United States

I Introduction&lt;br/&gt;Reactors have been widely used to ensure the stability of the power system by limiting short-circuit current, suppressing harmonic and compensating reactive power. There are some methods realize the vibration reduction of the reactor from the angle of the material filled at the air gap[1-2]. However, in order to improve this serious noise problem of the reactor, it is necessary to study the vibration reduction from the source of reactor vibration. Therefore, an optimization method based on topology optimization and GA(genetic algorithm) is proposed in this paper to reduce the segmented core reactor&#39;s vibration.&lt;br/&gt;II Topology Optimization of the Reactor&lt;br/&gt;This paper adopts the method of active vibration reduction and proposes a global topology optimization algorithm to get a novel design of the reactor. First, the electromagnetic-mechanical coupling model of the reactor is established to analyze the vibration and deformation of the reactor, which are shown in Fig1.(a-c). Second, the optimization problem is defined to obtain optimal configuration that minimum the vibration displacement of the reactor under a minimum bound of the total volume. Then, by introducing a SIMP(Solid Isotropic Material with Penalization) function into the definition of the permeability of the reactor&#39;s air gap part materials, the distribution of ferromagnetic material is represented. Meanwhile, the inductance value of the reactor can be kept stable in the optimization process through the joint GA. Finally, a novel reactor structure is obtained as shown in Fig2.(a) and the finite element numerical method is used to calculate the optimized parameter values before and after the optimization to verify the excellence of the design.&lt;br/&gt;III Conclusion&lt;br/&gt;The calculation result at sampling point A is shown in the Fig.2(b). It can be seen that the vibration displacement after optimization is reduced by about 15%, which verifies the effectiveness of the proposed method in this paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/11bed137395190e0772f919055bce856.jpg)

Fig.1. (a) Schematic diagram of the reactor,(b) Magnetic flux density modulus distribution nephogram,(c) Displacement distribution nephogram.

![](https://assets.underline.io/uploads/markdown_image/1/image/834fe5bd196aaa212c5cc13e1ff70c28.jpg)

Fig.2. (a) Value distribution of control variable p,(b) Comparison of vibration displacement at sampling point a.

2022 Joint MMM-INTERMAG

Reactor Vibration Reduction Using Global Topology Optimization Algorithms

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.

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 Reactors have been widely used to ensure the stability of the power system by limiting short-circuit current, suppressing harmonic and compensating reactive power. There are some methods realize the vibration reduction of the reactor from the angle of the material filled at the air gap[1-2]. However, in order to improve this serious noise problem of the reactor, it is necessary to study the vibration reduction from the source of reactor vibration. Therefore, an optimization method based on topology optimization and GA(genetic algorithm) is proposed in this paper to reduce the segmented core reactor's vibration. II Topology Optimization of the Reactor This paper adopts the method of active vibration reduction and proposes a global topology optimization algorithm to get a novel design of the reactor. First, the electromagnetic-mechanical coupling model of the reactor is established to analyze the vibration and deformation of the reactor, which are shown in Fig1.(a-c). Second, the optimization problem is defined to obtain optimal configuration that minimum the vibration displacement of the reactor under a minimum bound of the total volume. Then, by introducing a SIMP(Solid Isotropic Material with Penalization) function into the definition of the permeability of the reactor's air gap part materials, the distribution of ferromagnetic material is represented. Meanwhile, the inductance value of the reactor can be kept stable in the optimization process through the joint GA. Finally, a novel reactor structure is obtained as shown in Fig2.(a) and the finite element numerical method is used to calculate the optimized parameter values before and after the optimization to verify the excellence of the design. III Conclusion The calculation result at sampling point A is shown in the Fig.2(b). It can be seen that the vibration displacement after optimization is reduced by about 15%, which verifies the effectiveness of the proposed method in this paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/11bed137395190e0772f919055bce856.jpg)

Fig.1. (a) Schematic diagram of the reactor,(b) Magnetic flux density modulus distribution nephogram,(c) Displacement distribution nephogram.

![](https://assets.underline.io/uploads/markdown_image/1/image/834fe5bd196aaa212c5cc13e1ff70c28.jpg)

Fig.2. (a) Value distribution of control variable p,(b) Comparison of vibration displacement at sampling point a.

technical paper

1 INTRODUCTION Permanent magnet synchronous motors (PMSMs) are increasingly popular in various industrial applications due to their high power density and high efficiency. In high performance servo applications, it is essential to reduce the cogging torque to improve the positional accuracy. One of the most widely used methods is step skewing with uniform magnets. However, it suffers from unbalanced axial electromagnetic force[1]. Herringbone skew method can eliminate the unbalanced axial electromagnetic force because of the symmetrical structure, while it leads to higher cogging torque and torque ripple than the conventional step skew method[2]. In this paper, a nonuniform step skew method is proposed. The PMs of the nonuniform step skew motor have various thicknesses and various skew angles, and show an axisymmetric distribution meanwhile. The proposed method not only reduces the cogging torque to a greater extent than the uniform skew method, but also completely eliminates the unbalanced axial electromagnetic forces. 2 THE METHOD AND RESULTS Fig. 1 shows the rotor structure of the nonuniform step skew method. The 3D structure of motor with nonuniform step skew is shown in Fig. 1(a), and Fig. 1(b) shows the nonuniform distribution of magnet poles. Fig. 2 shows the FEM analysis results of cogging torque and output torque. Compared to the conventional, the uniform step skew and the herringbone skew motors, cogging torque of the nonuniform step skew motor is reduced by 86%, 44%, 80% and the torque ripple is reduced by 45%,15%,18%, without additional reduction in output torque. In addition, due to the symmetrical structure of PMs, the unbalanced axial electromagnetic force can be eliminated. 3 CONCLUSION This paper has proposed a nonuniform step skew method for cogging torque reduction in PMSMs. Based on FEM analysis results, the motor with nonuniform step skew has less cogging torque and torque ripple than the conventional, the skew and the herringbone skew motors, without more sacrifice of torque density. Moreover, the unbalanced axial electromagnetic force can be completely eliminated with the symmetrical structure.

![](https://assets.underline.io/uploads/markdown_image/1/image/39f6670df5abf00ab325198d12b169af.jpg)

Fig. 1. Rotor structure of the nonuniform step skew motor

![](https://assets.underline.io/uploads/markdown_image/1/image/eae04882046acc25709fb5dcdec79eac.jpg)

Fig. 2. Cogging torque and output torque

A Nonuniform Step Skew Method for Cogging Torque Reduction in Permanent Magnet Synchronous Motor

A finite element method (FEM) is widely used in electromagnetic design and analysis, though a long calculation time and a large computer capacity are generally required. To overcome this issue, the authors proposed a reluctance network analysis (RNA) [1], which is suitable for fast calculation of various electric machines’ characteristics, including transformers and motors. However, previous researches on the RNA were limited to closed magnetic circuits. To expand the application scope of the RNA for a wireless power transfer (WPT) and a particle accelerator, which have open magnetic circuits, this paper presents a novel RNA model for open magnetic circuits with the concept of the Kelvin transformation [2], [3], which has been utilized in the open boundary problems in the FEM. Fig. 1(a) – (c) show the proposed RNA model for open magnetic circuits in case of 2-D and 3-D axisymmetric analysis. As shown in Fig. 1(a), an analytical region including the point at infinity is separated into interior and exterior regions. Then, both regions are divided into multiple elements represented by unit magnetic circuits as shown in Fig. 1(b) and (c). Each reluctance can be determined by dimensions of the element and permeability of the material. In the quadrant regions, dimensions of an element lr, lq, lz, lj, lr’, lq’, lz’, and lj’ can be expressed by the following equations: lr = |r1-r2| (1) lq = r1|q1-q3| = r3|q1-q3| (2) lz = const. (3) lj = p(r1+ r2)sin(|q1+q3|/2) (4) lr’ = |a2/r1’- a2/r2’| (5) lq’ = a2/r1’|q1’-q3’| = a2/r3’|q1’-q3’| (6) lz’ = const. (7) lj’ = p(r1’ + r2’)sin(|q1’+q3’|/2) (8) On the other hand, in the rectangular regions, dimensions of an element are lx, ly, lz, lj, lx’, ly’, lz’, and lj’. Among them, lx and lx’ can be expressed by the following equations: lx = |x1-x2| = |r1-r2| (9) lx’ = |x1’-x2’| = |a2/r1’- a2/r2’| (10) Fig. 2(a) - (c) show specifications of an analytical object and simulation results, respectively. Here, flowing flux density on a red solid line is compared. It is understood that calculated values of the RNA and FEM are in good agreement. Also, the RNA results hardly change even if the value of a changes within the range of the necessary and sufficient spatial resolution.

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A Novel Reluctance Network Analysis Applicable for Open Magnetic Circuits

In this work, we present an analytical solution of the magnetic field for a uniformly magnetized cylinder tile (= cylinder volume element, see Fig. 1). The derivation is based on the method of magnetic surface charge densities for arbitrary Cartesian magnetization directions and cylinder tile geometries. In contrast to previous works [1,2], all resulting expressions have been reduced to analytical forms containing only elementary operations, trigonometric functions and elliptic integrals. All special cases (singularities and indefinite expressions) are discussed, allowing an exact evaluation of the magnetic field (up to numerical errors) at any point in space. The results are validated by comparison with finite elements solutions and numerical integration. It is further demonstrated how vectorized implementation of all special functions enables effective computational times in few tens of microseconds on simple mobile CPUs (e.g. Intel i5-8365U). An efficient implementation has been integrated in version 4 of the open-source Python-package Magpylib [3]. Possible applications of the derivations are the analytical study of the Halbach cylinder [4] (see Fig. 2), rotary encoder geometries [5,6], electrical machines [7], or any other magnetic geometry consisting at least partially of cylinder tiles. Furthermore, since a full cylinder can be completely subdivided into cylinder tiles, it allows the study of demagnetization effects while maintaining the rotational symmetry using the magnetostatic moment method [8].

![](https://assets.underline.io/uploads/markdown_image/1/image/bc3e0d7162c81197735b0ceeae63edec.jpg)

Geometry of cylinder tile with parametrization in cylindrical coordinates. Six surfaces contribute to the magnetic field.

![](https://assets.underline.io/uploads/markdown_image/1/image/3b7aea7898e7ab1db5d67b69b6e94d45.jpg)

a) Cross-section through a Halbach cylinder array consisting of twelve cylinder tiles with different magnetization directions. The magnetic field is evaluated along the dashed blue line. b) Field for different numbers of subdivided cylinder tiles as a function of the radial coordinate r.

Full Analytical Solution for the Magnetic Field of Uniformly Magnetized Cylinder Tiles

The current loop is a fundamental building block of cylindrically symmetric magnetic calculations. However the off axis field magnetic involves the subtraction of elliptical integrals and second kind which is computationally expensive, hard to manipulate in equations and difficult to visualize. By doing a different binomial series expansion on the original loop integral a series solution is created, which can be simplified to a set of approximations with useful characteristics: exactly correct at along the axis, and at distance, while at the current loop itself the relative error is limited, computational simple, easy to visualize the behavior and symmetry. Figure 1 shows for the radial field the first and second order fits where the parameters are optimized to give minimize the relative peak error. Note the symmetry that occurs by expressing as function of W, allowing maximum relative errors of 0.025 for 1st order and 2.9E-4 for second order (Figure 2 shows 2nd order relative error in all space). For axial field (Figure 1) the first order, due to the subtraction the accuracy is only modest but the 2nd order has a 9E-4 maximum relative error along, with zero error at the loop on the z=0 plane. The axial field relative error stays low within the loop but outside does, and stays low for any h until it falls to 1% of the z=0 plane value, then lose accuracy as z is near where the field direction reverses sign due to slight differences in the zero crossing predicated coordinate, increasing again in accuracy as z further from the reveal. Higher order approximations add more W terms and see increasing accuracy – for radial 3rd order 1.8E-5, 4th 2,3E-6, 5th 4.9E-7, while axil goes as 3rd 4.6E-5, 4th 6E-6. Note that the simplicity of these functions suggests new ways combining loops to optimize such things as field uniformity. The final paper will show how these approximations are created, plus more details on the error distributions of these and higher orders.

![](https://assets.underline.io/uploads/markdown_image/1/image/347293d7d5dee787ac314cbdc7b3a7ca.jpg)

Figure 1: Current loop off axis field approximations of order 1 and 2

![](https://assets.underline.io/uploads/markdown_image/1/image/6324541f77db2ddbf4cd30f9337b8df7.jpg)

Figure 2: Radial field 2nd order approximation relative error

Current Loop Off Axis field Approximations with Excellent Accuracy and Low Computational Cost

INTRODUCTION Nowadays, given the rapid growth of industry electronification, there is a huge demand for microchips [1,2]. Therefore, the lithography industry has a necessity to improve the wafer positioning systems such that the throughput of the lithography machine is increased. To reduce the mass and to remove the cable slabs of the traditional double stroke wafer positioning system [3,4], a single stage system has been proposed in [5], consisting of two vertically attached stationary coils and a moving magnet array. A 2D schematic of this Dual Layer Planar Motor (DLPM) is shown in Fig. 1. The employed Halbach array does not utilize the entire available space. Furthermore, the magnet-to-magnet force within the array results in an undesired mover bending. This paper presents a framework to evaluate different novel magnet array architectures for the DLPM. Different configurations are analysed such that the secondary magnetization patterns of the Halbach array fully utilize the available space. Furthermore, an optimization is performed to enhance the thrust and levitation forces, while curtailing the magnet array bending. METHOD The presented tool is semi-analytical: the magnet flux densities are obtained analytically based on the Magnetic Charge Modeling method and the coil bed flux densities through the evaluation of the Biot-Savart’s law for a current carrying volume. The electromagnetic forces are derived numerically utilizing the Maxwell Stress Tensor. CONCLUSION The proposed analytical modeling method presents a framework for fast and accurate optimization of the DLPM. The model has been verified with Finite Element Modeling (FEM). Moreover, the approach improves the computation time compared to FEM in a maximum of a factor of 20. The optimization achieves a higher levitation force to weight ratio, even with the added magnet mass, according to Fig. 2. This releases stress from the power supply design as a lower heat dissipation requirement suffices. In the full paper the modeling approach will be elaborated on and the optimization for the thrust force is going to be addressed.

![](https://assets.underline.io/uploads/markdown_image/1/image/83136282f635c94a2eeca509925f7d2f.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/2ea16e4c5bf4023d50093487fe9a46f2.jpg)

Efficient Modeling Framework for the Synthesis of a Novel Magnet Array for Planar Motors

Efficient Modeling Framework for the Synthesis of a Novel Magnet Array for Planar Motors

Magnetic gears (MGs) have emerged in the last 20 years with promising torque densities [1]-[5]. Some of their potential advantages are reduced maintenance and improved reliability. Axial field MGs offer multiple possible advantages over radial field configurations [6]. However, modeling axial field structures requires a 3D geometrical model to properly represent its performance. The purpose of this work is to present a 3D permeance network modeling technique for axial field flux focusing magnetic gears. Permeance network (PN) is a very well-known modeling technique. It’s based on the concept of flux tubes where the geometry is divided into blocks with a central node, and then the network equations are generated using Gauss’s law, which states that the sum of all fluxes into a node must be zero [7][8]. The unknowns to be solved then are magnetic scalar potential values at every node. The nodes are connected through a lumped component which can be permeance or a source. Figure 1 shows the MG structure to be modeled with the elementary permeance block. It consists of three parts, a high-speed rotor (HSR), a low-speed rotor (LSR), and a stationary part. The versatility of the model will be demonstrated by changing multiple variables and calculating different quantities. Figure 2 compares the magnetic induction components calculated using the 3D PN model and a mean radius 2D finite element method model (FEM). The calculated components are at the LSR air gap side. The permanent magnets and the ferromagnetic materials were considered in their linear case with µrf = 10000, µrpm = 1, and Br = 1.25T. The differences between the 2D model and the proposed 3D model are visible, where we can see a difference in the amplitudes and shapes which demonstrates the importance of the radial component in the axial field structures. The full work will demonstrate more the potential of the proposed model by performing linear and non-linear calculations and showing how this model can be very suitable for early design stages.

![](https://assets.underline.io/uploads/markdown_image/1/image/7fc302ac3e86324c301eb2a6485110d5.jpg)

Fig.1 Flux focusing axial field magnetic gear with p1=6, n2=24, and p3=18 pole-pairs.

![](https://assets.underline.io/uploads/markdown_image/1/image/742a5eb474df0b9d992e46d479febd24.jpg)

Fig.2 Axial and radial magnetic induction components calculated using the 3D PN and a 2D FEM model at the LSR airgap side

3D Permeance Network Modeling for an Axial Field Flux Focusing Magnetic Gear

Minimizing the mass in electric machines while maintaining superior performance has become a new requirement for the advancement of drive trains used in wind energy and electric mobility. Topology Optimization (TO) for light-weighting electric machines using traditional approaches typically explore a restricted design space allowed by standard parameterizable geometry and manufacturing, while advanced methods like cell-based density approaches[1] suffer from a lack of robust manufacturability constraints during the optimization process. To overcome these drawbacks and identify best designs in terms of material reduction, we explore a grid-independent, boundary TO [2] where the outer boundary shape of the magnet is parameterized using Bezier curves. This approach will allow us to explore a large design space of free-form magnet shapes using fewer variables compared to cell-based TO methods while ensuring smooth boundaries for additive manufacturing. We conduct a design of experiments (DOE) to study the effect of different magnet shapes on machine performance by varying the control points on the Bezier curves. Each resulting topology in the DOE is imported into a parametric 2-dimensional CAD environment to build solid models with magnets distributed in the region bounded by the curves and analyzed using magnetostatic finite element analysis (FEA). A machine-learning based surrogate model is constructed using the data from the DOE to quantify the relationship between the control points, air-gap torque and mass. The control points are then optimized to maximize the torque density. The final paper will demonstrate this approach for minimizing the electrical steel mass in the IEA-15MW radial flux direct-drive wind turbine generator [4]. Free-from boundary TO resulted in a smooth shape with up to 20 ton reduction in electrical steel mass.

A boundary topology optimization approach for lightweighting electric machines inspired by additive manufacturing

Isotropic bonded NdFeB magnets are used extensively in various automotive, office automation and home appliance motors. Due to their superior magnetic properties over ferrite magnet, better motor efficiency and increased power density can be achieved. The magnetizing fixture winding configuration is critical in achieving the desired magnet flux profile1 and motor performance2. In this paper the effect of various fixture winding configurations on resultant magnet flux profile and motor performance is presented for a 4-pole automotive accessory motor. Magnetizing fixtures are designed using different sizes of round conductors to achieve 3T magnetizing field on the magnet. It is observed that when the number of turns is kept the same, 29% reduction in conductor diameter made magnetization profile more radial. When this magnetized magnet is used in a motor, peak to peak cogging torque increased by 20%. Using two-dimensional (2D) finite element analysis (FEA), magnetization fixtures are designed for six different winding configurations. Figure 1 shows four of the winding configurations evaluated using 2D FEA, to arrive at magnetized magnet profile and corresponding motor performance. Table I summarizes the flux integral of magnetized magnet profile and corresponding motor performance. From this table it is observed that for a 4-pole magnet, the ‘1-column’ conductor arrangement where-in the conductors are arranged along the radial direction in slot resulted in radial flux profile and highest flux integral per pole, resulting in higher back-emf and cogging torque. On the other side the flux integral per pole is minimum and flux profile becomes more sinusoidal for ‘1 row’ arrangement in which the conductors are arranged along a concentric arc from fixture core surface. This arrangement results in wider transition zone, and in lowest back-emf and cogging torque. All the designed magnetization fixtures were built, and performance of motors with magnetized magnets is evaluated. The experimental results are in good agreement with the simulation results.

![](https://assets.underline.io/uploads/markdown_image/1/image/c0f124c58c77a78597be17e9042810b0.jpg)

Fig.1. Various conductor arrangements (a) 1 column (b) 2 column (c) 1 row (d) Delta


![](https://assets.underline.io/uploads/markdown_image/1/image/69382f57730e441285b7d4c469a7460d.jpg)

Table I Effect of conductor configuration on magnet flux and motor performance

Influence of magnetization fixture winding parameters on airgap flux profile and performance of isotropic bonded NdFeB motor

The flux leakages at axial end and radial inner side on the flux-concentrating rotor are more likely to generate, as shown in Fig.1, and therefore reduces the motor performance and degrades the utilization efficiency of PMs [1]-[3]. According to Table I, the modeling of the machines are carried out. Fig. 2 is the 2-D model. Fig. 3 shows the leakage flux by 3-D spoke type model. In Fig. 4, the air gap flux density, back electromotive forces (EMFs) and torque from the 2-D and 3-D analyses indicate a difference of approximately 10.33%, 5.88 % and 9.45%, respectively. Fig. 5 shows the rotor prototype and the test platform. The errors are 9.78% and 5.21% compared with the experiment respectively which indicates that the 3-D model is closer to the actual result than the 2-D model because the 3-D model takes leakage flux into account. The motor with novel rotors are shown in Fig. 6 and Fig. 7 which consumes the same weight of PMs with spoke-type model. The principle of reduction axial end leakage flux in Model 1 is shown in Fig. 6 (b). The details on the rotor of Model 2 is shown in Fig. 7 (b). The model in Fig. 6 is called Model 1. The model in Fig.7 is Model 2. Fig. 8 and Fig. 9 compare the rotor-end and rotor-interior leakage flux of two new structures and spoke-type motor under no-load. In the spoke type model the leakage flux is large, up to 0.9 T and 1.0 T. But it is only 0.38 T and 0.29 T in Model 1 and Model 2. The electromagnetic characteristics are compared in Fig. 10. Model 2 can further improve the air-gap flux density, back-EMF and torque, up to 1.499 T, 512.4 V and 68.7 Nm. In this paper, a novel rotor with hybrid radial and axial flux concentration considering leakage flux is designed. The 3-D models of concentrated winding PM motors with different rotors are built to analyze the leakage flux and electromagnetic performance. A 12-slot 10-pole motor with spoke type PM rotor is manufactured and experimented to verfy the significance of 3-D FEA takeing into account flux leakages. The 3-D FEA results of novel flux-concentrating PM rotors shows the axial leakage flux and rotor-interior leakage flux are decreased by 58.1% and 61.7%. The electromagnetic torque can be improved by 5.48% with phase current of 30 A.

![](https://assets.underline.io/uploads/markdown_image/1/image/79a32f6edee8bf86e65a0ad4807a2502.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/630d41826d47b1eabe8f216c0ae4e275.jpg)

A Novel Spoke-type PM Rotor with Hybrid Radial and Axial Flux Concentration for Reduction of Interpolar Leakage Flux

It is necessary to accurately estimate iron loss including magnetic hysteresis behavior to develop high-efficiency electric machines. In a previous paper, a novel magnetic circuit model incorporating the play model [1] and Cauer circuit theory [2] was proposed. The play model is one of phenomenological models of the dc hysteresis characteristic. The Cauer circuit theory can simply express the skin effect by formulating the frequency characteristic of the complex permeability. Its calculation accuracy was proved by comparing the measured and calculated results of hysteresis loops and iron loss under PWM excitation by using simple-shaped ring cores. This paper presents that the above model is extended a reluctance network analysis (RNA) model to analyze devices with more complicated shapes such as motors. The proposed method is experimentally validated by using an IPM motor, which is generally driven by a PWM converter, as an object of discussion. Fig. 1(a) and (b) show the proposed RNA model of the IPM motor. In the stator core, each reluctance expresses the dc hysteresis characteristic and classical and anomalous eddy current losses. The dc hysteresis characteristic is represented by the play model, the classical eddy current loss is represented by the Cauer circuit theory, and the anomalous eddy current loss is represented by a simple controlled source element, respectively. On the other hand, each reluctance in an air gap and a rotor is determined by dimensions of each element and permeability of the materials, and each MMF generated from winding currents and permanent magnets is concentrated as a form of controlled source. Fig. 2(a) - (c) show the measured and calculated characteristics of torque, iron loss, and efficiency, respectively. Here, the theoretically infinite number of stages of the ladder circuit part which denotes classical eddy current loss is truncated at 3rd stage for practical use. From these figures, it is understood that both results are in good agreement. Besides, as shown in Fig. 2(d), the proposed method can draw the local hysteresis loop in a certain divided element, which is generally difficult to measure and calculate, though its calculation accuracy should be experimentally verified.

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![](https://assets.underline.io/uploads/markdown_image/1/image/51d551699060515c03658bc9628a80f9.jpg)

Iron Loss Calculation Method of Interior Permanent Magnet Motor Taking Carrier Harmonics into Account Based on Reluctance Network Analysis Representing Dynamic Hysteresis Characteristics

Reactor Vibration Reduction Using Global Topology Optimization Algorithms

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A Nonuniform Step Skew Method for Cogging Torque Reduction in Permanent Magnet Synchronous Motor

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Reactor Vibration Reduction Using Global Topology Optimization Algorithms

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A Nonuniform Step Skew Method for Cogging Torque Reduction in Permanent Magnet Synchronous Motor

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