I came up with an expression for the reflected power of a resonator:

$$\left| \Gamma \right|^2 = 1 - \frac{1-\left(\frac{\kappa - 1}{\kappa + 1}\right)^2}{1 + 4 \left( \frac{Q_U}{(\kappa + 1)} \frac{\Delta \omega}{\omega_0} \right)^2}$$

Here, $\Delta \omega \equiv \omega - \omega_0$ and I have assumed $\left| \frac{\Delta \omega}{\omega} \right| \ll 1$. $\kappa$ is the coupling coefficient and $Q_U$ is the unloaded quality factor.

From this, we see $\left| \Gamma \right|^2 = 0$ when $\kappa = 1$ on resonance, as expected, which corresponds to critical coupling.

Here's my question: If we define the loaded quality factor as $Q_L \equiv \frac{\omega_0}{\delta \omega}$, where $\delta \omega$ is the full-width half-maximum of the resonance, then from this expression for the reflected power it can be seen that $Q_L = \frac{Q_U}{\kappa + 1}$.

I'm not seeing why this is the case. Can someone explain this to me?

• Can you please give a system diagram, or otherwise define $\kappa$ a little more? Presumably $\kappa$ is an element of the scattering matrix that describes the link between the port you are calculating $\Gamma$ for and the resonator. – WetSavannaAnimal Sep 13 '15 at 5:18
• The expression for the reflected power comes from analyzing an equivalent circuit model. $\kappa$ is defined in terms of the circuit elements. The model is a transmission line inductively coupled to a series RLC resonator. – grover Sep 13 '15 at 9:57