A band-pass filter passes signals within a specific frequency band centred on f₀ and attenuates frequencies above and below. The key parameters are centre frequency f₀ (where peak transmission occurs), quality factor Q (selectivity), and bandwidth BW (the −3 dB width of the passband). They are related by BW = f₀ / Q. Higher Q = narrower passband = sharper transitions; lower Q = wider passband = gentler transitions.
Passive series RLC
A resistor, inductor, and capacitor in series make the simplest band-pass: at f₀ = 1/(2π√LC) the L and C impedances cancel (resonance), leaving just R between input and output → maximum transmission. Off-resonance, the LC combination has high impedance and most of the signal voltage drops across it instead of R. Q = (1/R)·√(L/C) — Q grows when R is smaller relative to the LC characteristic impedance.
Passive parallel RLC
The dual topology: a parallel LC tank in series with the source resistance, output across the tank. At f₀ the tank has infinite impedance (parallel resonance) → all source voltage appears across the tank → maximum output. Off-resonance the tank impedance is low and the source resistance drops most of the voltage. Q = R·√(C/L) — Q grows when R is large relative to characteristic impedance (opposite of series).
Multiple-Feedback (MFB) active band-pass
The standard op-amp band-pass topology. Three resistors + two capacitors + one op-amp. The op-amp's inverting input is held at virtual ground; positive feedback through the second capacitor + R₂ creates resonance, and R₃ in the feedback path sets the Q. The MFB is well-suited for moderate Q (up to ~10) with practical components, and can have arbitrary mid-band gain G. For higher Q you need state-variable or biquad topologies. Equations for equal-capacitor design:
- R₁ = Q / (G · ω₀ · C)
- R₂ = Q / ((2·Q² − G) · ω₀ · C)
- R₃ = 2·Q / (ω₀ · C)
The constraint G < 2·Q² is needed to keep R₂ positive. At Q = 5 and G = 1, R₂ is comfortably positive; pushing G near the limit makes R₂ very large and noisy.
The step response of a band-pass filter
Unlike LP and HP, the band-pass step response starts at zero (not 1 like HP) — because the DC component of the step is blocked. The response then rings at frequency f₀ with envelope decaying as exp(−t·ω₀/(2Q)). The higher Q, the longer the ringing lasts. A Q = 5 band-pass at f₀ = 1 kHz will ring for about 5/(π·1000) ≈ 1.6 ms before settling. This is exactly why bell-like resonant filters give that characteristic "ringing" sound when struck with a transient.
What's the difference between Q and bandwidth?
They're two ways of expressing the same thing: Q = f₀ / BW. A high-Q filter has a narrow bandwidth relative to its centre frequency. Example: a Q=10 filter at 1 kHz has BW = 100 Hz, passing 950–1050 Hz (approximately). A Q=1 filter at 1 kHz has BW = 1 kHz, passing 500–1500 Hz. Q is dimensionless; BW is in Hz.
When should I use series RLC vs parallel RLC vs MFB?
Passive RLC is simple but requires inductors (bulky, non-ideal at audio). Use it for RF where inductors are easy. Series RLC has output voltage rolling off as we approach resonance from outside (band-pass shape with R in series); parallel RLC is the dual — it peaks at resonance with R as the source impedance. MFB active is the practical choice for audio band-pass (no inductors, op-amp does the work, arbitrary gain). For Q > 10, switch to state-variable or biquad topologies (not covered here).
Why is my computed R₂ negative in MFB mode?
You've violated the topology constraint G < 2·Q². At Q = 1 and G = 3, you'd need 2·Q² = 2 to exceed G = 3 — which fails, making R₂ negative (impossible). Solutions: (a) lower the gain G, (b) raise Q so 2·Q² grows past G. For G = 1, the constraint requires Q > 0.707, which is easy. For G = 5, Q must be > 1.58. For G = 10, Q must be > 2.24. The tool flags this with an "N/A" R₂ value when the constraint is broken.
Why does the band-pass step response start at 0 instead of 1?
A unit step contains DC (low-frequency content) plus an instantaneous edge (high-frequency content). The band-pass filter blocks BOTH — only frequencies near f₀ make it through. So at t=0 there's nothing in the passband and output is 0. As the step "settles" into its DC level, the filter still blocks it, so output stays near 0. The transient ringing you see is the BPF's natural response to the brief moment when the step's frequency content overlapped with the passband. Compare: HP step starts at 1 (the edge passes), LP step rises to 1 (DC passes), BPF stays near 0 (neither passes).
What's the relationship between band-pass and cascaded LP+HP?
For a wide band-pass (BW comparable to f₀), cascading a low-pass at f_high with a high-pass at f_low gives almost identical results — passband from f_low to f_high, 6 dB/octave rolloff on each side (for 1st-order pairs). For a narrow band-pass (high Q, BW << f₀), the cascaded approach falls apart: both filters have to be very steep to define a narrow band, and you'd need much higher-order LP+HP than a single 2nd-order BPF. For Q > 3 the dedicated BPF topology is much more efficient.
Why are my computed inductor values so large?
L scales as 1/(ω₀·f) — for low f₀ at audio (say 1 kHz), inductors are in the milli-henry to henry range, which is bulky and expensive. This is why audio band-pass filters almost never use passive RLC — the MFB or other active topologies replace the inductor with op-amp gain + RC components. For RF (MHz range), inductors are tiny (µH or nH) and passive RLC dominates. The same calculator works at any frequency; just expect impractical L values at low frequencies.
How does this compare to the notch filter calculator?
A notch filter is the complement: it REJECTS frequencies in a narrow band around f₀ and passes everything else. Mathematically, |H_BPF|² + |H_notch|² = 1 (for matched topologies). Use BPF when you want to isolate one frequency band; use notch (in the Notch Filter Calculator) when you want to remove one specific interfering frequency (mains hum at 50/60 Hz is the classic use case). Both have the same Q and f₀ parameters.