I am interested in series RLC circuits coupled to transmission lines. Coupling can be done either inductively or capacitively.

The external quality factor $Q_e$ is defined by coupling. I have found literature that explains clearly how to obtain a formula for $Q_e$ in the case of inductive coupling.

However, I was not able to find the same for capacitive coupling, and this is where I need your help.

Consider the following circuit:

enter image description here

(the image is taken from this article; I modified it to turn a component into a resistor.)

Using the definition of quality factor we can write:

$Q_e = \omega \frac{ \text {Energy stored in resonator}}{\text{Power loss due to coupling}} = \omega \frac{E_m + E_e}{P_L}$.

At resonance the electric and magnetic energy stored in the resonator are equal so we can equivalently write

$Q_e = \omega_0 \frac{2E_m}{P_L} = \omega_0 \frac{2E_e}{P_L},$

where $\omega_0$ is the resonance frequency.

From this point, I don't know how to write the energies and the power loss since I can't visualize currents and voltages in the circuit.

For instance, the total voltage in the RLC circuit must be zero, and there is a nonzero impedance $Z_{RLC} = R + j\omega L_R + 1/(j\omega C_R)$, implying that no current is flowing in there, thus rendering the whole RLC circuit pointless; I don't think this is the case (otherwise what's the whole point?).

I am sure I am making some rookie mistake. What are the different currents and voltages at play, and how to finally use them to calculate $Q_e$ ?



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