Q-Factor & Bandwidth Calculator

Calculate Q-factor, bandwidth, and selectivity for resonant RLC circuits.

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What Resonance, Q-Factor, and Bandwidth Mean

An RLC circuit resonates when the inductor and capacitor exchange energy at the same natural frequency, so their reactances cancel. At that frequency, the circuit's behavior is controlled mainly by resistance and by how much energy is lost each cycle. The resonant frequency depends only on inductance and capacitance: increasing either L or C lowers the frequency because the energy exchange takes longer.

The quality factor Q describes how lightly damped the resonance is. A large Q means relatively little energy is lost per cycle, so the response is narrow and sharply concentrated near resonance. A small Q means stronger damping and a broader response. For a series RLC circuit, resistance reduces Q because the resistor directly dissipates current energy. For a parallel RLC circuit in the form used here, a larger parallel resistance produces a higher Q because it represents less shunt loss.

f0 = 1/(2π√(LC))    Z0 = √(L/C)    Qseries = Z0/R    Qparallel = R/Z0    BW = f0/Q
SymbolMeaningWhy it matters / units
f0Resonant frequencyFrequency of strongest resonance, in hertz
LInductanceStores magnetic-field energy, in henries
CCapacitanceStores electric-field energy, in farads
RResistanceRepresents energy loss, in ohms
QQuality factorDimensionless measure of resonance sharpness
BWBandwidthWidth between half-power frequencies, in hertz

A useful interpretation is that Q compares the resonant frequency with the bandwidth. If f0 = 5 kHz and Q = 25, the bandwidth is about 200 Hz. High-Q circuits are useful when a narrow range of frequencies must be selected; lower-Q circuits respond over a wider range. For high Q, the two half-power frequencies lie approximately symmetrically around f0.

Worked Examples

Example 1: AM tuner: R=5Ω, L=250μH, C=100pF
f₀=1/(2π√LC)=1007kHz
Result: Q=Z0/R=316.2, BW=3.18kHz
For these stated R, L, and C values, the series-circuit Q is about 316
Example 2: Low-Q filter: R=100Ω, L=10mH, C=10μF
f₀=503Hz, Z0=31.6Ω
Result: Q=0.316 — overdamped
Not resonant — critically/overdamped
Example 3: Series RLC with a sharp resonance
R = 10 Ω, L = 100 mH, C = 1 µF → f0 = 503.3 Hz, Z0 = 316.2 Ω
Result: Q = 31.6, BW = 15.9 Hz
The small series resistance compared with the characteristic impedance gives a narrow resonance. The bandwidth is only about 3.2% of the resonant frequency.
Example 4: Parallel RLC using the same Q idea
R = 10,000 Ω, L = 10 mH, C = 0.1 µF → f0 = 5032.9 Hz, Z0 = 316.2 Ω
Result: Q = 31.6, BW = 159.2 Hz
For the parallel form, increasing R raises Q. This is the opposite resistance trend from a series RLC circuit, which is why the circuit type must be identified before using a Q formula.

Common Mistakes

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Entering millihenries or microfarads as if they were base units

The resonance formula requires henries and farads. For example, 100 mH = 0.100 H and 1 µF = 1 × 10−6 F. A missed prefix can move the calculated frequency by orders of magnitude.

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Using the series Q formula for a parallel circuit

Series RLC uses Q = √(L/C)/R, while the parallel form used here uses Q = R/√(L/C). Resistance therefore affects the two configurations in opposite directions.

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Assuming resonance means zero current

In a series RLC circuit, inductive and capacitive reactances cancel at resonance, leaving the resistance to limit current. The current is therefore largest, not zero, for an ideal voltage-driven series circuit.

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Treating Q as a frequency

Q is dimensionless. Bandwidth and resonant frequency are measured in hertz, and the relationship BW = f0/Q converts the dimensionless sharpness into a frequency width.

Frequently Asked Questions

Q-factor physical meaning?
Q = energy stored / energy dissipated per radian. High Q = low energy loss per cycle = sharp resonance = many oscillations before dying. Q = f₀/BW — bandwidth inversely proportional to Q.
Applications by Q range?
Q < 0.5: overdamped, no oscillation (RC filters). Q ≈ 0.7: Butterworth filter (maximally flat). Q = 1-10: audio crossovers. Q = 10-100: radio tuners. Q > 1000: crystal oscillators, atomic clocks.
What happens to resonant frequency if capacitance doubles?
Because f0 is proportional to 1/√C, doubling capacitance lowers the resonant frequency by a factor of √2. The new frequency is about 0.707 times the original value, not one-half of it.
Why is the bandwidth inversely related to Q?
A high-Q resonator loses little energy and responds strongly only near its natural frequency, producing a narrow frequency response. A low-Q resonator is more strongly damped and responds across a wider range, so its bandwidth is larger.
Is the resonant frequency affected by resistance?
The ideal expression f0 = 1/(2π√LC) depends only on L and C. In real circuits, resistance and component parasitics affect the exact response and damping, but the ideal LC resonance remains the standard starting point.
What does the −3 dB bandwidth represent?
The bandwidth is commonly measured between the two frequencies where power has fallen to one-half of its resonant value. Those are the half-power points, corresponding to an amplitude ratio of 1/√2 for quantities whose power is proportional to amplitude squared.

Formula Explorer connections

Interpretation: This relationship connects frequency, wavelength, speed, phase, intensity or resonance in an oscillating system. Assumption: Identify the medium, boundary conditions and reference frame. Linear waves, small amplitudes, nondispersive media or ideal resonance may be assumed.

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