Compute center frequency (f0), bandwidth (BW), lower & upper cut-off frequencies, and quality factor (Q) for a second‑order band‑pass filter. Interactive magnitude response graph helps visualize the passband, -3dB points, and filter selectivity.
A band‑pass filter (BPF) allows signals within a specific frequency range to pass while attenuating frequencies outside that range. It is characterized by the center frequency (f0) and the ‑3 dB bandwidth (BW). The ratio defines the quality factor: Q = f0 / BW. Higher Q implies narrower passband and better selectivity.
Transfer function of a second‑order band‑pass filter (normalized form):
H(s) = H0 · (ω0/Q)·s / (s² + (ω0/Q)s + ω0²)
Magnitude response: |H(ω)| = H0 / √(1 + Q² (ω/ω0 - ω0/ω)²)
Where ω0 = 2πf0, and H0 is the gain at center frequency (linear).
For a band-pass filter derived from a low-pass and high-pass cascade, the -3dB cut-off frequencies relate to the center frequency by geometric mean: f0 = √(fL·fH) and bandwidth BW = fH - fL. When designing active filters using Sallen-Key or Multiple-Feedback topologies, component values are directly linked to f0 and Q. This calculator provides essential parameters for further circuit synthesis.
Given a desired center frequency f0, quality factor Q, and passband gain H0 (linear), the Sallen-Key topology (unity gain version) can be realized with two resistors and two capacitors. A common design uses equal capacitors: C1 = C2 = C. Then the component values are given by:
R1 = Q / (2π f0 C H0)
R2 = Q / (2π f0 C (2Q² - H0))
R3 = 2Q / (2π f0 C)
Example: f0 = 1000 Hz, Q = 5, H0 = 1 (0 dB gain). Choose C = 10 nF. Then:
| R1 | = 5 / (2π·1000·10e-9·1) ≈ 79.6 kΩ |
| R2 | = 5 / (2π·1000·10e-9·(2·25 - 1)) ≈ 5 / (2π·1000·10e-9·49) ≈ 1.62 kΩ |
| R3 | = 2·5 / (2π·1000·10e-9) = 10 / (6.283e-5) ≈ 159 kΩ |
These values are standard approximations; practical designs may use closest E24/E96 series resistors. For high Q (≥10), component tolerance becomes critical.
For non-unity gain or other topologies (Multiple Feedback), refer to specialized filter design handbooks. The calculator above helps you quickly obtain f0, BW, and Q — the first step toward any active or passive BPF implementation.
Given f0 and BW, the lower and upper cut-off frequencies are calculated by solving the system: f0² = fL·fH and BW = fH - fL. This yields:
fL = ( -BW + √(BW² + 4f0²) ) / 2 and fH = fL + BW
Alternatively, using the quality factor: fL = f0·( √(1+1/(4Q²)) - 1/(2Q) ) and fH = f0·( √(1+1/(4Q²)) + 1/(2Q) ). Our implementation uses the exact geometric relation for high precision.
The interactive Bode magnitude plot shows relative gain (dB) vs frequency (log scale). The peak at f0 corresponds to the specified passband gain. The vertical purple lines mark the computed fL and fH where the response drops by 3 dB from the peak. The orange dashed line indicates the -3 dB reference. The graph updates in real time as you adjust parameters, helping you understand the trade-off between bandwidth and selectivity.
A loudspeaker designer needs a band-pass filter for a midrange driver covering 500 Hz to 4000 Hz. Using the geometric mean, f0 = √(500·4000) ≈ 1414 Hz, BW = 3500 Hz, Q ≈ 0.404. This moderate Q ensures a smooth passband. Our tool quickly validates that the -3 dB points match the desired crossover region. The frequency response confirms minimal ripple and proper roll-off, enabling high-fidelity audio reproduction.
| Topology | Advantages | Disadvantages | Best for |
|---|---|---|---|
| Sallen-Key (unity gain) | Simple, low component count, good for moderate Q (≤10) | Sensitive to component tolerances at high Q, limited gain flexibility | General purpose, audio, low-frequency BPF |
| Multiple Feedback (MFB) | Low sensitivity, stable Q, allows gain adjustment | More components, inverting output | High Q applications, precision filters |
| State-Variable (Biquad) | Independent tuning of f0, Q, and gain; low distortion | Requires multiple op-amps, complex | Laboratory instruments, variable filters |