France



I.INTRODUCTION PM machines have obtained wide applications in the last several decades because of the high efficiency, high torque and high power density. Most PM machines adopt rare-earth PMs as magnetic potential source for high performance. However, due to the unstable supply of rare-earth PMs, the price of PMs is becoming a major issue. Therefore, many machines with less or no PM have been developed in the last a couple of years. Although less or no PM machines such as hybrid PM motors and synchronous reluctance motors can reduce the usage of PMs significantly, they still suffer from poor torque density. Adopting consequent-pole(CP) structure in PM machines is another solution to save PMs[1]-[3]. However, the rotors in CP motors are difficult to design and there is seldom design experience for reference. Topology optimization(TO) is an approach for shape optimization. It searches optimal shape by making free modifications in the distribution of material, and has potentialities to provide novel design. In this paper, The TO method based on ON/OFF method and immune algorithm (IA) was applied on the rotor core of a consequent-pole permanent magnet synchronous motor (CP-PMSM) to improve its torque quality. II.STRUCTURE AND OPERATION PRINCIPLE OF CP-PMSMS Fig.1(a) shows the 1/4 models of PMSMs with surface-mounted PM(SPM) and CP. For a CP-PMSM, the magnetic direction of all the PMs are the same, and each PM pole is set between two silicon steel poles. The magnetic circuits of two motors are established to explain the working principle of the CP motor. Then, torque waveforms are simulated for comparison. The average torque of SPM motor and CP motor are 3.25N and 2.09N, respectively. The average thrust will not reduce half with the missing of half PM poles, which means the PM has higher utilization in the CP-PMSM. Additionally, the motors still have potential to be optimized for better torque quality. III.TOPOLOGY OPTIMIZATION BASED ON ON/OFF METHOD AND IMMUNE ALGORITHMS The authors have proposed a TO method based on ON/OFF method with IA [4]. The ON/OFF method divides the search space into numerous finite element meshes to represent material distribution, each finite element mesh is associated with a binary value, indicating either the presence or absence of material. The area to be optimized is the search space of ON/OFF method. Consequently, the optimized shape could be represented by the presence (silicon steel) or absence (air) of silicon steel material. IA is used to search the optimal shape by changing the material distribution in the meshes. The TO procedures of ON/OFF method based on IA can be summarized as follows:1)Generate initial antibodies of &lt;i&gt;N&lt;sub&gt;pop&lt;/sub&gt;&lt;/i&gt; candidate solutions, create random initial shapes;2)Smoothen surface and eliminate holes by filter process;3)Evaluate the objective function for each antibody, list them up from high value to low value;4)Generate clones for each antibody, the antibodies with high evaluation function will be cloned more to generate more offspring;5)Apply affinity maturation (small modifications) on material surface to the clones;6)Smoothen surface and eliminate holes by filter process;7)Evaluate each clone and list them up, if one clone is better than its parent, it will replace its parent;8)Keep &lt;i&gt;N&lt;sub&gt;s&lt;/sub&gt;&lt;/i&gt;%(0＜&lt;i&gt;N&lt;sub&gt;s&lt;/sub&gt;&lt;/i&gt;＜1) clones with high ranking and weed others out. Then add randomly generated antibodies as step 1-2 to keep the population size constant;9) Back to step 4 until the optimization targets are met. IV.OPTIMIZATION RESULT OF THE CP-PMSM The rotor core is chosen to be the search space of ON/OFF method. The search space is defined in 0°–45°range, the optimized shape of this range is reversing copied to 45°–90°range, so as to make the rotor symmetrical and reduce computation cost. Both the average torque and torque ripple were considered in the evaluation function in optimization. Fig.1(b) shows the evolution of the rotor topologies during optimization. At first, the profile of silicon steel poles emerged in the surface of rotor core, and there were many big holes inside the rotor. Then, the holes inside the core gradually decreased and became smaller, and the shape of the silicon steel poles became more obvious. Finally, the holes inside the rotor core almost disappeared, and the silicon steel poles gradually became a‘W’shape. Torque characteristics before and after optimization are compared in Fig.2. The components of output torque, i.e., electromagnetic torque and cogging torque, are simulated for analysis. It can be found that the output torque of the CP-PMSM is greatly improved after optimization, the average torque is increased from 2.09N to 2.32N, and the torque ripple is reduced from 126.6% to 13.7%. Moreover, the improvement of output torque is contributed by both the improvement of cogging torque and electromagnetic torque. In the future, we plan to extend this TO method to PM poles and stator core so as realize comprehensive TO.

**References:**

[1]S. U. Chung, J. W. Kim, and Y. D. Chun,“Fractional slot concentrated winding PMSM with consequent pole rotor for a low-speed direct drive: reduction of rare earth permanent magnet,”IEEE Transactions on Energy Conversion, Vol.30, No.1, pp. 103–109, Mar. 2015. [2]L. Wei and T. Nakamura, &quot;Optimization Design of a Dual-Stator Switched Flux Consequent Pole Permanent Magnet Machine With Unequal Length Teeth,&quot; IEEE Transactions on Magnetics, Vol. 56, No. 2, pp. 1-5, Feb. 2020. [3]S. Teymoori, A. Rahideh, and H. Moayed-Jahromi,“2-D analytical magnetic field prediction for consequent-pole permanent magnet synchronous machines,” IEEE Transactions on Magnetics, Vol.6, No.6, pp. 1–14, Jun.2016. [4]K. Watanabe, T. Suga and S. Kitabatake, &quot;Topology Optimization Based on the ON/OFF Method for Synchronous Motor,&quot; IEEE Transactions on Magnetics, Vol. 54, No. 3, pp. 1-4, March 2018.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/643715c3ce09d6801d0368701f2223ca.png)
Fig. 1.(a) Comparison of surface-mounted PM motor and consequent-pole motor. In consequent-pole motor, each PM pole is set between two silicon steel poles, so as to save PM volume. (b)Variation of rotor topologies during evolution(Evolutions takes 15 hours in total). The rotor shape varies sharply in the beginning generations and gradually tends to be fixed in a &#39;W&#39; shape.

![](https://assets.underline.io/uploads/markdown_image/1/image/d441845f21330cb640be24a1e761dd93.jpg)
Fig. 2.(a)Torque characteristics before optimization. (b)Torque characteristics  after optimization. It can be found the improvement of output torque is contributed by both the improvement of cogging torque and electromagnetic torque

INTERMAG 2021

 Topology Optimization of Consequent-Pole PMSM Using ON/OFF Method

I.INTRODUCTION PM machines have obtained wide applications in the last several decades because of the high efficiency, high torque and high power density. Most PM machines adopt rare-earth PMs as magnetic potential source for high performance. However, due to the unstable supply of rare-earth PMs, the price of PMs is becoming a major issue. Therefore, many machines with less or no PM have been developed in the last a couple of years. Although less or no PM machines such as hybrid PM motors and synchronous reluctance motors can reduce the usage of PMs significantly, they still suffer from poor torque density. Adopting consequent-pole(CP) structure in PM machines is another solution to save PMs[1]-[3]. However, the rotors in CP motors are difficult to design and there is seldom design experience for reference. Topology optimization(TO) is an approach for shape optimization. It searches optimal shape by making free modifications in the distribution of material, and has potentialities to provide novel design. In this paper, The TO method based on ON/OFF method and immune algorithm (IA) was applied on the rotor core of a consequent-pole permanent magnet synchronous motor (CP-PMSM) to improve its torque quality. II.STRUCTURE AND OPERATION PRINCIPLE OF CP-PMSMS Fig.1(a) shows the 1/4 models of PMSMs with surface-mounted PM(SPM) and CP. For a CP-PMSM, the magnetic direction of all the PMs are the same, and each PM pole is set between two silicon steel poles. The magnetic circuits of two motors are established to explain the working principle of the CP motor. Then, torque waveforms are simulated for comparison. The average torque of SPM motor and CP motor are 3.25N and 2.09N, respectively. The average thrust will not reduce half with the missing of half PM poles, which means the PM has higher utilization in the CP-PMSM. Additionally, the motors still have potential to be optimized for better torque quality. III.TOPOLOGY OPTIMIZATION BASED ON ON/OFF METHOD AND IMMUNE ALGORITHMS The authors have proposed a TO method based on ON/OFF method with IA [4]. The ON/OFF method divides the search space into numerous finite element meshes to represent material distribution, each finite element mesh is associated with a binary value, indicating either the presence or absence of material. The area to be optimized is the search space of ON/OFF method. Consequently, the optimized shape could be represented by the presence (silicon steel) or absence (air) of silicon steel material. IA is used to search the optimal shape by changing the material distribution in the meshes. The TO procedures of ON/OFF method based on IA can be summarized as follows:1)Generate initial antibodies of Npop candidate solutions, create random initial shapes;2)Smoothen surface and eliminate holes by filter process;3)Evaluate the objective function for each antibody, list them up from high value to low value;4)Generate clones for each antibody, the antibodies with high evaluation function will be cloned more to generate more offspring;5)Apply affinity maturation (small modifications) on material surface to the clones;6)Smoothen surface and eliminate holes by filter process;7)Evaluate each clone and list them up, if one clone is better than its parent, it will replace its parent;8)Keep Ns%(0＜Ns＜1) clones with high ranking and weed others out. Then add randomly generated antibodies as step 1-2 to keep the population size constant;9) Back to step 4 until the optimization targets are met. IV.OPTIMIZATION RESULT OF THE CP-PMSM The rotor core is chosen to be the search space of ON/OFF method. The search space is defined in 0°–45°range, the optimized shape of this range is reversing copied to 45°–90°range, so as to make the rotor symmetrical and reduce computation cost. Both the average torque and torque ripple were considered in the evaluation function in optimization. Fig.1(b) shows the evolution of the rotor topologies during optimization. At first, the profile of silicon steel poles emerged in the surface of rotor core, and there were many big holes inside the rotor. Then, the holes inside the core gradually decreased and became smaller, and the shape of the silicon steel poles became more obvious. Finally, the holes inside the rotor core almost disappeared, and the silicon steel poles gradually became a‘W’shape. Torque characteristics before and after optimization are compared in Fig.2. The components of output torque, i.e., electromagnetic torque and cogging torque, are simulated for analysis. It can be found that the output torque of the CP-PMSM is greatly improved after optimization, the average torque is increased from 2.09N to 2.32N, and the torque ripple is reduced from 126.6% to 13.7%. Moreover, the improvement of output torque is contributed by both the improvement of cogging torque and electromagnetic torque. In the future, we plan to extend this TO method to PM poles and stator core so as realize comprehensive TO.

**References:**

[1]S. U. Chung, J. W. Kim, and Y. D. Chun,“Fractional slot concentrated winding PMSM with consequent pole rotor for a low-speed direct drive: reduction of rare earth permanent magnet,”IEEE Transactions on Energy Conversion, Vol.30, No.1, pp. 103–109, Mar. 2015. [2]L. Wei and T. Nakamura, "Optimization Design of a Dual-Stator Switched Flux Consequent Pole Permanent Magnet Machine With Unequal Length Teeth," IEEE Transactions on Magnetics, Vol. 56, No. 2, pp. 1-5, Feb. 2020. [3]S. Teymoori, A. Rahideh, and H. Moayed-Jahromi,“2-D analytical magnetic field prediction for consequent-pole permanent magnet synchronous machines,” IEEE Transactions on Magnetics, Vol.6, No.6, pp. 1–14, Jun.2016. [4]K. Watanabe, T. Suga and S. Kitabatake, "Topology Optimization Based on the ON/OFF Method for Synchronous Motor," IEEE Transactions on Magnetics, Vol. 54, No. 3, pp. 1-4, March 2018.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/643715c3ce09d6801d0368701f2223ca.png)
Fig. 1.(a) Comparison of surface-mounted PM motor and consequent-pole motor. In consequent-pole motor, each PM pole is set between two silicon steel poles, so as to save PM volume. (b)Variation of rotor topologies during evolution(Evolutions takes 15 hours in total). The rotor shape varies sharply in the beginning generations and gradually tends to be fixed in a 'W' shape.

![](https://assets.underline.io/uploads/markdown_image/1/image/d441845f21330cb640be24a1e761dd93.jpg)
Fig. 2.(a)Torque characteristics before optimization. (b)Torque characteristics after optimization. It can be found the improvement of output torque is contributed by both the improvement of cogging torque and electromagnetic torque

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Introduction: Time-step finite element method is widely used in the transient simulation because the harmonic of magnetic fields and the nonlinear characteristics of the material can be processed simultaneously [1]. However, the steady-state solution of the transient simulation convergence slowly due to the time constant characteristic in field-circuit coupling computation [2-3]. Aiming at this problem, this paper proposes an acceleration method based on the estimation of initial values in time step simulation by time harmonic analysis. With the precalculated initial solution, the iteration steps for each point calculation and time for convergence to the steady state are effectively saved. Meanwhile, the advantages of the traditional time step method are well retained. A permanent magnet assisted synchronous reluctance motor (PM-assisted SynRM) is investigated by the proposed method. The comparison results with traditional FEM demonstrate the effectiveness of the proposed method in reducing the iterations step for searching a steady-state solution in the transient simulation. The proposed approach: Firstly, the governing equation for the electromagnetic field can be formulated as equation (1) in Fig. 1. The problem formula also includes the following two equations (2) and (3) in Fig. 1, which describe the connection of the circuit. We can establish field-circuit coupling equations through (1), (2), and (3). Secondly, in time-harmonic analysis, time dependence can be eliminated from the equations by using complex field quantities (4) in Fig. 1. By Galerkin discretization in time-harmonic, the weak formulas (1) and (3) are formed into (5) and (6) in Fig.1. The complex system of equations can be written in the matrix form (7), (8), (9), and (10) in Fig.1. Newton's method is adapted for the nonlinear solution. Taylor expansion is applied to the Jacobian matrix as (11) and (12) in Fig.1 to satisfy the Cauchy-Riemann differential equations. The unknown vector in the time-harmonic simulation is then solved by (13) in Fig. 1 iteratively and only a small number of steps is required for convergence. In the harmonic analysis, the vector potential a can be described as a periodic function in Fig1. (14). The achieved time-harmonic solution is then transferred into the time domain system (16) according to (15) as the initial value at t =0s. Since the time-harmonic simulation results are very close to the steady-state solution in the transient simulation, the iteration for each time-stepping solution can be fast with fewer steps than the traditional method. Also, the time step required for convergence to the steady-state is shortened further. Thereby the total calculation time significantly reduced. The effectiveness of the presented method is verified by the following simulation case. Numerical example: Fig. 2 (a) illustrates the mesh diagrams for the FE model of the PM-assisted SynRM. The time-step simulation is performed at a frequency of 100Hz, 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 cost for the 0.3s simulation of the traditional time-stepping method and the proposed method is 58 mins and 16 mins, respectively. As shown in Fig. 2 (b), the simulation convergence to steady-state at 0.15s, and the corresponding computation cost is 8.5 mins, which means the proposed method is 6 time faster. The good agreement of both methods in steady-state torque calculation in Fig. 2 (b) also demonstrates the accuracy of the proposed method. Conclusion: The effectiveness of the proposed method is verified by the PM-assisted SynRM example. Moreover, due to the universality of the model derivation, it is also applicable to other electrical machines e.g. induction machines.

**References:**

[1] Shinji Wakao, Masato Tokuhisa, and Atsushi Maeda, “ Acceleration of convergence characteristics of time-stepping FE analysis for rotating machines,” IEEE Trans. Magn., vol.37, no.5, pp. 3558–3561, 2001. [2] W. N. Fu, S. L. Ho, and Ping Zhou, “Reduction of computing time for steady-state solutions of magnetic field and circuit coupled problems using time-domain finite-element method,” IEEE Trans. Magn., vol. 48, no. 11, pp. 3363–3366, 2012. [3] Oszkár Bíró and Kurt Preis, “An efficient time-domain method for nonlinear periodic eddy current problems,” IEEE Trans. Magn., vol. 42, no. 4, pp. 695–698, 2006.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/bf560494131098c95568658e4ce25e3f.png)
Fig. 1

![](https://assets.underline.io/uploads/markdown_image/1/image/e07e77a8a3291152aeff66b33318b170.jpg)
Fig. 2

 An Acceleration Method for Reaching Steady-State Performance in Time-stepping Finite Element Analysis

I. Introduction Axial flux permanent magnet (AFPM) machines are getting popularity in various applications. Due to the complex structure of the AFPM machine, 3 dimensional finite element analysis (3D FEA) is normally employed [1]. In order to save the computation time, several 2D FE modelling approaches have been adopted and studied by researchers. 2D linear motor approach (LMMA) is commonly used not only in the performance calculation but also in the analytical study [2]. 2D inner rotor modelling approach (IRMA) and 2D outer rotor modelling approach (ORMA) are proposed and compared with LMMA in [3]. Generally, 2D FE modelling approaches use the geometrical parameters at the average radius of the machine. Considering the variation of geometrical parameters along radial direction, geometrical parameters at different radius are taken into account by the multi-layer 2D FE modelling approaches [3, 4]. However, errors are still existed between the results gotten by 2D FEA and 3D FEA. In order to shorten the computation time and to ensure the accuracy, a 3D FE modeling approach based on scaling air gap method is proposed in this paper. II. The Principle of the Scaling Air Gap Method Basing on the expression of reluctance, the changing of geometrical parameters of an element can be balanced by the adjustment of its relative permeability. In the 3D FE modelling of an AFPM machine, the resizing of the air gap has no influence on the sizes of the other parts. It provides an opportunity to customize the air gap with equivalent relative permeability. If the air gap length is scaled to k (k＞1) times, its relative permeability in axial direction should be adjust to k times and its relative permeability in circumferential and radial direction should be adjust to 1/k times. The electromechanical energy conversion happens in the air gap, so the meshing of the air gap region is very important. Sometimes the air gap may be over meshed due to its small dimension. At this time, the scaling air gap method provides a way to reduce the number of meshes and to refine the meshing quality. III. Comparison among FEA Approaches and Experiments All the 2D and 3D FEA approaches for AFPM machines are implemented to a double-stator single-rotor AFPM machine. In this paper, we focus on the computation time and accuracy of motor performance including back electromotive force (EMF), torque output, and torque ripple. All the 2D FEA is based on multi-layer model to consider the variation of geometrical parameters at different radii. In order to make the study comprehensive, motors with different slot-pole combinations and magnet shapes are employed. Taking a 30-pole 36-slot motor with fan-shape magnets as an example, the comparison based on mesh profile and computation time are presented in Table 1. The comparison of motor performance is shown in Figure 1. It illustrates that all the 2D FEA approaches have advantages in the computation time, but there is a decrease in accuracy. The proposed 3D FEA approach based on scaling air gap method can reduce the computation time with good accuracy compared with the traditional 3D FEA approach. For further verification of the proposed method, a prototype is manufactured and tested. IV. Conclusions A 3D FE modelling approach for calculating AFPM machines based on scaling air gap method is proposed in this paper. The axial length and relative permeability of the air gap can be changed based on the scaling air gap method. In this way, the mesh setting for the air gap will be more flexible and the mesh of the whole machine can be adjusted to be more reasonable. The proposed approach is proved to have shorter computation time than the traditional 3D FEA approach and to be more accurate than 2D FEA approaches. The accuracy of the proposed approach is also verified by experiments.

**References:**

[1] S. H. Won, J. Choi, and J. Lee, "Axial-Gap Type Permanent Magnet Motor Modeling for Transient Analysis," IEEE Trans. Magn., vol. 44, no. 11, pp. 4085-4088, Nov. 2008. [2] F. Zhao, T. A. Lipo, and B. Kwon, "A Novel Dual-Stator Axial-Flux Spoke-Type Permanent Magnet Vernier Machine for Direct-Drive Applications," IEEE Trans Magn., vol. 50, no. 11, pp. 1-4, Nov. 2014. [3] M. Gulec, M. Aydin, "Implementation of different 2D finite element modelling approaches in axial flux permanent magnet disc machines," IET Electr. Power Appl., vol. 12, no.2, pp. 195-202, 2018. [4] C. Bianchini, F. Immovilli, E. Lorenzani, etc., "Axial flux permanent magnet machine desing and optimization using multi-layer 2-D simulation," in Proc. Ann. Conf. Ind. Electron. Soc., 2013, pp. 2620-2625.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/2dc32f60a9d925a176112f4245eb869d.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/6097f8bf03832bd974868f3cf2ed1994.jpg)

 A Novel 3D Finite Element Modelling Approach for Calculating Axial Flux Permanent Magnet Machines Based on Scaling Air Gap Method

Introduction: Nowadays, with the rapid development of the computer science, numerical methods have been widely used during all the design stages of an electrical machine, such as the electromagnetic design, mechanical structure optimization, thermal estimation etc. We may select the main dimension of the motor, according to the existing electromagnetic design scheme of the induction motor with the same power level. This approach is suitable for the conventional induction motor design, because the main performance is highly determined by the main dimension [1]. However, different from traditional induction motors, the water friction losses of wet submersible motors are much greater than the wind mill losses of air-cooled induction motors. As shown in Fig 1, unlike traditional industrial motors, submersible motors have a larger ratio of diameter to shaft Length The water wear of the water-filled submersible motor accounts for a large proportion of the overall losses, so how to reduce the water wear of the submersible motor has become the key to improving the efficiency of the submersible motor. Many scholars have put forward the numerical calculation method of water wear of water-filled submersible motor, which are shown in Formula (1)-(4) in Figure 2 [2-3]. It can be seen that the main factors affecting the water wear of submersible motors are the surface roughness of the rotor, rotor speed, outer diameter of the rotor and motor shaft length. The literature [4] verified the influence of surface roughness on water wear. For a submersible motor with a given output power, the speed and surface roughness are constant. In this paper, the diameter and shaft length ratio of the motor are optimized to achieve the purpose of reducing the water wear in the design stage. Optimization Method: It can be known from the main relation formula of AC motor that the motor power with constant effective volume of rotor is also approximate. Therefore, keeping the effective volume of the rotor constant, reducing the outer diameter of the rotor and increasing the shaft length can ensure the output power and reduce the water wear of the motor. However, reducing the water wear through the above methods will also increase the electrical losses of the motor. In this paper, the electrical losses of a series of size motors are solved by electromagnetic simulation, and the curve of electrical losses as a function of the rotor outer diameter is fitted by the cubic spline interpolation. A new curve is obtained by adding the water wear and electrical losses of the motor, and the minimum of the curve is the optimal rotor outer diameter. Results: As shown at point M in Fig.2, when the rotor outer diameter is 0.183 m, the sum of electrical losses and water wear of the motor are the minimum. Compared with the loss when the outer diameter of the rotor is 0.225, the loss is reduced by 360 W. This method realizes the optimization of the diameter-axle-length ratio of the submersible motor and reduces the water wear of the submersible motor in the design stage. Detailed results and discussion will be given in the full text.

**References:**

[1] J. Pyrhonen, T. Jokinen, and V. Hrabovcova. Design of Rotating Electrical Machines, 2nd ed. Hoboken, NJ, USA; wiley, (2014). [2] S. Li and S. Liu, Pump Technol., vol. 33, pp. 21–25(1988). [3] H. Liu, M. Tan, and S. Yuan, Trans. Chin. Soc. Agricult. Eng., vol. 22, pp. 107–109(2006). [4] X. He , W. Jiao , C. Wang , et al. IEEE Access, vol. 7, pp. 105331-105341(2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/4e584412cd93d4f18a9970cc97483c8a.jpg)
Fig. 1 A submersible motor with large ratio of diameter to shaft length.

![](https://assets.underline.io/uploads/markdown_image/1/image/ec1237c7417ea9ffc06e457d88d0505b.jpg)
Fig. 2 Losses at different rotor outer diameters.

 Efficiency Improvement for Submersible Motors by Optimizing the Ratio of Diameter to Shaft Length

This paper proposes an equivalent magnetic network (EMN) model for predicting the electromagnetic performances of hybrid excitation machines. In addition to the influence of flux leakage and iron saturation, the proposed EMN model also takes the change of rotor position into consideration. A new node connection method is proposed to make the dynamic analysis easier. Moreover, the reasonable simplification of stator yoke and teeth further reduces the calculation time and ensures the accuracy of the proposed EMN model.In addition, the proposed model in the motor hybrid excitation part can consider the electromagnetic characteristics under different DC excitation conditions. Finally, the prototype is manufactured, and the effectiveness of the EMN model is verified by comparison with the finite-element analysis (FEA) and experimental tests. I.Introduction Hybrid excitation machines not only has many advantages of permanent magnet machine, but also has the advantages of smooth and adjustable air gap magnetic field, wide speed range in electric operation, and wide voltage regulation ability in power generation operation [1]. Therefore, it has received a lot of attention from scholars. In order to analyze the performance of the machine, the finite-element analysis (FEA) method is usually used [2]. However, the hybrid excitation machine is a three-dimensional magnetic circuit structure, which requires a large number of grids, which is very time-consuming. In order to reduce the calculation time, the analytical method is also used to reveal the machine performance [3]. However, the effect of iron saturation is mostly considered, resulting in low accuracy. The equivalent magnetic network method has attracted more and more attention due to its short calculation time and moderate accuracy. In this paper, a dynamic magnetic network analysis model is proposed to predict the electromagnetic performance of consequent pole hybrid excitation machines. The EMN model considers the magnetic flux leakage, the nonlinearity saturation of iron materials and the magnetic bridge, which has high calculation accuracy. The calculated results are compared with the FEA and experimental test results. II.PROPOSED EMN MODEL FOR HYBRID EXCITATION MACHINES Generally, the yoke and tooth magnetic circuit of the machine is regular, and the Simplified EMN modeling method can be used to establish the model. However, the tooth tip, air gap and magnetic bridge magnetic circuit are irregular, and complex magnetic network needs to be established to improve the accuracy and be used for dynamic analysis. At the same time, the DC excitation source should be considered when the hybrid excitation motor is built for three-dimensional magnetic circuit. Figure 1 shows the topology of the machine. The following is a brief introduction to the modeling process: A.Simplified Stator EMN Model The magnetic circuit of the yoke and teeth of the stator is regular. Therefore, a simplified stator EMN model can be used to reduce the calculation time without affecting the accuracy requirements. B. Air Gap EMN Model and Dynamic Analysis The tooth tip of the machine is seriously saturated due to the quasi-closed slot. It is necessary to consider the influence of saturation in order to establish the cross-shaped mesh model. The air gap is the key part of the model, which connects the stator and the rotor. In the dynamic analysis, the stator and rotor depend on the air gap EMN model node connection. It is important to consider the rotor position change with time for the dynamic analysis. Therefore, it is necessary to build cross-shaped mesh model in the air gap. This makes dynamic analysis more effective and faster. C. EMN Model of Rotor and Hybrid Excitation Part When modeling the rotor part, the saturation of the magnetic bridge part is serious, and the cross-shaped mesh model needs to be established. Under the influence of DC excitation, the iron-pole generates magnetic field in the hybrid excitation section of the machine. The permanent magnet and DC magnetic source can be considered separately in the modeling to avoid the complex situation caused by two magnetic sources in the model. It should be noted that the alternating existence of permanent magnet and iron-pole also makes the modeling difficult. III.Results Verification In order to verify the effectiveness of the proposed EMN model, a hybrid excitation machine prototype is developed. The experimental test platform is shown in Fig. 2. The electromagnetic performances, such as the air-gap field distribution, phase flux linkage and back-electromotive force (EMF), d-axis and q-axis inductances, and electromagnetic torque, are predicted by the proposed EMN method. Finally, the results obtained from the EMN method are the compared with those obtained by the FEA and experimental tests. IV.CONCLUSIONS In this paper, an equivalent magnetic network analysis model is proposed to predict the electromagnetic performance of hybrid excitation machine. The new node connection form is established in the tooth tip and air gap for considering dynamic analysis. The proposed EMN model considers the influence of flux leakage and Nonlinear saturation of iron materials. The calculated results are compared with the FEA and experimental test results, which proves the effectiveness of the proposed EMN model. The detailed modeling process and calculation results will be given in the full paper.

**References:**

 [1] Z. Q. Zhu, I. A. A. Afinowi, and Y. Guan, “Hybrid-Excited Stator Slot Permanent Magnet Machines—Influence of Stator and Rotor Pole Combinations,” in IEEE Transactions on Magnetics, vol. 52, no. 2, pp. 1-10, Feb. 2016, Art no. 8100710. [2] K. Wang, J. Li, and C. Liu, “Novel hybrid-pole rotors for consequent-pole PM machines without unipolar leakage flux,” IEEE Trans. Ind. Electron., vol. 66, no. 9, pp. 6811–6823, Sep. 2019. [3] S. G. Min, and B. Sarlioglu, “Fast and systematic design optimization of surface-mounted PM machines using advanced analytical models and subharmonic elimination methods,”IEEE Trans. Magn., vol. 55, no. 1, Jan. 2019, Art. no. 8100216.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/b80f9b7b361e759844d752bbbfdd52dc.jpg)
Fig. 1. Topology of hybrid excitation machine. (a) Topology of stator. (b) Topology of rotor.

![](https://assets.underline.io/uploads/markdown_image/1/image/bb6ec34b1f0fdd2d048e55985615523c.jpg)
Fig.2.Test platform of a hybrid excitation prototype.

 Electromagnetic Performance Analysis of a Hybrid Excitation Machine Based on Equivalent Magnetic Network

Nowadays some industrial applications require contactless motion transmission. The use of magnetic couplers is the most appropriate solution. The synchronous version of this solution, based in PM excitation, can be adopted in radial or axial topologies. In these topologies the most common shapes of permanent magnets are rectangular, sector and circular. The electromagnetic torque of a magnetic coupler, required to transmit motion, is usually calculated using numerical methods or even 3D analysis which takes into account all the geometric and physical effects. However, in the design process, you need a fast and precise model that does not require a lot of computation time. Firstly, a purely 3D analytical model for axial coupler (with sector shape magnets), based on magnetic scalar potential formulation, is developed. So, the magnetic field distribution in different regions (PM, air gap and iron disks) leads to solve the Laplace’s equation. In low-permeability sub domains, the fields due to the two PM rotors of the magnetic coupler are calculated separately. The solutions form of the partial differential equations expressed in cylindrical coordinates are determined using the method of separation of variables. Secondly, we extend the above developed model to calculate the field in the magnetic coupler with magnets of rectangular shape. So, it is necessary to divide the geometry of the rectangular magnet into a number of elementary geometric elements where each element is approximated by a sector-shaped one keeping the same volume of magnet. The approach adopted can be summarized as follows:- Subdivide each rectangular Magnet in N radial Elementary Blocks; - Approximate the elementary block by a sector shape with the same volume; - Compute the magnetic Field of each sector shape block with the developed model;- The total magnetic field, torque and axial force are computed by using the superposition principle. Hence, the total torque developed by the magnetic coupler is obtained by summation of all torques produced by the interaction of each block of the first inductor with all blocks of the second inductor. The results obtained by the proposed 3D analytical model are compared to those obtained by 3D finite elements and to those determined experimentally. A close agreement is noticed (relative error on the static torque don’t exceed 2 %) between the measured and the predicted torques, which demonstrates the validity of the proposed model. The low computing time of the developed model provides an efficient and quick tool for assessing the influence of the geometrical and physical parameters of a magnetic coupler as a part of a design optimization process.

**References:**

[1] L. Belguerras and L. Hadjout, "Study of HTS Magnetic Coupler Using Analytical and Numerical Computations" IEEE Trans. Applied Superconductivity, Vol. 24, No. 6, December 2014 [2] Jian Wang and all, "A General Analytical Model of Permanent Magnet Eddy Current Couplings", IEEE Trans. Magn, Vol. 50, No. 1, January 2014. [3] Hyeon-Jae, Shin and all, "Design and Analysis of Axial Permanent Magnet Couplings Based on 3D FEM" IEEE Trans. Magn, Vol. 49, No. 7, JULY 2013. [4] T. Lubin ,S. Mezani and A. Rezzoug, "Simple analytical expressions for the force and torque of axial magnetic couplings". IEEE Trans. on Energy Conversion, volume 27, N°2, pp. 536-546, 11 pages, Juin 2012.

**Images:**
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 Improved 3D Electromagnetic Analytical Model of an Axial Flux Magnetic Coupling with Rectangular Shaped Permanent Magnets

I.Introduction Hybrid magnetic bearings (HMBs) have several merits, such as no friction and abrasions, no lubrication and no sealing, high speed, high precision and long life. In HMBs, the permanent magnet is adopted to generate bias flux for the low power consumption and small volume. However, in existing HMBs, several suspension poles are designed in the same stator core. The suspension force is generated by the interaction of the bias flux and control flux. The HMB of this structure realizes the radial two-degree-of-freedom (2-DOF) suspension in one stator core, especially when the rotor is deviated from unbalance position, the suspension force coupling in the X-Y direction is severe and the control is complicated. To compensate for this deficiency, a novel DC HMB with crossed poles is studied. The suspension poles in the X and Y direction are designed different stator cores. Therefore, it can solve the radial 2-DOF coupling of traditional four-pole HMBs and simplify the control. A simple mathematical models are presented, which can make the maximum magnetic levitation force in the X and Y directions. The nonlinearity of suspension forces and the coupling of electromagnetic among this four-pole magnetic bearing are analyzed by MagNet 3D. The finite element simulation is carried out to verify the theoretical analysis. II.Configuration and Suspension Principle The configuration and magnetic circuit of a novel DC HMB with crossed poles are presented in Fig. 1. As shown in Fig. 1(a), the main parts of this HMB are two slice stators, namely X-stator and Y-stator, axial magnetized permanent magnets and two slice rotor irons. Two symmetrical suspension poles in the X direction are evenly distributed along the circumference of X-stator, and the suspension poles in the Y direction are crossed in the Y-stator. Radial control windings are wound on each stator. The magnetic circuit of radial suspension control flux and static bias flux is represented in Fig. 2(b). The solid lines denote the static bias flux generated by the permanent magnets. The dotted lines describe the radial suspension control flux generated by radial control current. If the rotor irons are displaced from the balance position in the radial position by the radial disturbances, the fluxes between the magnetic flux and the bias flux are overlapped, which results in force to bring the rotor to the radial balance position. According to the equivalent magnetic circuit method, the radial suspension forces in the X and the Y directions are deduced as follows: Fj=Fpm+kij×ij(1) where the subscript “j” denotes x and y. Fpm is the magnetic attraction force produced by the bias flux at the equilibrium position. kij is the radial force-current stiffness. Fpm and kij can be given as: Fpm=-S×μ0×Fm×Fm÷g3 Kij=2S×μ0×Fm×N÷g2(2) where Fm is the magnetomotive force. μ0 is vacuum permeability. S is radial magnetic pole area. g is radial air gap length. N is number of radial control coils. III.Models, Design and Analysis Based on above-mentioned analysis, electromagnetic performance of the presented HMB is manufactured by the MagNet software. The prototype is simulated to analyze the electromagnetic performance by the three-dimensional finite element method, and the different simulation cases are given in Fig. 2. Fig. 2(a) shows the distribution of the bias flux. The radial air-gap magnetic flux density of each pole is 0.6T symmetrically. In Fig. 2(b), the radial control winding in the X direction is electrified with 3A current, and the control flux density distribution in the X direction is generated. It can be seen from Fig. 2(c) that there is an increase of air-gap flux in the positive X direction and a decrease in the negative X direction by the superposition of the control flux in the X direction and bias flux. Therefore, only the suspension force in the X direction is produced. To verify the correctness of the mathematical models, the control current is applied to generate radial suspension force only in the positive X direction. The relationships between the suspension force and control current are shown in Fig. 2(d). The currents change from 0 to 3 A. The maximum suspension force reaches 260 N. It can be seen that the finite element simulation results are in good agreement with the theoretical values, and there is a linear force-current relationships. IV. Conclusion This paper aims to construct a novel DC HMB with crossed poles to simplify the structure, obtain the maximum bearing capacity and solve the radial two-degree-of-freedom coupling. The simulation results have shown that the suspension forces produced by the crossed poles have better linearity around the balance position and no movement. There is no coupling of two directions for the independent magnetic circuit. The research results verify the correctness of magnetic circuit, suspension mechanism and mathematical model.

**References:**

 [1]D. Ning, H. Kai, and J. Long, "Review on High Speed Permanent Magnet Machines Including Design and Analysis Technologies," in Proceedings of the CSEE, vol. 34, no. 27, pp. 4640-4653, Sep, 2014. [2]J. Ju, H. Zhu and C. Zhao, "Radial Force-Current Characteristic Analysis of Three-Pole Radial-Axial HMB," 2016 IEEE Vehicle Power and Propulsion Conference (VPPC), Hangzhou, 2016, pp. 1-6.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/b086123141f2db12e85d7de06a7be696.jpg)
Fig. 1. The configuration and magnetic flux path of the proposed HMB.

![](https://assets.underline.io/uploads/markdown_image/1/image/2114063a67fc96a8d89fdd14022a9115.jpg)
Fig. 2. Flux distribution and the force-current relationships. (a) Bias flux. (b) Control flux. (c) Resultant flux. (d) Force-current relationship.

 Modeling and Electromagnetic Performance Analysis of the Novel DC Hybrid Magnetic Bearing with Crossed Poles

The traditional finite element model of permanent magnet synchronous generator（PMSG） needs nonlinear iteration for each grid, and time-consuming. Besides, the performance of various components of the voltage regulator controller (VRC) will be affected by the operating environment. The traditional component library cannot meet the accurate digital twin effect. Taking an emergency power supply system for an unmanned aerial vehicle as an example, this paper uses the device-level modeling method to build a digital twin simulation model (DTSM) of a PMSG and its AC-DC regulated power supply system, which simulation speed can guarantee real-time output results in synchronization with the experiment, and has an agile digital twin effect. Finally, the calculated results are compared with the experimental values to verify the accuracy of the proposed digital twin device-level model, which is of great significance to the combined design of PMSGs and VRCs. Index Terms：Permanent magnet synchronous generator (PMSG); Digital twin ;Voltage regulator controller. I. Introduction The performance of permanent magnet synchronous generator (PMSG) is closely related to its control method and load characteristics. When the AC-DC step-down chopper voltage stabilization control strategy is adopted, it can ensure the output of stable DC power while reducing the three-phase line current of the generator, reducing the motor winding loss greatly, and optimizing the matching design of the generator and the power converter. The power stabilization system is stable and efficient. Therefore, the research of PMSG and its AC-DC conversion and voltage stabilization system (VSS) simulation can promote the development of PMSG and VSS technology, improve system design accuracy effectively, and realize the perfect match to improve the stability and reliability of the VSS [1][2][3][4]. II. Composition of VSS Circuit After studying the load characteristics and actual working environment, the power supply system adopts a PMSG combined with a three-phase uncontrolled rectifier bridge and a BUCK converter. The actual power supply system consists of PMSG, three-phase uncontrolled bridge rectifier circuit, power regulator circuit, filter circuit, feedback circuit, control circuit, drive circuit and auxiliary power supply circuit. III. Digital Twin Device-Level Simulation Modeling In order to improve the simulation speed of the simulation model and achieve a fast digital twin effect, this paper uses the ECE reduced-order model extraction method to convert the generator finite element model into a data matrix model, and establishes the PWM main control chip SG1525 and the device-level modeling method. The digital twin simulation model (DTSM) of the AC-DC VSS of the PMSG is built as shown in Fig.1. The simulation speed can ensure the real-time output results in synchronization with the experiment results, and realize the fast and accurate digital twin effect. IV. Experimental Verification The PMSG and its VSS were tested on a dedicated test bench, which is shown in Fi.g. 2. The oscilloscope is used to display the output DC voltage waveform in real time, and the DTSM is run on the computer at the same time. The simulation data is highly consistent with the measured data. V. Conclusion This paper proposed a high-precision DTSM based on the actual prototype of an emergency power supply system for an unmanned aerial vehicle. Firstly, the ECE reduced-order mathematical model is extracted by the alternating-direct-axis periodic current scanning method, which accelerates the simulation speed greatly. Secondly, according to the actual circuit structure of the emergency voltage stabilization control power converter, a DTSM of the entire VSS circuit is built. Finally, the accuracy of the DTSM is verified by experiments.The reduced-order model extraction and device-level modeling techniques used in this paper are in line with the development trend of digital twin technology. The research results can provide a reference for the design of PMSG and the optimal matching design of VSS.

**References:**

[1] A. Rassõlkin, V. Rjabtšikov and T. Vaimann. "Digital twin of an electrical motor based on empirical performance model,"2020 XI International Conference on Electrical Power Drive Systems (ICEPDS), Saint Petersburg, Russia, 2020. [2] N. Khaled , B. Pattel and A. Siddiqui. “Digital twin development and cloud deployment for a DC motor control embedded system”, 2020. [3] R. Goraj. “Digital twin of the rotor-shaft of a lightweight electric motor during aerobatics loads”, Aircraft engineering and aerospace technology, 2020. [4] N. Khaled , B. Pattel and A. Siddiqui. “Digital twin development for an inverter circuit for motor drive systems”, Digital Twin Development and Deployment on the Cloud, PP. 163-201, 2020.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/82e16107aa4d5c0d5770a5259dc1f38a.jpg)
Fig.1. The DTSM of the AC-DC VSS of the PMSG.

![](https://assets.underline.io/uploads/markdown_image/1/image/bccf3f35c4be885acb6e6910cdef3814.jpg)
Fig.2. The test bench of PMSG and its VSS.

 Quickly and High-Precision Digital Twin Device-Level Simulation Modeling of Permanent Magnet Synchronous Generator and Voltage Stabilizing System

I. Introduction Recently, permanent magnet (PM) motors have been widely applied in various application fields, such as electric vehicles, aerospace, and household appliances, owing to the inherent advantages of high torque density, high energy efficiency, and compact structure. With the increasingly severe and diversified application scenarios of PM motors, higher and more complicated requirements are raised for optimization design. It is noted that the motor system is actually a multi-disciplinary, nonlinear, strong coupled system, hence the objectives are often in conflict with each other in the design process. In previous studies, multi-objective optimization design is considered as an effective solution for obtaining high-performance PM motors, and some optimization methods are often utilized, such as a multi-objective genetic algorithm (MOGA) [1]. However, the traditional optimization method is a deterministic design approach and has not investigated the uncertainties in engineering manufacturing, such as material diversity, manufacturing error, and assembly inaccuracy. These will result in design parameter fluctuation affecting the reliability and quality of the machine. To deal with this problem, the robust optimization design approach is proposed and gradually becomes a hotspot in machine design research orientation. Generally, the design for the six sigma (DFSS) technique is an effective optimization approach, which is often adopted to provide direct and reliable evaluation for the assessment of robustness. In [2], a soft magnetic composite (SMC) motor is optimized based on DFSS, which takes robustness into consideration in not only motor but also integrated control schemes. In [3], the robust design of a surface-mounted permanent magnet (SPM) motor is realized by the particle swarm optimization (PSO) instead of genetic algorithm (GA) due to long computational time of DFSS. Yet, it is noted that in the DFSS method, the mean and variance of the optimization objectives should be optimized at the same time, which greatly improves the dimension and complexity of the motor optimization model. Thus, for the machine robust design, ensuring the optimization convergency is a big challenge. It is necessary to propose an optimization method to improve the efficiency and accuracy of DFSS, effectively achieving machine robust design. To address this problem, this paper proposes an efficient machine robust design approach, where the dimension of the robust optimization model can be reduced according to the assessment of the relationship between objectives. To better clarify the proposed method, an interior PM machine with a multi-layer flux barrier is investigated as a design example. By implementing the proposed optimization method, it can be found that the performance and design reliability of the machine can be effectively improved. II. Proposed optimization method As is shown in Fig. 1, the design concept from the traditional deterministic optimization to robust design optimization is described for better presentation. Fig. 2(a) gives the configuration of the design example, an interior PM motor with a multi-layer flux barrier. It can be seen that the motor adopts the 36/8 slot/pole combination and NdFeB-PM is embedded in the middle of each arc-shaped magnetic barrier. Fig. 1(b) shows the parameter model of the motor and structural parameters for optimization is marked. Based on the parametric model, the process of robust optimization is given as the flowchart in Fig. 3. Firstly, in order to reduce the computation of FEA, Latin hypercube sampling (LHS) was utilized to approximate random sampling from multivariate parameter distributions. Then, parameters are divided into two layers by the result of sensitivity analysis. In Layer 1, the parameters with low sensitivity are optimized for objectives that have relatively strong relevance with them by parameter sweeping. In addition, robust factors of the performance fluctuation caused by sensitive parameters are considered in Layer 2. And then, a non-parametric measure of conflict method is adopted for objective reduction, then, the objectives with high conflict have remained for multi-objective optimization. Finally, the filtered objectives are optimized by NSGA-II (Non-Dominated Sorting Genetic Algorithm-II). III. Analysis and results Based on the robust optimization method proposed above, the analysis results are obtained. According to spearman's rank correlation coefficient, five objectives with relatively high assessed conflict value are remained in the process of multi-objective optimization, and the result is shown in Fig. 4 (a). In addition, the solution set corresponding to all objectives based on parameters determined above is shown in Fig. 4 (b). The torque and torque ripple of the proposed interior PM motor are calculated and compared in Fig. 5 (a), it can be seen that the average torque has been improved from 23.56 Nm to 25.95 Nm, while the torque ripple decreases from 10.14% to 9.50%. In order to evaluate the robust performance, 1000 points are sampled separately around initial and optimized parameters by Monte Carlo analysis method, as is shown in Fig. 5 (b), the variance μ of the distribution is decreased from 0.43 Nm to 0.27 Nm. Additionally, the improvement of core loss performance is given in Fig. 6. In conclusion, it can be seen that deterministic and robust performances of the proposed motor are improved simultaneously. More detailed design process description and experimental verification will be given in the full paper.

**References:**

[1] X. Zhu, W. Wu, L. Quan, Z. Xiang, and W. Gu, “Design and Multi-objective Stratified Optimization of a Less-rare-earth Hybrid Permanent Magnets Motor with High Torque Density and Low Cost,” IEEE Trans. Energy Convers., vol. 34, no. 3, pp. 1178-1189, Sept. 2019. [2] G. Lei, T. Wang, J. Zhu, Y. Guo and S. Wang, "System-Level Design Optimization Method for Electrical Drive Systems—Robust Approach," IEEE Trans. Ind. Electron., vol. 62, no. 8, pp. 4702-4713, Aug. 2015. [3] V. Rafiee and J. Faiz, "Robust Design of an Outer Rotor Permanent Magnet Motor Through Six-Sigma Methodology Using Response Surface Surrogate Model," IEEE Trans. Magn., vol. 55, no. 10, pp. 1-10, Oct. 2019, Art no. 8107110.

**Images:**
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 Robust Multi-Objective Optimization Design of Permanent Magnet Motors Based on Objective-Dimension-Reduced Method

I. Introduction In recent years, permanent magnet brushless machines (PMBL) have been widely applied in various applications thanks to the merits of high efficiency, high power density, and free maintenance. For this type of machine, PMs are crucial components, whose quality has a significant influence on motor performance. However, there are many uncertainty factors in the PM material characteristic and manufacturing, which often lead to machine performance deviate from the design results. For example, for PM magnetic characteristics, there exists inconsistency between different batches of PM on magnetization magnitude and direction, which leads to PMs remanence and magnetization angle slightly deviate from a given reference. On the other hand, since some PM material machining precision is relatively lower, there is often a discrepancy between the actual and nominal size. During the assembly, this discrepancy might cause a gap between PM and core, which will seriously influence machine performance. Thus, how to design PMBL machines in a robust way to resist PM material uncertainty factor has attracted much attention. To address this problem, researchers have paid plenty of effort, proposing multiple optimization design methods. In reference [1], a robust optimization method is proposed for transverse flux PM motor, in which the influence of inconsistency of magnetic characteristics of PM material is considered. Then, in order to reduce the influence of PM machining error during the motor manufacturing process, a robust optimization method is utilized for a novel double-sides air-core PM linear synchronous motor. [2]. Besides, many engineering probability uncertainty analysis methods are employed to improve PM machine design robustness.[3] Yet, it is noted that these researches generally treated uncertainties as single random variables, lacking the consideration about the specific influence of uncertainties on PM material. In this paper, a robust optimization method is proposed to improve PMBL machine design reliability, where PM material uncertainty factors are sufficiently considered. To better clarify the optimization method, a double PM brushless machine (DPMBL) is investigated as a design example, where NdFeB-PM and ferrite-PM are adopted together to improve machine torque density and reduce the Rare-earth PM usage. The detailed design process and optimized results are presented specifically. By implementing the proposed optimization method, it can be found that the performance and design reliability of the DPMBL machine can be improved. II. Design case and design method Fig.1(a) gives the configuration of the investigated DPMBL machine. It can be seen that the machine adopts a 12/10 slot/pole combination, where two types of permanent, NdFeB and ferrite, are parallelly arranged in the rotor. Fig 1(b) shows the parametric rotor model, where the corresponding design variables are marked out. It is noted that there are two kinds of uncertainty factors in the PM material. The first one is the ferrite width, which might lead to a gap between steel and PM. The second one is the material characteristic of ferrite and NdFeB, including remeance and magnetization direction. Fig.2 describes the proposed robust optimization design process, where PM material uncertainty factors are treated as disturbance parameters. In the implement, the NdFeB and ferrite dimensions are set as design parameters, and the ferrite width, the width of the gap between ferrite and steel, the NdFeB and ferrite remeance, and magnetization direction are regarded as disturbance parameters. Based on multi-objective optimization and uncertainty analysis, the proposed approach conducts the optimization of the design parameters for machine performance enhancement. III. Results After optimization, the design scheme of the DPMBL machine is obtained. Fig 3 gives a comparison of output torque between before and after optimization. It can be seen that in the case of PM without any uncertainty factor, the average torque obtains an improvement from 35.87Nm to 38.21Nm and the torque ripple decreases from 11.26% to 8.78%. As the same PM uncertainty factor is applied to these two machines, the torque quality of the initial machine suffers an obvious decline while the optimized machine can maintain stability. To better validate the effectiveness of the optimization method, Fig 3(b) and Fig.3(c) give the torque and torque ripple distribution of a batch of machines with random PM uncertainty factors. It can be observed that the optimized machine possesses more concentrated performance. In conclusion, the proposed optimization method can resist the PM uncertainty factor and benefit the manufacturing of good quality PMBL machines. More detailed design process descriptions and experimental validation will be presented in the full paper.

**References:**

[1] G. Lei, T. Wang and J. Zhu, "System-Level Design Optimization Method for Electrical Drive Systems—Robust Approach," IEEE Trans. Ind. Electron., vol. 62, no. 8, pp. 4702-4713, Aug. 2015. [2] F. Dong, J. Zhao and J. Zhao, "Robust Optimization of PMLSM Based on a New Filled Function Algorithm with a Sigma Level Stability Convergence Criterion," IEEE Transactions on Industrial Informatics, doi: 10.1109/TII.2020.3020070. [3] B. Ma, J. Zheng and J. Zhu "Robust Design Optimization of Electrical Machines Considering Hybrid Random and Interval Uncertainties," IEEE Trans. Energy Convers, vol. 35, no. 4, pp. 1815-1824, Dec. 2020.

**Images:**

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 Robust optimization design for PMBL machine considering permanent magnet material uncertainties

I. Introduction Permanent magnet (PM) machines have been extensively investigated and employed in various industrial applications due to their inherit merits of high torque and high efficiency. Nevertheless, because of the ever-increasing price and unstable supply, the rare-earth material cost is a major concern [1]. In order to address above-mentioned issue, the electrical machines with no or less rare-earth PM material are widely developed [1]. Because half of magnetic poles are replaced by iron core, the consequent-pole PM (CPM) machines [2]-[13] exhibit higher PM utilization ratio and comparable torque capability compared with conventional surfaced-mounted PM (SPM) structures. Therefore, various CPM machines in terms of surface-inset [2] [3], interior [4]-[7] and hybrid pole [8]-[10] configurations are reported and extensively investigated, which shows that the CPM design exhibits better field-weakening capability, higher PM utilization ratio, similar torque capability and efficiency compared with SPM machines. Recently, the field modulation effect (FME) concept has been extended to the CPM machines [11]-[13]. It can be found that the CPM machines exhibit more air-gap field harmonics responsible for torque generation compared with the SPM counterparts [11]-[13], which is mainly due to its doubly salient structure, unipolar PM magneto-motive force (MMF) distribution. The split ratio is a key design parameter of the PM machines, which significantly reflects the torque capability, iron loss, efficiency and thermal behavior [14]. However, the influences of split ratio on the FMEs in CPM machines remain unreported hitherto. Therefore, in order to provide an in-depth understanding of their torque production mechanism, the influence split ratio on FME of the CPM machines is investigated in this paper. II. Machine Topology and Preliminary Electromagnetic Performance The topology of the proposed CPM with 12-stator-slot/14-rotor-pole is illustrated in Fig. 1(a). The fractional slot concentrated windings are adopted to reduce the end winding length and improve the efficiency. The split ratio γ can be defined as γ=rsi/rso (1) where rsi and rso are the stator inner and outer diameters, respectively. The average torque variations of the CPM machine with different split ratios are illustrated in Fig. 1(b). It can be seen that the average torque is rises first, and then drops steadily when split ratio exceeds 0.55, which is mainly due to the increased first and then decreased electrical load. In order to quantify the torque contributions of the various working harmonics, the Maxwell stress tensor method is employed. The electromagnetic torque can be expressed as Τe=∑(πlstkrg2/μ0)BrkBtkcos[θrk(t)-θtk(t)]=∑Tk(t) (2) where rg is the air-gap radius. lstk is the active stack length. Tk is the electromagnetic torque generated by the kth harmonics of radial and tangential air-gap flux densities, respectively. Thus, the contributions of the kth field harmonics to the total electromagnetic torque can be separated when the kth radial and tangential air-gap field harmonics are obtained. The average torque proportion can be defined as λ=(Tk/Te)×100% (3) The torque contributions of FME can be expressed as λFME=(1-TZr/Te)×100% (4) where TZr is the average torque generated by Zr order air-gap field harmonic. The corresponding torque proportions of the various air-gap flux density harmonics can be obtained (2)-(4), as shown in Figs. 2(c) and (d). Due to the slightly increased air-gap flux density amplitudes, the torque proportions of 5th, 19th and 21th are increased with split ratio, as shown in Fig. 2(c). Similarly, the torque proportion of 7th harmonic is decreased with split ratio, which is mainly due to its decreased 7th air-gap flux density amplitude. The corresponding torque proportion of FME can be obtained by (4), as shown in Fig. 1(e). Because of the decreased amplitude and torque proportion of the Zr order air-gap flux density, the torque proportion of FME is increased with split ratio, which indicates that the more pronounced FME can be obtained in relatively higher split ratio. III. Experimental Validation In order verify the FE simulation, a 12-slot/14-pole CPM machine prototype is manufactured, as shown in Figs 2. The test results are illustrated in Figs. 2(c) and (d), respectively. It can be seen that the measured back-EMF values are slightly lower than FE predictions, as shown in Fig. 2(d), which is mainly due to the fact that the end-winding effect and mechanical tolerance are not considered in FE analysis. The detailed analysis and experimental measurements will be presented in the full paper.

**References:**

[1] H. Tahanian, M. Aliahmadi and J. Faiz, IEEE Transactions on Magnetics, Vol. 56, Art. no. 900120, (2020) [2] S. U. Chung, S. H. Moon and D. J. Kim, IEEE Transactions on industrial Electronics, Vol. 63, p.302-309, (2016) [3] J, Li, K. Wang and F. Li, IEEE Transactions on industrial Electronics, Vol. 66, p.1736-1747, (2019) [4] M. Onsal, B. Cumhur and Y. Demir, IEEE Transactions on Magnetics, Vol. 54, Art. no. 8206005, (2018) [5] W. Liu and T. A. Lipo, IEEE Transactions on Industry Application, Vol. 54, p.5918-5929, (2018) [6] X. Ren, D. Li and R. Qu, IEEE Transactions on Energy Conversion, Vol. 33, p.2112-2121, (2018) [7] J. Li, K. Wang and H. Zhang, IEEE Transactions on industrial Electronics, Vol. 67, p.3374-3385, (2020) [8] K. Wang, J. Li and S. S. Zhu, IEEE Transactions on industrial Electronics, Vol. 66, p.6811-6823, (2019) [9] G. Dajaku, IEEE Transactions on Magnetics, Vol. 56, Art. no. 8200409, (2020) [10] G. Xu, W. Zhao and G. Liu, IEEE Transactions on Transportation Electrification, accepted, (2020) [11] Y. Li, H. Yang and H. Lin, in Proc. XIII International Conference Electrical Macine (ICEM), pp.65-71, (2018) [12] D. Jang and J. Chang, IEEE Transactions on Energy Conversion, Vol. 34, p.2019-2028, (2019) [13] Y. Li, H. Yang and H. Lin, IEEE Transactions on Energy Conversion, accepted, (2020) [14] Y. Shen and Z. Q. Zhu, IEEE Transactions on Magnetics, Vol. 47, p.4187-4190, (2011)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9549b4f93fc46a47e02517de642ae14b.png)
Fig. 1. (a) Configuration of 12-stator-slot/14-pole CPM. Torque characteristics of the CPM machine with different split ratio. (b) Average torque. Torque proportions of the various working harmonics (c) field modulated harmonics, (d) 7th, (d) FME.

![](https://assets.underline.io/uploads/markdown_image/1/image/a93893e2c93e7f9a334ca1037de1cbd4.jpg)
Fig. 2. 12-slot/14-pole CPM machine prototype and measured results. (a) Stator. (b) Rotor. (c) Measured open-circuit back-EMFs. (d) Comparison of the measured and FE predicted open-circuit fundamental amplitudes of back-EMFs.

Topology Optimization of Consequent-Pole PMSM Using ON/OFF Method

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