a Sallen-Key filter
/ SAL-en KEE /
The Sallen-Key filter is the most popular way to build one second-order filter section with a single op-amp. It is the workhorse you will meet first, prized for being simple, well-behaved, and easy to design from a table. Picture two RC stages feeding an op-amp, with a clever feedback wire from the output that conjures up resonance no plain RC ladder could manage.
In the low-pass version, the signal passes through two resistors in series, each followed by a capacitor, into the op-amp's non-inverting input, which is wired as a buffer or a gain stage. The trick is that the first capacitor connects not to ground but back to the op-amp's output. Because the output is a near-copy of the input in the passband, that feedback bootstraps the capacitor and creates the second-order resonant peak, the controllable Q, without any inductor. With equal R and C and unity gain, the cutoff is f_c = 1/(2 times pi times R times C) and the Q is set by the ratio of the components and the stage's gain.
Sallen-Key sections cascade neatly to make any order, and named designs (Butterworth, Bessel, Chebyshev) just pick each section's Q from a table. Its honest limits: it gives low to moderate Q comfortably but gets touchy and component-sensitive at high Q, the non-inverting input couples some input signal straight through at very high frequencies (poor far-stopband rejection), and like any active filter it lives within the op-amp's bandwidth. For high-Q or band-pass work the multiple-feedback or state-variable topology is often a better choice.
A unity-gain Butterworth second-order low-pass at 1 kHz: pick C, then with equal resistors the Butterworth Q of 0.707 is set by making the two capacitors a 2-to-1 ratio. For example R = 11.3 kilohm, C1 = 20 nF, C2 = 10 nF gives a flat passband and a 12 dB-per-octave roll-off above 1 kHz.
One op-amp, two Rs and two Cs make a second-order section; the component ratios set the Q.
Sallen-Key is easy and well-behaved at low-to-moderate Q, but at high Q it becomes very sensitive to component tolerances, and its non-inverting input lets high frequencies leak straight through, hurting deep-stopband rejection.