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 Zin=R+jωLj1ωCand power delivered to resonator is  Pin=VI*2=I22(R+jωLjωC).

Power dissipated on the resistor R is PR=12I2R. Magnetic energy stored at the inductor L is PL=14I2L. Electric energy stored at the capacitor C is PC=14I21ω2C.

So total power  for this circuit is Ptot=PR+2jω(PLPC). At the same time Ptot=I2Zin2. Here we can get the input impedance as Zin=2I2(PR+2jω(PL+PC)).

The condition of resonance is equality of energies stored at capacitor and inductor. So Zin=R, and input impedance is equal to resistance R. And ω0=1LC because PC=PL.

Resonant circuit is characterised with the quality factor Q=ωPc+PLPR, 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 Q0. In the other case resonator circuit is loaded and quality factor is Q.

In case of unloaded resonant circuit easy to show that Q0=ω0LR=1ω0RC.

Let’s consider input impedance in the area of resonance ω0+ω, where ωω0Zin=R+jωL(11ω2LC)=R+jωL(1ω02ω2), where ω0=1LC.

Mathematically we can show that ω2ω022ωω, so ZinR+j2RQ0ωω0.