Mutual inductance in BLDC motor in the DQ reference frame question

[Etesla]

Hi all,

I'm trying to understand what a schematic would look like capturing the terms Matlab has documented here:

https://www.mathworks.com/help/sps/ref/pmsm.html

I'm going from simple to complex. Starting simple now.

My first question is the following.

The matlab documentation states:
Ld = Ls + Ms - 3/2 Lm

Let's pretend it's a non salient motor and Lm = 0. Lets just worry about the locked rotor nothing is moving case.

In that case there seems to be an Ls and Ms that I have to worry about. I'm assuming Ms exists because there is a transformer like coupling between the stator windings, is that true?

If I set up an LTSpice model to inject a voltage vector with any magnitude and angle into a three phase motor model in a stepped voltage transient case, with no coupling between the motor coils, I see that my calculated Id rises as my Id voltage steps. I can also see that my 'Ld' and 'Rd' that model this correctly have equivalent values to the per coil inductance and resistance to a Y wound motor model. I'm assuming this is represented by a Ls = 40u, Ms = 0, Lm = 0, Rs = 10m case:



From the equation, I would assume that Ld would increase as the coupling between the coils increases, because Ms would become a positive nonzero number, Ls would stay the same, and Lm would still be zero. So Ld would increase. However if I add coupling between the three coils:


Then I observe that the calculated Id increases faster for the same Vd. To me that indicates that Ld decreased when I increased the coupling, instead of increasing.

What's incorrect about the ideas in my LTSpice model? Does Ms not come from a transformer like coupling between the stator windings? How would a person measure Ls and Ms experimentally when handed a black box motor with three terminals?

LTSpice model is attached.