Q.Define the quality factor (Q-factor) and the bandwidth of a resonant series LCR circuit. Derive the relation $Q=\dfrac{\omega_0 L}{R}$, and explain in words what a large Q-factor means physically.

Bandwidth. $\Delta\omega$ is the range of angular frequency, centred on $\omega_0$, over which the current stays at or above $I_{max}/\sqrt2$ (equivalently, the delivered power stays at or above half its maximum) -- a standard analysis gives $\Delta\omega=R/L$.

Q-factor definition.

Deriving the formula. Substituting $\Delta\omega=R/L$:

Since at resonance $\omega_0L=X_L=X_C=1/(\omega_0C)$, this is equally $Q=\dfrac{1}{\omega_0CR}=\dfrac1R\sqrt{L/C}$. …

Define $Q$ as $\omega_0/\Delta\omega$, substitute the bandwidth formula $\Delta\omega=R/L$, and interpre …

• Confusing bandwidth $\Delta\omega$ (which SHRINKS for a sharper resonance) with Q-factor (which GROWS for a sharper resonance) -- they move in opposite directions with …