How do you extract self-inductances, coupling coefficient (k), and quality factor from 4-port S-parameter file of a transformer?

So, I have made a transformer in CMOS that has 4 ports.

Coil 1: ports 1 and 2

Coil 2: ports 3 and 4

in this arrangement (pictorial):

Then, I put this in EMX and simulate it, giving me a 4-port S-parameter file (s4p). EMX has this nice feature that when you simulate a 4-port structure and select that it is a transformer, it will plot for you self-inductances, coupling coefficient and the quality factor. But I want to do it myself and not rely on it because it is a bit clumsy and more importantly, I want to understand it.

So far, I am aware that in case of 2 port s2p data, the following equation can be used:

But, I need to do this for an s4p file.

I also found this thread where the OP had solved the problem of self-inductances and the quality factor but had not figured out the coupling coefficient. I checked this and the results do not match what EMX plots.

Here's my replication of what OP shows in that thread:




First, I don't understand where this equation is coming from and the OP had not clarified this

CR_diff1=(S(1,1)+S(2,2)-S(1,2)-S(2,1)-2*(S(1,1)*S(2,2)-S(1,2)*S(2,1)))/(2-(S(1,1)+S(2,2)+S(1,2)+S(2,1)))

Can anyone shed light on how this is derived? Maybe sources in literature?

But, I must say the results from this are at least reasonable, even though it does not match what EMX shows for the same data....

Also, the issue of coupling coefficient still stand and would like to know how that might be done. If anyone know, I would appreciate if you could clarify or point me to the right literature.