France



I.Introduction&lt;br/&gt;The spoke-type permanent magnet (PM) machines are widely used in many industrial fields because of their excellent performance of concentrating flux from the PM. Accurate calculation of the magnetic field is the premise for predicting the machine performances [1]-[3]. The analytical calculation methods of the magnetic field commonly used for spoke-type machines include the lumped magnetic circuit (LMC) model [1], subdomain model [2], and the method combining the two above [3]. The rotor magnetic potential (RMP) model is another analytical method to accurately calculate the magnetic field [4], which has been widely used in interior radial PM machines but has not been applied to spoke-type machines. In this paper, the RMP models of the spoke-type machine are proposed. Furthermore, a method is provided by combining the proposed RMP model and the winding function, which can realize a fast and accurate calculation of the magnetic field in spoke-type machines. The modeling processes of the RMP model proposed in this paper are straightforward, and it has fewer variables to be solved. Moreover, the proposed method does not need to consider the change of the relative position between the stator and rotor caused by rotor rotation.&lt;br/&gt;II.The Proposed RMP Model&lt;br/&gt;The RMP model is used to reflect the rotor surface magnetic potential distribution. In this paper, the magnetic field is divided into d-axis magnetic field (the magnetic field generated by the interaction of PM and d-axis current component) and q-axis magnetic field (the magnetic field generated by the q-axis current component). When there is only the d-axis magnetic field, it can be seen from Fig. 1(a) that the pole cap region is not saturated. Therefore, the RMP of the pole cap region is equal to that of the corresponding PM surface. Due to symmetry, the magnetic potentials on the left and right PM surfaces are opposite to each other. Hence the zero-crossing point of the RMP is located in the middle of the bridge (point B in Fig. 1(a)). According to the flux lines distribution in the bridge region, the RMP in this region changes linearly to both sides as the distance from point B increases. Therefore, the d-axis RMP model is obtained, as shown in Fig. 1(c). When there is only the q-axis magnetic field, it can be seen from Fig. 1(b) that the flux lines do not pass through the PMs; hence the RMP in the pole cap region is equal to zero. In the bridge region, the flux lines are separated in the middle of the bridge. Since the d-axis magnetic field is larger than the q-axis magnetic field, the influence of the d-axis magnetic field should be taken into account in the q-axis RMP model. Therefore, the q-axis RMP reaches the maximum value in the middle of the bridge (point B in Fig. 1(b)) and decreases to both sides along the flux lines. Therefore, the q-axis RMP model is obtained, as shown in Fig. 1(d). It should be noted that the position of points A and B in Fig. 1(a) and Fig. 1(b) are same.&lt;br/&gt;III.The Solution of the Proposed RMP Model&lt;br/&gt;The d-axis and q-axis LMC models are established to solve the unknown parameters (&lt;i&gt;U&lt;/i&gt;&lt;sub&gt;rd1&lt;/sub&gt; and &lt;i&gt;U&lt;/i&gt;&lt;sub&gt;rq1&lt;/sub&gt;), as shown in Fig. 1(e) and Fig. 1(f). &lt;i&gt;R&lt;/i&gt;&lt;sub&gt;au&lt;/sub&gt; and &lt;i&gt;R&lt;/i&gt;&lt;sub&gt;ad&lt;/sub&gt; are nonlinear reluctances corresponding to the upper and lower bridge regions, respectively. &lt;i&gt;R&lt;/i&gt;&lt;sub&gt;gd&lt;/sub&gt; is d-axis air-gap reluctance. &lt;i&gt;R&lt;/i&gt;&lt;sub&gt;m&lt;/sub&gt; is PM reluctance. &lt;i&gt;F&lt;/i&gt;&lt;sub&gt;c&lt;/sub&gt; is PM magnetomotive force (MMF). &lt;i&gt;F&lt;/i&gt;&lt;sub&gt;d&lt;/sub&gt; is the equivalent stator MMF, and its value is &lt;i&gt;F&lt;/i&gt;&lt;sub&gt;d&lt;/sub&gt;=(&lt;i&gt;F&lt;/i&gt;&lt;sub&gt;d1&lt;/sub&gt;+&lt;i&gt;F&lt;/i&gt;&lt;sub&gt;d2&lt;/sub&gt;+&lt;i&gt;F&lt;/i&gt;&lt;sub&gt;d3&lt;/sub&gt;-&lt;i&gt;F&lt;/i&gt;&lt;sub&gt;d4&lt;/sub&gt;-&lt;i&gt;F&lt;/i&gt;&lt;sub&gt;d5&lt;/sub&gt;-&lt;i&gt;F&lt;/i&gt;&lt;sub&gt;d6&lt;/sub&gt;)/3. &lt;i&gt;F&lt;/i&gt;&lt;sub&gt;d1&lt;/sub&gt;-&lt;i&gt;F&lt;/i&gt;&lt;sub&gt;d6&lt;/sub&gt; are the MMF at the corresponding stator teeth generated by the d-axis current component. &lt;i&gt;R&lt;/i&gt;&lt;sub&gt;gq&lt;/sub&gt; is q-axis air-gap reluctance. &lt;i&gt;F&lt;/i&gt;&lt;sub&gt;q&lt;/sub&gt; is the MMF generated by the q-axis current component. It should be noted that when there is only q-axis magnetic field, the bridge region is not saturated. Therefore, the reluctance &lt;i&gt;R&lt;/i&gt;&lt;sub&gt;au&lt;/sub&gt; is only affected by the d-axis magnetic field. Moreover, there is a saturation overflow phenomenon in the bridge region (the width of the saturation region is wider than the actual bridge width). To reflect this phenomenon, the flux density value of point A in Fig. 1(a) is set as 1.6T (the magnetic potential drop in the core with flux density value less than 1.6T is ignored). Therefore, the length &lt;i&gt;l&lt;/i&gt;&lt;sub&gt;ro&lt;/sub&gt; of the saturation overflow region (as shown in Fig. 1(a)) changes with different excitation conditions. The value of &lt;i&gt;l&lt;/i&gt;&lt;sub&gt;ro&lt;/sub&gt; can be solved by using the continuity principle of flux, and then the value of abscissa &lt;i&gt;x&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt; of point A can be obtained. Since the dq-axis cross-coupling effect has been taken into account in the above models, the total RMP model can be obtained by adding the d-axis and q-axis RMP models directly. Then the magnetic field and related parameters can be calculated by combining winding function with the proposed RMP model.&lt;br/&gt;IV.FEA Verification&lt;br/&gt;The basic parameters of the machine are listed in Table I. The validity of the proposed method (ANA) is verified by FEA. The comparison results of no-load air-gap flux density and back EMF are shown in Fig. 2 (a) and Table II. Under different current effective value &lt;i&gt;I&lt;/i&gt; (20A to 200A) and different current angle &lt;i&gt;φ&lt;/i&gt; (0° to -90°), the corresponding results of the air-gap flux density fundamental amplitude (FA) and average torque are shown in Fig. 2 (b) and Fig. 2 (c), respectively. As shown in Fig. 2, the maximum error of air-gap flux density FA does not exceed 8%, and the maximum error of average torque does not exceed 2%.

**References:**

[1] Q. Chen, G. Liu, and W. Zhao, &lt;i&gt;IEEE Transactions on Magnetics.&lt;/i&gt;, vol. 49(9), pp. 5150–5157 (2013).&lt;br/&gt;[2] P. Liang, F. Chai, and Y. Li, &lt;i&gt;IEEE Transactions on Industrial Electronics.&lt;/i&gt;, vol. 64(5), pp. 3479–3488 (2017).&lt;br/&gt;[3] P. Liang, F. Chai, and Y. Bi, &lt;i&gt;Journal of Magnetism and Magnetic Materials.&lt;/i&gt;, vol. 417, pp. 389-396 (2016).&lt;br/&gt;[4] Q. Li, T. Fan, and X. Wen,&lt;i&gt; IEEE Transactions on Magnetics.&lt;/i&gt;, vol. 49(3), pp. 1193-1201 (2013).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3220580be1e7460840ec14da0e2fa0f4.jpg)
Fig. 1. (a) d-axis magnetic field (b) q-axis magnetic field (c) d-axis RMP model (d) q-axis RMP model (e) d-axis LMC model (f) q-axis LMC model

![](https://assets.underline.io/uploads/markdown_image/1/image/b959b7b8f4307f450d5a59f2a751f4a3.jpg)
Fig. 2. The comparison results (a) no-load (b) air-gap flux density at load (c) average torque

INTERMAG 2021

 Analytical Calculation for Magnetic Field in Spoke-Type Permanent Magnet Machines Based on a Rotor Magnetic Potential Model

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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 The spoke-type permanent magnet (PM) machines are widely used in many industrial fields because of their excellent performance of concentrating flux from the PM. Accurate calculation of the magnetic field is the premise for predicting the machine performances [1]-[3]. The analytical calculation methods of the magnetic field commonly used for spoke-type machines include the lumped magnetic circuit (LMC) model [1], subdomain model [2], and the method combining the two above [3]. The rotor magnetic potential (RMP) model is another analytical method to accurately calculate the magnetic field [4], which has been widely used in interior radial PM machines but has not been applied to spoke-type machines. In this paper, the RMP models of the spoke-type machine are proposed. Furthermore, a method is provided by combining the proposed RMP model and the winding function, which can realize a fast and accurate calculation of the magnetic field in spoke-type machines. The modeling processes of the RMP model proposed in this paper are straightforward, and it has fewer variables to be solved. Moreover, the proposed method does not need to consider the change of the relative position between the stator and rotor caused by rotor rotation. II.The Proposed RMP Model The RMP model is used to reflect the rotor surface magnetic potential distribution. In this paper, the magnetic field is divided into d-axis magnetic field (the magnetic field generated by the interaction of PM and d-axis current component) and q-axis magnetic field (the magnetic field generated by the q-axis current component). When there is only the d-axis magnetic field, it can be seen from Fig. 1(a) that the pole cap region is not saturated. Therefore, the RMP of the pole cap region is equal to that of the corresponding PM surface. Due to symmetry, the magnetic potentials on the left and right PM surfaces are opposite to each other. Hence the zero-crossing point of the RMP is located in the middle of the bridge (point B in Fig. 1(a)). According to the flux lines distribution in the bridge region, the RMP in this region changes linearly to both sides as the distance from point B increases. Therefore, the d-axis RMP model is obtained, as shown in Fig. 1(c). When there is only the q-axis magnetic field, it can be seen from Fig. 1(b) that the flux lines do not pass through the PMs; hence the RMP in the pole cap region is equal to zero. In the bridge region, the flux lines are separated in the middle of the bridge. Since the d-axis magnetic field is larger than the q-axis magnetic field, the influence of the d-axis magnetic field should be taken into account in the q-axis RMP model. Therefore, the q-axis RMP reaches the maximum value in the middle of the bridge (point B in Fig. 1(b)) and decreases to both sides along the flux lines. Therefore, the q-axis RMP model is obtained, as shown in Fig. 1(d). It should be noted that the position of points A and B in Fig. 1(a) and Fig. 1(b) are same. III.The Solution of the Proposed RMP Model The d-axis and q-axis LMC models are established to solve the unknown parameters (Urd1 and Urq1), as shown in Fig. 1(e) and Fig. 1(f). Rau and Rad are nonlinear reluctances corresponding to the upper and lower bridge regions, respectively. Rgd is d-axis air-gap reluctance. Rm is PM reluctance. Fc is PM magnetomotive force (MMF). Fd is the equivalent stator MMF, and its value is Fd=(Fd1+Fd2+Fd3-Fd4-Fd5-Fd6)/3. Fd1-Fd6 are the MMF at the corresponding stator teeth generated by the d-axis current component. Rgq is q-axis air-gap reluctance. Fq is the MMF generated by the q-axis current component. It should be noted that when there is only q-axis magnetic field, the bridge region is not saturated. Therefore, the reluctance Rau is only affected by the d-axis magnetic field. Moreover, there is a saturation overflow phenomenon in the bridge region (the width of the saturation region is wider than the actual bridge width). To reflect this phenomenon, the flux density value of point A in Fig. 1(a) is set as 1.6T (the magnetic potential drop in the core with flux density value less than 1.6T is ignored). Therefore, the length lro of the saturation overflow region (as shown in Fig. 1(a)) changes with different excitation conditions. The value of lro can be solved by using the continuity principle of flux, and then the value of abscissa x1 of point A can be obtained. Since the dq-axis cross-coupling effect has been taken into account in the above models, the total RMP model can be obtained by adding the d-axis and q-axis RMP models directly. Then the magnetic field and related parameters can be calculated by combining winding function with the proposed RMP model. IV.FEA Verification The basic parameters of the machine are listed in Table I. The validity of the proposed method (ANA) is verified by FEA. The comparison results of no-load air-gap flux density and back EMF are shown in Fig. 2 (a) and Table II. Under different current effective value I (20A to 200A) and different current angle φ (0° to -90°), the corresponding results of the air-gap flux density fundamental amplitude (FA) and average torque are shown in Fig. 2 (b) and Fig. 2 (c), respectively. As shown in Fig. 2, the maximum error of air-gap flux density FA does not exceed 8%, and the maximum error of average torque does not exceed 2%.

**References:**

[1] Q. Chen, G. Liu, and W. Zhao, IEEE Transactions on Magnetics., vol. 49(9), pp. 5150–5157 (2013). [2] P. Liang, F. Chai, and Y. Li, IEEE Transactions on Industrial Electronics., vol. 64(5), pp. 3479–3488 (2017). [3] P. Liang, F. Chai, and Y. Bi, Journal of Magnetism and Magnetic Materials., vol. 417, pp. 389-396 (2016). [4] Q. Li, T. Fan, and X. Wen, IEEE Transactions on Magnetics., vol. 49(3), pp. 1193-1201 (2013).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3220580be1e7460840ec14da0e2fa0f4.jpg)
Fig. 1. (a) d-axis magnetic field (b) q-axis magnetic field (c) d-axis RMP model (d) q-axis RMP model (e) d-axis LMC model (f) q-axis LMC model

![](https://assets.underline.io/uploads/markdown_image/1/image/b959b7b8f4307f450d5a59f2a751f4a3.jpg)
Fig. 2. The comparison results (a) no-load (b) air-gap flux density at load (c) average torque

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– Electric motors, magnetic resonance imaging and many other modern applications are built out of systems of multiple permanent magnets, which are arranged so that the magnetic field that they produce has the required properties. Here, starting from a given magnetic field, we invert the design procedure to predict a matching magnetic structure using deep learning. A convolutional neural network (CNN) learns the mapping from a desired magnetic field to a simple magnetic structure with an error of 0.22%. We further extend the method to work with multiple permanent magnets. Introduction Systems of multiple magnets are the basis for many modern applications like magnetic resonance imaging, frictionless bearings in flywheels or electric motors. In these applications permanent magnets of different shapes and magnetic properties are assembled to produce a magnetic field, whose field strength and homogeneity match the requirements for the given application. In order to provide better medical devices, store energy or pave the way to green mobility with electric vehicles, optimized structures of multiple magnets are essential. Traditionally, the design procedure for producing optimized structures of permanent magnets is by selecting a known design that produces a field close to the desired field and then optimizing the structure through modeling. As an example, consider the Halbach cylinder [1] where in real-world applications manufacturing tolerances mean that its actual field will not be exactly equal to the analytically calculated field. In this case one can e.g. use a genetic algorithm to incorporate the manufacturing variations of the single magnetic segments in a traditional design optimization approach [2]. However, this optimization approach is limited in scope, as it relies on a design structure to start from and is computationally intense. Problem Here we consider the general inverse problem of determining a magnetic structure that generates a desired magnetic field. We investigate this through the use of a deep neural network, with the network learning the mapping of Maxwell’s equations in an inverse manner. We want to predict the properties of a system of multiple permanent magnets from a part of the magnetic field, as illustrated in Fig. 1. We consider complicated structures of permanent magnets formed out of different simple geometries, e.g. prisms, tetrahedrons or cylindrical tiles. Previously, artificial intelligence has been used to approximate the solution of Maxwell’s equations for electromagnetic structures [3], in order to calculate the magnetic field in a computationally cheap manner. Here, however we invert this problem to use deep learning to predict magnet structures from the magnetic field. A major challenge is to find an invariant representation for systems of a varying number of magnets. Each of the permanent magnets is specified by its geometric appearance, location, and magnetic properties. The output space easily explodes with an increasing number of magnets, when a poor representation is chosen. Additionally, the problem becomes complex when predicting the properties of the magnetic system from a magnetic field, which does not contain the structure itself. There may exist several solutions for the given desired field. Method The magnetic field in a so-called “design area” is calculated point-wise, which results in a matrix of field values in each of the spatial dimensions. That tensor is fed into a neural network of multiple convolutional layers. As baseline model, we use a convolutional neural network (CNN) inspired by the ResNet architecture [4]. The output shape of the structural design depends heavily on the chosen magnetic system representation. The model parameters are then updated by backpropagation of the difference between the ground-truth and the predicted target structure. To train such models of millions of tunable parameters artificial neural networks require a large amount of data. We produce accurate field-structure-pairs fast and efficiently with MagTense [5], an open-source magnetostatic and a micromagnetism calculation framework. The learned network parameters are then able to infer magnetic structures of unseen magnetic fields. Initially we limit the input space to be a two-dimensional magnetic field by setting the z-component of the single magnet magnetizations to 0 T. We further only use prisms with a constant height and omit rotations of the geometries. The primarily two-dimensional “design area” lies in the plane, which cuts the single tiles in their center point. Results We show in [6] that the prediction of the properties of a single permanent magnet from its magnetic field has an average error rate of 0.22%, which is much below magnet manufacturing tolerances. Predictions of the trained CNN are hardly distinguishable from its ground-truth, when given a previously unseen magnetic field, as exemplified in Fig. 2. We discuss the error rates in extending this to multiple magnets and finally discuss how to represent magnetic structures with multiple permanent magnets of different shapes.

**References:**

[1] K. Halbach, Nuclear Instruments and Methods, Vol. 169, pp. 1-10 (1980) [2] K.K. Nielsen et al., Journal of Magnetism and Magnetic Materials, Vol. 514, p. 167175 (2020) [3] A. Khan, V. Ghorbanian and D. Lowther, IEEE Transaction on Magnetics, Vol. 55, no. 6 (2019) [4] K. He et al., CVPR, ”Deep Residual Learning for Image Recognition” (2016) [5] R. Bjørk and K.K. Nielsen, www.magtense.org, https://doi.org/10.11581/DTU:00000071 (2019) [6] S. Pollok, R. Bjørk and P.S. Jørgensen, “Inverse Design of Magnetic Fields using Deep Learning”, Under review (2021)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/67b4dca71c7e6adebff4b6367ae16294.jpg)
Fig. 1. Deep learning pipeline for inverse design of magnetic structures. The magnetic field in a so-called "design area" serves as input to the neural network, which predicts a systems of permanent magnets required to produce such a field.

![](https://assets.underline.io/uploads/markdown_image/1/image/f82853dcf9e607e8bc8bf12268bbdd30.jpg)
Fig. 2. Prediction of a trained convolutional neural network from an unseen magnetic field. When plotting the predicted single permanent magnet it is visually indistinguishable from its ground-truth.

 Convolutional Neural Networks for Inverse Design of Magnetic Structures

This paper proposes an efficient surrogate-assisted design optimization method based on multicriteria optimal Latin hypercube design (LHD) for multi-objective optimization of a permanent-magnet vernier machine (PMVM). By combining simulated annealing algorithm, the uniformity and orthogonality of spatial distribution of the LHD are optimized to efficiently capture data features over the optimization parameter ranges. Consequently, an accurate surrogate model with high generalization capability can be built with a small training set by using the optimal LHD. The optimal design of the PMVM is obtained by multi-objective design optimization based on the surrogate model. A prototype is made and tested to validate the proposed method. I. Introduction Due to high torque capability at low speeds, permanent-magnet vernier machines (PMVMs) are suitable for direct-drive applications such as wind power generation and electric drilling [1]. To deal with the issue of massive finite-element analysis (FEA) for multi-objective design optimization, this paper proposes a novel surrogate-assisted multi-objective design optimization method for PMVMs. A multicriteria optimal Latin hypercube design (LHD) employing simulated annealing (SA) algorithm is proposed for efficient date feature capturing that enables to use a small training dataset for obtaining an accurate surrogate model. The sparrow search algorithm (SSA)-optimized multi-output least squares support vector machine (MLS-SVM) [2] and the non-dominant sorted genetic algorithm-III (NSGA-III) [3] are combined to build the multi-output surrogate model and to perform the multi-objective design optimization. A prototype is made based on the obtained optimal design and tested for validation. II. PMVM and Sensitivity Analysis A surface-mounted PMVM with 18 slots and 15 pole pairs is studied as an example, whose topology is illustrated in Fig. 1 (a). The four optimization objectives are average torque, power factor, THD of back-EMF, and cogging torque. To improve the optimization efficiency, a global sensitivity analysis (GSA) of four optimization targets is performed using the Sobol’s method [4], the stack bars of total sensitivity coefficient (TSC) for optimization parameters are given in Fig. 1 (b), which shows the geometrical parameters that have the greatest influence on design objectives. Thus, five geometrical parameters including PM thickness (tPM) and width (wPM), yoke thickness (ty), slot width (ws), and stator split ratio (k=Rsi/Rso) are selected for optimization while stator outside diameter, axial length, air-gap length, and current density are fixed. III. Surrogate-Assisted Optimization with LHD The optimal LHD employs SA to minimize a composite indicator ψp that consists of the Morris-Mitchell index Φp and the correlation coefficient of the test factors ρ2 to guarantee the spatial distribution uniformity and orthogonality as given in (1), where ω1 and ω2 are the weight factors. Subsequently, a 270*5 LHD is built to capture the data feature over the optimization parameter ranges more efficiently as shown in Fig.2 (a), in which 20 sets of data are used for testing the regression results. minψp = ω1Φp + ω2ρ2(1) Finite element analysis (FEA) simulations are only performed within the optimal LHD to acquire the training data. The surrogate model based on MLS-SVM are trained using the data of the optimal LHD. The radial basis function (RBF) kernel is adopted for MLS-SVM and SSA is combined with MLS-SVM to find an optimal hyper-parameter set that produces a surrogate model with the highest average determination coefficient R2. The four-objectives optimization is performed by NSGA-III and an optimal design is selected from the Pareto front shown in Fig. 2 (b). A prototype is made based on the optimal design and tested, Figs. 2 (c) and (d), to validate the proposed method and the optimization. The proposed method needs only 270 FEA simulation cases during the whole design optimization procedure. For the conventional FEA-based parametric optimization method, even when only 5 levels are selected for each of the five geometrical parameters, 3125 FEA cases are required. Meanwhile, since the parameter space is discretized, the results of the conventional optimization method are likely to be locally optimal. For the method employing randomly generated LHDs [5], it requires nearly 500 FEA cases to achieve consistent R2s, which consumes 2 times the time of the proposed method. Thus, the proposed method can secure both efficient and accurate global design optimization of PMVMs. IV. Conclusions This paper proposes a novel optimal LHD-based surrogate assisted design optimization method for multi-objective optimization of PMVMs, which can perform high efficiency and high validity simultaneously. The proposed method and optimal design are validated by experiments on a prototype. Details of the design optimization and test results will be given in the full paper.

**References:**

[1] Y. Oner, Z. Q. Zhu and L. J. Wu, IEEE Trans. Ind. Electron., Vol. 63, No. 7, pp. 4105–4117 (2016). [2] S. Xu, X. An and X. Qiao, Pattern Recognit. Lett., Vol. 34, No. 9, pp. 1078–1084 (2013). [3] A. M. Usman, U. K. Yusof and S. Naim, IEEE Access, Vol. 8, pp. 76333-76356 (2020). [4] J. Yin, X. Zhu, L. Quan and C. Zhang, IET Electr. Power Appl., Vol. 11, pp. 72–79 (2017). [5] J. B. Kim, K. Y. Hwang and B. I. Kwon, IEEE Trans. Magn., Vol. 47, No. 5, pp. 1078-1081 (2011).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a02e0d4508835ed9b436b082204c4f95.jpg)
Fig. 1 Topology of PMVM and GSA results. (a) PMVM. (b) GSA results characterized by TSCs.

![](https://assets.underline.io/uploads/markdown_image/1/image/0710c9f0c14fbc22cb16e33d2590d647.jpg)
Fig. 2 Surrogate-assisted optimization. (a) Optimal LHD. (b) Pareto front of multi-objective optimization. (c) Rotor and stator of the prototype. (d) Test platform.

 Multicriteria Optimal Latin Hypercube Design-Based Surrogate-Assisted Design Optimization for a Permanent-Magnet Vernier Machine

1. Introduction: Optimal operation of electromagnetic drive systems requires coupled analysis of electromagnetic equipment with their control units. Accurate equivalent circuit models of the electromagnetic equipment are essential and are expected to include the high-frequency eddy currents in nonlinear laminated iron cores. Model order reduction (MOR) is a powerful methodology providing reduced models based on finite element analysis. Some of the MOR approaches are capable of generating equivalent circuit models such as Cauer Ladder Network (CLN). The CLN method is one of the promising MORs proposed for eddy current problems in linear mediums [1]. It was extended to the magnetically nonlinear problems using the first-order approximation method (FO-CLN) [2]. Another extension concerns the analysis of laminated cores using homogenized eddy current field as a multi-scale MOR for linear mediums [3]. This paper makes a progress by introducing the nonlinear multi-scale MOR based on a new piece-wise linearization technique. Homogenization on its nonlinear form has been always a big challenge and is a fortiori so in multi-scale MOR. The main contribution of this work is to linearize peace-wise the nonlinear homogenization using the first-order approximation methodology of FO-CLN to find the equivalent circuit model for nonlinear laminated iron cores. 2. Multi-Scale MOR: The CLN method is a "Macroscopic-Scale" MOR, which aims for efficient electromagnetic field calculation over the whole machine on the macroscopic scale. On the other hand, "Microscopic-Scale" MOR deals with the homogenization and expressing the components of electromagnetic equipment with fine structures, such as laminated steel sheets. Fig. 1 illustrates how the microscopic-scale MOR approximates a stacked laminated-core Fig.1(a) into a homogenized bulk material Fig.1(b); see e.g. [4] for a homogenization method that considers eddy currents and skin effect in laminated stacks using the Legendre polynomials. The homogenized eddy-current field is expressed as K'a=-σ'da/dt+j0, where K', σ', a and, j0 are the homogenized stiffness and conductivity matrices, magnetic vector potential, and external current vectors, respectively. On the multi-scale MOR, the standard CLN procedure is adapted with K' and σ'. 3. FO-CLN: Using the CLN, the time-dependent electromagnetic fields in a linear eddy-current problem are decomposed into time-independent modes weighted by time-dependent coefficients. On linear case, these modes are constant and the coefficients are derived from the network currents and voltages in Fig. 1(c). When the core has nonlinear magnetic characteristics, the modes vary with respect to saturation in the core with field-dependent permeability μ(B) In the FO-CLN method, the first magnetic mode is considered as the main and only source of the core saturation. The saturation level is measured by the current h1 in Fig. 1(c). The CLN is performed at different degrees of saturation which results in parametric CLN elements as in Fig. 1(c). In Fig.1 the FO-CLN method is shown as the transition between Fig. 1(b) to Fig. 1(c). 4. Nonlinear Multi-Scale MOR: The FO-CLN methodology is applied to the multi-scale MOR since it is based on the frozen permeabilities obtained during the calculation of the first magnetic mode. More details will be given in the full paper. 5. Numerical Example: The accuracy of the proposed method is verified over a laminated inductor with sheet thickness of 0.5mm, shown in Fig. 2(a). The currents from the conventional transient finite element method and it's equivalent CLN compared with the 0th order homogenization in Fig. 2(b), demonstrates the accuracy of the proposed method.

**References:**

[1] A. Kameari, H. Ebrahimi and T. Matsuo, IEEE Trans. Magnetics., Vol. 54, no.11 (2018) [2] H. Eskandari and T. Matsuo, IEEE Trans. Magnetics., vol.56, no.2 (2020) [3] H. Eskandari, J. Gyselinck and T. Matsuo, CEFC Conference, TA_O2 (2020) [4] J. Gyselinck, P. Dular, and L. Krahenbuhl, IEEE Trans. Magnetics., vol.52, no.3 (2016)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/6912dd17a44d4e0739b952fdd708318c.jpg)
Fig.1 (a) Eddy-current problem with fine structure, (b) Nonlinear homogenized eddy-current, (c) Equivalent parametrized CLN.

![](https://assets.underline.io/uploads/markdown_image/1/image/16d36cf0e58884690f6fea3ac0a2aefa.jpg)
Fig.2 (a) Iron-core inductor (b) Transient responses to sinusoidal excitation.

 Nonlinear Multi-Scale Model Order Reduction of Eddy-Current Problem

The finite element method (FEM) is widely used to model electromagnetic devices. This method ensures precise results at the expense of a consequent computation time, especially when dealing with parametric problems. Model order reduction methods had proven their efficiency in reducing the computation time for parametric problems. The proper orthogonal decomposition (POD) is a popular model order reduction approach [1]. Based on the solutions of the FE model for different values of parameters (called snapshots), the POD approximates the solution of the FE model in a reduced basis. The parametric FE model is projected onto a reduced basis, decreasing the order of the numerical model to be solved. Another approach consists in constructing a metamodel by interpolation of the parametric FE solution expressed into a reduced basis. A hybrid approach has also been proposed combining the POD with an interpolation-based method such as the radial basis functions (RBF) interpolation method [2]. Moreover, when the problem involves geometric parameters, an adaption of the FEM mesh is required. Remeshing the geometry at each modification can deteriorate the precision of the FEM model by introducing significant numerical noise. A solution to preserve the problem's consistency is to deform the initial mesh using the RBF interpolation [3][4]. The method consists of imposing a displacement on a set of nodes and interpolating the remaining nodes' position. Then, a metamodel of a FE solution taking into account geometric and electric parameters, can be defined by using the POD approach and the RBF interpolation. In this work, we propose to build a metamodel of a nonlinear magnetostatic parametric problem involving geometric parameters using the POD-RBF approach. The study case is a single-phase EI inductance, with one electrical parameter and two geometrical parameters. An initial geometry is meshed with 36859 elements; the number of nodes is 18553. The geometric parameters are the thickness of the airgap varying from 0.1 mm to 0.9 mm, the width of the central column of the magnetic core varying from 20 mm to 40 mm. The phase current varying from 0 to 50 A is also a parameter of the model. The mesh deformation is modeled using the RBF interpolation with a multi-quadratic function for two consecutive times to consider the two variating geometric parameters. The results of the original FEM model are taken as a reference. The POD-RBF approach is applied to the parametric FEM model. The method consists of first gathering a set of data, using the method of snapshots, by solving the original FEM model for a set of inputs. The snapshots matrix is decomposed using the singular value decomposition (SVD). The left orthogonal matrix of the SVD decomposition corresponds to the reduced basis, and the right orthogonal matrix is composed of the FE solution projected into the reduced basis for each snapshot parameter. An RBF interpolation is then performed on the right orthogonal matrix to interpolate the solution expressed in the reduced basis for the new inputs set. Finally, an approximation of the FE solution is defined by projecting the reduced solution into the FE basis. The choice of snapshots is made using a greedy algorithm [2], which consists of an iterative process. The metamodel is then improved at each iteration until the required precision is reached. The stop criterion to the greedy algorithm is set to an error between the FE solution and the metamodel solution lower than 0.1%. The evolution of error represented by Frobenius norm as a function of the iterations' number of the greedy algorithm is represented in "Fig. 1". In order to satisfy the selected criterion, 158 snapshots are required. The magnetic flux cartography as a function of phase current and the central column width for an airgap width fixed at 0.1 mm is represented in "Fig.2". The global mean absolute error between the flux cartographies obtained using the metamodel and the original FEM model is 1.7%, while the maximal local error is 4.96%. Then, the metamodel of the parametric FE solution can be used in an optimization process for different applications or to be coupled with other numerical models in order to simulate a system.

**References:**

[1] L. Sirovich, Quarterly of applied mathematics, vol. 45, no 3, pp.561-571, (1987). [2] T. Henneron, A. Pierquin and S. Clénet, IEEE Transactions on Magnetics, vol. 56, no. 1, pp. 1-4, (2020). [3] A. de Boer, M.S. van der Schoot and H. Bijl, Conference on Computational Fluid and Solid Mechanics, vol. 85, pp. 784-795, (2007). [4] T. Henneron, A. Pierquin and S. Clénet, IEEE Transactions on Magnetics, vol. 55, no. 6, pp. 1-4, (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/66ade7d4382d68fb036d292ea79e24d1.jpg)
Fig.1 Error as a function of the number of iterations

![](https://assets.underline.io/uploads/markdown_image/1/image/d0ae73399462ce7b097771fa6cc378a2.jpg)
Fig. 2 Flux cartography as a function of current and central column width

 Parametric Geometric Metamodel of Nonlinear Magnetostatic Problem Based on POD and RBF Approaches

I. INTRODUCTION Although inductive power transfer technology is under development with consideration for the protection of the human body and electromagnetic compatibility for practical use [1], further magnetic field strength reduction methods are necessary to accommodate the trend of increasing transmission power. Coil structures and shielding structures to suppress the magnetic field intensity [2,3]; however, further measures are required to build safer and higher-power inductive power transfer systems. Among the components comprised in inductive power transmission systems except the coils and shielding, the load is the only component that strongly affects the magnetic field intensity. The load impedance indirectly affects the magnetic field intensity because of the dependence of the amplitude and phase of the currents flowing through the transmitter and receiver on the load impedance. In this study, we design the load impedance based on a magnetic energy minimization problem. As the magnetic energy stored in the inductive power transfer system is positively correlated with the magnetic field intensity, solving the magnetic energy minimization problem leads to the load impedance at which the magnetic fields generated by the transmitter and the receiver cancel each other as much as possible. Because the proposed method does not assume the physical structures of the coils to be incorporated into the system, it can be applied along with existing magnetic field intensity suppression methods to achieve enhanced suppression. II. METHODOLOGY To solve the magnetic energy minimization problem for the load impedance Zload, the equivalent circuit and circuit parameters shown in Fig. 1 are used. Although minimizing the magnetic field intensity with the equivalent circuit is not straightforward because the magnetic field intensity depends on the three dimensional structures of the coils, the magnetic energy stored in the inductances can be minimized with the equivalent circuit. The magnetic energy EL1 stored in the inductance L1 of the transmitter can be calculated as EL1 = L1i12/2. Similarly, the magnetic energy Esum stored in inductances L1 and L2 and their mutual inductance M is expressed by Esum = L1i12/2 + L2i22/2 + Mi1i2. In addition, for the angular frequency and the phasor current in the transmitter I1, the phasor current I2 in the receiver is expressed as I2 = −jωMI1/(r2 + jωL2 + Zload). Zload can determine the amplitude and phase of I2 in the receiver with respect to I1 in the transmitter, thereby manipulate the magnetic energy. Under the condition of constant received power, we derive the Zload that minimizes the time average of Esum. We omit the derivation process here for conciseness; however, when the quality factor of the receiving coil is sufficiently high such that r2 approximates zero, the load impedance is formulated as: Zload = ω|k|√(1−k2)L2 + jω(k2−1)×L2, (1) where k is the coupling coefficient between the transmitting and receiving coils. The effectiveness of the derived Zload was confirmed using electromagnetic field simulations via Altair FEKO. The parameters of the transmitting and receiving coils were set assuming a power supply system for electric vehicles, with reference to [4]. Specifically, the outer diameter of the coil was 450 mm, the inner diameter was 115 mm, the number of turns was 50, the winding pitch was 3.35 mm, the diameter of the copper wire was 1.6 mm, and the operating frequency was 100 kHz. Simulation of a single coil yielded the inductance of 660.3 μH. The coupling coefficient, calculated by simulation with a transmission distance of 100 mm in the vertical direction, was 0.3470. The optimum Zload calculated from (1) was 135.0 − j364.9 Ω. When a load of 135.0 − j364.9 Ω is connected and 1 kW is transferred to the load, the magnetic field intensity is shown in Fig. 2(a) with a drawing range of 1 m × 1 m. When the Zload value for maximizing power efficiency [5] is connected and 1 kW is transferred, the magnetic field intensity is as shown in Fig. 2(b). Although the magnetic field intensity with the proposed method is not suppressed uniformly, it is significantly suppressed in the space between the transmitting and receiving coils. At the central point in this space, the magnetic field intensity is weakened by 16%. The power efficiencies are nearly equal at 98.77% and 98.84% for the cases of magnetic energy minimization and power efficiency maximization, respectively. Zload in (1) asymptotically approaches the value at which the power efficiency is maximized as the coupling coefficient decreases; therefore, it is considered that the power efficiency is not significantly impaired. III. CONCLUSION In this study, the load impedance that minimizes the magnetic energy per transmission power is formulated by equivalent circuit analysis to suppress the magnetic field intensity generated by an inductive power transmission system. The validity of the theoretical formula is confirmed by electromagnetic field simulations. It is shown that in a system with a coupling coefficient of 0.35, the magnetic field intensity can be suppressed by up to 16% with a deterioration of approximately 0.1% in power efficiency.

**References:**

[1] S.W. Park, IEEE Transactions on Magnetics, Vol.54, pp.1–8 (2018) [2] J. Kim, J. Kim, S. Kong, H. Kim, I.-S. Suh, N. P. Suh, D.-H. Cho, J. Kim, and S. Ahn, Proceedings of the IEEE, Vol.101, pp.1332–1342 (2013) [3] J. Park, C. Park, Y. Shin, D. Kim, B. Park, J. Cho, J. Choi, and S. Ahn, Applied Physics Letters, Vol.114, pp.1–5 (2019) [4] M. Kato, T. Imura, and Y. Hori, in Proceedings of 2012 IEEE International Telecommunications Energy Conference, pp.1–5 (2012) [5] I. Awai and T. Ishizaki, in Proceedings of 2012 IEEE MTT-S International Microwave Workshop Series on Innovative Wireless Power Transmission: Technologies, Systems, and Applications, pp.105–108 (2012)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/704edf48957b8dfd93e057c9402d600d.jpg)
Fig.1 Equivalent circuit of an inductive power transfer system.

![](https://assets.underline.io/uploads/markdown_image/1/image/6cf4ee86514d2a23304569a618a4176a.jpg)
Fig.2 Comparison of the magnetic field intensity using different load impedances when the received power is 1 kW. (a) The proposed load impedance for minimizing the magnetic energy, (b) the load impedance for maximizing power efficiency.

 Magnetic Energy Minimization for Suppressing Magnetic Field Intensity of Inductive Power Transfer Systems

—Two-degree-of-freedom rotary/linear flux-switching permanent magnet (2DoFRL-FSPM) machine, which capable of rotary, linear and helical motion, has a wide application in automated lines, grinders, robot arms, etc. In this paper, the back-EMF characteristics of 2DoFRL-FSPM machine are investigated by the 3D finite element model, including the effect of motion forms and coupling effect. In addition, a prototype machine of 2DoFRL-FSPM machine is manufactured and tested. I. Introduction With the increasing requirement of machine tools, grinders, robot arms, automated lines, etc., two-degree-of-freedom (2DoF) machines have attracted more and more attention. 2DOF machines capable of rotary, linear and helical motions through the use of a single motor, seem to be an excellent solution to the problems of large volume, mechanical errors of the traditional plans which realized by the combination of multiple motors. Amiri E was the pioneer of 2DoF machines and proposed the first 2DoF machine named two-armature rotary-linear induction motor, which stator adopts a rotary armature connecting with a linear armature in series [1]. Afterward, several topologies of 2DoF machines had been proposed and investigated. To realize 2DoF motion form, special structures such as multi stators, spiral grooves, and windings, Halbach structure were designed and applied in 2DoF machines [2,3]. Previously, a Halbach PM array and a multi-winding array were adopted in an air-cored linear and rotary Halbach permanent magnet actuator [4]. The motion forms of the machine were determined by the mode of power supply. Normally, a long mover was always adopted in most 2DoF machines, which use more permanent magnets and result in low utilization of permanent magnets because there are always some permanent magnets beyond the stator. To overcome the above problems, 2DoFRL-FSPM machine whose permanent magnets are arranged in the stator part was proposed to increase the utilization and reduce the cost. In this paper, the back-EMF characteristics of 2DoFRL-FSPM machine are investigated. A 3D FE model considering the end and coupling effects are constructed to predicted the back-EMFs in different motion forms. In addition, experiments concerning 2DoFRL-FSPM machine are conducted based on its prototype machine. II. Topology of 2DoFRL-FSPM Machine A 2DoFRL-FSPM machine consists of two split arc-shaped stators (rotary and linear stators) and a long solid salient-poles mover to achieve rotary, linear and helical motions. In order to analyze the back-EMF characteristics of 2DoFRL-FSPM machine, a 3D FE model, which can simulate multi-degrees of freedom motions, is constructed as shown in Fig. 1. III. Back-EMF characteristics The stators of 2DoFRL-FSPM machine are arc-shaped, hence, the longitudinal end effect exists in the two parts of 2DoFRL-FSPM machine, which results that the back-EMF of one phase is larger than that of the other two phases. Due to the mover with an array of salient-poles on its surface, the coupling areas between the rotary or linear stator and the mover are variable in helical motion. In addition, the coupling effect between rotary and linear stators exists in helical motion, which leads to an increase of back-EMF harmonics. IV. Experiment In order to verify the above back-EMF characteristics, a prototype machine of 2DoFRL-FSPM machine is manufactured and tested. The rotary stator, linear stator, mover, and assembled machine are shown in Fig. 2. The predicted results are almost consistent with the measured, i.e., the back-EMF of one phase is larger than that of the other two phases.

**References:**

[1] E. Amiri, P. Gottipati, and E. A. Mendrela, "3-D space modeling of rotary-linear induction motor with twin-armature,"in 2011 1st International Conference on Electrical Energy Systems,2011, pp. 203-206 [2] K. Guo, S. Fang, and H. Lin, "3-D Analytical Analysis of Magnetic Field of Flux Reversal Linear-Rotary Permanent-Magnet Actuator," IEEE T. Magn., vol. 6, no. 53, pp. 1-5, 2017. [3] P. Jin, S. Fang, and H. Lin, "Analytical Magnetic Field Analysis and Prediction of Cogging Force and Torque of a Linear and Rotary Permanent Magnet Actuator," IEEE T. Magn., vol. 10, no. 47, pp. 3004-3007, 2011. [4] P. Jin, Y. Yuan, and G. Jian, "Static Characteristics of Novel Air-Cored Linear and Rotary Halbach Permanent Magnet Actuator," IEEE T. Magn., vol. 2, no. 50, pp. 977-980, 2014.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/74e449380c0a2c5c236a5b877f3bbe1d.jpg)
Fig.1 3D FE model of 2DoFRL-FSPM machine.

![](https://assets.underline.io/uploads/markdown_image/1/image/cf6c643afeabfb7a60986fe8cf8c9638.jpg)
Fig.2 Prototype machine of 2DoFRL-FSPM machine.

 Back-EMF characteristics of Two-Degree-of-Freedom Rotary-Linear Flux-Switching PM Machine

Although e-bikes are widely used today, there is no detailed system design for their wireless charging. In this study, the effects of switching frequency, quality factor, current density of the windings and critical coupling value have been examined and their boundaries have been determined for low power contactless charge system. According to these design constraints; an optimized Inductive Power Transfer (IPT) system has been analytically designed, which transfers the required power (120W) for the charging of the e-tricycle scooter from 100 mm air-gap. In view of previous studies in the literature [1, 2], the required power for e-bikes can be transmitted contactless at a frequency of 20-360 kHz. As the operation frequency increases, the size and quantity of coils to be used will decrease. The weight and volume of the coils are constraints in the system, that’s why the operating frequency may increase as in small household appliances and biomedical applications. However, as in the case of electric vehicles where the power is relatively high; operating at high frequencies can be risky for human health. In addition, working at high frequencies, can increase switching losses and may cause noise in the communication system. Using ferrite cores can also reduce coil sizes [3]. But, the efficiency will decrease as core losses occur, and also the system weight will increase. On the other hand, in order to transfer the same power at low frequencies, the coil dimensions and number of turns increase. Therefore, an optimal design is required for IPT using air-core windings. The most important step affecting the system design performance is to calculate the real inductance values of the coils analytically. An IPT optimization used rectangular windings is described in ref [4] for 2kW and 200kW. But in ref [4], the average winding dimensions has been used in the calculation of self and mutual inductances. However, when the coil dimensions become smaller or the number of turns increases, the inductance calculation with this method gives unacceptable incorrect results. Therefore, we need to make separate calculations for each turn of spiral coils. For this, magnetic flux is calculated for each side of each winding in the spiral rectangular coil and all the results are summed for mutual inductance calculation [5]. Thus, the gaps between each winding can also be taken into account. The optimized IPT system is expected to operate under any operating conditions. The equivalent impedance of the batteries change during charge and discharge. The bifurcation event occurs when the load changes or in the misalignment condition. Therefore, primary and secondary quality factors should be carefully determined. Critical parameters are maximum switching frequency, the maximum current density of the coils, and quality factors in IPT design. Optimal system design is decided by the proper combination of critical parameters [4]. In this study, in addition to other optimization parameters, the critical coupling value is also considered in IPT optimization to avoid bifurcation [6]. The input impedance has been observed at different frequencies when the amplitude of load (Fig1a) and the coupling changes (Fig1b). Optimized design parameters for the inductive power transfer link are given in Figure 2. A controlled rectifier is used on the secondary-side to increase the overall system efficiency. Details of the optimum IPT design and experimental results of the system will be provided in the final version of the paper.

**References:**

 [1] N. Campagna, V. Castiglia, R. Miceli and F. Pellitteri, "A Bidirectional IPT system for Electrical Bicycle Contactless Energy Transfer," 2019 8th International Conference on Renewable Energy Research and Applications (ICRERA), Brasov, Romania, 2019, pp. 1063-1068, doi: 10.1109/ICRERA47325.2019.8996640. [2] A. Ahmad, M. S. Alam and R. Chabaan, "A Comprehensive Review of Wireless Charging Technologies for Electric Vehicles," in IEEE Transactions on Transportation Electrification, vol. 4, no. 1, pp. 38-63, March 2018. DOI: 10.1109/TTE.2017.2771619. [3] D. Iannuzzi, L. Rubino, L. Pio Di Noia, G. Rubino, and P. Marino, “Resonant inductive power transfer for an E-bike charging station,” Elect. Power Syst. Res., vol. 140, pp. 631–642, Jun. 2016. [4] J. Sallan, J. L. Villa, A. Llombart and J. F. Sanz, "Optimal Design of ICPT Systems Applied to Electric Vehicle Battery Charge," in IEEE Transactions on Industrial Electronics, vol. 56, no. 6, pp. 2140-2149, June 2009. DOI: 10.1109/TIE.2009.2015359. [5] P. Venugopal, S. Bandyopadhyay, P. Bauer and J.A. Ferreira, “A Generic Matrix Method to Model the Magnetics of Multi-Coil Air-Cored Inductive Power Transfer Systems,” Energies 2017, 10, 774. DOI: 10.3390/en10060774 K. Aditya and S. S. Williamson, "Design Guidelines to Avoid Bifurcation in a Series–Series Compensated Inductive Power Transfer System," in IEEE Transactions on Industrial Electronics, vol. 66, no. 5, pp. 3973-3982, May 2019, doi: 10.1109/TIE.2018.2851953.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/1c4a13f7a42e04630cb0f05a5228ccc9.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/00bdfebee85e7778ca10dd23a5e6f330.jpg)

 Optimal IPT System Design Using Spiral Rectangular Coils for E-Tricycle Scooters Charging

In recent years, it is strongly required to establish a practical calculation method for iron loss of electric machines including magnetic hysteresis behavior, in order to develop high-efficient electric machines. However, magnetic hysteresis is not considered in general-purpose numerical analysis programs due to a long calculation time. To overcome the above problem, this paper proposes a reluctance network analysis (RNA) [1]. The RNA expresses an analytical object by one reluctance network. All the reluctances can be determined by B-H curve of the material and dimensions. The RNA has some advantages such as a simple model, fast calculation, and easy coupling with external electric circuit and motion equation. The RNA has been applied to the calculation of characteristics of various electric machines including transformers and motors. In previous papers, a novel RNA model incorporating a play model [2], which is one of the phenomenological models of magnetic hysteresis behavior, was presented. So far, this method was applied to the analyses of a permanent magnet synchronous motor [3] and a variable inductor [4], and its calculation accuracy was proved by comparing the measured and calculated values. However, the voltage and current waveforms are almost sinusoidal in these devices. Therefore, in this paper, the versatility of this method is experimentally demonstrated by using a switched reluctance (SR) motor, which has stronger nonlinear magnetic properties and is excited by a square voltage waveform, as an object of discussion. In this paragraph, a deriving method for the RNA model of the SR motor incorporating the play model is described [5]. First, the analytical object is divided into multiple elements. To allow consideration of leakage flux flowing between the stator pole and yoke, and between the adjacent stator poles, surrounding space is also divided. In the core region, each reluctance is needed to be determined by considering magnetic hysteresis. Thus, the dc hysteresis is represented by the play model, which can express an arbitrary hysteresis loop by multiplying play hysterons with different widths by shape functions. Although the play model generally requires a large number of measured dc hysteresis loops with different maximum flux densities to derive the shape functions, the proposed method requires only one or two measured dc hysteresis loops because the Landau-Lifshitz-Gilbert (LLG) equation is used to calculate the dc hysteresis loops [6]. Fig. 1(a) shows the hysteresis loops calculated by the LLG equation. As shown in this figure, a large number of hysteresis loops which are used to derive the play model can be obtained without experiments. On the other hand, the classical and anomalous eddy current losses are denoted by the inductance and controlled-source, respectively. In addition, since an SR motor has a salient pole structure, the magnetic flux distribution around the stator and rotor pole tips change dynamically with rotation. Thus, the magnetic circuit around poles and the air gap is represented by variable reluctances depending on the rotor position angle as shown in Fig. 1(b), so that the local magnetic saturation and the fringing magnetic flux can be taken into consideration. Fig. 1(c) shows a schematic diagram of the proposed RNA model incorporating the play model. The controlled-sources at the stator poles are the MMFs generated from the winding current. Using the proposed RNA model, the characteristics of the SR motor are calculated when the excitation voltage is 9 V, the excitation starting angle is 0 deg., and the excitation width is 70 deg. Fig. 2(a) and (b) show the measured and calculated winding rms current density versus torque and torque versus rotational speed characteristics, respectively. As shown in these figures, the measured and calculated values are in good agreement. Next, Fig. 2(c) and (d) show the measured and calculated iron loss and efficiency characteristics, respectively. In the experiment, iron loss is measured by subtracting the sum of mechanical output and copper loss, and mechanical loss measured separately in advance, from electrical input. From these figures it is clear that the measured and calculated values are in good agreement. Furthermore, Fig. 2(e) indicates calculated hysteresis loops in certain divided elements of the RNA model. As shown in the figure, the magnetic hysteresis inside the core, which is generally difficult to measure and calculate, can be drawn by the proposed method. In the future, we plan to verify the validity of such local hysteresis loops in the experiment. This work was supported by Grant-in-Aid for JSPS Fellows (JP19J20572).

**References:**

[1] C. P. Steinmetz, “On the law of hysteresis”, Proc. IEEE, Vol. 72, No. 2, pp. 197-221 (1984). [2] D. C. Jiles and D. L. Atherton, “Theory of Ferromagnetic Hysteresis”, J. Magn. Magn. Mater., Vol. 61, No. 1-2, pp. 48-60 (1986). [3] G. Friedman, “New Formulation of the Stoner-Wolfarth Hysteresis Model and the Identification Problems”, J. Appl. Phys., Vol. 67, No. 9, pp. 5361-5563 (1990). [4] G. S. Park, S. Y. Hahn, K. S. Lee, and H. K. Jung, “Implementation of Hysteresis Characteristics Using the Preisach Model with M-B Variables”, IEEE Trans. Magn., Vol. 29, No. 2, pp. 1542-1545 (1993). [5] S. Bobbio, G. Miano, C. Serpico, and C. Visone, “Models of Magnetic Hysteresis Based on Play and Stop Hysteresis”, IEEE Trans. Magn., Vol. 33, No. 6, pp. 4417-4426 (1997). [6] H. Tanaka, K. Nakamura, O. Ichinokura, “Magnetic Circuit Model combined with Play Model Obtained from Landau-Lifshitz-Gilbert Equation”, Journal of Physics: Conference Series, Vol. 903, 012047 (2017). [7] K. Nakamura and O. Ichinokura, “Reluctance Network Based Dynamic Analysis in Power Magnetics,” IEEJ Trans. FM, Vol.128, No.8, pp. 506-510 (2008). [8] K. Nakamura, K. Kimura, and O. Ichinokura, “Electromagnetic and motion coupled analysis for switched reluctance motor based on reluctance network analysis,” Journal of Magnetism and Magnetic Materials, Vol. 290-291, pp. 1309-1312 (2005).

**Images:**

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![](https://assets.underline.io/uploads/markdown_image/1/image/c72332b985e27ffec00705765b6035b5.jpg)

 Reluctance Network Model of Switched Reluctance Motor Considering Magnetic Hysteresis Behavior

In recent years, quantitative analysis of iron loss taking magnetic hysteresis behavior into account is essential to development of high-efficiency electric machines. In a finite element method (FEM)-based design and analysis, iron loss of electric machines is approximately calculated based on Steinmetz’s equation [1], although its calculation accuracy is not necessarily sufficient in the case of PWM and dc-biased excitation conditions. Thus, various hysteresis models have been proposed so far [2]-[4], in order to accurately simulate the magnetic hysteresis behavior. However, most of the general-purpose FEM programs neglect magnetic hysteresis since its calculation takes a lot of time in general. Therefore, establishment of a simple and practical method for calculating iron loss including magnetic hysteresis behavior is strongly required. In a previous paper, a magnetic circuit model incorporating a play model [5], which is one of the phenomenological models of magnetic hysteresis behavior, was proposed [6]. It was clear that the proposed model can calculate the hysteresis loop of the magnetic reactor with high-speed and high-accuracy. However, there is a possibility for further improvement in the calculation accuracy of the minor loop generated from carrier harmonics under PWM excitation because the influence of the skin effect cannot be considered. Therefore, in this paper, the magnetic circuit model is improved to be that the dynamic hysteresis characteristics including the skin effect can be considered, so that the minor loop can be calculated with higher accuracy. The validity of the improved model is proved by using ring cores made of various materials, such as grain-oriented (GO) and non-oriented (NO) silicon steels, an amorphous alloy, and a soft magnetic composite (SMC). Fig. 1(a) shows the previously proposed magnetic circuit model incorporating the play model. As shown in the figure, the dc hysteresis is represented by the play model. The play model can express an arbitrary hysteresis loop by multiplying play hysterons with different widths by shape functions. To derive the shape functions, the play model generally requires a large number of measured dc hysteresis loops with different maximum flux densities [1]. On the other hand, the classical eddy current loss and the anomalous eddy current loss are denoted by the inductance and the controlled-source, respectively. Fig. 1 (b) shows the comparison of the measured and calculated hysteresis loops under PWM excitation for each core material. This figure reveals that the previous magnetic circuit model can calculate the major loop with high accuracy, while there are errors in the shape of the minor loops. The cause of the errors is not to consider the influence of the skin effect by carrier harmonics. Therefore, in order to calculate minor loops with higher accuracy, the magnetic circuit model is improved as shown in Fig. 2(a). In the previous model, the eddy current loss is given by single inductance element. On the contrary, in the proposed model, the influence of the skin effect is represented by a ladder circuit in which inductances and resistances are repeatedly connected in parallel and in series. This ladder circuit represents the frequency characteristic of the complex permeability at higher frequencies as the number of circuit stages is larger. The parameters Lm and Rm of the circuit elements are given by the following equations: Lm = (4/σd^2)×(l/S) (1) Rm = (1/μ)×(l/S) (2) where the magnetic path length is l, the cross-sectional area is S, the conductivity is σ, the thickness of the silicon steel is d, and the permeability is μ, respectively. The validity of the proposed model is proved by comparing the measured and calculated hysteresis loops under PWM excitation, as shown in Fig. 2(b). In the analysis, the ladder circuit is terminated by the second inductance Lm1. From this figure, it is clear that the minor loops can be calculated with much higher accuracy comparing with the previous model for GO and NO silicon steels and a SMC. On the other hand, the calculation accuracy hardly changes for an amorphous metal since the values of σ and d is quite small. The cause of the error of the minor loops is considered to be the overestimated anomalous eddy current loss, which is one of the issues to be solved in the future. In the future, we plan to apply the proposed method to the analysis of reactors and transformers, which are used for switching power supplies, in order to calculate the iron loss considering magnetic hysteresis behavior including the minor loop with higher accuracy. This work was supported by Grant-in-Aid for JSPS Fellows (JP19J20572).

**References:**

[1] C. P. Steinmetz, “On the law of hysteresis”, Proc. IEEE, Vol. 72, No. 2, pp. 197-221 (1984). [2] D. C. Jiles and D. L. Atherton, “Theory of Ferromagnetic Hysteresis”, J. Magn. Magn. Mater., Vol. 61, No. 1-2, pp. 48-60 (1986). [3] G. Friedman, “New Formulation of the Stoner-Wolfarth Hysteresis Model and the Identification Problems”, J. Appl. Phys., Vol. 67, No. 9, pp. 5361-5563 (1990). [4] G. S. Park, S. Y. Hahn, K. S. Lee, and H. K. Jung, “Implementation of Hysteresis Characteristics Using the Preisach Model with M-B Variables”, IEEE Trans. Magn., Vol. 29, No. 2, pp. 1542-1545 (1993). [5] S. Bobbio, G. Miano, C. Serpico, and C. Visone, “Models of Magnetic Hysteresis Based on Play and Stop Hysteresis”, IEEE Trans. Magn., Vol. 33, No. 6, pp. 4417-4426 (1997). [6] H. Tanaka, K. Nakamura, O. Ichinokura, “Magnetic Circuit Model combined with Play Model Obtained from Landau-Lifshitz-Gilbert Equation”, Journal of Physics: Conference Series, Vol. 903, 012047 (2017).

**Images:**
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 Dynamic Hysteresis Modeling Considering Skin Effect for Magnetic Circuit Analysis

1 Introduction High-torque-density permanent-magnet (PM) motors are widely used in many industrial applications. They often work in overload situations. The magnetic saturation effect should hence be considered in the optimization process. These motors can have many slots and pole-pairs combinations, which makes optimization difficult. In the initial design step, analytical models allow a quick selection of the best candidates. However, most analytical models, such as subdomain models [3], are linear. The error of such models is important when the motor works in an overload situation. Researchers have provided hybrid models, which combine a reluctance network or a finite element model (FEM) with a subdomain model [5], [4]. These hybrid models are general and precise for different motors. However, they are slow, which is a major drawback compared to analytical models. Another quick nonlinear model is the surrogate model [1], [2]. The traditional surrogate model uses design parameters as the input and performance parameters as the output. Machine learning is used to build a mapping between input and output. However, PM motors always have many design parameters which may induce a large design parameters space. For this reason, this method can only build a surrogate model of the PM motors in a small local region of the design space. To overcome these drawbacks, we combine the subdomain and surrogate model into a new surrogate model assisted with a subdomain model (SMASM), which allows a very quick optimization and takes magnetic saturation into account. 2 Surrogate model assisted with a subdomain model (SMASM) We use more general features generated by the subdomain model to replace the design parameters as the input. The ratio of performance parameters between the linear (subdomain model) and nonlinear model (FEM) is used as output. Then, we use the Gaussian process regression (GPR) to build a mapping between them. We build a traditional surrogate model and SMASM which use the design parameters and features generated by the subdomain model as input separately. The ratio of the flux-linkage generated by PMs in no-load conditions between the linear and nonlinear model is selected as the output. Then, we use GPR to build two surrogate models. Finally, the test datasets are used to compare the performance between the two models. The topology of the test motors are surface-mounted PM (SPM) motors is shown in Fig. 1. The design parameters of the test motor include: 1. ro, the outer radius of the stator, 2. t, the thickness of yoke, 3. h, the depth of slot, 4. ri, the inner radius of the stator, 5. a=ri-ra, the air-gap, 6. tPM=ra-rPM, the thickness of PM, 7. w1, the ratio of the tooth, 8. w2, the ratio of tooth-tip, 9. Br, the remanence of PM, and 10. μr, the relative permeance of PM. The parameters include t, h, ri, a, and tPM divided by ro to scale motors of different sizes into similar motors. The μr does not change too much in different SPM motors. In the traditional method, 8 design parameters are needed to be selected for the training dataset except for ro and μr. However, in the SMASM, 5 features are selected which can contain the most information on these design parameters. 2.1 Training dataset for SMASM For SMASM, we select five features which include: 1. Btooth, the maximum of the magnetic density in the tooth during the rotation, 2. Byoke, the maximum of the magnetic density in the yoke during the rotation, 3. Φ/r2, the maximum of the flux in the tooth during the rotation divided by the radius of yoke r2=ro-0.5t, 4. P/r2, the equivalent air-gap P=tPM/μr+a divided by the radius of yoke r2, and 5. h/r2, the depth of the slot divided by the radius of yoke r2. The ranges of the features are: 1. Btooth from 1.5 T to 2.5 T; 2. Byoke from 1.5 T to 2.5 T; 3. Φ/r2 from 0.023 to 0.027; 4. P/r2 from 0.03 to 0.05; 5. h/r2 from 0.1 to 0.15. 2.2 Training dataset for the traditional method For a fair comparison, the five design parameters which change the features in the SMASM are selected correspondingly in the traditional method. They include w1/r2, t/r2, Br, tPM/r2 and h/r2. The range of these design parameters are: 1. w1/r2 from 1.0 to 1.5; 2. t/r2 from 0.15 to 0.20; 3. Br from 1.1 T to 1.3 T; 4. tPM/r2 from 0.02 to 0.04; 5. h/r2 from 0.1 to 0.15. 2.3 Test dataset The test datasets of SMASM and the traditional method have eight features. Five of them are inherited from the training dataset separately. The other three design parameters are a/r2, w2, and w3. We fix them in the training dataset and change them in the test dataset. The range of these design parameters are : 1. a/r2 from 0.005 to 0.015; 2. w2 from 0.45 to 0.75; 3. w3 from 0.7 to 1.0. 2.4 Model verification The prediction error distribution is shown in Fig. 2. The average and maximum errors of the traditional method are 6.5% and 21.3%. The average and maximum errors of the SMASM are 1.1% and 3.5%. Despite that only 5 features are used, the performance of the SMASM is better. 3 Conclusion The SMASM uses five features to generate the surrogate model which can reduce the size of the training dataset and keep the accepted precision. If we sample 4 points for every feature, we can reduce the size of the training dataset from 4^8=65536 to 4^5=1024 sample points. Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant 51851110761. Corresponding author: P.-D. Pfister (e-mail: pierredaniel.pfister.public@gmail.com).

**References:**

[1] Y. Fujishima, S. Wakao, A. Yamashita, T. Katsuta, K. Matsuoka, and M. Kondo. Design optimization of a permanent magnet synchronous motor by the response surface methodology. Journal of Applied Physics, 91(10):8305–8307, May 2002. [2] J. Gong, H. Zhang, B. Zhao, D. Fu, and F. Gillon. Proposal of a Bi-Objective Kriging Adapted Output Space Mapping Technique for Electromagnetic Design Optimization. IEEE Transactions on Magnetics, 55(6):1–5, June 2019. [3] P.-D. Pfister, X. Yin, and Y. Fang. Slotted permanent-magnet machines: General analytical model of magnetic fields, torque, eddy currents, and permanent-magnet power losses including the diffusion effect. IEEE Transactions on Magnetics, 52(5):1–13, May 2016. [4] M. Shen, P.-D. Pfister, C. Tang, and Y. Fang. A hybrid model of permanent-magnet machines combining Fourier analytical model with finite element method, taking magnetic saturation into account. IEEE Transactions on Magnetics, pages 1–5, 2020. [5] L. Wu, H. Yin, D. Wang, and Y. Fang. On-Load Field Prediction in SPM Machines by a Subdomain and Magnetic Circuit Hybrid Model. IEEE Transactions on Industrial Electronics, 67(9):7190–7201, September 2020.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/cbe3e0c43aeec5d9394cefae85d17f1f.jpg)
The topology of the test motors

![](https://assets.underline.io/uploads/markdown_image/1/image/4f3b9ff43dd40b27703f7df3128cd6dc.jpg)
Flux linkage produced by the PMs calculated by the traditional and the SMASM methods: prediction error distribution

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