Analog Electronics

Active Filters: Sallen-Key

When passive RC filters are cascaded they load each other and the sharpness is lost. In an active filter the op-amp buffers the stages, and positive feedback places the poles exactly where you want them.

Sallen-Key builds a 2nd-order filter around a unity-gain follower with 2R+2C: the slope beyond cutoff is −40 dB/decade. The Q factor is set by the capacitor ratio; Q=0.707 (Butterworth) gives the flattest passband.

Higher orders are obtained by cascading 2nd-order stages with different Q values (e.g. 4th-order Butterworth = two stages with Q=0.54 and Q=1.31).

Formulas

fc = 1/(2π·R·√(C1·C2))

Q = ½·√(C1/C2) (equal R)

Slope: −40 dB/decade

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Test Yourself

What is the stopband slope of a 2nd-order low-pass filter?

Answer: −40 dB/decade — Each pole contributes −20 dB/decade; 2 poles → −40.

What should Q be for a Butterworth response?

Answer: 0.707 — Q=1/√2 gives the maximally flat amplitude response.

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