**Series RLC resonant circuit is used to model a resonator at frequencies close to resonance. Resonator is a system, demonstrating resonance behaviour. Resonance in a circuit occurs when power stored in the inductor is equal to power stored in capacitor.**

Example of this circuit is shown below.

Here input impedance ${Z}_{in}=R+j\omega L\u2013j\frac{1}{\omega C}$and power delivered to resonator is ${P}_{in}=\frac{V{I}^{*}}{2}=\frac{{\left|I\right|}^{2}}{2}(R+j\omega L\u2013\frac{j}{\omega C})$.

Power dissipated on the resistor $R$ is ${P}_{R}=\frac{1}{2}{\left|I\right|}^{2}R$. Magnetic energy stored at the inductor $L$ is ${P}_{L}=\frac{1}{4}{\left|I\right|}^{2}L$. Electric energy stored at the capacitor $C$ is ${P}_{C}=\frac{1}{4}{\left|I\right|}^{2}\frac{1}{{\omega}^{2}C}$.

So total power for this circuit is ${P}_{tot}={P}_{R}+2j\omega ({P}_{L}\u2013{P}_{C})$. At the same time ${P}_{tot}=\frac{{\left|I\right|}^{2}{Z}_{in}}{2}$. Here we can get the input impedance as ${Z}_{in}=\frac{2}{{\left|I\right|}^{2}}({P}_{R}+2j\omega ({P}_{L}+{P}_{C}\left)\right)$.

The condition of resonance is equality of energies stored at capacitor and inductor. So ${Z}_{in}=R$, and input impedance is equal to resistance $R$. And ${\omega}_{0}=\frac{1}{\sqrt{LC}}$ because ${P}_{C}={P}_{L}$.

Resonant circuit is characterised with the *quality factor* $Q=\omega \frac{{P}_{c}+{P}_{L}}{{P}_{R}}$, which is always proportional to the energy stored in a circuit and energy lost in a circuit.

The reasons of losses in a circuit can be :

- conduction losses;
- dielectric losses;
- radiation losses;
- external circuit losses.

If resonator does not have external circuit, it is called unloaded, and it’s quality factor is ${Q}_{0}$. In the other case resonator circuit is loaded and quality factor is $Q$.

In case of unloaded resonant circuit easy to show that ${Q}_{0}=\frac{{\omega}_{0}L}{R}=\frac{1}{{\omega}_{0}RC}$.

Let’s consider input impedance in the area of resonance ${\omega}_{0}+\u2206\omega $, where $\u2206\omega \ll {\omega}_{0}$: ${Z}_{in}=R+j\omega L(1\u2013\frac{1}{{\omega}^{2}LC})=R+j\omega L(1\u2013\frac{{{\omega}_{0}}^{2}}{{\omega}^{2}})$, where ${\omega}_{0}=\frac{1}{LC}$.

Mathematically we can show that ${\omega}^{2}\u2013{{\omega}_{0}}^{2}\approx 2\omega \u2206\omega $, so ${Z}_{in}\approx R+j\frac{2R{Q}_{0}\u2206\omega}{{\omega}_{0}}$.