use GBW ≥ 10 × Gain × fc × Q per stage — check this before blaming anything else.
• Single-stage implementation: A 4th- or 6th-order filter should never be built in one op-amp stage. Break it into cascaded 2nd-order sections.

Digital Filter Cutoff Frequency Is Wrong

Digital Butterworth cutoff errors are almost always one of two things:

• Missing pre-warping: The bilinear transform compresses frequencies near Nyquist. Without pre-warping, your designed cutoff lands lower than intended in the digital domain. Apply ωa = 2fs · tan(πfc/fs) before transforming.
• Wrong normalization: Tools like scipy and MATLAB expect the cutoff normalized to the Nyquist frequency (fc / (Fs/2)), not an absolute Hz value. Entering 500Hz into a tool expecting 0.5 produces a completely wrong filter.

Filter Shows Gain Peaking Just Below Cutoff

A correctly designed Butterworth filter shouldn’t peak at all. If you’re seeing a gain rise just before cutoff, Q is running higher than 0.7071 somewhere. Check component tolerances in each second-order section and verify the op-amp GBW is sufficient for that stage’s operating frequency.

Standard Component Values Cause Unacceptable Frequency Error

E-series values follow geometric progressions, which leaves real gaps between what you calculate and what you can actually buy. When resistors and capacitors each drift by even 1–5%, those deviations stack. Switch to E96 components, or combine standard values in series and parallel to hit the target resistance precisely.

Frequently Asked Questions