Title Analysis of Litz Using Homogenization-Based FEM in Conjunction With Integral Equation

Author(s) Hiruma, Shingo; Otomo, Yoshitsugu; Igarashi, Hajime

IEEE Transactions on Magnetics, 54(3), 7001404 Citation https://doi.org/10.1109/TMAG.2017.2772335

Issue Date 2018-03

Doc URL http://hdl.handle.net/2115/68966

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Eddy Current Analysis of Litz Wire Using Homogenization-based FEM in Conjunction with Integral Equation

Shingo Hiruma1, Yoshitsugu Otomo1, Hajime Igarashi1, Member, IEEE

1Graduate School of Information Science and Technology, Hokkaido University, 060-0814 Sapporo, Japan

A new method is introduced to evaluate the macroscopic permeability of a litz wire which is composed of stranded conductors. In this method, an integral equation is solved for the complex magnetization in the litz wire generated due to the proximity effect. The macroscopic permeability computed from the magnetization is used in the homogenization-based finite element analysis of eddy currents in a litz-wire coil. It is shown that the wire twist has little effect on the complex permeability.

Index Terms—Homogenization method, integral equation, litz wire, macroscopic permeability, Ollendorff formula

litz wire. Using the proposed method, we evaluate the I. INTRODUCTION dependence of the macroscopic permeability on the pitch of the N electric machines and devices, materials which involve litz wire. Moreover, the numerical results are compared with I small structures such as litz wire, steel sheet, dust core, and measured values. so on, are widely used to reduce the eddy current loss. In recent years, analysis of eddy current losses in these materials has been II. COMPLEX PERMEABILITY OF STRANDED required because of increase in speed of the switching A. Complex permeability of a round wire frequency of power electronics devices. When analyzing those materials by the space-discretization method such as the finite When a wire is immersed in time-harmonic magnetic field element method (FEM), the resultant equation can be extremely which is perpendicular to the wire axis, anti-parallel eddy large because they have to be discretized into huge number of currents are induced along the wire. Because the eddy currents elements that are smaller than the skin depth. generate the diamagnetic field, the effective permeability of the The homogenization-based FEM has been shown effective to wire becomes lower than its specific permeability. Moreover, such analysis [1-7]. In [5], complex permeability has been the eddy currents give rise to Joule losses. These effects can be introduced for the analysis of multi-turn coils. The macroscopic expressed by the complex permeability. The relative complex permeability of a coil region is obtained by substituting the permeability of a round wire is given by [5] complex permeability of a round conductor to the extended Ollendorff formula. The homogenization method allows us to J1z r  r (1) avoid discretization of the components in fine-structured zJ1z material into small finite elements because it is possible to treat it as uniform material. However, in the homogenization-based where 푧 = (1 − 푗)/훿 훿 denotes the skin depth 퐽1(푧) FEM, the stranded structure of a litz wire has been ignored. represents the first-order Bessel function and 휇푟 is the relative In [8], to consider the stranded structure, an integral equation permeability of the wire. Using this permeability, we can method to analyze the eddy current losses in the stranded wire consider the eddy currents losses due to the proximity effect has been proposed by the authors. In this method, the integral without fine discretization of the wire cross section. equation is solved for the magnetization in the wire generated by the proximity effect. In this paper, we show that the B. Periodic integral equation macroscopic permeability of the litz wire can be evaluated The anti-parallel eddy currents along stranded wires induced using the integral equation method. To do so, uniform magnetic by a uniform magnetic field can be obtained by the integral field is imposed to a litz wire, and the integral equation is solved equation method in which a wire is modeled as one-dimensional to evaluate the eddy currents in the litz wire. Then, the curve [8]. Assuming that wires have periodic structure, the macroscopic permeability is derived from the computed around the wire can be obtained by magnetic field for the following FE analysis. We discuss considering a unit cell under the periodic boundary condition. validation of the proposed method by comparing the results The basic idea of the integral equation method is that the anti- obtained by the proposed method and conventional FEM for a parallel eddy currents in the wire due to the proximity effect generate a dipole field. The source of the dipole field can be Manuscript received April 1, 2015; revised May 15, 2015 and June 1, regarded as a magnetization in the wire. Therefore, if the 2015; accepted July 1, 2015. Date of publication July 10, 2015; date of current distribution of the magnetization is obtained, the anti-parallel version July 31, 2015. Corresponding author: S. Hiruma (e-mail: eddy currents in the wire can be determined. [email protected]). If some authors contributed equally, write here “Y. Otomo and H. Igarashi contributed equally.” IEEE TRANSACTIONS In the following, we derive the integral equation for self- ON MAGNETICS discourages courtesy authorship; please use the consistently determining the distribution of the magnetization Acknowledgment section to thank your colleagues for routine contributions. in stranded wires. Let us consider the complex magnetization Digital Object Identifier (inserted by IEEE).

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푴 perpendicular to the wire axis generated by the anti-parallel a unit cell eddy currents. Considering the eddy currents, the magnetization d is expressed in terms of the magnetic field 푯 and 휇̇ as

M   r 1H (2)

The magnetic field 푯 is composed of the components due to the 푗휔푡 푯0푒 complex magnetization vector 푴 distributed along the wire and 푗휔푡 푯0푒 the uniform magnetic field 푯0. Then we obtain the following Fig.1. Litz wire immersed in time-harmonic magnetic field. integral equation from (2) for 푴 [8]:

   Mx    τ   GMdv  H x  τ (3a)  0    r 1     c   1  Mx Mx RR M    3 (3b) G  3 5  4  R R  a unit cell

푯푒푗휔푡 where Ω푐 is the wire domain in which the magnetization 푴 exists, 풙 and 풙′ are the observation and source points in Ω , 흉 Fig.2. External magnetic field 푯 is imposed to a bundle of litz wires. If we 푐 remove a unit cell, there left Lorentz field which is generated by 푯 and the is the tangential unit vector parallel to the wire axis at 풙, 푹 = magnetization of the other unit cells. 풙 − 풙′, 푅 = |푹|, and G denotes the operator defined by (3b). for the Lorentz field: Then, we consider the stranded wires which have periodic structure in the direction of the wire axis as shown in Fig.1. Due to the periodic structure, the magnetization satisfies Hloc  H  N Mave (6)

Mx  nd   Mx (4) where 푯 represents the external magnetic field, which is ̅ imposed on the entire cells, 푁 denotes the effective diamagnetic where 푛 denotes integer and 풅 is the distance vector between constant of a unit cell, and 푴푎푣푒 is the averaged magnetization the adjacent unit cells. The integral equation for the wire with over a unit cell. The computation of 푁̅ is given in Appendix A. periodic structure is, thus, can be expressed as Then the averaged magnetic flux density over the unit cell can be written as

   Mx   B   H  M  (7)  τ   GM dv  H x  τ (5a) ave 0 ave  n 0    r 1     nZ unit     On the other hand, by introducing the macroscopic permeability 1  Mx Mx R R  〈휇̇〉, 푩푎푣푒 can be written as GM    3 n n (5b) n  3 5  4  Rn Rn  B    M (8) ave r 0 ave where Ω푢푛푖푡 represents the repeating unit of Ωc , 푹푛 = 풙 − ′ Eliminating 푯 from (6), (7), and (8), we have 풙 − 푛풅, 푅푛 = |푹푛|. The summation in (5a) is terminated at adequate number considering both accuracy and computational 1 time. In the following computations, we evaluate 11 terms.   (9a) r M C. Macroscopic permeability 1  ave 0 B To perform homogenization of a bundle of litz wires, we ave B   H  N M  M (9b) derive its macroscopic permeability from 휇̇푟. To do so, we use ave 0  loc ave ave  푴 obtained by solving (5a) as will be described below. 푗휔푡 Let the time-harmonic uniform magnetic field 푯0푒 , Note that this permeability is valid under the assumption that angular frequency ω, be imposed on the unit cells as shown in the magnetic field imposed on the litz wire is uniform. Fig.1. We can obtain the distribution of 푴 along the wires by solving the periodic integral equation (5a). The uniform III. NUMERICAL RESULTS magnetic field 푯0 in (5a) can be regarded as Lorentz field 푯loc A. Validation of the proposed method when we consider the unit cell which contains a bundle of To validate the proposed method, a 3D model of a litz wire stranded wire as shown in Fig.2. The following relation holds shown in Fig.3 is analyzed by conventional FEM and the

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proposed method. The model is composed of 16 wires wire . mm radius 푎 =0.15 mm pitch 15 mm conductivity σ = 5.76×107 S/m and relative permeability 휇r = 1.0. 15 mm In the FE analysis, the uniform magnetic flux density is imposed to the unit cell. The boundary conditions are . mm summarized in TABLE.1. The macroscopic permeability is computed from the following equation: 푦 푥

2 B 푧 j ave dv     unit (10) r 2 Fig.3. A unit cell of litz wire model composed of 16 wires. B 2 j dv  E dv unit  unit 1.2 0.00 1.0 -0.02 0 1 2 0.8 -0.04 where 푩푎푣푒 denotes the averaged magnetic flux density over -0.06 the domain. 0.6 -0.08 TABLE.1. BOUNDARY CONDITION OF FE MODEL 0.4 -0.10 boundary planes boundary condition partReal 3DFEM 0.2 integral equation -0.12 푦푧 Dirichlet part Imaginary -0.14 3DFEM 푥푧 Neumann 0.0 -0.16 integral equation 0 1 2 푥푦 Periodic a/d a/d Fig.4. Real (left) and imaginary (right) parts of macroscopic permeability 〈 〉 The results are shown in Fig.4. The horizontal axis is the wire 휇̇ are plotted against the normalized wire radius. radius normalized to the skin depth δ. We can see that 〈휇̇〉 0.00 obtained by the proposed method is in good agreement with that 0 0.5 1 1.5 2 obtained by 3D FEM when 푎 < 훿 . The ratio of the -0.05 computational time of the proposed method to that of FEM is about 1.4×10−2 for one frequency sample. The proposed -0.10 10 [mm] method is, therefore, more useful to evaluate the dependence of -0.15 15

〈휇̇〉 on number of wires, wire pitch and strand structure. Imaginary part 20 30 -0.20 B. Dependence of 〈휇̇〉 on pitch 50 100 The dependence of 〈휇̇〉 on the pitch of a litz wire is evaluated -0.25 Ollendorff a/d from (9a). We consider a litz wire composed of 49 wires, radius Fig.5. Imaginary part of macroscopic permeability of 49 stranded wires is 7 푎 = 0.15 mm, the conductivity 휎 = 5.76×10 S/m, relative plotted against the normalized wire radius. permeability 휇r = 1.0. TABLE.2. PARAMETERS OF WPT COIL The imaginary part of 〈 〉 is plotted against frequency in 휇̇ Strand number 50 Fig.4. It is found that 〈휇̇〉 scarcely depends on the pitch under Radius [mm] 0.025 the condition that 푎 < 훿 which holds for usual litz wires. In Turn number 40 Fig.5, 〈휇̇〉 computed from Ollendorff formula [5][9] Inner diameter [mm] 30 Outer diameter [mm] 74 Thickness [mm] 0.8 2r 1 r  1 (11) 2   1r 1 the computed AC resistance agrees well with the measured values except at high frequencies where the capacitance among in which wires are assumed to be in parallel is also plotted, the wires would not be negligible. where 휂 represents the filling ratio. From this result, we conclude that the proximity effect losses can be efficiently V. DISCUSSION evaluated from (11) for the litz wire shown in Fig.3. Namely, In this paper, we have discussed the eddy currents losses due the wire twist has little effect on 〈휇̇푟〉 when 푎 < 훿. to the proximity effect. We find that wire twist has little effects on 〈휇̇푟〉 whose imaginary part represents the AC loss due the IV. EXPERIMENTAL RESULT proximity effect when 푎 < 훿 . However, there exists other We consider the AC resistance of the wireless power transfer factors that can cause AC losses in the wires. The main factors (WPT) coil composed of the litz wire shown in Fig.6(a). The are the and circulation currents. Here, we discuss the parameters are summarized in TABLE.2. The AC resistances latter because the former can be sufficiently reduced when computed by FEM and measured values are plotted against using litz wires. The latter effect has not been included in the frequency in Fig.6(b). The macroscopic permeability analysis of eddy current losses of litz wire [10-12]. 〈휇̇〉 obtained from (9a) is used in the FE analysis. We can see The circulation currents are the currents which flow along

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two wire conductors electrically connected at their terminals. 1.50 Ollendorff For example, we consider the twisted wires shown in Fig.7. 1.45 Proposed Measured When magnetic flux interlinkages across the wires, the 1.40 circulation currents flow along the wires due to Faraday’s law. 1.35 If the wires are ideally well twisted and the magnetic field is [] 1.30 uniform, the circulation currents cancel out each other. In ESR practice, however, the wires are not always ideally twisted and 1.25 the magnetic field can vary along the wires. The circulation 1.20 0 500 1,000 currents, therefore, can contribute to the AC losses. In [13], the Frequency [kHz] current sharing has been determined by solving the circuit (a) WPT coil (b) AC resistance Fig.6. WPT coil and its AC resistance. equation resulted from 2.5D PEEC method. This method leads to, however, a huge equation system when the litz wire contains circulation currents a number of strands. The AC losses due to the circulation currents could be evaluated in the post-processing; the wire is modeled as a curve or segmented lines and the eddy currents along them are 푗휔푡 computed from the interlinkage flux. This remains for future 푩0푒 work.

VI. CONCLUSION Fig.7. Circulation currents in a pair of wires. In this paper, we have proposed a new method to obtain the macroscopic permeability 〈휇̇〉 of a litz wire using integral equation method. By using this method, we can evaluate 〈휇̇〉 REFERENCES without heavy computational burden. It has been shown that [1] A. D. Podoltsev, I. N. Kucheryavaya, B. B. Lebedev “Analysis of the macroscopic permeability of the litz wire obtained by the Effective Resistance and Eddy-Current Losses in Multiturn Winding of proposed method is in good agreement with that obtained by the High-Frequency Mangetic Components ” IEEE Trans. Magn., vol.39, no.1, January 2003. Ollendorff formula. Thus, it is concluded that the eddy current [2] Xi Nan C. Sullivan “An Equivalent Complex Permeability Model for losses due to the proximity effect can be accurately evaluated Litz-wire windings,” in IEEE Ind. Appl. Soc. Ann. Meeting, 2005, by the Ollendorff formula. This result has also been verified by pp.2229-2235. [3] J. Gyselinck P. Dular “Frequency-Domain Homogenization of Bundles experiment. We have also discussed the circulation current of Wires in 2-D Magnetodynamic FE Calculations ” IEEE Trans. Magn., which gives rise to additional AC losses. The evaluation of the vol. 41, no. 5, pp. 1416-1419, May, 2005. circulation current losses in the litz wire is our future work. [4] H. Waki, H. Igarashi, T. Honma “Estimation of Effective Permeability of Magnetic Composite Materials Magnetics ” IEEE Trans.Magn., vol. 41, no. 5, pp. 1520 – 1523, 2005. ACKNOWLEDGEMENT [5] H. Igarashi “Semi-Analytical Approach for Finite Element Analysis of This work was supported in part by KAKENHI 15H02976. Multi-turn Coil Considering Skin and Proximity Effects ” IEEE Trans. Magn., vol.53, no.1, January 2017. [6] Y. Sato, H. Igarashi, “Homogenization Method Based on Model Order APPENDIX A Reduction for FE Analysis of Multi-turn Coils ” IEEE Trans. Magn., vol.53, no.6, June 2017. We consider here the effective diamagnetic constant of a [7] Y. Sato, H. Igarashi, “Time-domain Analysis of Soft Magnetic Composite stranded wire. By definition, it would be expressed as Using Equivalent Circuit Obtained via Homogenization ” IEEE Trans. Magn., vol.53, no.6, June 2017. [8] S. Hiruma, H. Igarashi, “Fast Three-Dimensional Analysis of Eddy in H0  H Current in litz Wire Using Integral Equation ” IEEE Trans. Magn., vol.53, N  ave (A1) no.6, June 2017. in M ave [9] F. Ollendorff “Magnetostatik der Massekerne ” Arch. f. Elektrotechnik., 25, pp. 436-447, 1931. [10] Q. Deng, J. Liu, D. Czarkowski, M. K. Kazimierczuk, M. Bojarski, H. 풊풏 where 푯푎푣푒 is the averaged magnetic field over the stranded Zhou W. Hu “Frequency-Dependent resistance of Litz-Wire Square wires. The averaged magnetic field is computed from Solenoid Coils and Quality Factor Optimization fo Wireless Power Transfer ” IEEE Trans. Ind. Elec., vol. 63, no. 5, May 2016. [11] D. C. Meeker “An improved Continuum Skin and Proximity Effect in Model for Hexagonally Packed Wires ” Journal of Computational and in Mave Have  . (A2) Applied Mathematics, Volume 236, Issue 18, 2012, Pages 4635-4644, r 1 ISSN 0377-0427, http://dx.doi.org/10.1016/j.cam.2012.04.009. [12] C. R. Sullivan R. Y. Zhang “Analytical Model for Effects of Twisting on Litz-Wire Losses ” Control and Modeling for Power Electronics 푖푛 where 푴푎푣푒 is the magnetization, computed by solving (5a) (COMPEL), 2014 IEEE 15th Workshop on. IEEE, 2014. with real-valued permeability, averaged over the conductor. [13] T. Guillod, J. Huber, F. Krismer J. W. Kolar, Litz Wire Losses: Effects of Twisting Imperfections, Proceedings of the 18th IEEE Workshop on Control and Modeling for Power Electronics (COMPEL 2017), Stanford, California, USA, July 9-12, 2017.