Calculate trace characteristic impedance from electrostatic fields

I have been doing some studying this week. I'm trying to use a 2D electrostatic field solver to calculate the even & odd mode impedance, as well as the isolated impedance, propagation velocity, etc etc.

the approach I have taken is using Agros2D (an opensource FEM package, its actually quite nice to use http://www.agros2d.org/)
I modeled a surface microstrip differential pair with the dimensions described on this page: http://www.pcdandf.com/pcdesign/index.php/magazine/10472-simulation-1512
because it has the results from 6 different commercial impedance calculators and all the field based solvers are in good agreement.

To calculate the various capacitances and inductances I did 6 different simulations, 3 with the FR4 substrate er=4.0 and three with the FR4 substrate = 1.0:
FR4 substrate = 4.0
electrode 1=1v, electrode 2=0v
electrode 1=0v, electrode 2=1v
electrode 1=1v, electrode 2=1v

FR4 substrate = 1.0
electrode 1=1v, electrode 2=0v
electrode 1=0v, electrode 2=1v
electrode 1=1v, electrode 2=1v

Because of symmetry, simulation 1 & 2 will always yield the same result

I then found the the energy stored in the dielectric material using a volume integral of the FR4 and the Air space in the model. For er=4.0 I found three energies as follows:
We1: electrode 1=1v, electrode 2=0v
We2: electrode 1=0v, electrode 2=1v
We3: electrode 1=1v, electrode 2=1v

This means that the mutual capacitance (C12) and the capacitance to ground (C10 & C20) are found by:
Cm_FR4 = We1+We2-We3
C10_FR4 = We1-We2+We3
C20_FR4 = -We1+We2+We3
This was derived from this source: www.fieldp.com/documents/mutualcapacitance.pdf

I repeat the process of calculating C12, C10 & C20 using the data from the simulations where the FR4 was replaced with air (er=1). This is done because the magnetic field is not disrupted by FR4, it has a relative permeability of ur=1 and the inductances will be found using this field solution.
Cm_Air = We1+We2-We3
C10_Air = We1-We2+We3
C20_Air = -We1+We2+We3

Next step is to calculate the effective capacitance as per section 4.3.1 of "Advanced Signal Integrity for High-Speed Digital Designs" (hampoo.net/bbs/getattachfile/456)
For Odd mode:
Ceff_FR4 = C10_FR4 + 2*C12_FR4
Ceff_Air = C10_Air + 2*C12_Air
And Leff = 1/(2.998e8[m/s]^2 * Ceff_Air)
then Z0,odd = sqrt(Leff/Ceff_FR4)

This process is repeated for the Z0,even:
Ceff_FR4 = C10_FR4
Ceff_Air = C10_Air
And Leff = 1/(2.998e8[m/s]^2 * Ceff_Air)
then Z0,odd = sqrt(Leff/Ceff_FR4)

And also for Z0,isolated:
Ceff_FR4 = C10_FR4 + C12_FR4
Ceff_Air = C10_Air + C12_Air
And Leff = 1/(2.998e8[m/s]^2 * Ceff_Air)
then Z0,odd = sqrt(Leff/Ceff_FR4)




Wow, if your still reading, thank you! haha, that was a long story. Anyways, that what I'm currently doing and I'm seeing results that are very similar to the reference webpage above. The problem is that since I don't have the software packages myself, I can't be sure I'm interpreting the information that is given correctly. The page says in this image:
http://www.pcdandf.com/pcdesign/images/stories/ArticleImages/1512/bogatin3top.jpg
52.3 < Z0 < 52.9
48.8 < Zodd < 49.5
55.8 < Zeven <56.5

but my simulations, no mater what I try - increasing polynomial order, increasing mesh size, adaptive solutions based on minimum energy etc - my result is always just a little off.
I am getting:
Z0 ~= 52.3
Zodd ~= 47.7
Zeven ~= 58.0

Z0 has good agreement, but Zeven and Zodd are off.

My question to anyone who has read this far:
Do you have any suggestions as to why my calculations might be off like this? Or any ideas what I might try in order to improve the results?

Thanks to everyone who read this far!

Unfortunately you didn't report the geometry so nobody can check the results.

Commercial impedance calculators are mostly using empirical formulas and tables rather than solving the EM problem. EM solvers on the other hand are physically exact but limited by mesh resolution and numeric precision. In so far it's not unusual to get slightly different results.

Sorry, yes I suppose I didn't directly post the geometry. This link:
https://www.pcdandf.com/pcdesign/index.php/magazine/10472-simulation-1512
Shows results from 6 different software packages, 3 of which are 2D field solvers, 2 are 3D field solvers and the last one is an implementation if an anytic solution. They specify the geometry (assuming I am interpreting it correctly) as:
Trace width: 5 mil
Trace separation: 5 mil
Trace thickness: 0.7 mil
Substrate thickness: 3 mil
Relative permittivity: 4.0

I posted a pic of my simulation to give you an idea of what the geometry looks like. If anyone would like to simulate this if they have access to any other software, I'd love to see the results.

Thanks for the response!

bump bump! It's Monday morning, anyone have any thoughts on this?

I agree with FvM, your results are close enough that your process might be considered correct. Perhaps try some some more cases, and see if you get values with larger errors?