You quoted my appnote, so let me try to explain the different concepts.
Let's start with inductance. Below is the simplified equivalent circuit model for on inductor in Silicon technology. When we want to extract inductance, we need to define what inductance value we mean: the core Ls value? The effective value with parasitics? For what port connection, because that gives different effective values?
If we are a PDK modelling engineer, we might want the core Ls value (excluding paraitics). For circuit design, we are more interested in an effective series inductance value that includes all the effects (includes parasitics). In Silicon technologies, inductors in tank circuit are usually driven in
differential mode, so we want the effective value for this configuration between 1+ and 1- :
The equation from my appnote converts the 2-port parameters (one port on each end of the inductor) into 1-port parameter measured between the two terminals. The result for inductance is the same as directly simulating as a 1-port, and then using L = imag(Z_ind)/(omega). However, the advantage of using 2-port EM data in circuit simulation is that it is valid both differential and common mode excitation (and a mixture of both). That's why I prefer simulating as a 2-port (data for circuit sim), and instead of doing another 1-port sim for plotting differential inductor parameters that is calculated by Zseries = (Z11*Z22 - Z12*Z21)/Z21
I have a question that does the Z11 from stoz() function different from the Zin, which is calculated from S11 and another question is what Zseries means and why do we use this impedance to calculate the inductance. I found that the inductance, calculated from Zseries and Zin are quite similar, and these results are more different from the inductance, determined from Z11_stoz.
As explained above, the circuit use in Si technology is differential drive of the inductor. If you measured into port 1, with port 2 terminated into 50 Ohm, the effect of parasitics is different. That is why evaluting inductor parameters should be done for the actual model of operation, i.e. differentially between the inductor terminals for differential circuit use.
Hope this makes sense now?
Best regards
Volker