How is the resonant frequency of a series RLC band-pass defined?

The resonant frequency is f_0 = 1/(2 pi sqrt(L C)). At resonance the reactances of L and C cancel, leaving only the resistance R, so the output across R is maximum. The phase shift is exactly 0 degrees at f_0, meaning the output is in phase with the input.

What does the Q-factor (quality factor) represent?

Q = (1/R) sqrt(L/C) = omega_0 L / R characterizes the sharpness of the resonance. The -3 dB bandwidth follows BW = f_0 / Q, so larger Q gives a narrower, more selective passband. Radio tuners run at Q = 50 to 100, while general band-pass stages typically use Q = 1 to 3.

Why does R control the Q-factor?

R is the loss element of the network. It determines the fraction of energy stored in L and C that is dissipated per cycle. Smaller R means less loss and a sharper resonance (higher Q). In real circuits, the inductor's series resistance r_L and stray resistance add to the design R, so practical Q is often lower than the ideal value.

How is a band-pass different from low-pass and high-pass?

A low-pass passes only frequencies below the cutoff and a high-pass only those above. A band-pass selects a central band around f_0 and attenuates both sides. Taking the output across R in a series RLC gives a peak at f_0 with a -20 dB/decade roll-off on each side, ideal for radio tuning, sensor front ends, and audio band selection.

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Signal Processing Simulator

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

Inductance L

Capacitance C

μF

Observation frequency f

Results

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Resonant frequency f_0

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Q-factor

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Bandwidth BW

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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):

$$H(j\omega) = \dfrac{R}{R + j\!\left(\omega L - \dfrac{1}{\omega C}\right)}$$

Resonant frequency, Q-factor and bandwidth:

$$f_0 = \dfrac{1}{2\pi\sqrt{LC}},\quad Q = \dfrac{1}{R}\sqrt{\dfrac{L}{C}} = \dfrac{\omega_0 L}{R},\quad \mathrm{BW} = \dfrac{f_0}{Q}$$

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:

$$H(j\omega) = \dfrac{V_{out}}{V_{in}} = \dfrac{R}{R + j\!\left(\omega L - \dfrac{1}{\omega C}\right)}$$

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.

$$|H(f)| = \dfrac{R}{\sqrt{R^2 + (\omega L - 1/(\omega C))^2}},\quad Q = \dfrac{1}{R}\sqrt{\dfrac{L}{C}},\quad \mathrm{BW} = \dfrac{f_0}{Q}$$

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.

How to Use

Set resistance R (ohms) using slRVal slider; typical audio filters use 10–100 Ω for impedance matching
Set inductance L (millihenries) via slL_mHVal; for RF circuits, 1–10 mH is standard
Set capacitance C (microfarads) using slC_uFVal; audio-band filters typically use 1–100 µF
Adjust probe frequency F (hertz) with slFVal to observe gain and phase response at that point
Read resonant frequency f₀, Q-factor, −3 dB bandwidth BW, and magnitude gain directly from output labels

Worked Example

Design a band-pass filter for a 1 kHz audio application: R=50 Ω, L=10 mH, C=2.5 µF. Simulator yields f₀≈1005 Hz, Q≈6.3, BW≈159 Hz (−3 dB cutoff points at 925 Hz and 1084 Hz), gain at resonance ≈0 dB (unity). Probe at 500 Hz shows −18 dB attenuation; at 1005 Hz shows peak response. This matches LC tank impedance Z_L=2πfL and 1/(2πfC) equations.

Practical Notes

Higher Q narrows bandwidth but increases peak impedance; Q=ωL/R, so reduce R to sharpen selectivity in RF tuned circuits (1 kHz to 100 MHz range)
Series RLC resonance occurs when inductive reactance equals capacitive reactance; verify f₀=1/(2π√LC) matches simulator output
Real components have parasitic resistance; add equivalent series resistance (ESR) to R value for accurate crystal or inductor models
Phase shift crosses −90° (capacitive leading) through 0° (resonance) to +90° (inductive lagging); observe phase plot when tuning notch filters