RF Electronic Devices and Systems

# Series RLC resonant circuit

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

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

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;
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 ${\omega }_{0}+∆\omega$, where $∆\omega \ll {\omega }_{0}$${Z}_{in}=R+j\omega L\left(1–\frac{1}{{\omega }^{2}LC}\right)=R+j\omega L\left(1–\frac{{{\omega }_{0}}^{2}}{{\omega }^{2}}\right)$, where ${\omega }_{0}=\frac{1}{LC}$.
Mathematically we can show that ${\omega }^{2}–{{\omega }_{0}}^{2}\approx 2\omega ∆\omega$, so ${Z}_{in}\approx R+j\frac{2R{Q}_{0}∆\omega }{{\omega }_{0}}$.