Quality Factor and Bandwidth: Sharpness of Resonance

The quality factor, written as Q, is a number that tells how sharp and selective the resonance of an LCR circuit is. A high Q means the circuit responds strongly only in a very narrow band of frequencies close to the resonant frequency and lets nearby frequencies pass only weakly. The frequency range over which the response stays significant is called the bandwidth.

Q and bandwidth matter every time a signal must be picked out from a crowd. A radio receiver in a city must separate stations that sit close together; too low a Q and two stations overlap, too high a Q and part of the wanted signal itself is cut off. Mobile networks, Wi-Fi, television and radar all depend on getting this sharpness exactly right.

Key Concepts

• Quality factor (Q): a dimensionless measure of the sharpness of resonance and of voltage magnification at resonance.
• Resonant frequency (f₀): the central frequency where current is maximum in a series circuit.
• Half-power frequencies: the two frequencies at which the current falls to 1/√2 of its maximum and the power falls to half.
• Bandwidth (Δf): the difference between the two half-power frequencies, measured in hertz.
• Selectivity: the ability of a circuit to accept one frequency and reject neighbouring frequencies; directly set by Q and bandwidth.

Formulas

For a series LCR circuit, Q = (1/R) √(L/C), and equivalently Q = 2πf₀L / R = 1 / (2πf₀CR), where R is in ohms, L in henrys, C in farads and f₀ in hertz. Q also equals the voltage magnification at resonance: Q = VL / Vsupply = VC / Vsupply.

The bandwidth is Δf = f₀ / Q = R / (2πL), in hertz. This simple relation shows the whole trade-off: increasing resistance lowers Q and widens the bandwidth, while a better coil with less resistance raises Q and narrows the response.

Worked Example

A series LCR circuit has L = 0.2 H, C = 50 μF and R = 10 Ω. Find the resonant frequency, the quality factor and the bandwidth.

1. f₀ = 1 / (2π√(LC)) = 1 / (2π√(0.2 × 50 × 10⁻⁶)) = 1 / (2π × 3.16 × 10⁻³) ≈ 50.3 Hz.
2. Q = (1/R) √(L/C) = (1/10) × √(0.2 / (50 × 10⁻⁶)) = 0.1 × √4000 = 0.1 × 63.2 = 6.32.
3. Bandwidth Δf = f₀ / Q = 50.3 / 6.32 ≈ 7.96 Hz, about 8 Hz.

The circuit responds appreciably only within roughly 46 Hz to 54 Hz. If R were reduced to 2 Ω, Q would rise to about 31.6 and the bandwidth would shrink to about 1.6 Hz, a much sharper but more fragile tuning.

Common Misconceptions

• A higher Q is always better. Wrong. Very high Q gives a bandwidth so narrow that the sidebands of a modulated signal, which carry the actual information, get cut off. Speech and music would sound muffled. Every application chooses a suitable Q, not the maximum.
• Bandwidth is the range where current is maximum. Wrong. Bandwidth is defined between the half-power points, where current is 70.7 percent of maximum. The response is a smooth peak, not a flat top.
• Q depends only on the inductor. Wrong. Q = (1/R)√(L/C) depends on all three components and on frequency. Doubling C, for example, lowers the resonant frequency and changes Q even with the same coil.

Real-World Applications

Radio receivers use Q to balance selectivity against sound quality, with different values for AM and FM bands. Quartz crystal oscillators, with Q values in the lakhs, keep clocks and computer processors stable. Band-pass filters in mobile phones separate the closely packed channels of 4G and 5G networks. Medical MRI machines and radar systems use high-Q resonant circuits for sensitivity, and engineers deliberately lower Q with resistance where a flat, wide response is safer, such as in audio amplifiers.

The quality factor measures how sharply a circuit resonates, and bandwidth, equal to f₀ divided by Q, measures the frequency window it accepts. High Q means sharp tuning and high voltage magnification but a narrow window; the right choice depends on the job the circuit must do.