France



Introduction&lt;br/&gt;MnZn ferrites are widely used in power electronics for high frequency transformers and inductors since they offer a good compromise between low core losses, permeability and saturation flux density. For properly designing magnetic components, the effective permeability and the core losses must be determined. Different phenomena, like magnetic hysteresis or eddy-currents, lead to core losses. State-of-the-art loss models are often based on the Steinmetz equation or extensions such as Generalized Steinmetz Equation [1]. These approaches aim to model the overall losses at once using a power law based on three (or more) coefficients. A given set of parameters is only accurate around the operating point used for determining these parameters, for the various phenomena scale differently with frequency and flux density. This results in reduced operating ranges in which the methods are accurate. In contrast, with the Bertotti method [2] hysteresis and classical eddy current losses as well as an extra term, called excess losses, are considered separately. The three terms are calculated as with the Steinmetz model but with specific coefficients for each loss term. Even if using three loss components widens the frequency and flux density ranges, the parameters remain frequency dependent and the excess losses lack a proper physical explanation. Therefore, this work presents a physical approach based on the combination of an electrical model of the material structure with the flux distribution resulting from solving Maxwell equations. The new model is used to explain and estimate the frequency dependency of complex permeability as well as the eddy-current and excess core losses that can be then combined with hysteresis losses predicted by classical methods.&lt;br/&gt;&lt;br/&gt;Equivalent electrical model of material structure&lt;br/&gt;The sintering process leads to a meta-material made of grains that are insulated one from each other. Such a structure reduces significantly eddy-current losses since the effective conductivity is small due to the insulating layer between grains. However, the apparent conductivity and permittivity of ferrite depend on frequency because the grain boundary interface acts like a capacitor (Fig 1.a). Its impedance reduces at higher frequencies what short-circuits the high resistivity of the boundary (cf. [3] for NiZn ferrites and [4] for MnZn ferrite).&lt;br/&gt;&lt;br/&gt;As proposed in [3], a two series cells model could be used to describe the grain and the boundary (Fig. 1.b) for calculating the overall conductivity and permittivity. Each cell is composed of a parallel conductance and capacitance, representing for one cell the admittance of the grains and for the other cell the admittance of the boundaries. By taking the limit of the admittances when the cross section d&lt;i&gt;A&lt;/i&gt; and the length d&lt;i&gt;l&lt;/i&gt; converges to zero, one can obtain the apparent conductivity and permittivity. These quantities are then a function of the frequency and of the material parameters, i.e.: conductivities and permittivities of the boundary and of the grain.&lt;br/&gt;&lt;br/&gt;Model of flux distribution vs. frequency&lt;br/&gt;The dielectric and ferromagnetic nature of ferrite results in standing waves in the core when the frequency increases. This affects the flux density distribution and induces eddy-currents. To model those effects, two approaches can be considered: lumped elements circuit [5] or full wave equation solution [6]. The first one assumes that induced currents flow only around the cross section in concentric loops, which is satisfied if only the first mode is excited.&lt;br/&gt;A more general approach is to solve the wave equations, as proposed in [6] for toroid and pot cores. The solution given in [6] converges quickly with respect to the number of considered spatial harmonics, reducing the computational cost.&lt;br/&gt;&lt;br/&gt;Proposed model&lt;br/&gt;Based on the electrical model of the material structure and on the solution for the flux distribution, a new core loss model is presented in this paper. This model allows to compute the complex permeance of a toroid core, even if the electrical material parameters of the material change with frequency. The effective complex permeability is derived by dividing the complex permeance by the ratio of the effective cross section over the effective magnetic length of the core. The overall core losses are calculated by adding the losses associated with the imaginary part of the permeance to the hysteresis losses. The latter are obtained from low frequency measurements and are then scaled linearly with frequency (similar to Bertotti model).&lt;br/&gt;&lt;br/&gt;Experimental validation&lt;br/&gt;The proposed approach is validated on a R36 toroid core made of N87. The complex effective permeability and the core losses are measured between 10 kHz and 1 MHz for various flux densities, at room temperature using an IWATSU SY-8219 BH Analyzer. The model is then fitted with complex permeability measurements using MATLAB curve fitting toolbox.&lt;br/&gt;&lt;br/&gt;Results and discussion&lt;br/&gt;As shown in Fig 2, the presented model captures accurately the variations of both the complex permeability and the core losses. Furthermore, it can be concluded that the so-called excess losses are due to additional eddy-current losses due to the excitation of wave modes in the core and to the increasing effective conductivity with frequency.

**References:**

[1] J. Li, T. Abdallah and C.R. Sullivan, Conference Record of the 2001 IEEE Industry Applications Conference. 36th IAS Annual Meeting, IEEE, (2002)&lt;br/&gt;[2] G. Bertotti, IEEE Transactions on Magnetics, Vol. 24, p. 621-630 (1988)&lt;br/&gt;[3] C. G. Koops, Physical Review, Vol. 83, p. 121-124 (1951)&lt;br/&gt;[4] M. Kacki, M.S. Rylko, J.G. Hayes and al., 2020 IEEE Applied Power Electronics Conference and Exposition (APEC) (2020)&lt;br/&gt;[5] C.R. Sullivan, PSMA 2019 Magnetics Workshop (2019)&lt;br/&gt;[6] W. Hauser, Modellbildung für strukturabhängige Effekte in Ferritkernen (2018)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7b6236946e32bac2c3dcdb7b998f4ba2.jpg)
Fig. 1: a) The granular microstructure of Ferrites leads to a capacitive behaviour that can be represented by a 1D model. b) Equivalent two cells model for the permittivity and conductivity of ferrites.

![](https://assets.underline.io/uploads/markdown_image/1/image/07e6234e666009d4a326fcb26db45258.jpg)
Fig. 2: Loss density against frequency for various flux density values.

INTERMAG 2021

 New Core Loss Model for Ferrite Cores Based on a Meta-Material Approach

Introduction MnZn ferrites are widely used in power electronics for high frequency transformers and inductors since they offer a good compromise between low core losses, permeability and saturation flux density. For properly designing magnetic components, the effective permeability and the core losses must be determined. Different phenomena, like magnetic hysteresis or eddy-currents, lead to core losses. State-of-the-art loss models are often based on the Steinmetz equation or extensions such as Generalized Steinmetz Equation [1]. These approaches aim to model the overall losses at once using a power law based on three (or more) coefficients. A given set of parameters is only accurate around the operating point used for determining these parameters, for the various phenomena scale differently with frequency and flux density. This results in reduced operating ranges in which the methods are accurate. In contrast, with the Bertotti method [2] hysteresis and classical eddy current losses as well as an extra term, called excess losses, are considered separately. The three terms are calculated as with the Steinmetz model but with specific coefficients for each loss term. Even if using three loss components widens the frequency and flux density ranges, the parameters remain frequency dependent and the excess losses lack a proper physical explanation. Therefore, this work presents a physical approach based on the combination of an electrical model of the material structure with the flux distribution resulting from solving Maxwell equations. The new model is used to explain and estimate the frequency dependency of complex permeability as well as the eddy-current and excess core losses that can be then combined with hysteresis losses predicted by classical methods. Equivalent electrical model of material structure The sintering process leads to a meta-material made of grains that are insulated one from each other. Such a structure reduces significantly eddy-current losses since the effective conductivity is small due to the insulating layer between grains. However, the apparent conductivity and permittivity of ferrite depend on frequency because the grain boundary interface acts like a capacitor (Fig 1.a). Its impedance reduces at higher frequencies what short-circuits the high resistivity of the boundary (cf. [3] for NiZn ferrites and [4] for MnZn ferrite). As proposed in [3], a two series cells model could be used to describe the grain and the boundary (Fig. 1.b) for calculating the overall conductivity and permittivity. Each cell is composed of a parallel conductance and capacitance, representing for one cell the admittance of the grains and for the other cell the admittance of the boundaries. By taking the limit of the admittances when the cross section dA and the length dl converges to zero, one can obtain the apparent conductivity and permittivity. These quantities are then a function of the frequency and of the material parameters, i.e.: conductivities and permittivities of the boundary and of the grain. Model of flux distribution vs. frequency The dielectric and ferromagnetic nature of ferrite results in standing waves in the core when the frequency increases. This affects the flux density distribution and induces eddy-currents. To model those effects, two approaches can be considered: lumped elements circuit [5] or full wave equation solution [6]. The first one assumes that induced currents flow only around the cross section in concentric loops, which is satisfied if only the first mode is excited. A more general approach is to solve the wave equations, as proposed in [6] for toroid and pot cores. The solution given in [6] converges quickly with respect to the number of considered spatial harmonics, reducing the computational cost. Proposed model Based on the electrical model of the material structure and on the solution for the flux distribution, a new core loss model is presented in this paper. This model allows to compute the complex permeance of a toroid core, even if the electrical material parameters of the material change with frequency. The effective complex permeability is derived by dividing the complex permeance by the ratio of the effective cross section over the effective magnetic length of the core. The overall core losses are calculated by adding the losses associated with the imaginary part of the permeance to the hysteresis losses. The latter are obtained from low frequency measurements and are then scaled linearly with frequency (similar to Bertotti model). Experimental validation The proposed approach is validated on a R36 toroid core made of N87. The complex effective permeability and the core losses are measured between 10 kHz and 1 MHz for various flux densities, at room temperature using an IWATSU SY-8219 BH Analyzer. The model is then fitted with complex permeability measurements using MATLAB curve fitting toolbox. Results and discussion As shown in Fig 2, the presented model captures accurately the variations of both the complex permeability and the core losses. Furthermore, it can be concluded that the so-called excess losses are due to additional eddy-current losses due to the excitation of wave modes in the core and to the increasing effective conductivity with frequency.

**References:**

[1] J. Li, T. Abdallah and C.R. Sullivan, Conference Record of the 2001 IEEE Industry Applications Conference. 36th IAS Annual Meeting, IEEE, (2002) [2] G. Bertotti, IEEE Transactions on Magnetics, Vol. 24, p. 621-630 (1988) [3] C. G. Koops, Physical Review, Vol. 83, p. 121-124 (1951) [4] M. Kacki, M.S. Rylko, J.G. Hayes and al., 2020 IEEE Applied Power Electronics Conference and Exposition (APEC) (2020) [5] C.R. Sullivan, PSMA 2019 Magnetics Workshop (2019) [6] W. Hauser, Modellbildung für strukturabhängige Effekte in Ferritkernen (2018)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7b6236946e32bac2c3dcdb7b998f4ba2.jpg)
Fig. 1: a) The granular microstructure of Ferrites leads to a capacitive behaviour that can be represented by a 1D model. b) Equivalent two cells model for the permittivity and conductivity of ferrites.

![](https://assets.underline.io/uploads/markdown_image/1/image/07e6234e666009d4a326fcb26db45258.jpg)
Fig. 2: Loss density against frequency for various flux density values.

technical paper

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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:**

![](https://assets.underline.io/uploads/markdown_image/1/image/ac8bb34a39dad2b2f59bcfb3313e1f0a.jpg)

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 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:**
![](https://assets.underline.io/uploads/markdown_image/1/image/0a98330396b77a10e8ab775bd0ae4576.jpg)

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 Dynamic Hysteresis Modeling Considering Skin Effect for Magnetic Circuit Analysis

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

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

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

 A surrogate model assisted with a subdomain model for permanent-magnet machine

I. INTRODUCTION Coreless axial-flux permanent-magnet synchronous machines (AFPMSMs) have many advantages such as high power density, simple structure, high overload capability, no cogging torque and so on. It has been used more and more widely in many applications, such as flywheel energy storage systems, wind energy power generation, electric vehicles, and ultra-precision positioning and processing areas [1-3]. However, it is difficulty to accurate modeling of AFPMSM as 3D electromagnetic problems, such as the curvature effect and fringing effect. The electromagnetic characteristics, such as air-gap flux density, back electromotive force (EMF), and electromagnetic torque, are critical in motor design. Precise design methods and fast performance estimations for coreless AFPMSM would be necessary. The most accurate method to design the AFIPMM is the 3D finite-element analysis, which is exceedingly time-consuming [4-5]. To solve this problem, a three-dimensional analytical approach, based on the volume integral method and second order vector potential (SOVP), is developed to calculate the coreless AFPMSM’s steady-state and transient performances. II. ANALYTICAL MODEL The 3-D geometric model of the coreless AFPMSM is shown in Fig. 1 (a). This coreless AFPMSM consisting of one stator with ironless winding and two rotors with Halbach magnet array mounted on nonmagnetic supports. Such a Halbach rotor can be modeled using equivalent current sheets, thereby enabling the field at any point in 3-D space to be calculated using the Biot–Savart law [3]. For instance, the axial magnetized magnet can be created by the radial and circumference current filament model, as shown in Fig. 1(b), and then a volume integral approach is proposed to obtain the field for the current sheet shown as in Fig. 1(c) and ironless winding. The calculation of the magnetic field using the integral equations is obtained in very short times thanks to the use of particular volumetric basis elements for which the solution is obtained analytically. The field solution from volume integral approach is then used as the magnetic charge source term for an equivalent magnetic charge model. The replace function of the field for wingding and Halbach rotor were given by Fourier series, thus the program difficulty (not shown here due to space limits) for three-dimensional distribution of the magnetic field will be reduced. The 3-D torque equations are derived using a magnetic charge sheet approach that enables the final torque equation to be expressed by a single equation without the need for a large number of summations. III. RESULTS AND VALIDATION The accuracy of the approach was validated 3D finite element analysis shown as in Fig 1 (d). The flux density calculated with the proposed approach are in good accordance with that obtained by 3D FEM simulation.Currently, we have designed and established a coreless AFPMSM prototype in laboratory as shown in Fig.2. Performance of the AFPMSM such as the flux density and torques are measured with the prototype. The experiment results are validated by FEA and proposed approach results and they will be included in the full paper. IV. CONCLUSIONS A three-dimensional analytical approach based on the volume integral method and magnetic charge sheet for the flux density and torque calculations in a coreless AFPMSM. These results will be compared to the 3D FEA and experiment results in the full paper.

**References:**

[1] Z. Kohari, Z. Nadudvari, L. Szlama, M. Keresztesi, and I. Csaki, IEEE Trans. Appl. Supercond., vol. 21, no. 3,pp. 1497–1501. (2011). [2] T. F. Chan, L. L. Lai, IEEE Trans. Energy Convers., vol. 22, no. 1, pp. 86–94. (2007). [3] N. Taran, V. Rallabandi, G. Heins, and D. M. Ionel, IEEE Trans. Ind. Appl., vol. 54, no. 6, pp. 5907–5917. (2018). [4] Xu L, Xu Y, Gong J. . IEEE Transactions on Magnetics,vol. 54, no. 11, pp. 1-5.(2018). [5] Daghigh A, Javadi H, Torkaman H. .IEEE Transactions on Magnetics,vol.52,no.9.(2016)．

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9f6c252054a481f964baaafc996e4475.jpg)
Fig.1（a）Structure of coreless AFPMM（b）Equivalent current sheet model for a axial magnet piece ( c）Contour plot for the magnetic field by proposed method (d)Comparison of FEA and calculated results for flux density

![](https://assets.underline.io/uploads/markdown_image/1/image/74e017f083f724d22d670552f199140e.jpg)
Fig.2 Experimental setup to test the coreless AFPMM

 3-D Analytical Modeling and Optimization of Axial Flux Coreless PM Motor

Introduction In recent years, SMPMSM is quietly famous compared with other conventional motors in industrial applications due to higher efficiency, higher reliability, low cost, and smooth torque [1]–[3]. In general, several objective optimizations need to be consider, which required an optimization technique for an electrical motor. This paper deals with an optimal design of a surface-mounted permanent magnet synchronous machine (SMPMSM) with an exact analytical subdomain model by using a genetic algorithm method. To analyze the characteristic of permanent magnet (PM) motors, the classical optimization method, such as the finite element method (FEM), is intensively used. However, FEM has several time problems that require a longer computational time to evaluate the performance of PM motors. This problem can be overcome by using a genetic algorithm (GA) method combined with a subdomain model (SD), which developed an improved performance of SMPMSM, for instance, total harmonic distortion (THDv) and phase back-EMF. In this design, a three-phase 12-slot/8-pole PM motor is established with an exact SD model with radial magnetization patterns. Then, the GA ensemble with SD model to search the optimality of SMPMSM machine design. In the final analysis, the optimal new design of SMPMSM demonstrated by comparing with the initial design that is investigated by FEM. The result of induced back-EMF, and magnetic flux density of optimal design is compared with the initial design to show the advantages of GA optimization method. Analytical Model, Genetic Algorithm and Parameter of optimization The purpose of this analysis is to investigate the optimization design of a three-phase 12-slot/8-pole SMPMSM with a radial magnetization pattern. This optimization problem has been solved by obtains the minimum THDv as a fitness function. In order to process the optimization design of SMPMSM, several parameters need to be optimized i.e magnet pole-pitch, αp and slot-opening angle, boa. The roulette method is used to select the objective function of optimization with 50 maximum generations per iteration. The cross and mutation probability need to be set which requires linear and uniform recombination which is 0.89 and 0.025 respectively. The parameters of 12-slot/8-pole SMPM motor as follows: radius of the rotor, Rr = 23mm, the radius of PM surface Rm = 26mm, the inner radius of the stator Rsb = 42.5mm, axial length, la = 50mm, air-gap length, lg = 1mm, remanence of the magnet, Br, 1.12T, relative coil permeability, µr = 1.05 with the number of pole pairs, Np is 8 and number of stator slots, Ns is 12. The result, Discussion and Conclusion The phase back electromotive force has a significant relationship with flux-linkage because phase back electromotive force is derived from the rate of change of flux-linkage. In Fig. 1. show the waveform of phase back electromotive force against rotor position of initial and optimal design in radial magnetization patterns. The initial design of phase back electromotive force waveform shows more trapezoidal compare with the optimal design which has a sinusoidal waveform. The comparison of total harmonic distortion between the initial and optimal design of SMPMSM are demonstrated in Fig. 2 below. The result from Fig.2. shows the highest total harmonic distortion of 12-slot/8-pole motor is an initial design which is 11.34% compared with the optimal design where 1.3%. The magnitude of harmonic in phase back electromotive force |En| of initial design shows 22.24 V compare to optimal design which is 17.4 V. This shows a good agreement that the optimal motor design gives lowest total harmonic distortion compare with the initial design. The optimizes of SMPMSM motor with the following parameters are given in Table 1.

**References:**

[1] S. Wang et al., “Multilevel optimization for surface mounted PM machine incorporating with FEM,” IEEE Trans. Magn., vol. 45, no. 10, pp. 4700–4703, 2009, doi: 10.1109/TMAG.2009.2024875. [2] K. Sindhya, A. Manninen, K. Miettinen, and J. Pippuri, “Generator Using Interactive,” vol. 64, no. 12, pp. 9776–9783, 2017. [3] H. Li, Z. Chen, and H. Polinder, “Optimization of multibrid permanent-magnet wind generator systems,” IEEE Trans. Energy Convers., vol. 24, no. 1, pp. 82–92, 2009, doi: 10.1109/TEC.2008.2005279.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/26ad63fb00869f77cdf94d41e4a9e659.jpg)
The comparison of phase back electromotive force between initial and optimal design of 12-slot/8-pole SMPMSM under radial magnetization pattern.

![](https://assets.underline.io/uploads/markdown_image/1/image/c8181ba507618ea87f3870d3996c04bf.jpg)
The comparison of total harmonic distortion between initial and optimal design of 12-slot/8-pole SMPMSM under radial magnetization pattern.

 Genetic algorithm optimal design of SMPMSM using analytical subdomain model

Closed core magnetic circuit components have long been the most difficult circuit component to miniaturize. Loss of cross sectional core area and available space to wrap windings trades off inductance as the component becomes smaller. For both circuit compactness and high frequency performance, there has been interest in the embedding of magnetic core materials within planar substrate (printed circuit board, PCB) and fabricating the windings lithographically [1-5]. This allows precise winding fabrication with control of capacitances, as well as being able to reclaim the circuit board area of the formerly surface mounted package. However, high turn numbers are difficult to achieve using PCB via fabrication methods. In standard PCB via fabrication, via diameters are limited by the aspect ratio of the hole that needs to be drilled. In-situ via fabrication in full 1.6mm thickness boards is limited to about a 1:8 aspect ratio. This translates to a ~200 micron via diameter, with a required via-to-via gap of about the same. This limits the number of turns that can be wrapped around the embedded core. Wire-wound magnetic components can use wire with diameters of 50 microns, and gaps equal to the varnish thickness of a couple of microns; wire-wound components can fit significantly more winding turns than embedded components using PCB via fabrication, and thus will have a much higher inductance for the same sized core. In this digest, an alternative ex-situ via forming method is presented, which uses embedded z-conductive via plugs, to fabricate embedded transformers. The effective aspect ratio is theoretically unlimited, but is 1:40 in the process presented here, competitive with wire-wound components. The general embedding process has six steps (Fig. 1): 1. Mill pocket for via plug, 2. Insert via plug and embed with epoxy, 3. Planarize both sides to reveal plug faces, 4. Mill pocket for magnetic core material, 5. Embed magnetic core with epoxy and planarize, 6. Double sided photolithography to form windings. The ex-situ fabricated via plug is the enabling part of the process; the plug is z-conductive in that it electrically conducts along the plug’s length, but is electrically isolating over the plug’s cross-section. When the plug is embedded in the PCB, the plane of the PCB is considered the xy plane, and the thickness of the PCB is in the z-direction. The major advantage of this process compared to alternative ex-situ via fabrication methods is how there is no precise pattern in the z-conductive plug. The plug, ideally, creates an xy continuum of z-direction conduction. Interlayer electrical connections occur wherever a circuit pattern on the top layer overlaps the circuit pattern on the bottom layer; precise alignment to the via plug is not required. Thus, the resolution of the lithography, not the via fabrication, is what determines the maximum number of turns on the winding. To fabricate the z-conductive plugs, commercial litz wire (660 strand bundle/ 46 American wire gauge strands) is epoxy cast into a bar, then cut into 1.6mm thick segments (“plugs”, fig. 2b and c). The 46 gauge strands effectively create 40µm diameter vias (fig. 2c), with a strand-to-strand gap of a few microns. A racetrack core [1, 6] size of 2.1mm outer diameter, 1.0mm inner diameter, leg length of 6.25mm, and a thickness of 1.0mm (fig. 2) is used for embedding. After step 5 (fig 1), the samples are polished with 1 micron slurry to reduce surface roughness and repair surface smearing of the copper litz wire. The transformer winding design has 14 turns per winding (200 µm lines, fig. 2). The lithography process developed is an etch process, where copper is deposited over the entire sample, after step 5, by e-beam evaporation or wet chemical plating. A photolithography process using microelectronics tools was developed, but is difficult to perform given the bulky board materials and the samples having a surface that was not precisely planar. One non-functioning 14-turn sample was completed using photolithography. As a novel alternative approach, spray paint was applied to a sample as an etch mask, and a 3D laser writer, which is less sensitive to surface non-uniformity, was used to ablate the negative image of the 14-turn winding pattern. The winding pattern was successfully etched using this robust approach, but windings were found to be consistently shorting. To help trace the path of the current shorting through the windings, a magneto-optical garnet was used, where a DC current was applied through the shorting path, and the garnet was placed on the windings and observed through a polarizing light microscope. The garnet indicates optically the magnetic fields emanated by the current, showing the shorting current path (Fig. 2d). To correct the shorting defect, high DC current was applied to thermally decompose the assumed high-resistance shorting anomaly. The approach effectively opened the shorting path, but collateral damage occurred destroying the device. In summary, we present this work as innovative both in approach and diagnostic technique. Support for this project has ended and work will not continue, but others working with the same miniature, high-frequency, high-inductance application requirements should consider this approach.

**References:**

References:: [1] D. Bowen, D. Basu, C. Krafft and I. Mayergoyz, "Fabrication and Evaluation of PCB-Embedded Broadband Signal Transformers With Custom Machined Racetrack-Shaped Ferrite Cores for Ethernet Applications," in IEEE Transactions on Magnetics, vol. 53, no. 11, pp. 1-5, Nov. 2017, Art no. 8401805 [2] D. Bowen, A. Lee, C. Krafft and I. D. Mayergoyz, "Design Control of Performance in Nested and Interleaved Winding Printed Circuit Board Transformers for Ethernet Applications," in IEEE Transactions on Magnetics, vol. 49, no. 7, pp. 4013-4016, July 2013. [3] D. Bowen, A. Lee, C. Krafft and I. D. Mayergoyz, “Zero-footprint Ethernet transformers using circuit-board embedded ferrites,” Journal of Applied Physics 115, 17E716 (2014) [4] J. Quilici, “Embedded magnetics technology overview,” see http://www.radial-e.com/?page_id=2for Radial Electronics Technology Brief. [5] “Planarmag Product Technology From TE Brings Automated Embedded Magnetic Products To The Ethernet Market,” Electronic Specifier, Feb. 2011, retrieved from https://www.electronicspecifier.com/communications/new-planarmag-product-technology-from-te-brings--automated-embedded-magnetic-products-to-the-ethernet-market Jan. 2020 [6] D. Bowen, C. Krafft and I. D. Mayergoyz, “Inductance optimization of miniature Broadband transformers with racetrack shaped ferrite cores for Ethernet applications,” AIP Advances 7, 056663 (2017)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d566179c3f82a8d154a69afc5295cd28.jpg)
Figure 1. Process for embedding a Z-conductive plug to fabricate a magnetic circuit component.

![](https://assets.underline.io/uploads/markdown_image/1/image/bea5a391e76230077b3160e1c07ae413.jpg)
Figure 2. 14-turn racetrack core transformer (a) Layout (b) Top Image (c) Image of turns overlapping via plug (d) Magneto-optic image of current flowing between primary and secondary windings

New Core Loss Model for Ferrite Cores Based on a Meta-Material Approach

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