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

RLC series resonant circuit.  

Here Zin=R+jwLjwC.  Input resonator power is Pin=VI2=ZinI22=I22(R+jwLjwC). Power, loss on the resistor R is Ploss=I2R2 , power stored at the inductor is PL=I2R4. Power stored at capacitor is PC=I24w2C. Then Pin=Ploss+2jw(PLPC)  and Zin=2PinI2 .

Resonance happens when PL=PC, then Zin=R when w0=1LC.

Resonant circuit can be characterised by quality factor Q=PL+PCPlossw.

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 Q0.


Resonator behaviour near it’s resonance frequency Zin=R+jwL(w2w02w2). w2w02=(ww0)(w+w0)2ww. Then Zin=R+jwL2www2=R+2jLw.

Then Zin=R(1+jQww0). Quality factor can also be a characteristics of a resonator bandwidth 1Q0.

RLC parallel resonant circuit.

Here 1Zin=(1R+1jwLjwC).  As for the case above we calculate input power for resonator Pin=VI2=12V2(1R+1jwLjwC). Resistor power losses are Ploss=V22R. Energy stored in capacitor PC=V2C4, power stored in inductor PL=V24w2L.  As for the first example Pin=Ploss+2jw(PLPC). And Zin=2PinI2.

When we have a resonance,Zin=2PlossI2 . Resonant frequency w0=1LC. Resonance in the parallel circuit is called anti-resonance.

Considering circuit frequencies close to resonant, and making similar calculations,  Q0=Rw0LKnowing that 11+x1x, then Zin=12jC(ww0) and bandwidth 1Q0.

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 RL, then quality factor for external load is Qe=w0LR , series connectionRLw0L, parallel connection.

Then total quality factor is Q=QeQ0Qe+Q0.