Quality factor, bandwidth, and voltage resonance

Bandwidth

Around resonance, the bandwidth is defined as the range of angular frequencies [omega1, omega2] for which the current amplitude remains greater than Im_max/sqrt(2) (i.e., half the maximum power). It can be shown that:

Delta_omega = omega2 - omega1 = omega0/Q = R/L

The larger Q is (small R), the narrower the bandwidth: the resonance is said to be selective or sharp. The smaller Q is, the wider and 'flatter' it is.

Summary table

Quantity	Expression	Effect of a large Q
omega0	1/sqrt(L*C)	peak position (unchanged)
Delta_omega	omega0/Q	narrower peak
Im_max	Em/R	higher peak

Voltage resonance across C or L

The voltage across the capacitor (or the coil) does not resonate exactly at omega0 as soon as Q is not very large. For U_C, the resonance angular frequency is omega_C = omega0sqrt(1 - 1/(2Q^2)), which exists only if Q > 1/sqrt(2). Moreover, the overvoltage across C or L at resonance can exceed Em by a factor close to Q when Q >> 1.

Concrete example

A circuit with Q = 10 shows an overvoltage across the capacitor about 10 times higher than the supply voltage Em: this is the principle exploited, with care, in certain tuning or selective filtering circuits.

Common pitfall

Confusing current resonance (always exactly at omega0) with voltage resonance across C (slightly shifted, and which may not even exist if Q is too low) is a very common mistake in exams.