BAPIN II: Sistema de Inversión Pública
DOTACIÓN DE PERSONAL
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The flux densities in the middle cross-section of the stator pole at full alignment with the rotor magnet pole are evaluated by the anisotropic 3-D FEA model for different magnet shapes and positions, which provides the elementary informa- tion from innermost layer to outermost layer as shown in Figure 2.39. Due to anisotropy of the lamination, the model shows that the flux density is generally higher at the inner side than at the outer side, and the difference gets smaller with increasing magnet span angle.
Figure 2.39: Flux density in the middle cross-section of stator pole for different magnet shapes.
It is known that both the amplitude and polarity of the cogging torque would vary along with the magnet pole arc width for conventional AFPM machines
Chapter 2. AFPM SAT Machine for In-Wheel Direct Drive Applications
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Figure 2.40: Cogging torque profiles for different magnet shapes: (a) analytical results; (b) 3-D FEA results.
2.9. Torque Ripple Reduction
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Figure 2.41: P-P values of cogging torque for different magnet shapes.
with radially constant magnet pole-arc to pole-pitch ratios [Fei & Luk (2008a)]. Here, the magnet shape optimization technique employed would result in radial variation of the magnet pole-arc to pole-pitch ratio. Similar to the triangular mag- net skewing [Aydin et al. (2007)], cogging torque can be radially counteracted by the proposed technique as long as the variation range covers the values that would bring opposing cogging torque polarities. The variations of the cogging torque with different rotor magnet shapes are predicted by the proposed analyt- ical approach together with the 3-D magnetostatic FEA model at different rotor positions, as shown in Figure 2.40. Moreover, the comparison of P-P cogging torque variation is illustrated in Figure 2.41. The results show that the magnet shape has a significant impact on the cogging torque. From Figure 2.40, it can also be clearly seen that the shape of cogging torque profiles by analytical means is almost the same as the one from 3-D FEA. On the other hand, Figure 2.41 shows large deviations of the P-P cogging torque values between the analytical and 3- D FEA results. This is largely because the analytical model is developed based on an approximate geometric shape of the stator pole shoe instead of the actual
Chapter 2. AFPM SAT Machine for In-Wheel Direct Drive Applications
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Figure 2.42: Phase back EMF characteristics for different magnet shapes: (a) fundamental and 3rd harmonic amplitudes; (b) THD and BHD. ("AN" - analyitcal, "FE" - 3-D FEA)
2.9. Torque Ripple Reduction
shape. However, the maximum and minimum amplitudes of both approaches occur at approximately the same magnet span angles. The polarity of the cog- ging torque alters when the magnet span angle passes the minimum trough, at about 46 degrees, which is the optimal magnet span angle for minimum cogging torque. Moreover, as a measure of time saving in computation, just about half an hour was devoted to calculate all the cogging torque profiles in 2.40 by the analytical approach, whilst more than 38 hours would be needed for a 3-D FEA approach running on a standard high performance computer. Thus, the analytical model represents an improvement factor of more than 76 over an exclusively FEA modeling approach.
The phase back EMF is also studied by both the analytical approach and the 3-D magneto transient FEA model, as shown in Figure 2.42. Both the analytical and FEA results show the fundamental component of the back EMF remains al- most unchanged for different magnet span angles because the same amount of magnet material is utilized. The total harmonic distortions (THD) are relatively large due to the high third harmonic components. Nevertheless, triplen har- monics would be eradicated internally in three phase machines. Consequently, the belt harmonic distortions (BHD) are somewhat small, as depicted in Figure 2.42(b), which means the machine would benefit from operating in brushless ac mode. In 2.42(a), both fundamental and third harmonics components from the analytical and 3-D FEA models are in reasonable agreement and the discrepan- cies are largely due to approximation made for the stator pole shoe shape in the analytical model. However, in Figure 2.42(b), the analytical results of THD and BHD are rather deviated from the 3-D FEA ones. Furthermore, the THD and BHD variation profiles from analytical model are fairly smooth and exhibit some modest correlations between the THD/BHD and magnet span angle, while the ones from 3-D FEA do not show any clear correlations. Due to limitation of the 3-D transient FEA, simulation errors originated from insufficient meshes might incur, especially in the higher harmonics of the back EMF. As would be expected, the optimal magnet span angle for the smallest BHD of back EMF can be found as just about 52 degrees from both analytical and 3-D FEA results. However, more elaborate 3-D transient FEA models with higher number of meshes are always essential in order to achieve higher accuracy.
Chapter 2. AFPM SAT Machine for In-Wheel Direct Drive Applications
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Figure 2.43: Flux density distributions from anisotropic 3-D FEA: (a) no load condition; (b) full load condition.
2.9. Torque Ripple Reduction
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Figure 2.44: Torque characteristics for different magnet shapes: (a) average torque output; (b) P-P torque ripple. ("AN" - analyitcal, "FE" - 3-D FEA)
Chapter 2. AFPM SAT Machine for In-Wheel Direct Drive Applications -30 -20 -10 0 10 20 30 0 12 24 36 48 60 72
Rotor Position (degree)
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30 Degree 50 Degree 0 5 10 15 20 25 30 35 1 3 5 7 9 11 Harmonics Amplitude (V) 30 Degree 50 Degree (a) (b)
Figure 2.45: Comparison of phase back EMF profiles and harmonic components between the original and optimal magnets: (a) phase back EMF profiles ; (b) phase back EMF harmonic components.
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Figure 2.46: Comparison of the cogging torque profiles between the original and optimal magnets.