RLC Resonance

Finding the resonance of an RLC Circuits is often a critical part of circuit analysis. Because voltages and currents in AC circuits are often time dependent, there will be a particular frequency for which these values are maximized.

For a simple RLC Circuit, the resonance frequency can be found by finding the frequency at which the reactance of the capacitor is equal to the reactance of the inductor (See Complex Analysis). $\omega L=\frac{1}{\omega C}$ Solving for $\omega$, we find that $\omega$ must be equal to ${\omega }_0=\sqrt{\frac{1}{LC}}$ This frequency yields the highest value of voltage and current in an RLC circuit.

Quality Factor

The quality factor is a unitless quantity that describes the quality of the peak resonance. The sharper the peak (and lower the bandwidth), the higher the quality factor.

The bandwidth of the resonance peak is defined as the range of angular frequencies $\omega$ over which $P_{avg}$ is greater than one-half the maximum value of $P_{avg}$.

Mathematically, the quality factor is defined as: $Q=\frac{{\omega }_0}{\Delta \omega }$ where ${\omega }_0$ is the resonance frequency. We can rewrite Q in terms of circuit parameters as: $Q=\frac{{\omega }_{0}L}{R}$ which can be further rewritten as: $Q=\frac{1}{R}\sqrt{\frac{L}{C}}$