Active Band Pass Filter

A speech intelligibility processor for a PA system must amplify only the 300 Hz to 3.4 kHz voice band and reject everything else. The active band-pass filter does this in one stage using a single op-amp.

Core Concept

A band-pass filter passes a range of frequencies centered on a peak called the center frequency f0, and attenuates frequencies both above and below that range. The width of the pass-band is defined by the bandwidth BW = fH - fL, where fH and fL are the upper and lower -3 dB frequencies.

The multiple feedback (MFB) topology is the standard single op-amp second-order BPF. It uses two capacitors and three resistors with the op-amp in an inverting configuration. The feedback path through the capacitor provides the high-pass character, and the RC input network provides the low-pass character. The two combine to create a bandpass response centered at f0.

The quality factor Q = f0/BW determines how selective the filter is. A narrowband speech filter with f0 = 1 kHz and BW = 200 Hz has Q = 5. TL071 or NE5534 op-amps work well here. The MFB topology is limited to Q less than about 20 before gain-bandwidth limitations of the op-amp degrade the response.

Key Equations

Center frequency: f0 = 1 / (2π × sqrt(R1 × R2 × C1 × C2)). For equal capacitors C1 = C2 = C: f0 = 1 / (2πC × sqrt(R1 × R2))

Quality factor: Q = f0 / BW = (1/2) × sqrt(R2/R1) for equal capacitors in MFB topology.

Bandwidth: BW = fH - fL = f0 / Q. BW = 1/(πC × R2) in the equal-C MFB.

Pass-band gain at f0: |A0| = R2 / (2 × R1) for the equal-C MFB configuration (inverting, so output is 180 degrees from input at f0).

Given:
  R1 = R2 = 16 kΩ
  C1 = C2 = C = 10 nF = 10 × 10^-9 F
  Op-amp: TL071

Why this formula:
  Equal-C MFB BPF: f0 = 1/(2πC × sqrt(R1 × R2))

Formula:
  f0 = 1 / (2π × C × sqrt(R1 × R2))

Substitution:
  sqrt(R1 × R2) = sqrt(16k × 16k) = 16,000
  f0 = 1 / (2π × 10×10^-9 × 16,000)
     = 1 / (2π × 1.6×10^-4)
     = 1 / 1.0053×10^-3

Calculation:
  f0 = 994.7 Hz ≈ 995 Hz

Q = (1/2) × sqrt(R2/R1)
  = (1/2) × sqrt(16k/16k)
  = (1/2) × 1 = 0.5

BW = f0 / Q = 995 / 0.5 = 1990 Hz

Pass-band gain = R2/(2×R1) = 16k/(2×16k) = 0.5 (-6 dB)

Final Answer:
  f0 ≈ 995 Hz, Q = 0.5, BW = 1.99 kHz, Gain = 0.5

Exam Tip: GATE problems on active BPF often ask you to calculate Q, BW, and f0 separately. Know that BW = f0/Q and fL = f0/Q is wrong: the correct relations are fL = f0 × (sqrt(1 + 1/(4Q^2)) - 1/(2Q)) and fH similarly, but for GATE you usually use BW = f0/Q and fL × fH = f0^2 (geometric mean relation). The product fL × fH = f0^2 is the key identity tested most often.

Key Properties

• Active BPF passes a band of frequencies centered on f0 and attenuates signals both above and below this band.
• Center frequency f0 satisfies fL × fH = f0^2, meaning f0 is the geometric mean of the lower and upper -3 dB frequencies.
• Quality factor Q = f0/BW. Higher Q means narrower bandwidth and more selective filtering.
• MFB single op-amp topology is practical for Q up to about 10 to 20; above that, state-variable or biquad topologies are used.
• TL071 and NE5534 are common op-amp choices. NE5534 has lower noise, making it better for audio BPF applications.
• Pass-band gain in MFB equal-C topology is R2/(2R1). Gain cannot be set independently of Q without redesigning the topology.
• Phase shift at f0 is 180 degrees (inverting) for MFB. Phase approaches 0 degrees at very high and very low frequencies.

Quick Revision

• Active BPF has a peak at f0 and rolls off at -20 dB/decade each side (second-order gives -40 dB/decade total outside pass-band).
• f0 = 1/(2π × sqrt(R1R2C1C2)).
• Q = f0/BW. High Q = narrow band, selective filter.
• fL × fH = f0^2 (geometric mean property of bandpass filter).
• MFB gain at f0 = R2/(2R1) for equal-C design.
• MFB is inverting: output is 180 degrees from input at center frequency.
• State-variable topology allows independent control of Q, gain, and f0.
• Exam trap: Students confuse arithmetic mean and geometric mean. f0 is NOT (fL + fH)/2. It is sqrt(fL × fH). These are equal only for symmetric linear-scale bandpass, not for the standard BPF where the response is symmetric on a log frequency scale.