Sharpness of Resonance: Q Factor

Even away from a series LCR circuit's precise resonant frequency $\omega_r$, the current amplitude $i_0=\dfrac{e_0}{\sqrt{R^2+(X_L-X_C)^2}}$ does not fall to zero immediately -- for a given resistance R, it decreases only gradually as $\omega$ moves away from $\omega_r$ on either side. To quantify exactly HOW sharply peaked (how frequency-selective) a given resonance curve is, we identify the two 'half-power' frequencies -- the two values of $\omega$, one above and one below $\omega_r$, at which the current amplitude falls to $\dfrac{1}{\sqrt2}$ times its own peak (resonant) value; since power is proportional to the square of current amplitude, current falling to $1/\sqrt2$ of its peak corresponds to POWER falling to exactly HALF its peak value, which is why these are called the half-power frequencies.

Calling these two frequencies $\omega_1=\omega_r+\Delta\omega$ (above resonance) and $\omega_2=\omega_r-\Delta\omega$ (below resonance) -- symmetric about $\omega_r$ -- the difference $\omega_1-\omega_2=2\Delta\omega$ is called the BANDWIDTH of the circuit: a circuit whose current stays close to its maximum over only a very narrow range of $\omega$ around $\omega_r$ has a small bandwidth and is described as sharply resonant / highly selective, while a circuit whose current falls off only gradually over a wide range has a large bandwidth and is less selective. …