Band-Pass Filter Simulator — Series RLC Resonance and Q-factor
Explore the frequency response of a series RLC band-pass filter built from R, L and C. Inspect the resonant frequency, Q-factor, bandwidth and gain with a live Bode plot and circuit waveforms.
Parameters
Resistance R
Ω
Inductance L
mH
Capacitance C
μF
Observation frequency f
Hz
Results
—
Resonant frequency f_0
—
Q-factor
—
Bandwidth BW
—
Gain at observation f
Phase at observation f: —
Series RLC band-pass circuit
Bode plot (magnitude)
Theory & Key Formulas
Transfer function (output taken across R):
$$H(j\omega) = \dfrac{R}{R + j\!\left(\omega L - \dfrac{1}{\omega C}\right)}$$
At $f = f_0$, $|H| = 1$ (0 dB) and the phase is exactly 0°. Away from $f_0$ the response rolls off at -20 dB/decade on each side, reaching -3 dB at $f_0 \pm \mathrm{BW}/2$.
What is a series RLC band-pass filter?
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A band-pass blocks both lows and highs and only lets a middle band through? How can a simple RLC do that?
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Roughly speaking, in a series RLC the inductor reactance $\omega L$ and the capacitor reactance $1/(\omega C)$ point in opposite directions. At low frequencies C blocks the current; at high frequencies L blocks it. Only at the resonant frequency f_0 do they cancel out so the impedance reduces to R, the current peaks, and the voltage across R reaches its maximum.
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How sharp does Q = 10 feel compared with Q = 1?
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At Q = 1 the bandwidth equals f_0 itself (e.g. f_0 = 1.6 kHz gives BW ≈ 1.6 kHz), so the peak is wide and gentle. At Q = 10 the BW shrinks to 160 Hz, ten times sharper. An AM radio tuner runs at Q = 50–100 so it can separate adjacent stations, while audio tone controls usually sit at Q = 0.7–1.5 for a natural-sounding bell.
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Is achieving a high Q in practice difficult?
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Yes. The inductor always has a DC resistance r_L that effectively adds to R. A 10 mH coil may already carry 2–5 Ω, so even a design R of 1 Ω becomes several Ω in practice. For very high Q you use air-core solenoids or ferrite cores, or replace LC with an active band-pass topology such as Sallen-Key or multiple feedback.
Physical model and key equations
Driving the series RLC with a sinusoidal voltage $V_{in}$ and taking the output $V_{out}$ across R, the network impedance is $Z = R + j(\omega L - 1/(\omega C))$, so the transfer function becomes:
The resonance condition $\omega L = 1/(\omega C)$ yields $\omega_0 = 1/\sqrt{LC}$, that is $f_0 = 1/(2\pi\sqrt{LC})$. At $f_0$ the magnitude is unity and the phase is zero.
The two -3 dB points $f_L, f_H$ where the gain falls to $1/\sqrt{2}$ are separated by BW, and they satisfy the geometric mean $f_0 = \sqrt{f_L f_H}$.
Real-world applications
Radio tuners: AM and FM receivers use LC resonant tanks to select a single carrier frequency, sweeping f_0 with a variable capacitor or varactor.
Sensor front ends: Reject 50/60 Hz mains hum and broadband noise while passing only the band of interest (for example, a 1 kHz vibration channel).
Audio equalization: Boost or cut a midrange band (e.g. around 1 kHz). A graphic EQ is essentially a parallel set of fixed-Q band-pass cells.
Vibration and condition monitoring: Isolate a specific mechanical mode for condition-based maintenance. Higher Q improves resolution at the cost of slower response.
Common misconceptions and caveats
Myth 1: the passband gain is flat at 1. Only the peak at $f_0$ reaches unity. The two -3 dB edges sit at 0.707, and the response is a smooth bell shape. For a flat passband choose a higher-order Butterworth or Chebyshev design.
Myth 2: more Q is always better. Raising Q narrows the band but slows the transient response (time constant ~ Q/f_0) and increases ringing. AM demodulation and pulse passing favor moderate Q (1 to 5).
Myth 3: setting R = 0 gives an ideal band-pass. In theory Q tends to infinity, but the inductor's DC resistance, wiring losses and source impedance set a practical ceiling. Loading effects also limit the achievable Q.
FAQ
A series RLC with the output across R has maximum current and maximum output at f_0. A parallel RLC has maximum impedance at f_0 and, driven from a current source, produces a similar bell. In the parallel case Q = R sqrt(C/L), so increasing R increases Q — the opposite of the series version.
In a passive RLC the simplest way is to vary R, but that also changes the output level so a gain trim is needed. An active filter (state-variable or biquad) lets you tune gain, f_0 and Q independently and even control them digitally.
Pick a practical L (say 10 mH), then compute C = 1/(L (2 pi f_0)^2). Finally choose R = (1/Q) sqrt(L/C) = omega_0 L / Q, subtracting the inductor's DC resistance r_L to size the external R.