De-Embedding Symmetric Structures

[MattiJo]

Dear all,

i am currently characterizing passive structures, such as Vias and MIM-Capacitors.
For that purpose, I have two symmetric structures, for example in the case of the vias:

(VIA DOWN) (30 um BURIED LINE) (VIA UP)

and

(VIA DOWN) (60 um BURIED LINE) (VIA UP)

In the case of MIM-capacitors, replace (VIA DOWN) by (MIM CAP DOWN) or (VIA UP) by (MIM CAP UP), respectively.
I already de-embedded Pads and feedlines successfully. Next I want to calculate the s-parameters of one via out of my two symmetric structures.

I tried it in the following way:

1) Convert both S-Matrixes to ABCD-Matrices (containing scattering chain/transmission parameters)

2) In transmission parameters, we can display both matrices in the following way:
T30 = (VIA)*(LIN30)*(VIA)
T60 = (VIA)*(LIN60)*(VIA) = (VIA)*(LIN30)*(LIN30)*(VIA)

3) Calculate: T30 * (T60)^-1 * T30
= (VIA)*(LIN30)*(VIA) * (VIA)^-1*(LIN30)^-1*(LIN30)^-1*(VIA)^-1*(VIA)*(LIN30)*(VIA)
= (VIA)*(VIA) = (VIA)^2

4) Take the matrix square root.

5) Convert the result back to S-Parameters.

I tried that procedure in Matlab. But this way I faced several problems:

- The square root of a matrix is regularly not unique. (In my case, i have 2x2-matrices, so I tried all four square roots)
- The result magnitude (S21 and S11) is accurate (similar to my EM-Simulation). But the phase result is very noisy, much more compared to my measurements. I cannot recognize any phase curve in all the roots.

I think my procedure is not the best one. I hope anyone of you has an idea, how to get accurate results for my model.

Many Thanks to everybody in advance!
Best regards,

MattiJo