Let’s consider series and parallel RLC circuits with lumped parameters. They can be used for modelling resonators.

RLC series resonant circuit.

Here ${Z}_{in}=R+jwL\u2013\frac{j}{wC}$. Input resonator power is ${P}_{in}=\frac{VI}{2}=\frac{{Z}_{in}{I}^{2}}{2}=\frac{{I}^{2}}{2}(R+jwL\u2013\frac{j}{wC})$. Power, loss on the resistor $R$ is ${P}_{loss}=\frac{{I}^{2}R}{2}$ , power stored at the inductor is ${P}_{L}=\frac{{I}^{2}R}{4}$. Power stored at capacitor is ${P}_{C}=\frac{{I}^{2}}{4{w}^{2}C}$. Then ${P}_{in}={P}_{loss}+2jw({P}_{L}\u2013{P}_{C})$ and ${Z}_{in}=\frac{2{P}_{in}}{{I}^{2}}$ .

Resonance happens when ${P}_{L}={P}_{C}$, then ${Z}_{in}=R$ when ${w}_{0}=\frac{1}{\sqrt{LC}}$.

Resonant circuit can be characterised by quality factor $Q=\frac{{P}_{L}+{P}_{C}}{{P}_{loss}}w$.

Resonator losses can have different nature, including conductor losses, dielectric losses, radiation losses and others. Quality factor of a resonator is called unloaded $Q$ and assigned as ${Q}_{0}$.

${Q}_{0}=\frac{{w}_{0}L}{R}=\frac{1}{{w}_{0}RC}$.Resonator behaviour near it’s resonance frequency ${Z}_{in}=R+jwL\left(\frac{{w}^{2}\u2013{w}_{0}^{2}}{{w}^{2}}\right)$. ${w}^{2}\u2013{w}_{0}^{2}=(w\u2013{w}_{0})(w+{w}_{0})\approx 2w\u2206w$. Then ${Z}_{in}=R+jwL\frac{2w\u2206w}{{w}^{2}}=R+2jL\u2206w$.

Then ${Z}_{in}=R(1+j\frac{Q\u2206w}{{w}_{0}})$. Quality factor can also be a characteristics of a resonator bandwidth $\frac{1}{{Q}_{0}}$.

RLC parallel resonant circuit.

Here $\frac{1}{{Z}_{in}}=(\frac{1}{R}+\frac{1}{jwL}\u2013jwC)$. As for the case above we calculate input power for resonator ${P}_{in}=\frac{VI}{2}=\frac{1}{2{V}^{2}}(\frac{1}{R}+\frac{1}{jwL}\u2013jwC)$. Resistor power losses are ${P}_{loss}=\frac{{V}^{2}}{2R}$. Energy stored in capacitor ${P}_{C}=\frac{{V}^{2}C}{4}$, power stored in inductor ${P}_{L}=\frac{{V}^{2}}{4{w}^{2}L}$. As for the first example ${P}_{in}={P}_{loss}+2jw({P}_{L}\u2013{P}_{C})$. And ${Z}_{in}=\frac{2{P}_{in}}{{I}^{2}}$.

When we have a resonance,${Z}_{in}=\frac{2{P}_{loss}}{{I}^{2}}$ . Resonant frequency ${w}_{0}=\frac{1}{\sqrt{LC}}$. Resonance in the parallel circuit is called anti-resonance.

Considering circuit frequencies close to resonant, and making similar calculations, ${Q}_{0}=\frac{R}{{w}_{0}L}$. Knowing that $\frac{1}{1+x}\approx 1\u2013x$, then ${Z}_{in}=\frac{1}{2jC(w\u2013{w}_{0})}$ and bandwidth $\frac{1}{{Q}_{0}}$.

Cases above are for unloaded circuit. Let’s consider the case if the circuit is loaded as outlined below.

Let’s say the resonance circuit is loaded with the load resistor ${R}_{L}$, then quality factor for external load is ${Q}_{e}=\left\{\begin{array}{l}\frac{{w}_{0}L}{R},seriesconnection\\ \frac{{R}_{L}}{{w}_{0}L},parallelconnection\end{array}\right.$.

Then total quality factor is $Q=\frac{{Q}_{e}{Q}_{0}}{{Q}_{e}+{Q}_{0}}$.