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I have an inductor coupled via coupling capacitance C_c to a transmission line with impedance Z_L. Given that the inductor has some internal losses and a capacitance, I imagine that for all intents and purposes it can be modeled as an RLC circuit. This gives a simple circuit shown below:

enter image description here

What I am interested in here is an analytic expression for the external quality factor of the circuit in terms of C_c, C, L, Z_L, and perhaps R but I imagine that only goes into the internal quality factor.

Now, from the definition of the quality factor, we can write that $$Q_E = \omega \frac{E}{P}$$

where omega is the frequency, E is the energy stored in the RLC circuit, and P is the power lost in the transmission line, essentially due to the coupling. But I have to admit, at this point I'm already starting to get stuck in relating it to C_c, C, L, Z_L. Could someone help me get started on how to continue?

Coplanar Waveguide Resonators for Circuit Quantum Electrodynamics by Goppl et al from 2008 looks at something along these lines, but this is not an identical system. They transform their circuit using Norton equivalences, but I don't think that is the way to go. However, if one were to take their figure 5c and make some adjustments, perhaps that helps?

enter image description here

Taking the above with R* = R_L, C* = 0, and replacing C by the series capacitance of C_c and C, would that not be identical? In that case the external quality factor is given in the paper to be Z_L/2 times C_c C / (C_c + C) times the resonant frequency of the RLC circuit. I am not 100% sure this is correct; could someone help me out there?

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    \$\begingroup\$ Long winded and unclear. Simplify and don't expect people to understand what it is you are trying to do or expect them to be interested. Draw your inductor equiv circuit and aks a simple question. \$\endgroup\$ – Andy aka Apr 10 '18 at 15:06
  • \$\begingroup\$ @Andyaka Fair enough! The context would have made more sense on the physics stackexchange, where people are interested in the physics. I'll remove it. \$\endgroup\$ – user129412 Apr 10 '18 at 15:13
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The simpler answer is for a parallel filter, Q(fo)=R/X(f0) or real to reactive impedance ratio. Simply scale the frequency range to your desired range. Once you get the ballpark values, a calculator will give you better accuracy. However the material dielectric , Er(effective) properties, must be modelled correctly as a function of f.

This Q is trivial to lookup on a RLC nomograph and the series cap acts as a capacitive voltage transformer similiar to an inductive impedance transformer.

Transmission line properties can be extracted from Saturn's (free) PCB designer.exe tool for the geometry and PCB substrate material chosen and stripline, microstrip or coplanar geometry chosen. Beyond this use COMSOL or EM CAD designer tools. Phase velocity , \$v_{ph} = \dfrac{1}{ \sqrt{LC}}\$

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  • \$\begingroup\$ Thanks for these suggestions! In practice I would really benefit from an analytic expression though, which I will have to continue looking for I suppose. For example in the paper I linked they derive Q_E = C/(2 omega_n Z_l C_c^2) with omega_n just 1/sqrt(LC) for their circuit, which is quite similar but not similar enough for me to see the missing pieces \$\endgroup\$ – user129412 Apr 10 '18 at 15:33

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