# Resonance **The phenomenon of increased amplitude when the frequency of a waveform in an applied periodic force is similar to or matches the natural frequency of a system.** ## Electrical resonance ### RLC circuits **The condition where the [[impedance]] is purely resistive.** At resonance, - The source voltage is held *completely* across $R$. - The [[Power Angle and Power Factor#Power Factor|power factor]] is unity; voltage and current in [[phase]]. - The [[transfer function]] $\mathbf{H}(\omega)=\mathbf{Z}(\omega)$ is at a *minimum*. - The inductor and capacitor voltage may *exceed* the source voltage. >[!Resonant or natural frequency]+ >The value of $\omega$ that causes resonance is the *resonant* or *natural* frequency, $\omega_{0}$. For *both* series and parallel RLC circuits, it is given by >$\omega_{0}=\frac{1}{\sqrt{LC}}$ >[!Half-power frequencies]+ >The *half-power frequencies* are the frequencies where the dissipated power is *half* the maximum or half the value *at resonance*. At half-power frequencies, >$|\mathbf{H}(\omega)|=\frac{1}{\sqrt{2}}$ >The half-power *bandwidth*, $B$, is the frequency range in which the magnitude of the transfer function is at least $\frac{1}{\sqrt{2}}$. >$B=\omega_{2}-\omega_{1}$ >[!Quality factor]+ >RLC circuits also have a *quality factor*, $Q$, which is a dimensionless quantity describing the *selectivity* of the circuit. The *higher* the quality factor, the smaller the range which approaches resonance, and the longer a circuit resonates. >$Q=\frac{\omega_{0}}{B}$ For a *high-Q* circuit, where $Q\ge 10$, the half-power frequencies are practically *symmetrical* around the resonant frequency and can be approximated as >$\begin{align*} \omega_{1}\approx\omega_{0}-\frac{B}{2} \\ \omega_{2}\approx\omega_{0}+\frac{B}{2} \end{align*}$ #### Series Resonance The *half-power frequencies* are $\begin{align*} \omega_{1}&=-\frac{R}{2L}+\sqrt{\left(\frac{R}{2L}\right)^{2}+\frac{1}{LC}} \\ \omega_{2}&=\frac{R}{2L}+\sqrt{\left(\frac{R}{2L}\right)^{2}+\frac{1}{LC}} \end{align*}$ The *quality factor* is given by $Q=\frac{\omega_{0}L}{R}=\frac{1}{\omega_{0}CR}$ The *bandwidth* can be found with $B=\frac{R}{L}$ The half-power frequencies can also be expressed in terms of the quality factor. $\omega_{1},\omega_{2}=\omega_{0}\sqrt{1+\left(\frac{1}{2Q}\right)^{2}}\pm\frac{\omega_{0}}{2Q}$ ![[RLCSeriesCircuit.svg|600]] #### Parallel Resonance The *half-power frequencies* are $\begin{align*} \omega_{1}&=-\frac{1}{2RC}+\sqrt{\left(\frac{1}{2RC}\right)^{2}+\frac{1}{LC}}\\ \omega_{2}&=\frac{1}{2RC}+\sqrt{\left(\frac{1}{2RC}\right)^{2}+\frac{1}{LC}} \end{align*}$ The *quality factor* is given by $Q=\omega_{0}RC=\frac{R}{\omega_{0}L}$ The *bandwidth* can be found with $B=\frac{1}{RC}$ As in an series RLC circuit, the half-power frequencies can also be expressed in terms of the quality factor. $\omega_{1},\omega_{2}=\omega_{0}\sqrt{1+\left(\frac{1}{2Q}\right)^{2}}\pm\frac{\omega_{0}}{2Q}$ ![[RLCParallelCircuit.svg|600]]