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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.&lt;br/&gt;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.

2022 Joint MMM-INTERMAG

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

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.

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.

technical paper

I. INTRODUCTION Variable flux memory (VFM) machines were proposed to address the constant air-gap flux and thus sacrificed efficiency in high-speed operations of conventional permanent magnet (PM) machines [1]. The magnetization state (MS) of low coercive force (LCF) PMs can be intentional regulated by d-axis current pulses [2]. The properties and dimensions of LCF PMs are essential in VFM machines, which straightforwardly affect the intentional MS manipulation, torque density and unintentional demagnetization (UD) withstanding [3]. Two VFM machines with different LCF PMs, i.e. SmCo (Hc=335kA/m) and AlNiCo (Hc=112kA/m), are investigated and the guideline of PM design in VFM machines for EV tractions is proposed. II. MACHINE TOPOLOGY and PERFORMANCE Fig. 1 shows the VFM machine cross section, whose outline dimension is inherited from Prius2010 traction machine [4]. Two machine cases employing SmCo and AlNiCo as LCF PMs respectively, together with NdFeB PM to realize hybrid PM configuration for torque improvement, are presented. The NdFeB PM volumes are identical in the two cases while the LCF PM thickness is 4mm for the SmCo case but 12mm for the AlNiCo case, which guarantees an equal PM MMF. As magnetization characteristics shown in Fig. 2, the MS range is 55.11% in the AlNiCo case with the limitation of rated current (236A), which is much higher than 23.15% in the SmCo machine. However, with 2 p.u. current limitation, the MS range of the SmCo case increases to 54.07%, which is close to the AlNiCo case. In addition, the torque of the SmCo case is higher than the AlNiCo case by &gt;10%. III. CONCLUSION Although the identical initial PM MMFs are presented, the AlNiCo machine always requires lower de/re-magnetizing currents than the SmCo case. Nevertheless, the MS ranges are similar in both cases with 2 p.u. current limitation, while the SmCo machine exhibits a higher torque density. More details of the electromagnetic characteristics, LCF PM optimization, operation condition effects, and experimental validation will be demonstrated in the coming full paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/9f796652d2ae704bf4bb7ae10908e29a.jpg)

Fig. 1 Cross section of VFM machine.

![](https://assets.underline.io/uploads/markdown_image/1/image/1dbcfb3ab8d3cad6ba3f64b3b37e48ff.jpg)

Fig. 2 Fundamental phase back-EMFs at 1500 r/min after different d-axis excitations.

Investigation on Low Coercive Force Magnets in Variable Flux Memory Machines for Traction Applications

Open-phase failure is typical in multiphase machines [1]. During the open-phase faults and to support the desired developed torque in a six-phase interior permanent magnet motor (IPMSM), various fault tolerant control (FTC) methods have been proposed [2]-[3]. Although the degraded torque of a faulty IPMSM can be compensated via FTC, the noise and vibration (NV) behavior of the machine may still be affected. NV characteristics of healthy and open-phase IPMSMs have been studied before [4]-[6]. However, these characteristics have not been studied in FTC mode for an open-phase IPMSM. This paper, thus, predicts the NV performance of a laboratory six-phase IPMSM in a single open-phase fault under a negative sequence current compensated FTC mode, and the results are compared with the ones under healthy condition. The IPMSM is analyzed using finite element analysis (FEA) software and the single phase fault is remedied by FTC algorithm [2]. Fig. 1(a) shows the IPMSM and its section cut. The currents in healthy and faulty conditions are shown in Fig. 1(b) while phase A is open in faulty condition. Torque comparison is shown in Fig. 1(c). As seen, an average torque of 12.5 Nm is produced under both conditions. However, as presented in Fig. 2(a), radial electromagnetic force density under faulty condition is larger than the one under healthy condition. The acceleration and deformation in Figs. 2(b) and (c) show that the largest difference between the accelerations under two conditions happens at 6,800 Hz causing the sound pressure level (SPL) of the faulty IPMSM to be 14.1% larger than the healthy IPMSM as in Fig. 2(d). Furthermore, the maximum deformation under fault is 11.8% larger than the one under healthy condition. As demonstrated, although the IPMSM under faulty condition with FTC can produce comparable torque as healthy IPMSM, machine NV aspects need attentions. Experimental results and detailed NV analysis including modal analysis will be provided in the full paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/f696d21c3a89261ad880a06f12b0d39d.jpg)

Fig. 1. IPMSM characteristics. (a) The investigated IPMSM and its section cut. (b) Input six phase currents. (c) Torque under two conditions.

![](https://assets.underline.io/uploads/markdown_image/1/image/b8ef2fc43e78a009486f87b8deae8b66.jpg)

Fig. 2. NV performance. (a) Radial EM force density. (b) Radial acceleration. (c) Deformed housings. (d) SPL.

Noise and Vibration Prediction of a Six-Phase IPMSM in a Single Open-Phase Failure Under a Negative Sequence Current Compensated Fault Tolerant Control Mode

Comprehensive parameter identification and analysis are critical to develop accurate machine model. As reported in [1],[2], only two parameters can be estimated properly at the same time since the rank of the machine model is 2. However, in real-time application, machine parameters vary nonlinearly due to the change in operating conditions [3]–[5].Therefore, more accurate knowledge of motor parameters requires more undetermined coefficients in machine modelling, which results in severer rank deficiency problem. To overcome the rank deficiency and achieve a better estimation accuracy of dq-axis inductance modelling, this paper designs an experimentally testing pattern illustrated in Fig. 1 for PMSM and proposes a decoupled estimation scheme in (1) considering system uncertainties including magnetic saturation, core loss, temperature effect and voltage source inverter nonlinearity. (UqId-UdIq)/ωe+(DdIscos(θ)+DqIssin(θ))△V/ωe=IqIqLq+Id(Λ+LdId)+IqIqLqΛ/Ri (1) where variables I, U, L, λ, D with subscript ‘d’ and ‘q’ are current, voltage, inductances, flux linkage and distorted coefficients on dq-axis frame accordingly. Λ and ΔV are PM flux linkage and distorted voltage drop due to dead time effect of voltage source inverter. Ri is the core loss resistance. As mentioned previously, the proposed estimation approach takes core loss effect into account. Therefore, given different speed conditions in (1), the differential model can be achieved by subtracting (1) at the first speed condition denoted as subscript 1 from the one at the second speed condition denoted as subscript 2, which contains the information of iron loss. In the full paper, an efficient least square method incorporating decoupled estimation scheme is employed to converge the undetermined coefficients in the proposed estimation model. The machine parameters including resistance, PM flux linkage and inductances will be estimated and reported. The estimated inductances illustrated in Fig. 2 will be validated through virtual signal injection based MTPA control and more experimental results will be presented to demonstrate the effectiveness of the proposed method.

![](https://assets.underline.io/uploads/markdown_image/1/image/49bc237555843438394366800f84a779.jpg)

Fig.1.Illustration of experimental data collection

![](https://assets.underline.io/uploads/markdown_image/1/image/2df751a5afa6d8760802b04a4e37dbbd.jpg)

Fig.2.Estimated inductance maps.(a)d-axis(b)q-axis

Decoupled Estimation Scheme for PMSMs Towards Accurate Inductance Modelling

Recently, Ma et al. reported a hybrid rare-earth and ferrite permanent magnet (PM)-based spoke-type motor that can generate a comparable torque to the motor used in Prius 2010 with 60% lower cost [1]. However, the paper investigated only the motor having 10% NdFeB PM in the total PM volume on the top of the ferrite when the motor employs 3-phase integrated slot distributed winding (ISDW). This paper investigated the effect of the NdFeB PM position with different volume ratios for 3 different windings, including 3-phase (3Φ) ISDW, 3Φ fractional slot concentrated winding (FSCW), and 6Φ FSCW, on the motor performance using ANSYS Maxwell 2D. Fig. 1 shows the layout of 3 windings and 2 rotors: one with NdFeB PM on the top of the ferrite (MTop) and the other with NdFeB PM on the bottom (MBot). Fig. 2 shows a change in the maximum torque (Tm) and torque ripple (Trip) of the motor with 3 windings and 2 rotors with the height (Hm) of the NdFeB PM. The Tm of both MTop and MBot decreases as the Hm decreases for all windings. This is because the Tm is proportional to the PM volume. On the contrary, a decreasing trend of Trip with respect to an increase in the Hm is observed for both rotor types and windings except for 3Φ ISDW. Further, for the negative Hm, both Tm and Trip of the MTop and MBot show no difference for all windings. On the other hand, for the positive Hm, the MBot shows 0-100 Nm higher Tm but 0.57-18 % lower Trip than the MTop for all windings. These results suggest bottom NdFeB PM for high Tm and the top NdFeB PM for low Trip. We will discuss performance metrics such as irreversible demagnetization, mechanical analysis, and maximum speed in this paper. This work was supported in part by the NSF-I/UCRC for EV-STS under Grant number 1650564.

![](https://assets.underline.io/uploads/markdown_image/1/image/eb8c2b729ce69c6e48fb43468a1ec197.jpg)

Fig. 1. Stator view: (a) 3Φ ISDW, (b) 3Φ FSCW, and (c) 6Φ FSCW; Rotor view with (d) MTop and (e) MBot with (f) detailed description of the magnet.

![](https://assets.underline.io/uploads/markdown_image/1/image/3acedc1513aba35e89496d5cd5980874.jpg)

Fig. 2. (a) Maximum torque and (b) torque ripple of motor having 3Φ ISDW, 3Φ FSCW, and 6Φ FSCW and NdFeB top and bottom on the ferrite.

Performance Evaluation of Hybrid Spoke-type Ferrite-NdFeB Permanent Magnet Motor

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.

![](https://assets.underline.io/uploads/markdown_image/1/image/5443a706e016bc514b6319e51c6d51e5.jpg)


![](https://assets.underline.io/uploads/markdown_image/1/image/252ae9506a33d0dae7adcde2c8161a91.jpg)

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

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

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