The machine parameters obtained from the analytical analysis are recalculated with FEM.
ANSYS Maxwell software is used in the analysis. Torque values at different frequencies, electromotive forces and nominal currents are compared in Table 3.7.
Table 3.7. The comparison of torque, current and emf values calculated with analytical and numeri-cal methods.
As the results of the finite element saturation analysis have been used in analytical analysis, the torque values in the constant torque region are approximately equal. The torque values in field weakening region have some error. The highest error in torque, 7.8 %, is found at 450 Hz frequency. Without the saturation analysis the torque values would have much higher errors. The numerical analysis estimates 1 % lower electromotive force, which is acceptable.
The difference between the nominal currents obtained with different methods is less than 2.5
%. The difference is mainly because of the difference in the back emf. The cogging torques of the machines are evaluated with FEM. the results are shown in Fig. 3.12.
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Fig. 3.12. The cogging torque analysis. On the left side is the coil winding and on the right side the hairpin winding.
The difference between the cogging torques of the machines is minimal. Both machines pro-duce approximately 30 Nm peak-to-peak cogging torque, which is approximately 3.4 % of the nominal torque. The rotors of both machines are skewed by one slot pitch, which reduces the cogging torque.
The loss components and the total losses of the machines calculated with the MATLAB model and with FEM are compared in Table 3.8. As the iron losses in analytical model are based on the loss density of the selected electrical steel type at 200 Hz and 1.5 T, the calcu-lated iron losses are accurate. The iron losses are assumed to be proportional to the square of the load current and to the power of 1.5 of the frequency. The conductors in the analytical model are not affected by the time-varying magnetic field leaking across the slots. This re-duces the copper losses in the analytical model. Permanent magnet losses are analyzed three-dimensionally, as the magnets are split in both directions. The magnet splitting reduces the resulting eddy currents, as illustrated in Fig. 3.13. The magnet losses are assumed to be equal for both machines based on the analytical analysis, and thus the magnet losses are analyzed only for the coil winding to reduce the computing time. Analytically calculated mechanical losses and no additional losses are used in the total loss and efficiency calculation of the numerical analysis. According to the numerical analysis, the efficiency of the machine with the coil winding is 97.7 %, while the efficiency of the machine with the hairpin winding is 96.6 %. The differences between the efficiencies calculated with the analytical and numeri-cal method are 0.4 and 0.2 percentage points respectively. Because of this, the analytinumeri-cally calculated efficiencies are assumed to be accurate.
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Table 3.8. Machine losses calculated with analytical and numerical methods at the nominal torque with MTPA control strategy.
Coil winding, analytical
Coil winding, numerical
Hairpin, analytical
Hairpin, numerical
Copper losses 𝑃Cu [W] 2269 2403 3318 4719
Iron losses 𝑃Fe [W] 1609 1520 1604 1520
Mechanical losses 𝑃ρ [W] 748 - 748 -
PM losses 𝑃PM [W] 198 317 198 317
Additional losses 𝑃ad [W] 1078 0 1078 0
Total losses 𝑃tot [W] 5902 4988 6946 7304
Efficiency 𝜂 [%] 97.3 97.7 96.8 96.6
Fig. 3.13. The resulting eddy currents in a surface permanent magnet when the magnet is split in axial and radial directions. The machine dimensions are as described in Table 3.4.
When the numerically calculated losses are compared, the only difference is the copper losses. The difference between the copper losses is even bigger in the numerical analysis compared to the analytical analysis. Transient simulation method has been used to determine the copper losses in the numerical analysis. The method estimated a resistance factor of 2.3 for the hairpin winding at the nominal frequency. The resistance factors obtained from the analytical and numerical analyses are compared in Fig. 3.14. According to this comparison,
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the analytical model underestimates the resistance factor by a factor of 1.6. This difference is compensated with the additional losses in the analytical analysis.
Fig. 3.14. The resistance factors obtained from the analytical and numerical analyses as a function of frequency.
As an option for full copper winding, also previously described hybrid winding with pressed aluminum coil next to the slot opening is analyzed. Current density distribution in a full copper stator winding and in an aluminum-copper hybrid stator winding are analyzed and compared. The current density distribution in full copper winding is shown in Fig. 3.15.
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Fig. 3.15. The current density distribution in an aluminum-copper hybrid stator winding and in a full copper stator winding.
In a hairpin winding with eight layers the hybrid winding provides better current density distribution, but only in the layers with the aluminum conductors. The current density dis-tribution in other layers is not affected significantly. The difference in the copper losses is approximately 5 %, and thus the hybrid winding improves the efficiency of the machine slightly. The complexity of the manufacturing is increased with the hybrid winding, but at the same time the material price is lowered. Because of the increased manufacturing diffi-culty, the copper winding is the preferable option in this case.