question regarding mode responsible for phenomena

[yefj]

Hello,As you can see bellow there is resonance because the transmission line is the circomfrance of the ring.
in the solution they set the phase such that it has integer number of wavelength which is the condition for ressnance.
but i dont undestand the mode of operation here.
the ring so PEC the ground is PEC ,the dielectric walls is PMC.
So i assume we have TEM MODE?
but the e^{j*beta*a*phi} expression is very crutial in understanding.
is e^{j*beta*a*phi} comes from the E-field?there is no such expression in TEM mode.
what is the source of the e^{j*beta*a*phi} came from? is it E-field ,H field, of is it the pointing vector.
saying tat its the wave is not very much specific because it doesnt allow me to recognise what mode we operate on.
Thanks.

 

[PlanarMetamaterials]

Beta is the phase constant of the assumed TEM mode; all field components are assumed to incur phase at this rate. a*theta is the arc length along the ring (radius*angle). Therefore, e^{j*beta*a*phi} simply describes the wave motion of the TEM mode along the ring (ignoring effects from the width w of the ring).

An equivalent expression in rectangular coordinates is e^{j*beta*x}.

 

[yefj]

Hello Planarmetamaterial , i have problem imagining this TEM mode circulating around the ring.
I was trying to read about ring resonator and i see only whispring gallery mode.
Its not a TEM mode.Could you help me with a manual on how to design or imagine the dinamics of TEM mode.
Thanks.

 

[PlanarMetamaterials]

By TEM, we usually mean quasi-TEM -- in the microstrip case it's a conductive ring printed on top of a dielectric and backed by another solid conductive plane.

There's not much to the design -- it appears you have most of what you need already. A quick search for "microwave ring resonator" turned up this basic paper, which investigates various forms and excitation schemes.

 

[yefj]

Hello Planarmetamaterials,I have tried to implement the attached thesis
I have got a result shown bellow which is somehow matched the ampirical formula as shown bellow.
In the formula i get 8.4 sections instead of 8 as shown bellow. where did i go wrong in the formula?

The thesis gives me only Ez component formula (shown bellow) of the E-field, but as you can see bellow
the behavior of E-field is radial firected and has angular dependancy.
Could you please help me to find more precise formulas of the E-field so i could recread the photo bellow?
Thanks.

https://oaktrust.library.tamu.edu/bitstream/handle/1969.1/578/etd-tamu-2004A-ELEN-Hsieh-1.pdf?sequence=1&isAllowed=y

At F=9.9GHz N=(9.9*4*pi*30*0.5*sqrt(1.8))/(300)=8.41

At F=12.327GHz N=(9.9*4*pi*30*0.5*sqrt(1.8))/(300)=10.48

 

[PlanarMetamaterials]

The mode you're interested in is a quasi-TEM microstrip mode. There's not going to be any [realistically simple] analytical expression for the fields, sorry. You could spend the next month deriving some more exacting expressions, but what's the point?

The thesis uses approximations (such as assuming only the z-component of the fields), which yields relatively straightforward and somewhat accurate expressions. In real life engineering, these are often all we can really hope for -- I suggest that your results are sufficiently accurate for almost all purposes.