Novel Quasi-Three-Dimensional Modeling of Axial Flux In-Wheel Motor With Permanent Magnet Skew

This paper presents a characteristic analysis of axial flux permanent magnet machines (AFPMMs) for in-wheel electric vehicles. Preferentially, a novel quasi-3-D model is developed for the fast and accurate design of AFPMMs. In electromagnetic field analysis, combined with field reconstruction method, the computation time of 2-D solutions is significantly reduced. With the use of time sweeping of the basis function, only the static finite element (FE) analysis is performed to calculate the air-gap flux distribution at the entire rotor position, whereas the conventional 2-D solutions require a transient FE analysis. In the shape sweeping process of the basis function, the virtual air-gap Section method is introduced to take into account that the ratio of slot opening to slot pitch is different depending on the radius of analysis plane, which causes errors in the analysis results of the conventional quasi-3-D method. The virtual air-gap sections are obtained by interpolation of the spatial field between the mapped cylindrical planes. The proposed technique reduces the number of 2-D analysis planes required for high accuracy in the conventional quasi-3-D method, and it can also predict the air-gap magnetic flux distribution for skewed permanent magnets without additional FE analysis. Finally, using the magnetic fields calculated in the proposed method, the electromagnetic performances of the AFPMM are calculated, such as load torque, cogging torque, attraction force, and back-EMF. The proposed method was used for the design of the AFPMM for a 5 kW in-wheel motor. The validity of analytical results is confirmed by 3-D FE and experimental results.

In electromagnetic field analysis of the proposed method,  planes. Then, the magnetic flux distributions on the virtual 99 air-gap section are reconstructed. The reconstructed magnetic 100 field is used to calculate the electromagnetic performances 101 such as load torque, cogging torque, attraction force, and 102 back-EMF. 103 In the previous research, numerous methods have been 104 proposed for the reduction of cogging torque, such as frac-105 tional slot overlapping windings [ [39], [40], [41]. 108 Although many techniques for reducing cogging torque have 109 been developed to contribute to the improvement of con-110 trol quality, the trade-off between manufacturing cost and 111 effectiveness remains a necessary task. Among those tech-112 niques, the permanent magnet (PM) skew has been widely 113 used because deforming the shape of the stator is difficult 114 to manufacture and ultimately increases the manufacturing 115 cost. In particular, it should be taken into account that in-116 wheel motors are limited in axial length due to the spatial 117 requirements in electric vehicles. Fortunately, combined with 118 the fact that PMs in AFPMMs are flat, the PM skew reduces 119 the cost and effort in fabrication [33], [42], [43], [44], [45]. 120 However, when using the PM skew technique to reduce the 121 cogging torque, it should be taken into account that other 122 characteristics vary according to the changed skew angle [46]. 123 Hence, the determination of the optimal PM skew angle is 124 the result of a compromise between the characteristics in the 125 optimization process. In the conventional quasi-3-D analysis, 126 in order to select the optimal skew angle, the FE analysis of 127 the AFPMM with the skew angle is inevitable whenever the 128 skew angle is changed. In the proposed method, the magnetic 129 field according to the skew angle can be predicted without 130 additional FE analysis by using the field reconstruction of the 131 basis function. Compared with the conventional quasi-3-D 132 FE analysis, the proposed method provides faster design 133 optimization with high accuracy.
The rest of this paper is organized as follows. In section II,  The armature reaction field, the sum of the magnetic fields 150 generated from the excitation current of the individual slot 151 coils, is expressed as: where B A is the total armature reaction field, B A,k is the winding as follows:  The 2-D analysis plane is located at a diameter of 108 mm.  and v is the motor speed of the AFPMM. B z,A and B θ,A are 200 the z-direction and θ-direction components of the armature 201 reaction field, respectively. The slot winding is excited with 202 a current of 28A and the respective current angles are 0, 2π/3 203 and −2π/3. It can be seen that the armature reaction field has 204 a period of 2 pole pitches. In the conventional quasi-3-D method, it is assumed that the 209 electromagnetic properties between neighboring 2-D analy-210 sis planes are the same. The spatial field between the 2-D 211 analysis planes, however, has an enlarged shape along the 212 radial direction. Therefore, the length of the air-gaps along the 213 radius is different. Also, because of the parallel slot openings, 214 the ratio of slot opening to slot pitch is not constant, which 215 results in different electromagnetic properties in the air-gaps 216 in the spatial field. For these reasons, a sufficient number 217 of 2-D analysis models are needed to reduce the error with 218 the analysis results of the 3-D FE method. The enhanced 219 quasi-3-D shown in Fig. 4 is the method proposed in this 220 paper, and the magnetic flux distribution on virtual air-gap 221 sections between 2-D analysis planes are obtained through 222 spatial interpolation.
A,k is the armature reaction field for the γ th virtual The open-circuit field is expressed as the sum of the magnetic 268 fields generated by individual PMs as follows.

282
where v is the motor speed, and τ p is the pole pitch. As one of many techniques for reducing cogging torque, 293 the PM skew is well known for its high cost-effectiveness.

294
In particular, in the AFPMM for in-wheel motor, it is difficult 295 to apply techniques that require shape modification or end 296 winding in the stator due to space and structural constraints.    to predict the change in magnetic flux distribution according 342 to a specific PM skew angle without additional FE analysis, 343 unlike the conventional quasi-3-D method. The electromagnetic torque in the proposed quasi-3-D analy-347 sis is calculated using Maxwell stress tensor (MST). The total 348 air-gap magnetic flux density required for force calculation is 349 obtained as 350 B r (t, r, θ, z) = B r,A (t, r, θ, z) + B r,O (t, r, θ, z) (t, r, θ, z) = B θ,A (t, r, θ, z) + B θ,O (t, r, θ, z) (14) 352 B z (t, r, θ, z) = B z,A (t, r, θ, z) + B z,O (t, r, θ, z) (15) 353 where B r , B θ , and B z are the r-direction, θ-direction, and  (1) and (7).

359
Based on the MST, the force distribution in the cylindrical 360 coordinate system is expressed as follows: where F is the force distribution, S is the surface area in 366 middle of the air-gap,n is the normal vector of S, ↔ T is the 367 MST, and µ 0 is the vacuum permeability. The torque at each 368 air-gap section is calculated as follows:   Table 2. The 383 electromagnetic torques in Fig. 9 only correspond to the 2-D 384 analysis plane in (18), and the total torque in (19) is shown in 385 Fig. 10. The magnetic flux distribution for torque calculation 386 was obtained with 6 2-D analysis planes and 55 virtual air-387 gap sections. The virtual air-gap sections were obtained from 388 the spatial field interpolated with 11 lattices. The cogging 389 torque was calculated in (20) and the reconstructed open-390 circuit field in (7) was used. Using the reconstructed armature 391 reaction field (1) and open-circuit field (7), the load torque 392 was calculated in (18)-(19). In Fig. 9, it can be seen that 393 the torque in each 2-D analysis plane increases as the radius 394 increases, and the period is the same, but the shape of the 395 waveform is not similar. This is because the ratio of slot 396 opening to slot pitch is different and the magnetic flux density 397 distribution over the air-gap is different. 398 Fig. 10 shows the cogging torque and load torque of the 399 AFPMM with different PM skew angles. The PM skew angles 400 are 0, 2, 4, and 6 degrees, respectively, and the analysis 401 results for each model are compared in Table 3. For the 402 VOLUME 10, 2022   It is noted that the cogging torque is reduced due to PM skew, 406 but the load torque is also reduced. Based on the non-skewed where F att. is the total attraction force. when ignoring 419 the radial magnetic flux density in the quasi-3-D analysis 420 (B r = 0), it can be simplified as follows:  Fig. 11 (a) shows the attraction force according to different 423 2-D analysis planes calculated using (22). The total attrac-424 tion force is compared with the 3-D FE analysis result in 425 Fig. 11 (b). It is noted that the attraction force increases closer 426 to the outer diameter, and the result of the total attraction force 427 matches well with the 3-D FE analysis result with an error of 428 less than 1.1%.   (15) and (14), respectively. 451 The magnetic flux distribution corresponds to the air-gap 452 at the center radius of the AFPMM, 108.75mm, and the rotor 453 is in the initial position. The difference between the proposed 454 quasi-3-D FE analysis and 3-D FE analysis results is less 455 than 1.7%.

456
Fig. 14 shows a prototype of the AFPMM for a 5 kW 457 in-wheel motor. The specifications of design parameters are 458 described in Table 4. The material for the stator and rotor 459 back-irons is 35JN230. The material of PM is N42SH. The 460 test set is shown in Fig. 14 (c). Fig. 15 shows the back EMF 461 under no-load condition, for the non-skewed PM model and 462 the model with the PM skew angle of 6 degrees, respectively. 463 The proposed quasi-3-D analysis results were calculated 464 in (25) using the open-circuit field in (7). Each analysis result 465 has a mean error of 3.2% and 4.1%, respectively, compared 466 to the measurement result.

467
In Fig. 16, the cogging torque for the model without PM 468 skew and the model with a skew angle of 6 degrees is shown, 469 VOLUME 10, 2022     by 15.1% and 3.2%, respectively, compared to the measured 473 results. In the case of the load torque, the results are compared 474 in Fig. 17, and it can be seen that the errors of the results are 475 acceptable as less than 1.7% and 2.3%, respectively.  provides a reduction of 98.87% compared to 3-D FE analysis. 540 This paper is expected to contribute to the effective analysis 541 of the characteristics of AFPMMs used as in-wheel motors. 542 The comparative study of several optimization techniques 543 (e.g., overhang structures, new geometries of PM skew) is 544 recommended as valuable future work. 545