Sharpness of Resonance: Bandwidth and Q-Factor

Defining the bandwidth. The bandwidth $\Delta\omega$ of a resonant circuit is defined as the width of the frequency range, centred on $\omega_0$, over which the current stays at or above $I_{max}/\sqrt2$ -- equivalently, the range over which the power delivered (which is proportional to $I^2$) stays at or above HALF its maximum value, which is why the two boundary frequencies $\omega_1$ and $\omega_2$ (with $\Delta\omega=\omega_2-\omega_1$) are often called the half-power frequencies. A standard result of this analysis (not required to be re-derived at this level) gives

showing directly, and consistently with Section 7.11's qualitative argument, that a SMALLER $R$ gives a SMALLER bandwidth -- i.e. a sharper, more narrowly-peaked resonance.

Defining the quality factor. The quality factor (Q-factor) of a resonant circuit is defined as the ratio of the resonant angular frequency to the bandwidth:

Deriving $Q$ in terms of circuit elements. Substituting $\Delta\omega=R/L$ from above,

and, since at resonance $\omega_0 L=X_L=X_C=1/(\omega_0 C)$, this can equally well be written as

all three forms being algebraically equal at resonance. …