# Mapping geometric to electric parameters for a PMSM machine

I am looking at a PMSM machine described by the following equations:

$$\frac{di_d}{dt} = \frac{\omega_e L_q}{L_d}i_q - \frac{R_s}{L_d}i_d + \frac{u_d}{L_d}$$ $$\frac{di_q}{dt} = -\frac{\omega_e (i_dL_d + \Psi_m)}{L_q}i_q - \frac{R_s}{L_q}i_q + \frac{u_q}{L_q}$$ $$\frac{d\omega}{dt} = \frac{\frac{3p}{2}((L_d-L_q)i_d + \Psi_m)i_q - T_L - \beta \omega_m}{J}$$

where $$\ \omega_e\$$ is the electrical speed, related to the rotor speed as $$\\omega_e=p\omega_m\$$. $$\p\$$ denotes the number of pole pairs. $$\u_d\$$ and $$\u_q\$$ are the stator voltages in the d-q frame and $$\i_d\$$ and $$\i_q\$$ are the currents. $$\T_L\$$ is the load torque. $$\L_d\$$ and $$\L_q\$$ are the stator d-axis and q-axis inductance. $$\R_s\$$ is the equivalent resistance of each stator winding. $$\\Psi_m\$$ is the permanent magnet flux linkage. $$\J\$$ is the moment of inertia of the rotor. $$\\beta\$$ is the viscous damping coefficient.

I was wondering if there is any method to map geometric parameters like: air gap, shaft diameter, tooth width etc. to parameters like: resistance, inductance, flux etc. with analytical equations? I have read about magnetic equivalent circuits but whenever I dig deeper in the literature I find that people use FEM methods to obtain those parameters. Are there approximation methods, so I could a) Define geometric parameters for my machine; b) Link geometric parameters to electric parameters; c) To model my system as in the described differential equations above? Any help is appreciated.

• I fear this isn't standard fare. And you are likely on your own. I think it would make a great master's thesis, though. And yes, numerical approximation (FEM/FEA, etc.) are appropriate tools if you can create the necessary partial differentials needed. And you cannot just punch those out, blindly. But given a good understanding of how FEM works and what it needs to get you answers that will matter to you and a good understanding of your own problem, I'm sure things can be paired up to give reasonable answers. I just doubt anyone here has the cheat-sheet for this.
– jonk
Oct 21, 2022 at 4:53