Active Filters: Low-Pass, High-Pass and Band-Pass Guide.

An active filter is a frequency-selective circuit built around an op-amp with resistors and capacitors. Unlike passive filters, it provides voltage gain and low output impedance, so later stages do not load it.

Almost every electronic system must keep wanted frequencies and reject unwanted ones: audio splits bass from treble, instruments remove mains hum, radios select one station. Active designs do this without the bulky inductors passive filters need at low frequencies.

Key Concepts

• Low-pass filter: passes frequencies below cutoff, attenuating higher ones; removes noise and prevents aliasing.
• High-pass filter: passes frequencies above cutoff, blocking lower ones including DC; removes drift and rumble.
• Band-pass filter: passes a band between lower and upper cutoffs; built by cascading high-pass and low-pass sections.
• Cutoff frequency (f_c): where output power is half the passband value (a 3 dB drop).
• Order: the number of reactive stages: first-order rolls off at 20 dB/decade, second-order at 40 dB/decade.
• Sallen-Key topology: a popular second-order stage using one op-amp, two resistors and two capacitors.

Formulas

• First-order cutoff: f_c = 1/(2 pi R C) with R in ohms and C in farads; it designs the basic RC stage.
• Non-inverting stage gain: A_v = 1 + R_f/R_in, setting the op-amp's boost.
• Second-order roll-off: 40 dB/decade (12 dB/octave), twice as steep as first-order, for sharper passband-stopband separation.
• Band-pass bandwidth: BW = f_H − f_L between the 3 dB points; Q = f_0/BW with f_0 the center frequency.

Worked Example

Design a first-order low-pass filter: f_c = 1 kHz, gain 2, C = 10 nF. R = 1/(2 pi × 1000 × 10e-9) = 1/6.2832e-5 = 15,915 ohms; use the standard 16 kΩ. With R_in = R_f = 10 kΩ: A_v = 1 + 10k/10k = 2. Check: f_c = 1/(2 pi × 16000 × 10e-9) = 994.7 Hz, within 1% of target. Signals below ~1 kHz pass with gain 2; higher frequencies fall at 20 dB/decade.

Common Misconceptions

• Active filters need no power supply. The op-amp must be powered; without supply rails the circuit does nothing.
• Higher order is always better. Steeper filters need matched components; tolerances can distort the response or cause peaking.
• The op-amp works perfectly at any frequency. A real op-amp's gain-bandwidth product limits performance; keep the cutoff well below it.

Real-World Applications

• Audio systems: crossover networks and tone controls split and shape bass, midrange and treble.
• Medical instruments: ECG and EEG front ends use active filters to remove 50 Hz mains interference while keeping the tiny biological signal.
• Data acquisition: anti-aliasing low-pass filters sit before every analog-to-digital converter to block frequencies above half the sampling rate.
• Communications: intermediate-frequency stages in radios select the wanted channel with band-pass filtering.

Active filters combine RC frequency selectivity with op-amp gain and buffering. With one cutoff formula and one gain formula, you can design practical stages for audio, measurement and communications.