Finite Element Model
Torque, phase flux-linkage and voltage, eddy current, and hysteresis losses were calculated using FEMM 4.2. For the motor model, I simply used automatic meshing.
I computed torque in two different ways. For the optimization routine, the torque was calculated in the following way. I fixed the current phase angle, and measured the total system co-energy with the rotor in two positions 12 degrees apart. The torque is thus given by ΔCE/Δθ. This works because the stator symmetry is over 12 degrees, and the torque does not change significantly over 12 degrees difference between phase and rotor angle. Torque for the higher fidelity time series simulation was calculated using the built in "Torque via weighted stress tensor" integration in FEMM.
I computed torque in two different ways. For the optimization routine, the torque was calculated in the following way. I fixed the current phase angle, and measured the total system co-energy with the rotor in two positions 12 degrees apart. The torque is thus given by ΔCE/Δθ. This works because the stator symmetry is over 12 degrees, and the torque does not change significantly over 12 degrees difference between phase and rotor angle. Torque for the higher fidelity time series simulation was calculated using the built in "Torque via weighted stress tensor" integration in FEMM.
Stator AC Loss Model
To calculate eddy current and hysteresis loss in the stator, I used the built in harmonic AC solver in FEMM 4.2. To save computation time, rather than calculating the AC losses for each simulation in the optimization, I solved for the losses in a simplified geometry (a simple tube with 50 mm OD, and 42 mm ID, the same dimensions as the stator back iron) over the entire space of frequencies and magnetic field strengths. Thus, by measuring the average magnetic field of the back-iron during a static simulation, I could use this data as a look up table for eddy current and hysteresis losses at any given frequency. I also checked the difference between the look-up losses and the calculated loss for the actual motor geometry, and the look up table consistently predicts loss powers ~15% greater.
Stator Loss Lookup Data
Note that losses increase ~quadratically in frequency, but completely non-linearly in field strength. This is due to the particular B-H curve used (Here this is for the M-19 steel in the FEMM 4.2 default library).
Not included in the model
I should mention that for this project, I did not model the eddy current / hysteresis losses in the rotor. This is because the field in the rotor is primarily DC, with only a very small AC component. Ultimately this should be included, but is not great enough to really be a large contribution. Perhaps more troublesome are the higher intensity oscillations in flux at the very tip of the rotor, where the variation in flux is at a much higher magnitude (but for a very small volume). While, again, I suspect these losses would be low, I actually do not know how would even model that using FEMM. There are other solvers, however that might be more suited to answer this question.
Another aspect not modeled are field harmonics / non-sinusoidal fields in the stator. The above loss model assumes the entire field is oscillating at the fundamental frequency, which is not actually true. While the back iron field is mostly in the fundamental, the field in the stator teeth has large components in much higher harmonics, and could contribute to additional loss not captured in my model.
The following image shows both the high magnitude field in the stator teeth and the field gradient at the tip of the rotor.
Another aspect not modeled are field harmonics / non-sinusoidal fields in the stator. The above loss model assumes the entire field is oscillating at the fundamental frequency, which is not actually true. While the back iron field is mostly in the fundamental, the field in the stator teeth has large components in much higher harmonics, and could contribute to additional loss not captured in my model.
The following image shows both the high magnitude field in the stator teeth and the field gradient at the tip of the rotor.
B-Field Near Tip of Rotor