How it works
An active filter uses an op-amp with RC networks to achieve frequency-selective amplification. First-order low-pass filter: Vout/Vin = −(Zf/R1) where Zf = Rf ∥ (1/jωC); cutoff frequency fc = 1/(2πRfC), roll-off 20 dB/decade. Second-order Sallen-Key LPF uses two RC sections around a non-inverting op-amp, achieving 40 dB/decade roll-off with transfer function H(s) = ω0²/(s² + (ω0/Q)s + ω0²). Quality factor Q determines response shape: Q = 0.707 (1/√2) gives the maximally flat Butterworth response. For high-pass, capacitors and resistors are swapped in the RC networks. Band-pass filters cascade LPF and HPF stages, or use the state-variable topology.
Key points to remember
First-order filter: fc = 1/(2πRC), roll-off 20 dB/decade. Second-order: roll-off 40 dB/decade, cutoff fc = 1/(2π√(R1R2C1C2)) for Sallen-Key. Butterworth response requires Q = 0.707 — maximally flat passband, no ripple. Chebyshev filters allow passband ripple (typically 0.5 dB to 3 dB) for sharper roll-off at the same order. At fc, gain is −3 dB (≈ 0.707 of passband gain) for all filter types — this definition must be stated in exam answers. Active filters avoid inductors, making them compact and integrable; inductor-based passive filters become impractical below 1 kHz due to physical size. The gain-bandwidth product of the op-amp limits the upper frequency of active filter designs.
Exam tip
The examiner always asks you to calculate the cutoff frequency of a first-order active LPF and state the roll-off rate — write fc = 1/(2πRC) and state −3 dB point with 20 dB/decade roll-off explicitly in your answer.