Hi Loucy -- Very nice questions.
Our vias are volume current, as described in the Sonnet documentation.
Consider an infinite length uniform transmission line (or at least one that is long compared to wavelength). The "length" is the dimension that is large compared to a wavelength. The width and thickness are << wavelength. Basically, in a uniform transmission line, the currents along the width and thickness must be very small exactly because their size is << wavelength. Doesn't matter what your ground return is.
Let's explore this in more detail. If you look at a plot of Zo or Eeff for thick lossless microstrip, you will see that it is very close to TEM. This is why the microstrip mode is called a "quasi-TEM" mode. In fact a quasi-static analysis of microstrip comes up with almost exactly the right answer over a good share of the useful single mode frequency range.
However, at high frequency, even lossless zero thickness microstrip has large dispersion. I think you will agree that zero thickness microstrip can not support Z-directed (vertical) current. Therefore, Z-directed current does not cause dispersion in this case. In fact, the dispersion is caused by non-TEM E and H fields surrounding the line. The non-TEM fields are caused by the non-uniform dielectric (substrate below, air above).
Lossy transmission lines of any kind do have very large dispersion at low frequency. This is due to series R. The R in series with L (per unit length) causes the circuit theory Zo (=sqrt(L/C)) and velocity of propagation (=1/sqrt(LC)) to become complex and very dispersive at low frequency. But you can get this information out of pure circuit theory which allows only for longitudinal current (lossless substrate case). And you can see it in EM analysis for zero thickness transmission lines...no Z-current at all. (At low frequency this extreme dispersion does not matter: line lengths are such a tiny percentage of wavelength.)
You point out the case of width and thickness being about equal. We found, as described in my paper, that this specific case requires more sheets in order to calculate Zo to the same accuracy. (For volume meshers, it requires a finer mesh.) This is because the fields from the sides of the line contribute more to Zo, and fields from the top and bottom sides contribute less to Zo as compared to the wide case. Makes sense.
Let's look at the lossless case first. Now we don't have to worry about skin depth. In this case, we still find that nearly all the current is still longitudinal. Very easy to check. Select dimensions for a test line. Analyze it with any surface or volume mesh analysis, perfectly lossless. Look at the current. You will see it is nearly all longitudinal. You can do a 2-sheet circuit in SonnetLite (free from
www.sonnetsoftware.com). You can view 3-D currents in Version 10. No need to take my word for it. In fact, please don't take my word for it. If it is important to you, try it and see for yourself. Any EM analysis will give pretty much the same answer. If you use an iterative volume mesher, just be sure you have converged the much more sensitive current distribution, not just the S-parameters.
In the lossy case, we need to extend the current distribution inside the conductor. If there were significant Z-directed current, we would have to extend the vias inside. We tried it and looked at the result. No Z-current. So, we don't do that now because we know it isn't there. Doing it would be a waste of time. It is absolutely OK if you think I am wrong. Please try it and see for yourself. Maybe you can find an exception. I would welcome such information. This is the way science works.
What is important in the lossy case, as I said, is that the (longitudinal) current must extend inside the conductor. You can get a very reasonable approximation to the skin depth by making the cell size equal to the skin depth. This extends the current into the condutor uniformly for one skin depth. If you need a better representation of the current, then use a smaller mesh. But that costs more analysis time. Make sure you are getting increased accuracy you need by doing and A/B test. Don't waste hours of analysis time getting accuracy you don't need.
In the case of equal width and thickness, espeically with tightly coupled lines, we find that having the current extend into the body of the line can be important. Keeping it all on the surface results in increased error.
Having said all this, if you or anyone else can provide even one well converged plot of fundamental mode microstrip showing eddy currents swirling around, (along with sufficient information to duplicate the result!) you will have my full attention. A fundamental mode microstrip has width, thickness, and substrate thickness << wavelength. Please do not bother posting a current density plot that still has a lot of numercial noise in it. Anyone can see anything they want in a plot like that, it is like looking at clouds.
Loucy, if you don't mind, can you share with us where you first heard this "eddy current" concept for lossy microstrip? I'd be very interested in finding out where it came from. I guarentee it did not come from capable EM researchers.
Has anyone else been presented with this eddy current concept? Please feel free to share with us where you heard it. Would love to explore it's family tree.