RF Electronic Devices and Systems

RLC circuit analysis and quality factor

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

RLC series resonant circuit.

Here ${Z}_{in}=R+jwL–\frac{j}{wC}$.  Input resonator power is ${P}_{in}=\frac{VI}{2}=\frac{{Z}_{in}{I}^{2}}{2}=\frac{{I}^{2}}{2}\left(R+jwL–\frac{j}{wC}\right)$. 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 ${P}_{in}={P}_{loss}+2jw\left({P}_{L}–{P}_{C}\right)$  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 ${Z}_{in}=R+jwL\left(\frac{{w}^{2}–{w}_{0}^{2}}{{w}^{2}}\right)$. ${w}^{2}–{w}_{0}^{2}=\left(w–{w}_{0}\right)\left(w+{w}_{0}\right)\approx 2w∆w$. Then ${Z}_{in}=R+jwL\frac{2w∆w}{{w}^{2}}=R+2jL∆w$.

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

RLC parallel resonant circuit.

Here $\frac{1}{{Z}_{in}}=\left(\frac{1}{R}+\frac{1}{jwL}–jwC\right)$.  As for the case above we calculate input power for resonator ${P}_{in}=\frac{VI}{2}=\frac{1}{2{V}^{2}}\left(\frac{1}{R}+\frac{1}{jwL}–jwC\right)$. 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 ${P}_{in}={P}_{loss}+2jw\left({P}_{L}–{P}_{C}\right)$. 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 $\frac{1}{1+x}\approx 1–x$, then ${Z}_{in}=\frac{1}{2jC\left(w–{w}_{0}\right)}$ 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 .

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