United States

I. Introduction&lt;br/&gt;Hybrid excited machines have attracted wide attention due to several advantages, such as easy adjustment of air gap magnetic field, high power density and high efficiency in wide speed range [1]. Nowadays, hybrid excited multitooth flux-switching permanent magnet (HE-MFSPM) machines have become a research hotspot [2]. Based on the literature [3], three novel HE-MFSPM machines are proposed in this paper. Fig. 1 shows the topologies of the proposed three HE-MFSPM machines, called Model I, Model II and Model III respectively. The three models are basically the same except that PM with radial magnetization is added in each dummy slot in Model II, while a fault tolerant tooth is employed and wound with field winding in Model III. Significantly, the amount of PM of the Model I is the least, and the amount of PM of Model II is equal to that of Model III.&lt;br/&gt;II. Performance comparisons&lt;br/&gt;Fig. 2 (a) presents the overload capability of three proposed HE-MFSPM machines supplied with the same excitation current density (&lt;i&gt;J&lt;/i&gt;&lt;sub&gt;f&lt;/sub&gt;), i.e., 5A/mm&lt;sup&gt;2&lt;/sup&gt;. It can be seen that when the current density of armature winding (&lt;i&gt;J&lt;/i&gt;&lt;sub&gt;a&lt;/sub&gt;) increases from 2A/mm&lt;sup&gt;2&lt;/sup&gt; to 10A/mm&lt;sup&gt;2&lt;/sup&gt;, the torque of Model III increases the most, showing the best saturation characteristics. Fig. 2 (b) shows the speed range of the three proposed HE-MFSPM machines. It should be noted that all the three structures run at the same speed and the same rated voltage, i.e., 700r/min and 380V. It can be observed that Model III has the highest average torque in the constant torque region and has the widest speed range in the constant power region.&lt;br/&gt;Overall, all the proposed three HE-MFSPM machines exhibit good ability of flux regulation. But Model III is superior to Model I and Model II since it exhibits best overload capability, highest average torque and widest speed range. Moreover, fault-tolerant ability and modularity exist in Model III because of the existence of fault-tolerant teeth. The other performance, such as back-EMF, cogging torque and electromagnetic torque, will be further discussed in the full paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/2dadc16756df96c8c823592f1ae3ce2c.jpg)

Fig.1. Topologies of three HE-MFSPM machines. (a) Model I. (b) Model II. (c) Model III.


![](https://assets.underline.io/uploads/markdown_image/1/image/4adae74452ff0be4d22d493315020533.jpg)

Fig.2. Performance comparisons of three models. (a)Overload capability. (b) Speed range

2022 Joint MMM-INTERMAG

Comparative Study on Three Novel Hybrid Excited Multitooth Flux-switching Permanent Magnet Machines

#### Welcome to the 67th Annual Conference on Magnetism and Magnetic Materials, MMM 2022!

The Conference was held from October 31 to November 4 and offers both in-person and on-demand content.

MMM 2022 is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a tutorial, and several special sessions. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

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Please register for the MMM 2022 conference to join the event.

MMM 2022

This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

I. Introduction Hybrid excited machines have attracted wide attention due to several advantages, such as easy adjustment of air gap magnetic field, high power density and high efficiency in wide speed range [1]. Nowadays, hybrid excited multitooth flux-switching permanent magnet (HE-MFSPM) machines have become a research hotspot [2]. Based on the literature [3], three novel HE-MFSPM machines are proposed in this paper. Fig. 1 shows the topologies of the proposed three HE-MFSPM machines, called Model I, Model II and Model III respectively. The three models are basically the same except that PM with radial magnetization is added in each dummy slot in Model II, while a fault tolerant tooth is employed and wound with field winding in Model III. Significantly, the amount of PM of the Model I is the least, and the amount of PM of Model II is equal to that of Model III. II. Performance comparisons Fig. 2 (a) presents the overload capability of three proposed HE-MFSPM machines supplied with the same excitation current density (Jf), i.e., 5A/mm2. It can be seen that when the current density of armature winding (Ja) increases from 2A/mm2 to 10A/mm2, the torque of Model III increases the most, showing the best saturation characteristics. Fig. 2 (b) shows the speed range of the three proposed HE-MFSPM machines. It should be noted that all the three structures run at the same speed and the same rated voltage, i.e., 700r/min and 380V. It can be observed that Model III has the highest average torque in the constant torque region and has the widest speed range in the constant power region. Overall, all the proposed three HE-MFSPM machines exhibit good ability of flux regulation. But Model III is superior to Model I and Model II since it exhibits best overload capability, highest average torque and widest speed range. Moreover, fault-tolerant ability and modularity exist in Model III because of the existence of fault-tolerant teeth. The other performance, such as back-EMF, cogging torque and electromagnetic torque, will be further discussed in the full paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/2dadc16756df96c8c823592f1ae3ce2c.jpg)

Fig.1. Topologies of three HE-MFSPM machines. (a) Model I. (b) Model II. (c) Model III.


![](https://assets.underline.io/uploads/markdown_image/1/image/4adae74452ff0be4d22d493315020533.jpg)

Fig.2. Performance comparisons of three models. (a)Overload capability. (b) Speed range

poster

The 15th Joint MMM-INTERMAG Conference (2022 Joint) is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a plenary session, and an evening session, with about 1500 presentations overall. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

The 2022 Joint Conference will offer both in-person and prerecorded on-demand content.

For tickets to the event, please register at the following link: https://magnetism.org/register

Please register for the Joint MMM-Intermag conference to join the event.

Although the inner rotor permanent magnet (IRPM) motor and the outer rotor permanent magnet (ORPM) motor have different rotor structures, the analysis method of magnetic field is the same. An analytical method for calculating the no-load magnetic field of ORPM motor in the stator static coordinate and the rotor motion coordinate is presented in [1], this analysis method is based on the transformation of coordinate system, and does not consider the change of rotor structure. The magnetic field distribution calculation method of surface mounted permanent magnet motor is presented in [2], but unified inner/outer rotor structural permanent magnetic motor magnetic field calculation method is not given. Equivalent calculation of IRPM motor and ORPM motor with the same structure size is of great significance for motor design and performance analysis. In this paper, an equivalent analytical calculation theory of IRPM motor and ORPM motor is presented. The IRPM motor and ORPM motor with the same geometric size are established, as shown in Figure 1. The effects of curvature effects on magnetic field calculation is considered [3], and establish a unified calculation model. The complete unified computational theory derivation will be presented in the full paper. In order to verify the correctness of the equivalent calculation theory proposed in this paper, the finite element method (FEM) model and equivalent calculation model of the two motors are established according to Fig. 1. The FEM results and equivalent model results of air gap flux density component, output torque and other electromagnetic performance are compared. The comparison results show that the unified calculation model established in this paper is correct and has high calculation accuracy. Fig. 2 shows the comparison between the FEM results of the normal and tangential components of no-load air gap magnetic flux density and the equivalent calculation model. The complete electromagnetic performance comparison results will be given in the full paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/3b8f6fd695489b2e21fba76a685772ef.jpg)

Equivalent Analytical Calculation Theory of Inner Rotor and Outer Rotor Permanent Magnet Motor

With the advantages of high efficiency, high power density and wide constant power speed range, interior permanent magnet synchronous motor (IPMSM) is widely used in industrial, home applications, electric vehicles and other fields [1]. The magnetic equivalent circuit (MEC) model and finite element model (FEM) have been widely used in motor design. However, the MEC model has the disadvantage of low calculation accuracy, and the FEM model has the shortcomings of calculation time and occupy more computational resources [2]. The subdomain model is an optimal method for the initial design and optimization of the motor, which achieves a good compromise between the precision and the time of the magnetic field calculation [3]. In this paper, a hybrid model for magnetic field calculation considering rotor slots effect is proposed, which is improved the method in [3]. To take armature reaction into account, the IPMSM rotor is usually assumed to be a toothless model to simplify calculations [3], as shown in Fig.1. However, this simplified analytical model is difficult to accurately consider the armature reaction magnetic field. Considering that the relative permeability of the permanent magnet, a simplified and improved model of rotor structure is presented in this paper. The rotor is equivalent to a slotted rotor, when solving armature reaction magnetic field, the permanent magnet is equivalent to air. In order to verify the correctness of the subdomain model proposed in this paper, the analytical calculation results and the FEM results of the air gap magnetic flux density of the general simplified model and the simplified model in this paper are compared. The results show that the accurate subdomain model proposed in this paper has better accuracy. On the other hand, the computation time of the proposed subdomain model and FEM are compared. This paper proposed an improved simplified subdomain model, which reduced the computation time remarkably while maintaining high accuracy. The complete analytical method derivation will be presented in the full paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/8f394af37c8ab0480c5a6e56276fd270.jpg)

An Improved Hybrid Method for Magnetic Field Calculation in Spoke PM machine

For non-contact, low-loss and controllable support, active magnetic bearings (AMB) are increasingly used in turbomachineries, continuously improving the operating speed and power density of turbomachineries [1,2]. However, as an active support system, improper design of the AMB controller may arouse modal resonance of the rotor-AMB system and lead to poor system performance. It is necessary to analyze the dynamic characteristics of the rotor-AMB system for the AMB controller design [3]. With disk structure, the rotor-AMB system has coupling vibration modes between the disks and the shaft. Traditional analytical modeling methods of rotor dynamics can hardly analyze the dynamics characteristics of the multi disks rotor-AMB system. To solve the dynamic modeling problem, this paper establishes the dynamic model of the multi disks rotor-AMB system of maglev turbomachineries using the two-dimensional finite element method based on Euler Bernoulli beam theory. As shown in Fig. 1, four virtual spring dampers are introduced in the radial direction to connect the disk and the shaft. The connection characteristics of the spring dampers are described by the angular stiffness and angular damping coefficient. The partition coefficient determines the radial relative position of the spring dampers and distributes the mass and inertia between the disk and the shaft, and the coupling model of the disks and the shaft is established. The angular stiffness and angular damping coefficients are identified from the system modes test data, and the division coefficient is obtained using multi-objective genetic optimization algorithm. Finally, a test rig is built as shown in Fig. 2(a) and the effectiveness of the proposed dynamic modeling method of the multi disks rotor-AMB system is verified through experiments as shown in Fig. 2(b).

![](https://assets.underline.io/uploads/markdown_image/1/image/a9dd064fec52630508d3b0718dfa755a.jpg)

Fig.1 Multi disks rotor system and coupling vibration model

![](https://assets.underline.io/uploads/markdown_image/1/image/7222e5b4114ddba636a2a24193328130.jpg)

Fig.2 Test rig and comparison between dynamic model results and experiments

Dynamic Modeling of Multi Disks Rotor System of Maglev Turbomachineries

Sleeve or bolting is applied to a permanent magnet synchronous generator (PMSG) to prevent defects caused by shattering. Bolting is more useful than the sleeve in manufacturing processes; however, the magnetic flux is reduced by the material characteristics of bolting. In addition, permanent magnet machines exhibit an overhang structure because the end effect causes leakage flux. Therefore, the axial length of the rotor must be considered in 3D analysis according to the overhang structure. However, the analysis is time-intensive for the 3D model with diverse and complex design variables. In this study, we performed a semi-3D analysis of a PMSG considering the bolting and overhang structure. The analysis time of a PMSG must be reduced because of the complex structure. Hence, we propose a semi-3D analysis for the 2D model by considering the operating point of the magnet in the 3D analysis. The operating point of the permanent magnet was calculated using magnetic energy analyses of the 2D and 3D models. Variations in the operating point were applied to the 2D analysis models, which were separated according to the permanent magnets and bolting regions. Then, a circuit parameter was derived using the semi-3D analysis and applied to the equivalent circuit of the generator. Characteristic analysis was performed using the superposition method. The results of the semi-3D model analysis were compared with those of the 3D model analysis. Figs. 1(a), (b), and (c) show the manufactured model. Fig. 2(a) shows the analysis model. Fig. 2(b) shows the variation in the operating point of the permanent magnet. Figs. 2(c) shows the output characteristic results from the comparison between the 3D and semi-3D models. We confirmed that the results of the semi-3D and 3D model analyses are congruent. Detailed data and experimental data is presented in the full paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/b3d3b3536eccb58e234ae1715b043e4c.jpg)

Figs. 1. Manufactured model: (a) experimental set, (b) stator, (c) rotor

![](https://assets.underline.io/uploads/markdown_image/1/image/8a5fb512170e0ca742075f05c20bed37.jpg)

Figs. 2. (a) Analysis model, (b) demagnetization curve, and load power characteristic curve: (c) power-voltage curve

Semi 3D Analysis of a Permanent Magnet Synchronous Generator Considering Overhang Structure Using 2D Finite Element Analysis and Equivalent Magnetic Circuit

I. Introduction and Machine Modeling Fig. 1 shows the topology of a 6/5 wound field flux-switching machine (WFFSM), which is simple and reliable [1],[2]. However, the armature reaction and the excitation magnetic field are seriously coupled in WFFSM, resulting in a complicated mathematical model. To solve this problem, the mathematical model of 6/5 WFFSM is derived based on the precise inductance parameters considering magnetic saturation calculated by the frozen permeability method in this paper. The average electromagnetic torque equation is as follows: Te(av) =3/2[Pr Mf Im if cos(β−θmb)] − 9/4[Pr Lm1 Im2 sin(2β−θb(2))] (1) In this equation, Te(av) is composed of constant components of the excitation torque and the reluctance torque, which are related to the current parameters Im , if , β , and the inductance parameters Mf , θmb , Lm1 , θb(2). II. Verification of Mathematical Model Results Fig. 2(a) and (b) show the comparisons of the derived torque model and finite-element analysis (FEA) results under different current densities and different electrical angles, respectively. It can be seen that the derived model results are in good agreement with the FEA results. Fig. 2(c) shows the error between the model and FEA results in average electromagnetic torque. In most areas, the error is within 0.2%. When the current density is 10 A/mm2, the error reaches a maximum of 0.42% due to magnetic saturation. Fig. 2(d) shows the comparison between the model and the finite-element interpolation of the maximum electromagnetic torque. The error between them is no more than 0.45%. More derivations and verifications of the mathematical model of 6/5 WFFSM will be shown in the full paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/303c502f39e665f492642f770254637f.jpg)

Fig. 1. Topology of a 6/5 WFFSM.

![](https://assets.underline.io/uploads/markdown_image/1/image/20f8007d70d71600a0b4749b4befd1fd.jpg)

Fig. 2. Verification of mathematical model results.
(a) The torque under different current densities when β=0°. (b) The torque at different electrical angles when the current density is 5A/mm2. (c) The error between the model and FEA results in average electromagnetic torque under normal operation. (d) The electrical angle of maximum electromagnetic torque under different current densities.

Mathematical Modeling of Wound Field Flux-switching Machines Considering Magnetic Saturation

The doubly salient brushless DC generator (DSBLDCG) is suitable for aviation generator due to its simple and robust structure, and adjustable magnetic field [1]. The harsh environment of aviation operation puts forward a high demand on the thermal design of the generator winding. Hollow conductor with good heat dissipation performance can be used to replace conventional solid conductor in generator for winding cooling [2][3]. The harmonic content of the magnetic field in the slot of DSBLDCG is high and the radial component also exists, which makes its AC loss characteristics complex. Therefore, it is necessary to calculate the AC loss of the hollow conductor for DSBLDCG, analyze the influence of the conductor structural parameters on AC loss and optimize the structure of hollow conductor, so as to reduce the copper loss of the generator and facilitate the further cooling design. Fig. 1 shows the 2D finite element 1/2 section diagram of the 12-stator-pole/8-rotor-pole (12/8) DSBLDCG with hollow conductor. There are no windings or permanent magnets on the rotor, and the stator is wound with armature windings made of hollow conductors. Fig. 2 shows the magnetic flux density waveforms at different positions in the slot. The magnetic flux density can be decomposed into Bx in the x direction and By in the y direction. As the distance from the slot opening position increases, the amplitude of Bx decreases gradually, and By is mainly concentrated near the slot opening position. As shown in Fig. 3, the hollow conductor can be decomposed into four parts for the calculation of AC loss. The calculation formulas are shown as Equations (1) to (3). Fig. 4 shows the AC loss of hollow conductors at position I, II, III, and IV. It can be seen that as the distance from the slot opening position increases, the AC loss of the hollow conductor decreases. Take the hollow conductor at position I closest to the slot opening position as an example, the effect of conductor thickness on AC loss is compared and analyzed, as shown in Fig. 5. Considering the copper loss and cooling design of the generator comprehensively, 0.3mm section thickness of hollow conductor is more appropriate.

![](https://assets.underline.io/uploads/markdown_image/1/image/5474f84f7f9fda1adb57cdbc366a14cf.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/f157468289b7a84cbc9959820b16f0e1.jpg)

AC Loss Calculation and Analysis of Hollow Conductor for Doubly Salient Brushless DC Generator

Sliding electrical contact (SEC) exists extensively in the electromagnetic (EM) launchers (e.g., railguns). In an EM railgun, the armature slides on the rails at hypervelocity, and the real contact interface is never perfectly smooth. Due to the elevations and depressions of the contact interface, the sliding pair only contacts at discrete asperities, called ‘a-spots’ by Holm, at microscale [1]. The contact microscopic morphology of the armature and the rail is shown in Fig. 1. The fact that the current is only flowing through the a-spots results in larger electrical resistance in the contact zone than bulk electrical resistance and the generation of separation force due to current constriction effects [2]. Thus, it is crucially important to build a detailed model of SEC considering contact resistance to compute the interfacial current density and temperature distributions accurately. A 2-D EM railgun with a C-type armature model is simulated by the multiphysics coupling finite element method. The calculation method of contact resistance is based on Holm’s electric contact theory to simulate the real contact zone. Due to symmetry, only a half of the model is modeled, whose finite element mesh on the rail and armature is shown in Fig. 2. The copper rail and aluminum armature are used in the simulation. The driving current waveform is with a peak current of 100 kA and a rising time of 0.5 ms. An initial contact pressure of 10 MPa is applied on the apparent contact area. The armature is initially stationary and then accelerated up to 1000 m/s. The start-up behavior of the railgun is modeled to study the distribution of initial contact pressure, and then to analyze the influence of the unevenly distributed contact pressure on the interface current and temperature distributions. In this paper, the current and temperature distributions in an EM raligun have been studied by a theoretical analysis and numerical simulation. As the result shows, the skin effect and the influence of contact resistance is quite obvious when velocity is high, which provide a guidance for reducing the wear and damage of the contact surface.

![](https://assets.underline.io/uploads/markdown_image/1/image/28cc6b9adae3fc3940de66d9eaa13b65.jpg)

Fig. 1. Contact interface at the microscopic scale.


![](https://assets.underline.io/uploads/markdown_image/1/image/94aebe19ddc01fd3bee6f33998f658de.jpg)

Fig. 2. Finite element mesh of the rail and armature.

Electromagnetic Simulation of Current and Temperature Distribution in Railgun Based on Moving Contact Resistance Model

The hybrid electric vehicle composed of diesel engine and generator can greatly alleviate the “range anxiety” problem [1]. The concentrated winding PM motor with high power and small volume is adopted for the large transport or mining machinery vehicles. In order to solve the heat dissipation problem of concentrated winding PM motor and realize efficient cooling, the heat dissipation structure of stator core with auxiliary teeth without any additional facilities is introduced [1-4]. In this paper, the influence of auxiliary teeth in stator core on concentrated winding PM motor is studied from electromagnetic and thermal aspects. It can be used as an effective scheme for the thermal design of concentrated winding PM motor. Different from the slot type of general motors, a part of the stator core is extended into the slot to form the auxiliary teeth, as shown in Fig. 1. The purpose of adding the auxiliary teeth is to make the heat inside the slot winding have more paths to transfer to the outside. The stator core is mainly divided into three structures, as shown in Fig. 2. Fig. 5 shows the waveforms of the air-gap flux density. Fig. 7 shows the back EMF. The cogging torques are shown in Fig. 8. Fig. 9 shows the flux density contour. Fig. 10 shows a comparison of the torques of the three models. It can be seen the loss change from Fig. 11 and Fig. 12. Fig. 13 shows the change of torque from model 2 to model 3. Fig. 14 and Fig. 15 show the loss changes from model 2 to model 3. Fig. 16 shows the temperature distribution under the rated condition. The actual stator core with auxiliary teeth (Model 2) is shown in Fig. 17(a). In this paper, the influence of the size of stator core auxiliary teeth on concentrated winding PM motor is studied from electromagnetic and thermal aspects. The results show that the auxiliary teeth will weaken the performance of the motor and make the stator temperature drop and the rotor temperature rise. The auxiliary cooling teeth of stator core provide a new guide for the thermal design of concentrated winding PM motor.

![](https://assets.underline.io/uploads/markdown_image/1/image/e693d60f4b87e0da9aeb708253acfaa4.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/fa13280bf506c6232a888a058cef06ff.jpg)

Electromagnetic Thermal Coupling Optimization of Concentrated Winding PM Motor with Auxiliary Teeth for Electric Vehicle

Turbomachineries have large single-unit power and high efficiency, they play an important role in many areas, such as ship propulsion and thermal power generation and so on [1]. With the increase in demand, high power is an important development direction for turbomachineries in the future. However, now mechanical bearings are most used in turbomachineries, and the wear and loss at high speed severely restrict the speed. For this reason, the magnetic bearing (MB) is chosen to support the rotor to overcome the problems caused by the direct contact with the rotor [2]. However, limited by the operating principle of the steam turbine, the axial magnetic bearing (AMB) needs to overcome the large axial thrust generated by the steam flowing in turbomachineries. The traditional topology and design methods will cause the AMB to be too large to be integrated with the steam turbine. Therefore, this paper proposes an asymmetric axial magnetic bearing (AAMB) topology with large bearing capacity in a single direction. As shown in Fig. 1(a), the AAMB has a pair of stators of differential bearing and a stator of unidirectional bearing, both of which are fixed on the bearing seat and share a thrust plate. In normal operation, the unidirectional bearing works in the saturation zone of the material, it generate force and overcome the main axial thrust generated by steam. The differential bearing works in the linear region of the material, it is used to adjust the bearing capacity and overcome the remaining thrust. As shown in Fig. 1(b), The reluctance network method is used to establish the magnetic circuit model of the AAMB. And then the load capacity, dimensions and other constraints are considered, the shortest axial length is taken as the optimization object to design the AAMB, and the design scheme is obtained, and compared with the traditional differential bearing scheme, the main parameters of AAMB are shown in Fig. 2(a). Finally, The finite element model is built as shown in Fig. 2(b), the bearing capacity of the AAMB is verified by finite element method (FEM). The results show that the AAMB meets the bearing capacity requirements, and the overall axial length is less than half of the traditional design scheme.

![](https://assets.underline.io/uploads/markdown_image/1/image/3cf0b6ae6d5cd3f342de8321298524e9.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/4a7cbd7cb78b70bb7fd45e911c113de7.jpg)

Asymmetrical Axial Magnetic Bearings for Turbomachineries

Asymmetrical Axial Magnetic Bearings for Turbomachineries

This work paper presents a methodology based on discrete multi-material topology optimization (TO) for the design of electrical machine cores using finite element analysis. Our methodology is based on the Bi-directional Evolutionary Structural Optimization (BESO) heuristics to remove inefficient elements [1] or add magnets from a meshed model based on how they increase the magnetic energy. Determining the average torque with the calculation of its instantaneous values can be very expensive. The proposed methodology improves the average torque density while formulating cost function on calculated energy difference in the d-axis and q-axis. While usual multi-material topology optimization is formulated using a density-based method [2], this paper determines magnets distribution discretely. The magnetization magnitude can be considered as a discrete variable, and the challenge is to determine its direction. Thus, we deduce magnetization direction from the flux density distribution in the rotor; the magnet magnetization direction is chosen to be normal to B. The main advantage of the proposed methodology is that no “gray” materials are created, and the manufacturing of the machine with the proposed distribution of magnets can be realistically considered. The optimization problem is formulated in order to maximize the magnetic energy difference subject to xe = 0 or 1, xm= 1 or 0, ∑(xe.Ve)-Viron0 ≤ 0 and ∑(xm.Vm)-Vmagnets0 ≤ 0 where, xe,Ve are the iron presence and volume of an iron element e, and xm ,Vm have the same definitions for the magnets. Viron0, Vmagnets0 are the constraints on volumes. The basic diagram is presented in the figure below. The methodology is applied to design several stator structures as given in Fig. 1. The final optimization design of each structure is a classical synchro-reluctant machine with inserted magnets is presented in Fig. 2. The computation time is less than 10 min with 8640 mesh elements.

![](https://assets.underline.io/uploads/markdown_image/1/image/e1674d7217687fe9a3bb5451d806095b.jpg)

Figure 1. Methodology chart

![](https://assets.underline.io/uploads/markdown_image/1/image/67fc42d85f9f647459216a39a05335fd.jpg)

Figure 2. Optimized 36 slots electrical machines, flux density distribution and vectors in the flux barriers, magnets distribution

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Equivalent Analytical Calculation Theory of Inner Rotor and Outer Rotor Permanent Magnet Motor

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