Free Sallen-Key Active Filter Tool

1. Sallen-Key Topology & Second-Order Roll-Off

Invented by R. P. Sallen and E. L. Key of MIT Lincoln Laboratory in 1955, the Sallen-Key active filter is the world's most widely deployed second-order ($2$-pole) active filter architecture. A single operational amplifier provides a steep $-40 ext{ dB/decade}$ ($-12 ext{ dB/octave}$) roll-off without inductors.

2. Comparing Butterworth, Bessel, and Chebyshev Alignments

The damping ratio ($zeta$) and quality factor ($Q = rac{1}{2zeta}$) define the filter's frequency and time-domain behavior:

• Butterworth ($Q = 0.7071$, $zeta = 0.7071$): "Maximally Flat". Delivers flat passband frequency response with zero ripple. The $-3 ext{dB}$ point occurs exactly at cutoff $f_c$. Ideal for ADC anti-aliasing and audio preamplifiers.
• Bessel ($Q = 0.5774$, $zeta = 0.8660$): "Linear Phase". Produces flat group delay across the passband, resulting in zero ringing, zero overshoot, and perfect preservation of square waves and digital pulses.
• Chebyshev ($Q = 0.86$ to $1.0$): Provides the steepest initial transition band roll-off in exchange for a small amount of passband ripple ($0.5 ext{ dB}$ to $1.0 ext{ dB}$).

3. The Equal-Resistor Design Strategy

In the unity-gain equal-resistor design ($R_1 = R_2 = R$): $$C_1 = rac{2 Q}{omega_c R}, quad C_2 = rac{1}{2 Q cdot omega_c R}$$ Notice that the capacitor ratio is exactly $C_1 / C_2 = 4 Q^2$. For a Butterworth filter ($Q = 0.7071$), $4 Q^2 = 2.0$, meaning $C_1$ is exactly twice the value of $C_2$ ($C_1 = 2 cdot C_2$). This makes sourcing standard precision film capacitors straightforward (e.g. $C_1 = 20 ext{ nF}$ using two $10 ext{ nF}$ caps in parallel, and $C_2 = 10 ext{ nF}$).