Butterworth Filter 2nd Order: Sallen-Key Design & Component Math

A butterworth filter 2nd order is an analog signal processing circuit that provides a maximally flat frequency response in the passband with a roll-off rate of -40 dB/decade (-12 dB/octave) past the cutoff frequency. In a real circuit, it changes the signal path by eliminating the passband ripple found in Chebyshev designs while providing a significantly steeper high-frequency rejection than a basic 1st-order RC filter, making it the default choice for audio crossovers and ADC anti-aliasing.

Before we calculate component values, it is critical to understand where the Butterworth response sits among other standard filter alignments. The defining characteristic of the Butterworth is its Q-factor of exactly 0.707 ($1/\sqrt{2}$), which yields zero ripple in the passband but introduces a slight phase non-linearity near the cutoff.

Filter Response Comparison: Butterworth vs. The Rest

When designing an active filter, the topology (like Sallen-Key) dictates the circuit layout, but the alignment (Butterworth, Bessel, Chebyshev) dictates the component ratios. Here is how the 2nd-order alignments compare in real-world performance.

Designing a 1 kHz Sallen-Key Low-Pass Filter

The most common topology for a 2nd-order active filter is the Sallen-Key. For a Butterworth alignment, the easiest approach is the equal-component Sallen-Key (where $R_1 = R_2$ and $C_1 = C_2$), which requires setting the op-amp's non-inverting gain to exactly 1.586.

Target Specifications:

• Cutoff Frequency ($f_c$): 1,000 Hz
• Filter Type: Low-Pass, 2nd-Order Butterworth
• Topology: Sallen-Key (Equal Components)

Step 1: Select the Capacitors

Capacitor values dictate the impedance scale of the filter. We want to avoid values so low that stray PCB capacitance matters, or so high that we need electrolytic capacitors. Let's choose 10 nF for both $C_1$ and $C_2$.

Step 2: Calculate the Resistors

Using the standard Sallen-Key cutoff formula for equal components:

f_c = 1 / (2 * π * R * C)

Rearranging to solve for R:

R = 1 / (2 * π * 1000 Hz * 10e-9 F) = 15,915 Ω

The closest standard 1% (E96 series) resistor value is 15.8 kΩ. Using 15.8 kΩ shifts our actual cutoff frequency to 1,006 Hz, which is well within acceptable tolerances for audio and sensor applications.

Step 3: Set the Op-Amp Gain for Butterworth Q

In an equal-component Sallen-Key filter, the Q-factor is determined entirely by the amplifier gain ($A_v$) according to the formula:

Q = 1 / (3 - A_v)

For a Butterworth response, Q must be 0.707. Solving for $A_v$:

0.707 = 1 / (3 - A_v) => A_v = 1.586

The gain is set by the feedback resistors $R_f$ and $R_i$ in the non-inverting configuration: A_v = 1 + (R_f / R_i).
Let's choose 10.0 kΩ for $R_i$.
1.586 = 1 + (R_f / 10,000) => R_f = 5,860 Ω
The closest standard 1% value is 5.90 kΩ, yielding a real-world gain of 1.59 and a Q of 0.709—virtually indistinguishable from a perfect Butterworth on an oscilloscope.

Step 4: Select the Op-Amp

For a 1 kHz audio filter, a dual JFET-input op-amp like the TI OPA2134 or TL072 is ideal due to low voltage noise and high input impedance. If you are filtering a DC sensor signal (like a load cell) on a single 5V supply, use a precision CMOS rail-to-rail op-amp like the ADA4522 or OPA350.

Where You Meet This in Practice

The 2nd-order Butterworth is the workhorse of analog signal conditioning. Here is where you will actively deploy it on the bench:

• ADC Anti-Aliasing: When feeding a signal into a 16-bit SAR ADC (like the ADS1115), you must filter out frequencies above the Nyquist limit. A 2nd-order Butterworth at 1 kHz provides -40 dB/decade attenuation, ensuring high-frequency noise doesn't fold back into your digital sampling band.
• Audio Crossovers: In active speaker crossovers, the Butterworth alignment is used to split frequencies between woofers and tweeters. (Note: A 4th-order Linkwitz-Riley crossover is simply two 2nd-order Butterworth filters cascaded in series, yielding a -6dB intersection point for perfect acoustic summing).
• PID Control Loop Conditioning: When reading a noisy thermocouple or RTD via an instrumentation amplifier, a 2nd-order Butterworth low-pass filter removes high-frequency EMI without introducing the phase-delay ringing that would destabilize a tight PID feedback loop.

Common Confusions and Real-World Gotchas

Even experienced engineers trip over a few specific edge cases when moving from simulation to physical PCB layout.

Confusing Butterworth with Bessel for Pulse Signals

People commonly confuse the Butterworth with the Bessel filter. If you are filtering a PWM signal to extract a DC analog voltage, or passing square waves, do not use a Butterworth. The Butterworth's Q of 0.707 causes a slight step-response overshoot (ringing). You need a Bessel filter (Q = 0.577), which has a slower roll-off but a perfectly linear phase response, preserving the square edges of your pulses without ringing.

The Op-Amp GBW Trap

Active filters rely on the op-amp having sufficient open-loop gain at the cutoff frequency. A common mistake is using a general-purpose op-amp like the LM358 (GBW = 1 MHz) for a 100 kHz Butterworth filter. At 100 kHz, the LM358's internal gain has dropped so low that it can no longer maintain the required 1.586 closed-loop gain, causing the filter to peak wildly or oscillate. Rule of thumb: Your op-amp's Gain Bandwidth Product (GBW) must be at least 100 times the filter's cutoff frequency. For a 100 kHz filter, you need an op-amp with a minimum 10 MHz GBW, like the OPA350 (39 MHz).

Cascading for Higher Orders

If you need a 4th-order Butterworth (-80 dB/decade), you cannot simply cascade two identical 2nd-order Butterworth stages. The combined -3dB points will stack, shifting your overall cutoff frequency down and creating a droop in the passband. To build a 4th-order Butterworth, you must cascade two 2nd-order stages with different Q-factors (Stage 1: Q = 0.541; Stage 2: Q = 1.306). Consult standard filter coefficient tables for these values.

For deeper mathematical derivations of the Sallen-Key topology and pole-zero mapping, refer to the Texas Instruments SLOA024 Application Report on active filter design, or the comprehensive breakdown of second-order active networks at All About Circuits.