What Is a Butterworth Filter? Circuit Theory and Sallen-Key Design

A Butterworth filter is an electronic signal processing circuit designed to provide a maximally flat frequency response in the passband, meaning it passes desired frequencies with zero amplitude ripple before smoothly rolling off unwanted signals. In a real circuit or installation, inserting this filter changes how your system handles noise: it strips out high-frequency interference (in a low-pass configuration) or low-frequency DC drift (in a high-pass configuration) without altering the volume, amplitude, or phase relationships of the frequencies you actually want to measure or hear. Beginners frequently confuse it with Chebyshev filters (which trade passband ripple for a steeper roll-off) or Bessel filters (which prioritize linear phase response and step-response over amplitude flatness).

The Core Math: Maximally Flat Magnitude

The defining characteristic of a Butterworth filter is its magnitude response. Unlike other filter topologies that might boost frequencies near the cutoff point (peaking) or attenuate them prematurely (droop), the Butterworth response is mathematically optimized to be as flat as possible up to the -3 dB cutoff point.

The attenuation rate in the stopband is dictated by the filter's order (the number of poles). Each pole contributes -20 dB/decade (or -6 dB/octave) of roll-off. Therefore, a 2nd-order Butterworth filter attenuates unwanted signals at -40 dB/decade, while a 4th-order drops them at -80 dB/decade. The transfer function magnitude $|H(j\omega)|$ for an $n$-th order low-pass Butterworth filter is defined as:

$|H(j\omega)| = \frac{1}{\sqrt{1 + (\frac{\omega}{\omega_c})^{2n}}}$

Where $\omega_c$ is the cutoff frequency in radians per second. Notice that at $\omega = \omega_c$, the denominator becomes $\sqrt{2}$, yielding a gain of $1/\sqrt{2}$, which is exactly -3.01 dB, regardless of the filter order $n$.

Worked Numeric Example: 1 kHz Sallen-Key Low-Pass

Let’s design a 2nd-order (12 dB/octave) unity-gain Sallen-Key Butterworth low-pass filter. This is a standard active topology used to prevent aliasing when feeding an analog sensor signal into a microcontroller ADC, such as the 12-bit SAR ADC on an ESP32.

We will use a modern, low-noise, rail-to-rail op-amp like the Texas Instruments OPA1678, powered by a single 3.3V supply to match the ESP32's logic levels.

The Sallen-Key Topology Wiring

For a unity-gain Sallen-Key low-pass filter, the signal path and feedback network are wired as follows:

1. Input Path: The analog signal passes through Resistor 1 ($R_1$) to Node A, then through Resistor 2 ($R_2$) to the op-amp's non-inverting input (+).
2. Feedback Cap: Capacitor 1 ($C_1$) connects from Node A directly to the op-amp's output pin. This feedback path creates the complex poles required for the Butterworth response without needing inductors.
3. Ground Cap: Capacitor 2 ($C_2$) connects from the non-inverting input (+) to ground.
4. Unity Gain: The inverting input (-) is tied directly to the output pin.

Calculating Component Values

Our target cutoff frequency ($f_c$) is 1,000 Hz. For a 2nd-order Butterworth response in a unity-gain Sallen-Key topology, the standard design equations require a specific capacitor ratio. If we set $R_1 = R_2 = R$, the capacitor values must satisfy $C_1 = 2 \times C_2$ (specifically, $C_1 = 1.414C$ and $C_2 = 0.707C$ where $C = \frac{1}{2\pi R f_c}$).

Let’s choose a standard 1% resistor value of $R = 15.0\text{ k}\Omega$.

First, find the base capacitance $C$:

$C = \frac{1}{2 \pi \times 15,000 \times 1,000} = 10.61\text{ nF}$

Now, calculate the specific capacitor values:

• $C_1 = 1.414 \times 10.61\text{ nF} = \mathbf{15.0\text{ nF}}$
• $C_2 = 0.707 \times 10.61\text{ nF} = \mathbf{7.50\text{ nF}}$

Where You Meet This in Practice

While filter theory often lives in textbooks, the Butterworth response solves specific, painful hardware problems on the workbench.

1. ADC Anti-Aliasing Front-Ends

If your ESP32 samples a load cell at 10 kHz, the Nyquist limit is 5 kHz. Any noise above 5 kHz (like switching regulator ripple at 1.2 MHz) will fold back down into your baseband, appearing as phantom low-frequency noise. A 4th-order Butterworth low-pass filter set to 2 kHz provides a flat response for your actual load cell data (usually < 50 Hz) while aggressively crushing the high-frequency switching noise before it reaches the ADC sample-and-hold circuit.

2. Active Audio Crossovers

In bi-amplified speaker systems, a Butterworth filter is used to split the audio spectrum. Because the passband is maximally flat, a 2.5 kHz Butterworth low-pass feeding a woofer ensures that the vocal range (300 Hz - 3 kHz) is not artificially colored or boosted right before the crossover point, preserving the natural timbre of the audio.

3. Biomedical Sensor Conditioning

When reading ECG (electrocardiogram) signals, the raw data is plagued by 50/60 Hz mains hum and high-frequency EMG (muscle) noise. A band-pass Butterworth filter (combining a high-pass at 0.5 Hz to remove baseline wander, and a low-pass at 40 Hz to remove muscle noise) extracts the clean QRS complex without introducing passband ripple that could mimic cardiac arrhythmias.

Butterworth vs. Chebyshev vs. Bessel

Choosing the wrong filter type is a common reason for failed signal conditioning. Use the Analog Devices Filter Wizard to simulate these differences, or refer to the comparison matrix below to make your topology decision.

Choose Butterworth when: You need to preserve the exact amplitude of signals in the passband and don't mind a moderate transition band.
Choose Chebyshev when: You have a strict requirement to kill a specific nearby interference frequency and can tolerate passband ripple.
Choose Bessel when: You are filtering digital square waves or time-critical pulse data where phase shift would distort the pulse shape.