France



Permanent magnet machines are now-a-days very popular in electric traction applications requiring a wide constant-power speed range. However, the field weakening action, necessary in higher speeds creates important air gap field distribution distortion and increased losses leading to particular configurations search, such as conjunction with variable reluctance rotors [1], pole shoe configurations [2], flat embedded magnet topologies [3], convenient stator windings and permanent magnet arrangements [4], axial flux topologies [5].&lt;br/&gt;   In the present paper a convenient configuration of stator windings in permanent magnet motors is proposed involving appropriate additional end winding magnetic circuits in order to achieve inductance management. Such a technique can be applied in all permanent magnet motor configuration cases and enables increased flux weakening capability in high rotor speeds, resulting in substantial efficiency improvements. The optimal design of these extra iron cores can be modeled by adopting a particular reduced scalar potential formulation and the resulting finite element model can be facilitated considering appropriate two dimensional configurations. The overall analysis is performed through co-simulation of the additional magnetic circuits simultaneously with the permanent magnet synchronous machine active part.&lt;br/&gt;   The proposed methodology enables appropriate handling of increased flux weakening effects in traditional surface mounted permanent magnet motors. It exploits the end zone part of the windings, which does not contribute in the machine torque and performance. The main concept involves introduction of convenient magnetic circuits in the end zone parts of stator coils. These magnetic circuits enable magnetic coupling with auxiliary windings which are connected to appropriate varistors and are energized in desired electromotive force values developed at specific speed ranges through convenient selection of turns ratios. The magnetic material proposed for the additional magnetic circuits is a particular Ferrite ensuring low losses in order not to compromise the machine efficiency. Moreover, the increase of the stator windings leakage inductance is reduced when the auxiliary windings are not energized and is greatly increased when the varistors are activating them.&lt;br/&gt;    The representation of such configurations necessitates in general a 3D finite element modeling of the whole machine, which is very demanding in computational means. However, an important reduction in computation requirements can be achieved by adopting an appropriate reduced scalar potential formulation, involving one unknown per node. A further calculation simplification can be obtained by implementing an appropriate two dimensional axisymmetric representation for the additional magnetic circuits in conjunction with a two dimensional Cartesian representation of the motor active part and by adopting a direct coupling of field and circuit equations [6].&lt;br/&gt;   The methodology developed has been applied in a permanent magnet synchronous machine case with surface permanent magnet rotor configuration and rated power of 100 kW. The motor end winding geometry for one pole part illustrating the additional ferrite toroidal magnetic circuits positioning is shown in Fig. 1. The obtained efficiency maps in cases without and with the additional magnetic circuits are shown in Figs. 2a and 2b, respectively, illustrating the important efficiency improvement resulting in high speed ranges.&lt;br/&gt;   It may be noted that the proposed technique can offer great services in field weakening action and respective efficiency improvements at high speed ranges of all permanent magnet machine configurations implemented in electric traction applications.

**References:**

&lt;br/&gt;[1] C. Tang, W.L. Soong, T.M. Jahns, et al., IEEE Transactions on Industry Applications, Vol. 51, p.3761 (2015)&lt;br/&gt;[2] N. Zhao and N. Schofield, IEEE Transactions on Magnetics, Vol. 53, p. 8112607 (2017)&lt;br/&gt;[3] S. Chaithongsuk, B. Nahid-Mobarakeh, J.P. Caron et al., IEEE Transactions on Industrial Electronics. Vol. 59, p.2484 (2012)&lt;br/&gt;[4] M.S. Toulabi, J. Salmon and A.M. Knight, IEEE Transactions on Industry Applications, Vol. 53, p.1892 (2015)&lt;br/&gt;[5] A. Hemeida, M. Taha, A. Abdallh, et al., IEEE Transactions on Energy Conversion, Vol. 32, p.111 (2017)&lt;br/&gt;[6] E. Karamanis and A. Kladas, 2020 International Conference on Electrical Machines (ICEM), p.332 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/f741435f33fa0354a4568d7a094b7f78.jpg)
Fig. 1 Stator windings end zone geometry of one pole part of the machine illustrating the additional magnetic circuits positioning.

![](https://assets.underline.io/uploads/markdown_image/1/image/df6370cc71b7cdd695a934844fa86ae8.jpg)
Fig. 2 Efficiency maps of the permanent magnet motor in cases (a) without additional end winding magnetic circuits (b) with additional end winding magnetic circuits.

INTERMAG 2021

 Particular Reduced Scalar Potential Formulation for End Winding Magnetic Circuits modeling enabling increased Field Weakening in PM motors

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Permanent magnet machines are now-a-days very popular in electric traction applications requiring a wide constant-power speed range. However, the field weakening action, necessary in higher speeds creates important air gap field distribution distortion and increased losses leading to particular configurations search, such as conjunction with variable reluctance rotors [1], pole shoe configurations [2], flat embedded magnet topologies [3], convenient stator windings and permanent magnet arrangements [4], axial flux topologies [5]. In the present paper a convenient configuration of stator windings in permanent magnet motors is proposed involving appropriate additional end winding magnetic circuits in order to achieve inductance management. Such a technique can be applied in all permanent magnet motor configuration cases and enables increased flux weakening capability in high rotor speeds, resulting in substantial efficiency improvements. The optimal design of these extra iron cores can be modeled by adopting a particular reduced scalar potential formulation and the resulting finite element model can be facilitated considering appropriate two dimensional configurations. The overall analysis is performed through co-simulation of the additional magnetic circuits simultaneously with the permanent magnet synchronous machine active part. The proposed methodology enables appropriate handling of increased flux weakening effects in traditional surface mounted permanent magnet motors. It exploits the end zone part of the windings, which does not contribute in the machine torque and performance. The main concept involves introduction of convenient magnetic circuits in the end zone parts of stator coils. These magnetic circuits enable magnetic coupling with auxiliary windings which are connected to appropriate varistors and are energized in desired electromotive force values developed at specific speed ranges through convenient selection of turns ratios. The magnetic material proposed for the additional magnetic circuits is a particular Ferrite ensuring low losses in order not to compromise the machine efficiency. Moreover, the increase of the stator windings leakage inductance is reduced when the auxiliary windings are not energized and is greatly increased when the varistors are activating them. The representation of such configurations necessitates in general a 3D finite element modeling of the whole machine, which is very demanding in computational means. However, an important reduction in computation requirements can be achieved by adopting an appropriate reduced scalar potential formulation, involving one unknown per node. A further calculation simplification can be obtained by implementing an appropriate two dimensional axisymmetric representation for the additional magnetic circuits in conjunction with a two dimensional Cartesian representation of the motor active part and by adopting a direct coupling of field and circuit equations [6]. The methodology developed has been applied in a permanent magnet synchronous machine case with surface permanent magnet rotor configuration and rated power of 100 kW. The motor end winding geometry for one pole part illustrating the additional ferrite toroidal magnetic circuits positioning is shown in Fig. 1. The obtained efficiency maps in cases without and with the additional magnetic circuits are shown in Figs. 2a and 2b, respectively, illustrating the important efficiency improvement resulting in high speed ranges. It may be noted that the proposed technique can offer great services in field weakening action and respective efficiency improvements at high speed ranges of all permanent magnet machine configurations implemented in electric traction applications.

**References:**

 [1] C. Tang, W.L. Soong, T.M. Jahns, et al., IEEE Transactions on Industry Applications, Vol. 51, p.3761 (2015) [2] N. Zhao and N. Schofield, IEEE Transactions on Magnetics, Vol. 53, p. 8112607 (2017) [3] S. Chaithongsuk, B. Nahid-Mobarakeh, J.P. Caron et al., IEEE Transactions on Industrial Electronics. Vol. 59, p.2484 (2012) [4] M.S. Toulabi, J. Salmon and A.M. Knight, IEEE Transactions on Industry Applications, Vol. 53, p.1892 (2015) [5] A. Hemeida, M. Taha, A. Abdallh, et al., IEEE Transactions on Energy Conversion, Vol. 32, p.111 (2017) [6] E. Karamanis and A. Kladas, 2020 International Conference on Electrical Machines (ICEM), p.332 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/f741435f33fa0354a4568d7a094b7f78.jpg)
Fig. 1 Stator windings end zone geometry of one pole part of the machine illustrating the additional magnetic circuits positioning.

![](https://assets.underline.io/uploads/markdown_image/1/image/df6370cc71b7cdd695a934844fa86ae8.jpg)
Fig. 2 Efficiency maps of the permanent magnet motor in cases (a) without additional end winding magnetic circuits (b) with additional end winding magnetic circuits.

poster

Magnetic skyrmions are chiral, topologically-protected magnetic spin textures known for their rich physics and potential for future data storage devices. Recently, magnetic multilayers with interfacial Dzyaloshinskii-Moriya interactions (DMI) have attracted great attention because they can support stable Néel-type magnetic skyrmions at room temperature. Here, we report on micromagnetic simulations of the effect of initial magnetic field treatment on skyrmion formation in [Pt/Co/Ir]n magnetic multilayers. The multilayer system was modeled with MuMax3 using parameters determined from experimental measurements on [Pt/Co/Ir]n=3,5,8 multilayer thin films. The simulations show that the application and subsequent removal of a nearly in-plane magnetic field can transition the originally labyrinth domains into bubble skyrmions, which agrees well magneto-optical Kerr effect imaging results. Systematic simulations of the magnetic configuration of the [Pt/Co/Ir]n as a function of angle and magnitude of the applied magnetic field reveal that the formation of bubble skyrmions occurs for field magnitudes that are close to but below the in-plane saturating field and that an additional out-of-plane component will help the skyrmion formation. Skyrmion formation is observed over a wider magnetic field range as the repetition number is increased, which corresponds to increasing importance of the demagnetization field coupling. Work at Bryn Mawr College was supported by the National Science Foundation CAREER DMR-1053854 and DMR-1708790. Work at Colorado State University was supported by the National Science Foundation DMR-1709525 and work at Argonne National Laboratory was supported by the U.S. Department of Energy, Office of Science, Materials Science and Engineering Division.

**References:**



**Images:**

 The effects of field history on magnetic skyrmion formation in [Pt/Co/Ir]n multilayers.

Heat Assisted Magnetic Recording (HAMR) technology has reached an areal density of over 2Tb/in2, but the Signal to Noise Ratio (SNR) as well as the overall recording quality clearly decreases with the recording bit length (BL). In HAMR, recording patterns typically have a crescent shape due to the moving heat spot. The curvature of this crescent shape is of great significance as it is a significant source of transition broadening of the readback signal[1][2]. Various methods have been applied to determine and modify this curvature for better recording performance[3][4][5]. Thus, investigating how this curvature varies with different bit lengths and studying the change of this curvature during the HAMR writing process may help improve HAMR performance at the highest linear densities. In this digest, a detailed study of the dependence of curvature in HAMR transitions on bit length is presented. With the help of micromagnetic simulations, HAMR transition patterns for three different bit lengths, 10.5nm, 15nm and 21nm, were obtained and analyzed. For each bit length, 128 bits were written on multiple media and overlapped. Curvatures were then calculated using these averaged patterns. AC-erased thermal Exchange Coupling Composite (ECC) media was used, where a superparamagnetic layer with high Curie temperature forms the writing layer and a doped FePt layer with relatively low Curie temperature serves as the storage layer[6]. The change in curvature during the cooling process was also tracked for pure FePt media. Curvatures of DC-erased thermal ECC media were tested as well for comparison. Introduction of intergranular exchange was also implemented to find potential optimizations of HAMR system designs. Micromagnetic simulations in this study are guided by the Landau-Lifshitz-Gilbert equation. Renormalization group theory is applied to enable inclusion of atomic level phenomena[7]. Media dimension is 384nm x 48nm x 9nm. For a certain bit length, the 128 written bits were overlapped and averaged, as shown in Fig. 1. The left edge of the -1 bit (blue-colored) was chosen for curvature calculation. This curve was fitted to a parabolic curve and the reciprocal value of its second order coefficient was taken as its curvature. In HAMR, the cooling process could potentially modify the written bits as magnetic properties of the media materials, such as saturation magnetization and magneto-crystalline anisotropy, change greatly with temperature. Curvatures before and after the cooling process for both AC-erased thermal ECC media and pure FePt media with BL10.5nm, 15nm and 21nm are presented in Fig. 2. For thermal ECC media, curvatures before the cooling process for all the three bit lengths were close to each other, as shown with the cyan solid line in Fig. 2. After the bits were cooled down to room temperature, a decrease in curvature was seen for both BL10.5nm and 15nm while no obvious curvature change was observed for BL21nm. The BL10.5nm curvature experienced a bigger drop than the BL15nm curvature. This means that curvatures are reduced more during the cooling process for smaller bit lengths. The reason lies in the distance between the bit edge and the heat spot. When a -1 bit is just written, the left edge of this bit is about twice the bit length away from the heat spot. For BL10.5nm case, this distance is 21nm, which is much smaller than that for BL15nm and BL21nm cases. This implies that for BL10.5nm case, this bit edge is under a much higher temperature and spins here have a greater chance to be switched due to their reduced anisotropy fields. When the magnetic field switches as the head moves on, some spins at the center of this left bit edge area would get flipped, which will flatten the curve and thus cause the curvature to decrease. This leads to the interesting conclusion that the enhanced erase-after-write found in the thermal ECC media is actually beneficial owing to the resulting reduction in curvature. FePt media shows a very similar erase-after-write effect, with the 10.5 nm bits showing before and after cooling curvatures very close to the thermal ECC media. Interestingly, the 21 nm bits, while showing reduced erase-after-write as expected, have overall larger values of the curvature in comparison to the thermal ECC media, possibly owing to the differing magnetostatics. Curvatures for DC-erased thermal ECC media are found to be larger than those for AC-erased media in all the three BL cases. This is because the +1 background of DC-erased media provides stronger dipole interaction than the averaged 0 background of AC-erased media. Spins have a higher switching probability, especially in the middle of the track. For AC-erased thermal ECC media, when intergranular exchange was introduced, a decrease in curvature was seen in all the three BL cases. This results from the weaker intergranular exchange effect received by two transition corner areas than by the central areas due to the averaged 0 background. The two corners are then removed and the curvature is reduced.

**References:**

[1] Liu, Z. and Gilbert, I., IEEE Transactions on Magnetics, 55(3). (2019) [2] Tseng, C. Y. and Hurben, M. J., IEEE Transactions on Magnetics, 47(10), 2974–2976. (2011) [3] Hashimoto, M. and Salo, M., INTERMAG 2006 - IEEE International Magnetics Conference, 43(7), 448. (2006) [4] Zhu, J. G. J. and Li, H., IEEE Transactions on Magnetics, 53(2). (2017) [5] Scholz, W. and Batra, S., IEEE Transactions on Magnetics, 41(10), 2839–2844. (2005) [6] Liu, Z., Jiao, Y. and Victora, R. H., Applied Physics Letters, 108(23). (2016) [7] Victora, R. H. and Huang, P. W., IEEE Transactions on Magnetics, 49(2), 751–757. (2013)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/cb39a9a43bf6f4499492fde625d24769.jpg)
Fig. 1 Averaged HAMR transition patterns with BL 10.5nm.

![](https://assets.underline.io/uploads/markdown_image/1/image/bda63439307c1ae3b1ec0e0301bca139.jpg)
Fig. 2 Curvature change during HAMR cooling process. Solid lines are for thermal ECC media while dashed lines are for FePt media.

 Dependence of Curvature in HAMR Transitions on Bit Length

I.Introduction Spoke-Type permanent magnet synchronous motors (STPMSMs) are widely applied in the industry, especially for the electric vehicle where the high efficiency and high power density are required [1]. As a basic parameter of motor design, the inductance affects the performance, such as flux-weakening capability and control strategy [2]. It is necessary to accurately calculate the inductance. Various researches on inductance with analytical method have been carried out. However, the effect of saturation is normally neglected in the past. This paper presents an analytical method to calculate the d- and q-axis inductances accounting for saturation. II.Proposed Method The proposed analytical method is based on the subdomain model. Firstly, the STPMSM is divided into several types of simple subdomains. The analytical expression of each subdomain is derived and the field solution is obtained by the boundary conditions. Then, the lumped magnetic circuit model (LMCM) is established to calculate the stator reluctance. According to the iterative method, the flux linkage in the case of saturation is calculated. At last, the d- and q-axis inductances under saturation can be obtained. The solution process is summarized in Fig. 1. III.Results and Analysis The finite element method (FEM) is employed to validate the accuracy of the proposed model. Fig.2 shows the d- and q-axis inductances of a 18-slot/16-pole STPMSM. As the current increases, the effect of saturation on the inductance is more pronounced. Compared with the analytical method without saturation, the result of the proposed method is closer to that of FEM. Hence the proposed method is more advantageous for the inductance calculation. IV.Conclusion This paper presents an analytical method for inductance calculation which combines the advantages of subdomain model and LMCM. The effect of saturation on the inductance is fully considered, which improves the accuracy of the inductance calculation. The validity of the analytical technique is verified by FEM. In addition, the method can also be used in the inductance calculation for other types of motors.

**References:**

[1] W. Zhao, J. w. Kwon, X. Wang, T. A. Lipo, and B. i. Kwon, "Optimal Design of a Spoke-type Permanent Magnet Motor with Phase-group Concentrated-coil Windings to Minimize Torque Pulsations," IEEE Trans. Magn., vol. 53, no. 6, Jun. 2017, Art. no. 8104604. [2] J. Y. Kim, D. W. Kang, T. C. Jeong, D. H. Jeong, J. Lee, and S. G. Lee, "Optimal Rotor Design of an 150 kW-Class IPMSM by the 3-D Voltage-Inductance-Map Analysis Method," IEEE Trans. Magn., vol. 53, no. 11, NOV. 2017, Art. no. 8206805.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9e5feb6a11a5aab0b29176ccc159f03d.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/75139eb242956e0a98ed217fc6897d4c.jpg)

 Analytical Computation of Inductance for Spoke-Type Permanent Magnet Synchronous Motor Accounting for Saturation

I. Introduction Form-wound winding is one of the common forms of electrical machine windings, which owns many advantages, such as simple structure and good insulation. It is widely used in medium and large capacity electrical machines. In order to estimate the AC loss of the form-wound winding, researchers have proposed many analytical models and finite element models [1-6]. Among these analytical methods, many researches focused on the uneven current distribution inside the wire bars. Meanwhile, the circulating currents between parallel wire bars of one coil is usually neglected due to the torsion at the end winding. In fact, the torsion at the end winding cannot completely eliminate the circulating current, especially in the case of multi-bar parallel windings and high frequency conditions. It is a challenging task to establish a complex external circuit in order to simulate connections of wire bars. In this paper, an analytical calculation method to estimate the AC cooper loss of the form-wound winding, taking the circulating current effect into consideration, is presented. Firstly, the current of each wire bar is calculated by building the electric circuit model. Consequently, the current density distribution inside each wire bar is obtained by introducing the previous 1-D analytical method. Finally, the accuracy of the model is validated by FEM simulations. II. Analytical Model of Circulating Current Assuming that the main flux difference between parallel wire bars of one coil, the uneven current distribution inside the wire bar, the MMF drop inside the core and the magnetic field x component are neglected. The voltage equation of the pth wire bar of the ith coil is written as Eq. 1. Considering all voltage equations of wire bars in the same coil, two adjacent equations can be subtracted from each other, thus eliminating the induced voltage caused by main flux, as shown in Eq. 2. With total current constraint shown in Eq. 3, currents can be obtained. The mutual leakage inductances of the wire bars are the key parameters. As shown in Fig. 1 (b), the leakage inductance matrix of bars in one slot is firstly derived through the analytical method. By introducing the directional incidence matrix C between coil bars and slot bars, the mutual leakage inductance matrix of coil bars can be finally obtained, as shown in Fig. 1(c). It’s worth to mention that the positive and negative sides of one coil need to be considered separately to simulate the torsion at the end winding. III. Analytical Model of Uneven Current Distribution inside Bars The currents of bars calculated above are considered as inputs of the analytical model for calculating the current density inside bars. Assuming that the length of the slot opening is infinite and the flux lines are parallel to the slot bottom, the computation of the uneven current distribution inside bars can be simplified to 1-D problem. Based on the Maxwell equations and continuous boundary conditions and current constraints, the partial differential equation of the magnetic field can be derived and solved by analytical method [7]. IV. FEM Validation A 12-pole 72-slot PM generator with six phases is taken for FEM validation. The simplified structure of one coil is shown in Fig. 2(a). Due to the torsion at the end winding, the orders of bars placements in positive side and negative side of one coil are opposite.The currents of bars in one coil is shown in Fig. 2(b). It shows a good agreement between FEM and the proposed method results. Furthermore, the calculation error increases with frequency and the maximum amplitude error of current vectors is about 6.58% (@1000Hz).The current density distribution is shown in Fig. 2(c). The proposed method shows a high accuracy compared with the FEM result. Fig. 2(d) shows the variation curves of the AC loss factor with frequency, which are obtained by the FEM method, the conventional analytical method and the proposed method, respectively. These results show that the conventional method ignoring circulating current effect can cause significant error, especially in high-frequency conditions. Obviously, the accuracy of the AC copper loss estimation has been greatly improved by the proposed method in this case. V. Conclusion An analytical method for estimating form-wound winding loss considering circulating current effect was presented, the accuracy is validated by FEM method. For the case in this paper, the new proposed method greatly improved the accuracy compared with the conventional method. The proposed method provided a fast and accurate copper loss calculation approach for electrical machine optimization design.

**References:**

[1]P. L. Dowell, Proceedings of the Institution of Electrical Engineers, Vol. 113, No.8, p. 1387-1394 (1966) [2]A. S. Thomas, Z. Q. Zhu, and G. W. Jewell, IEEE Transactions on Magnetics, Vol. 45, No. 10, pp. 4748-4751 (2009) [3]L. J. Wu, Z. Q. Zhu and D. Staton, IEEE Transactions on Magnetics, Vol. 48, No. 7, p. 2138-2151 (2012) [4]Y. Wang, J. Pries and K. Zhou, IEEE Transactions on Magnetics, Vol. 56, No. 4, p. 1-9 (2020) [5]P. Mellor, R. Wrobel and N. Simpson, The Energy Conversion Congress and Exposition, p. 5563-5570 (2014) [6]H. Hamalainen, J. Pyrhonen and J. Nerg, IEEE Transactions on Magnetics, Vol. 49, No. 6, p. 2967-2973 (2013)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/58d991873f052535077a8fb8d9ba17d6.jpg)
Fig.1 Brief derivation of circulating current and inductance matrix

![](https://assets.underline.io/uploads/markdown_image/1/image/e0ba56abe6c49f44d04351c473c1e8ba.jpg)
Fig.2 Test model and results comparison

 Analytical Method of Form-Wound Winding Loss Calculation Considering Circulating Current Effect

1. Introduction Nowadays, Finite Element Model (FEM) is considered as the most accurate approach to design and study electrical machines as it is able to consider many physical phenomena. However, its computation time can constitute a real drawback when several machines have to be tested thus making analytical approaches more suitable. Analytical model that involves winding functions and airgap permeance remains one of the more accurate and widely used approach. It is based on different assumptions including that of a regular stator and rotor slot pitches. As such, analytical expressions leads to an accurate representation of the air gap permeance [1]–[3]. However, recent studies show that spatial harmonic content of the stator Magneto Motive Force (MMF) can be reduced using non-regular slot pitches [4], [5]. Therefore, to be able to quantify the interest of such proposition or even optimize the location of the slots with limited time calculation, an analytical approach which makes it possible to take into account any location of the slots is needed. To achieve that, a generic analytical expression to model the air gap permeance of electrical machines is proposed. It enables to accurately calculate the air gap permeance in all cases i.e. with regular and non-regular stator and/or rotor slot pitches. Results obtained using the proposed approach are compared to the ones given by FEM in the case of an example of slotless rotor electrical machine with regular and non-regular stator slot pitches. 2. Analytical model permeance Quantifying the performance of an electrical machine is based on an accurate calculation of the magnetic flux density in the air gap whose permeance is the key quantity. To obtain the latter through a general analytical expression, the air gap can be split into three areas as showed in the Figure 1: - Mechanical air gap, δe, of constant thickness all around the circumference of the machine - Additional air gap due to stator slots, δs, which depends on θs - Additional air gap due to rotor slots, δr, which depend on θs and θ θs represents a given position in the airgap with respects to a fixed stator axis and θ is the relative position of the rotor with respects to the stator. At steady state with a constant speed Ω, θ = Ω.t. The total air gap thickness g(θs, θ) is then the sum of these three air gaps allowing to obtain the permeance through the calculation of g-1(θs, θ). Air gap permeance calculation is usually made considering one regular slot pitch but if the latter is non-regular, it is more relevant to consider the entire air gap circumference. In the proposed approach, each additional air gap due to a stator slot i or rotor slot j is represented by an elementary function δs,ielem(θs,αi) or δrjelem(θs,θ,αj) where αi and αj are the spatial shift phases related to the positions of the stator slot i and the rotor slot j with respect to reference stator and rotor axes respectively.The superposition of the elementary functions of all the slots enables to obtain the complete functions of δs(θs) and δr(θs,θ) and hence the global air gap as g(θs, θ) = δe + δs(θs)+ δr(θs,θ). 3. Application In order to highlight the accuracy of the proposed model, a simple structure, i.e. slotless rotor, has been chosen. The proposed approach has then been used in the case of regular (Figure 2.a) and non-regular (Figure 2.b) slot pitch cases as proposed in the literature [4], [5]. The comparison between the magnetic flux density in the air gap obtained by both approaches under the assumption of linear behavior of the magnetic material are given in the Figure 2.c and 2.d. Results are very similar and show the accuracy of the proposed analytical expression of the air gap permeance. In the extended version, the analytical calculation of the air gap permeance will be presented in detail. Then, the proposed approach will be used in the case of non-regular rotor slot pitch and in the case of both non-regular stator and rotor slot pitches. Results will be given and discussed.

**References:**

[1] D. Zarko, D. Ban, et T. A. Lipo, « Analytical calculation of magnetic field distribution in the slotted air gap of a surface permanent-magnet motor using complex relative air-gap permeance », IEEE Transactions on Magnetics, vol. 42, no 7, p. 1828-1837, juill. 2006, doi: 10.1109/TMAG.2006.874594. [2] Z. Q. Zhu, D. Howe, E. Bolte, et B. Ackermann, « Instantaneous magnetic field distribution in brushless permanent magnet DC motors. I. Open-circuit field », IEEE Transactions on Magnetics, vol. 29, no 1, p. 124-135, janv. 1993, doi: 10.1109/20.195557. [3] B. Gaussens, E. Hoang, O. de la Barriere, J. Saint-Michel, M. Lecrivain, et M. Gabsi, « Analytical Approach for Air-Gap Modeling of Field-Excited Flux-Switching Machine: No-Load Operation », IEEE Transactions on Magnetics, vol. 48, no 9, p. 2505-2517, sept. 2012, doi: 10.1109/TMAG.2012.2196706. [4] J. Marault, F. Gillon, M. Hecquet, et A. Tounzi, « Optimization of the MMF spatial harmonic content to design electrical machine winding », International Journal of Applied Electromagnetics and Mechanics, vol. Preprint, no Preprint, p. 1-16, janv. 2020, doi: 10.3233/JAE-209508. [5] N. Bekka, M. E. H. Zaïm, N. Bernard, et D. Trichet, « A Novel Methodology for Optimal Design of Fractional Slot With Concentrated Windings », IEEE Transactions on Energy Conversion, vol. 31, no 3, p. 1153-1160, sept. 2016, doi: 10.1109/TEC.2016.2552546.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3648c4c383f1913c4405fe942990185c.jpg)
Fig 1. Air gap permeance function

![](https://assets.underline.io/uploads/markdown_image/1/image/a457905edd7422a022008b7f3cdcb912.jpg)
Fig 2. 18 slots, 10 poles simplified structure, (a) cross section with regular slot pitch, (b) ) cross section with non-regular slot pitch, (c) air gap flux density waveform for regular slot pitch, (d) air gap flux density waveform for non-regular slot pitch

 A general analytical expression to model the air gap permeance of electrical machines

Introduction: Winding loss generally takes a great proportion of the total loss of electrical machines. In many cases, e.g., the high-speed machines excited with high-frequency currents, the effect of the AC loss increases further[1]. Hence, the precise computation of the winding loss is critical in the design optimization process of electrical machines. In a traditional finite element model, the refined mesh is required for each strand, which results in a huge computation burden. To overcome this difficulty, this paper proposed a domain decomposition (DD) type of finite element approach based on the Mortar element method. The linear and nonlinear computation for the inside and outside slot domain is separated and coupled through the interface potentials by the Mortar element method. Therefore, even though a set of uniform refined mesh is generated for each slot domain, the dimension of the nonlinear system is not increased. Then the whole system can be solved efficiently without losing detailed information in each slot. A 37kW permanent magnet assisted synchronous reluctance motor (PM-assisted SynRM) is investigated by the proposed method. The comparison results with traditional FEM proves the feasibility of the presented method for efficient winding loss computation. The proposed DD approach: Fig. 1 (a) and (b) illustrate the topology of the example with divided linear and nonlinear domains, which can be formulated as equations (1) and (2) in Fig. 2. According to Faraday's law of electromagnetic induction, the circuit equation in jth stranded conductor equation is obtained as (3) in Fig. 2. Then the relationship between the stator circuit and the external power supply has the formulation expressed as (4) in Fig. 2 [2]. By Galerkin discretization, the weak formulas (1), (2), (3), and (4) are formed into a strong formulation (5) with non-conforming boundary conditions. Fig. 1 (c) demonstrates the domain division with one single set of slot and tooth in the stator, while other symmetrical stator segments are divided by the same rule. Note that the rotor and air gap of the model belong to the nonlinear domain. The linear domain Ωa and nonlinear domain Ωb mesh independently with non-conforming grids, which means the grids on the boundary are discontinuous. The Mortar element method is adopted to couple the separated domains in the field computation through the information processing on the interface Γ [3]. The node vector potentials belonging to the divide subdomains on Γ have a relationship (6) in Fig. 2 where H and D are line integral of the shape functions over Γ. The unknown potential vector in (5) can be represented in matrix form (9) in Fig. 2 or with a compact form (10). Substitute (10) into the equation (5), i.e., SA=F, the system formula can be expressed as (11) in Fig. 2. We rewrite (11) in the matrix form (12) in Fig. 2. Considering the linear domain in the model, the following relations can be obtained when decoupling the third and fourth rows from the system matrix (13) in Fig. 2. Due to the linearity, the product of left two matrices in the subproblem can be pre-calculated directly. Then (13) is simplified into (14) in Fig. 2. Where the left constant K matrix is the obtained product. Substitute relation (14) into the first row, second row, and fifth row of (12), the matrix or vectors of the system are finally simplified as (15), (16), (17) in Fig. 2, and the final compact system matrix is set up as (18). Finally, since the stiffness matrix in the linear domain can be pre-calculated as a constant, we obtain the values of aout, abnd1, i through the compact system (18) with the Newton Raphson method. The compact system (18) is generally of a lower dimension than the full problem, which means less effort is required for the nonlinear computation. Next, abnd2, ain, ustr are solved from (6) and (13) by using the solutions of the system (18). Although the system (18) matrix is very dense, it is still much faster than solving the whole problem, which is verified from the following simulation case. Numerical example: Fig. 1 (d) and (e) illustrate the mesh diagrams for the traditional FE model (conforming mesh) and DD FE model (non-conforming mesh), respectively. The time-step simulation is performed at speed of 3000 rpm and the motor is power by a sinusoidal voltage source with 350V RMS line voltage and -28.3 load angle. Two simulation models run in Matlab on the same workstation. The total elements in Fig. 1 (d) and (e) are 96234 and 105153, while the nodes, i.e., the unknown node potential amount are 48612 and 53778. By adopting the model reduction process presented above, the number of magnetic vector potentials required for the nonlinear computation is reduced to the node amount in the nonlinear region. For this case, the magnetic potential number of the DD FE model in the nonlinear system is reduced from 53778 to 3506, which is 13 times less than the number (48612) in the traditional model. The computation time of the traditional model and DD model are 38376s and 2834s. The proposed method was more than 13 times faster than the traditional FE. The AC current density map used in the simulations is shown in Fig. 1 (f). Fig. 1 (g) illustrates the AC loss comparison. The good agreement demonstrates the accuracy of the proposed method. Conclusion: The effectiveness of the proposed DD method is verified by the example. Moreover, due to the universality of the model derivation, the proposed method is also applicable to other electrical machines.

**References:**

[1] Pavel Ponomarev, Ilya Petrov, and Nicola Bianchi, “Losses in Stator Slot Windings of Permanent Magnet Synchronous Machines, ” doi:10.13140/RG.2.1.2081.9368, 2015. [2] A. Lehikoinen, N. Chiodetto, and E. Lantto, “Monte Carlo Analysis of Circulating Currents in Random-Wound Electrical Machines,” IEEE Trans. Magn., vol.52, pp.1-12,2016. [3] Orlando Jose Antunes, and J. P. A. Bastos, “Using Hierarchic Interpolation with Mortar Element Method for Electrical Machines Analysis,” IEEE Trans. Magn., vol. 41, pp. 1472-1475, 2005.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/00206775246b05e4a839be2a4dc73dc5.png)
Fig. 1

![](https://assets.underline.io/uploads/markdown_image/1/image/042e81d15cdaf336adcaf2dada7c2d14.png)
Fig. 2

 A Mortar Method Based Domain Decomposition Approach for Winding Loss Computation of Electrical Machines

I. INTRODUCTION Differential evolution (DE) is a powerful evolutionary algorithm that is widely used to solve global optimization problems because of its ease of use, simple structure, efficiency, and robustness. It has been pointed out that the optimization methods, especially of electrical machine designs, are shifting from response surface method toward varieties of DE algorithm [1]–[4]. However, some previous work indicated that DE faces significant difficulty particularly with problems that are not linearly separable [5]. It has been observed that DE sometimes has a limited ability to move its population through large distances across the search space if the population is clustered in a limited portion of it, and in such cases it also has a tendency to find the local rather than global optima due to this lack of movement abilities. Sutton et al. found strong evidence that DE becomes more reliant on differential mutation, and less so on orthogonal steps produced by crossover when the function is non-separable. This paper proposes a hybrid multi-objective optimization method for electrical machines with directed graph (or digraph, DiG) and DE. The proposed method, called DiG-DE, is able to keep the balance of exploration and exploitation in search. The operation of the proposed method is verified, and its performance is compared with that of the standard DE method, for electric machine optimization problem. According to the results, the proposed DiG-DE can solve the problem faster than classic DE. Additionally, for the same computing time, the DiG-DE can always find better candidate designs than classic DE. II. APPLICATION TO ELECTRICAL MACHINE OPTIMIZATION To verify its superiority in terms of optimization for electrical machine design, the proposed DiG-DE is applied to the optimal design of a special modular external rotor switched reluctance motor and compared with classic DE. Fig. 1 displays the fitness of all designs found by classic DE and DiG-DE. From the distribution point of view, the designs found by DiG-DE are more densely distributed near the target Pareto front. All the designs are found by classic DE, but DiG-DE would select those designs that are closer to Pareto front. With the accumulation of selection preferences over time, there is a difference in evolutionary speed. This indicates that the DiG-DE is able to provide higher selective pressure than classic DE. It has been noted that the selective pressure will lead to the loss of diversity. In DiG-DE, the population size of parents is increasing in every generation, and the population grows exponentially, which compensates for this loss of diversity. Fig. 2 shows the candidate designs found by DiG-DE and classic DE for four different solution times. It may be observed that DiG-DE’s solutions are closer to the target Pareto front as compared with the classic DE’s. After a long solution times, the results obtained by the two methods tend to be close. Therefore, there is sufficient evidence to show that the proposed DiG-DE has faster convergence and higher search efficiency than classic DE. III. CONCLUSION In this work, a new algorithm, DiG-DE, based on differential evolution, which uses a digraph to rank candidate designs is proposed and compared with classical DE for electric machine optimization. In contrast with the classic DE approach in which the parents produce the same number of children in every generation, the DiG-DE method shows features that are closer to natural evolution. Under the pressure of selection, the individuals with higher rankings in DiG-DE are more likely to be selected, resulting in more offspring. The conduced comparative studies show that this unique selective scheme gives DiG-DE faster convergence and higher search efficiency than classic DE.

**References:**

[1] Y. Duan, and D.M. Ionel, IEEE Trans. Ind. Appl., vol. 49, p. 1268–1275 (2013). [2] C.T. Krasopoulos, I.P. Armouti, A.G. Kladas, IEEE Trans Magn., vol. 53 (2017). [3] Y. Wang, D. Ionel, D.G. Dorrell, and S. Stretz, IEEE Trans. Magn., vol. 51 (2015). [4] M.E. Beniakar, A.G. Sarigiannidis, P.E. kakosimos, and A.G. Kladas, IEEE Trans. Magn., vol. 50 (2014). [5] S. Das, and P.N. Suganthan, IEEE Trans. Evol. Comput., vol. 15, p. 4–31 (2011).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/bf9eb36c051fe04dc836b61a6297562f.jpg)
Fig. 1 All designs by classic DE and DiG-DE.

![](https://assets.underline.io/uploads/markdown_image/1/image/ba11a4241f17cf8f61d112cfd47d29ce.jpg)
Fig. 2 Solutions of classic DE and DiG-DE computed within (a) 4 hours, (b) 6 hours, (c) 10 hours, and (d) 17 hours.

 A Novel DiG-DE Algorithm for Multi-Objective Design Optimization of Electrical Machine

Urban air mobility (UAM) motors must be designed for light weight and high power density. This is because reducing an aircraft's curb weight increases the aircraft's efficiency and cruising range. UAM motors are basically high torque low speed machines, so they are designed with high poles [1]. The fractional slot structure allows for higher pole numbers with fewer slots and provides a concentrated slot winding, thus reducing the weight of the stator. In addition, the Halbach array rotor can reduce or eliminate back iron cores, thus reducing the weight of the rotor. Therefore, the fractional slot Halbach motor can reduce weight, making it a very suitable for UAM applications. Fig. 1 (a) shows the basic layout of UAM motor, and the design target specifications are as follows: 40-pole 48-slot, DC voltage 600 V, maximum power 50 kW, maximum torque 180 Nm, maximum speed 3 krpm, power density 4 kW/kg. The rotor has a two-segment Halbach array, so only two kinds of magnetization are used: radial and circumferential. The wider the radial magnet, the higher the field density. For this, radial magnets are designed wide and circumferential magnets are narrow. Then the height of the circumferential magnet is not enough to provide complete flux return in the back. The design shown in Fig. 1 (a) has a shallow back iron core at the bottom of permanent magnets (PMs). It supports flux return that is not managed by circumferential PM. Therefore, it is necessary to efficiently determine the ratio of radial to circumferential PMs, and the ratio of magnet to back iron. There are three design variables for optimal design of the rotor. One is the radial magnet coverage ratio α=τr/τp, the other is the PM height ratio δ=hm/hr, the other is the rotor height ratio γ=hr/τp, where τr is the angle for radial magnets, τp is the pole pitch angle, hm is the magnet height, and hr is the rotor height. The design goal is to minimize the weight of the rotor while meeting the output torque requirements. To meet the design requirements in the proposed two-segment Halbach motor, three design variables α, δ, and γ must be found that ensure that wr(γ) is mimimized at the same time Te(α,δ,γ)&gt;180 Nm. Te is the electromagnetic torque and wr is the weight of the rotor. In finite element analysis (FEA)-based design, the number of models to be analyzed increases exponentially as the number of design variables increases. Therefore, FEA design is not suitable for such multivariate optimization. The subdomain can find the optimal design much faster because the result can be obtained by simply changing the design variables. In this paper, the subdomain method was combined with the magnetic equivalent circuit (MEC) analysis to account for the core saturation occurring under load conditions [2]. Fig. 1 (b) shows the domain setup of the subdomain analysis, and (c) shows the MEC model [3]. Fig. 1 (d) shows the airgap flux density compared to the FEA results. It more accurately matches the FEA results when saturation is considered through the MEC. With the combined method, we can find the minimum rotor height ratio γ(α,δ) that satisfies the torque specification Te≥180 Nm, which is shown as dots in Fig. 2 (a). Furthermore, γ(α,δ) can be approximated as a concave function with minimum values at a certain point (α*,δ*) and is: γLSM(α,δ)=p00+p10α+p20α2+p01δ+p11αδ+p02δ2, where pxy is the coefficient of αxδy in γLSM. Be applying the least square method (LSM), we can find the best fitting coefficients. γ≈XP, where P is the matrix of coefficients pxy and X is the matrix of variables α and δ. Let the cost be defined by J=1/2*(γ−XP)T(γ−XP). Then, the cost minimizing solution is obtained as P=(XTX)-1XTγ. The resulting function γLSM(α,δ) with the fitted coefficients are depicted as planes shown in Fig. 2 (a). The minimum point can be found by solving ∂γLSM/∂α=0 and ∂γLSM/∂δ=0, and is obtained such that α*=(2p10p02-p01p11)/(p112-4p20p02), δ*=(2p20p01–p10p11)/(p112-4p20p02). While satisfying the required torque, the minimum rotor height is obtained as γ*=0.37 when α*=0.61 and δ*=0.85. That is, when the rotor height is hr=0.37τp=6.5 mm, the desired torque can be achieved with the minimum rotor weight. Fig. 2 (b) shows rotor weights, and (d) shows an optimized design using the proposed method. To verify the optimized design algorithm, the proposed Halbach motor is compared with SPMSM and a coreless Halbach motor without rotor back irons. For comparison, SPMSM uses the same amount of magnets as the proposed Halbach motor and raises the height of the back iron until it meets the torque specification. For coreless Halbach motors, radial and circumferential magnets are designed to have the same arc according to the existing design. Note that the stator remains the same for accurate comparison. Fig. 2 (d) shows the torque waveform through FEA when Iph=90 Arms. The torque of the proposed Halbach motor is 187.4 Nm, and it has the highest torque. At this time, the rotor weight is 2.54 kg, which has a high power density of 4.17 kW/kg. The SPMSM and coreless Halbach motors have similar torque, but for SPMSM, the rotor height must be increased to 10.5 mm to achieve similar torque levels. Thus, the power density decreases to 3.71 kW/kg.

**References:**

[1] T. F. Tallerico, Z. A. Cameron, and H. Hasseeb, "Outer Stator Magnetically-Geared Motors for Electrified Urban Air Mobility Vehicles," AIAA/IEEE Electric Aircraft Technologies Symposium (EATS), pp. 1-25 (2020) [2] M. Lee, B. Koo and K. Nam, "Analytic Optimization of Halbach Array Slotless Motor Considering Stator Yoke Saturation," in IEEE Trans. on Magn., (2020) [3] S. Wu, X. Zhao, and C. Jiu, "Multi-Objective Optimal Design of a Toroidally Wound Radial-Flux Halbach Permanent Magnet Array Limited Angle Torque Motor," in IEEE Trans. on Ind. Electron., vol. 64, no. 4, pp. 2962-2971 (2017)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d1fa395a5fcb12da14a7c867c6af6d30.jpg)
(a) A pole pair of Halbach motor model, (b) simplified subdomain model, (c) MEC model, and (d) comparison of airgap flux density.

![](https://assets.underline.io/uploads/markdown_image/1/image/29ff00ba04ee4b43adae646d5bd993fe.jpg)
(a) γ(α,δ) that satisfies Te>180 Nm and fitted curve γLSM(α,δ) using LSM, (b) rotor weights versus δ for different α, (c) flux density of the optimized motor, and (d) torque comparison when Iph=90 Arms.

 Low Weight Halbach UAM Motor Design Based on Subdomain Analysis

1. Induction: The accurate estimation of the harmonic losses caused by the zig-zag leakage flux which occupies a high proportion of stray losses is of great significance in the efficiency calculation as well as the optimization design for the wet submersible induction machine [1]. To precise estimate the zig-zag leakage flux losses at the nominal load, it is necessary to determine losses induced by a particular order of the air-gap flux density harmonic. 2. Analysis and Calculation: In this paper, a two-dimensional finite element method is presented to separate zig-zag leakage flux losses produced by the stator slot harmonic, rotor slot harmonic and interaction harmonic between them, respectively. The air-gap magnetic flux density is decomposed by the two-dimensional fast Fourier transform at a sampling frequency of 5000 Hz. The hybrid excited virtual permanent magnet harmonic machine model is applied to reconstruct the air-gap magnetic field and excited the stator and rotor separately, in which the iron core node magnetic flux density is extracted and decomposed [2]. Each node harmonic iron loss is acquired through the modified Bertotti’s iron loss model with the consideration of high frequency power losses as a function of the magnetic flux density. The losses caused by a single harmonic are obtained by subtracting the losses resulted from the hybrid excitation [3]. Fig. 1(a) shows rotor iron losses density distribution induced by the -17th harmonic, which denotes the first order stator slot harmonic. The rotor is equipped with the closed slot because of the machine’s water-filled operating environment, where the mixed mesh subdivision through triangular mesh and rectangular mesh are employed. It can be seen in Fig. 1(a) that the -17th harmonic penetrates into the rotor surface where the mapped mesh exists. Meanwhile, Fig. 1(b) and Fig. 1(c) display rotor bar and stator winding eddy-current density distributions respectively, where the current density engendered by the -17th harmonic close to the air gap is higher than the rest in other positions. 3. Results: The zig-zag leakage flux losses calculated includes the surface and pulsation losses in the stator and rotor as well as eddy-current losses in the stator winding and squirrel cage. Different proportion of losses caused by corresponding harmonics have important reference for the optimization of the power density of the machine. Table I displays the losses caused by the first and second order stator slot harmonics, together with the first order rotor slot harmonic (-20th and 22nd). In Table I, ‘Nan’ expresses that the corresponding values are small or approach zero. It can be represented that the losses reduce with the order of harmonic, therefore, the iron losses in the stator can be neglected. Meanwhile, the rotor losses generated by the rotor slot harmonic is not listed in the table, because the rotor slot harmonic rotates synchronously with the rotor itself. It can be concluded that the eddy-current losses in the squirrel cage caused by -17th harmonic account for the largest proportion among the overall zig-zag leakage flux losses. Hence, to improve the power density of the machine, effective measures (e.g. rotor skew, semimagnetic wedge) can be taken to reduce the -17th harmonic. The method shown in this paper separates the zig-zag leakage flux harmonic losses and lays a fundamental for the machine design as well as the optimization.

**References:**

[1] R. Hagen, T. Knopik, and A. Binder, IEEE Int. Elect. Mach & Drives Conf., p. 1325-1330 (2009) [2] C. Di, I. Petrov, and J. J. Pyrhönen, IEEE Trans. Magn., Vol. 54, p. 1-12 (2018) [3] C. Di, I. Petrov, and J. J. Pyrhönen, IEEE Access, Vol. 7, p. 27746–27755 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c851afdc272d06b04905972692e636a5.jpg)
Fig. 1 Losses density and current density distributions produced by -17th harmonic at the nominal load. (a) Rotor iron losses. (b) Rotor bar eddy-current. (c) Stator winding eddy-current.

![](https://assets.underline.io/uploads/markdown_image/1/image/44204e85ef9b8db76828a38e27b94f87.jpg)
Table I 
Main Harmonic Amplitude and Zig-Zag Leakage Losses

 Calculation and Decomposition of Zig-Zag Leakage Flux Losses in a Wet Submersible Induction Machine by the Virtual Permanent Magnet Harmonic Machine Model

I. INTRODUCTION In recent years, due to the merits of easy thermal management, alleviated torque ripple and robust rotor, flux reversal permanent magnet (FRPM) machines have attracted more and more attention [1]. However, since the large leakage flux leads to the limitation of its output, a series of novel topologies are proposed, such as the consequent-pole FRPM machine [2], the FRPM machine with asymmetric-stator pole configuration [1] and the FRPM machine with Halbach array magnets in rotor slot (HAM-FRPM) [3], etc. This study presents an analytical sub-domain model for the characteristic evaluation of the HAM-FRPM machine. By dividing the HAM-FRPM machine into seven sub-domains and solving the Maxwell equations in polar coordinate for each region according to the boundary conditions, the vector magnetic potential can be obtained. Subsequently, the radial and tangential flux density can be calculated. Finally, the analytical results are compared with the finite element method (FEM) to verify the accuracy. II. ANALYTICAL SUB-DOMAIN MODEL One 12-stator slot/17-rotor tooth HAM-FRPM machine with double-layer winding is investigated in this paper to verify the proposed method. As shown in Fig.1, the permanent magnet (PM) arrangement in the stator of the HAM-FRPM machine is identical to that of the consequent-pole FRPM machine. In addition, the PMs in each rotor slot are the same, which contain a radially magnetized middle PM and two tangentially magnetized side PMs. In this analytical sub-domain modeling, some essential assumptions have been made in advance: (1) The permeability of iron is regarded as infinite. Also, the relative permeability of PMs and coils is assumed to be 1; (2) The end effect and the magnetic field component on the axial direction are neglected; (3) The geometrical structure side of the HAM-FRPM machine is parallel to either r or θ direction in polar coordinate. As shown in Fig. 2, the HAM-FRPM machine is divided into several regions: stator slot (Region i), stator slot opening (Region l and Region u), PM (Region s and Region g), air gap (Region I) and rotor slot (Region j). Maxwell equation is satisfied in each region as equation (1), which can be expressed as a Poisson equation or a Laplace equation and further solved by the separation of variables method. According to the law of magnetic flux continuity and Neumann condition, the unknown coefficients can de derived. Finally, the radial flux density Br and tangential flux density Bθ can be obtained by (2). where J and M are the current density vector and magnetization vector of magnets, respectively. μ0 is the magnetic permeability of vacuum. III. RESULTS AND DISCUSSION The air gap flux density comparisons between the analytical prediction and FEM results are exhibited in Fig.3-4, which present a good agreement between the analytical results and FEM solution for both radial and tangential air gap flux density. Hence, the proposed analytical model is verified to be used to predict the performances of the HAM-FRPM machines. Fig.5-6 show the back-EMF and torque waveforms, which validate the correctness of analytical model. More detailed derivation and analysis will be shown in the full paper.

**References:**

[1] H. Yang, H. Lin, Z. Q. Zhu, S. Lyu, and Y. Liu, "Design and Analysis of Novel Asymmetric-Stator-Pole Flux Reversal PM Machine," IEEE Transactions on Industrial Electronics, vol. 67, no. 1, pp. 101-114, 2020. [2] Y. Gao, R. Qu, D. Li, J. Li, and G. Zhou, "Consequent-Pole Flux-Reversal Permanent-Magnet Machine for Electric Vehicle Propulsion," IEEE Transactions on Applied Superconductivity, vol. 26, no. 4, pp. 1-5, 2016. [3] K. Yang, F. Zhao, Y. Wang, and Z. Bao, "Consequent-Pole Flux Reversal Permanent Magnet Machine with Halbach Array Magnets in Rotor Slot," IEEE Transactions on Magnetics, pp. 1-1, 2020.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/df027abcb712ed2aa1b4402da9ebc363.jpg)

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