A Surface impedance equation for resonators

[annaphys]

Hi all,
I am reading this thesis and have two questions to section 5.1.5: https://rsl.yale.edu/sites/default/files/files/RSL_Theses/reagor-thesis-20151202.pdf
First question is: Where did equation 5.19 come from?
Second question is: If you look at this paper https://arxiv.org/pdf/1308.1743.pdf, they mention that (just above eqn 1) X_s = w*u_0*lamda + deltaX_s. But in the Z_s formula we are only given deltaX_s: Z_s = R_s + j*deltaX_s (in section 5.1.5 of the thesis as well just above eqn 5.19) and in Dr. Reagor’s thesis only X_s is calculated (eqn 5.26b): X_s propotional to R_n*sigma_2^v. So how do I find what deltaX_s is? Would I really move w*u_0*lam to the other side and voila get deltaX_s?

 

[Baluncore]

First Answer.
Maybe from the reference, have you checked ?
[121]. J. Turneaure, J. Halbritter, and H. Schwettman, “The surface impedance of superconductors and normal conductors: The mattis-bardeen theory,” Journal of Superconductivity 4, 341–355 (1991). Cited on pages 80 & 124.

 

[DaveE]

OK, first, I probably can't answer your question.

But, I'm not even going to try, and I suspect many others, who probably could help, won't either. This isn't so much a question as a mini research project. We don't even know what your asking without going to other pages and searching for equation 5.19 in section 5.1.5. The equations you've buried into your text without even a new line are difficult to read.

We are all volunteers here. You'll get more, and better, answers if you don't make us work too hard. If you're not willing to post an easily understood question, why should we think it's worth investing the time into figuring out if we have something to contribute?

Hint: LaTex is good for equations, also the greek characters in text. For example, which is easier for you to read? X_s = w*u_0*lamda + deltaX_s,

or
X_s = w*u₀*λ + ΔX_s

or
$X_s = w u_0 \lambda + \Delta X_s$

sorry, I didn't click on your links, maybe others will.

 

[annaphys]

First Answer.
Maybe from the reference, have you checked ?
[121]. J. Turneaure, J. Halbritter, and H. Schwettman, “The surface impedance of superconductors and normal conductors: The mattis-bardeen theory,” Journal of Superconductivity 4, 341–355 (1991). Cited on pages 80 & 124.

Yes of course I've checked that reference and many others.

 

[f95toli]

I do in fact know quite a bit about this particular topic; but that equation is new to me and I am not surprised you can't find a good reference.
it is certainly not a "textbook" equation that you get directly from the Mattis-Bardeen equations.
It might be something people use in the accelerator cavity community, so perhaps you should have a look in some books about that topic?
Coaxial cavities are not commonly used in cQED which probably explains why some of this looks a bit unusual if you like are more used to planar/on-chip resonator.

 

[annaphys]

I do in fact know quite a bit about this particular topic; but that equation is new to me and I am not surprised you can't find a good reference.
it is certainly not a "textbook" equation that you get directly from the Mattis-Bardeen equations.
It might be something people use in the accelerator cavity community, so perhaps you should have a look in some books about that topic?
Coaxial cavities are not commonly used in cQED which probably explains why some of this looks a bit unusual if you like are more used to planar/on-chip resonator.

Do you know about the eqn 5.19 from Reagor's thesis and also the regular Z_s equation right about that?

 

[f95toli]

No, 5.19 is not an equation I've seen before.
All the other equations I am familiar with; it is just "normal" Mattis-Bardeen theory which you should be able to find in any number of references
I am not quite sure I understand your question about Zs; it is just the surface impedance.
Do you understand how superconductors are modeled in RF circuits? I.e. how one accounts for the kinetic inductance etc.

 

[annaphys]

No, 5.19 is not an equation I've seen before.
All the other equations I am familiar with; it is just "normal" Mattis-Bardeen theory which you should be able to find in any number of references
I am not quite sure I understand your question about Zs; it is just the surface impedance.
Do you understand how superconductors are modeled in RF circuits? I.e. how one accounts for the kinetic inductance etc.

Yes. I am dealing with thin film superconductors and there are different regimes one can use to plot the Z_s. It's just confusing that none of these papers and theses delve into how to do the actual plotting. This thesis is quite helpful https://thesis.library.caltech.edu/2530/1/thesismain_0610.pdf. If you look at section 2.2-2.4 it discusses this topic. Section 2.3 delves into thin film regime but then doesn't give a good explanation of how to plot. Section 2.4 seems to be my only hope as I can then perhaps use the complex conductivity to plot the thin limit.

 

[f95toli]

Again, I am not entirely sure what you are trying to do.
There is -in general- no closed form of these equations; you have to solve them numerically. That is, "plotting Zs" is -in general- not as simple as just plotting an expression. However, from what I remember (its literally been over a decade since I did it) it is not very difficult to do this using e.g. Matlab.

You can derive analytical solutions in certain limits and Gao's thesis does include some expressions, are these not applicable in your case?