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Title Identification of Magnetization Characteristics of Material From Measured Data

Author(s) Maruo, Akito; Igarashi, Hajime; Sato, Yuki; Kawano, Kenji

IEEE Transactions on Magnetics, 55(6), 7300205 Citation https://doi.org/10.1109/TMAG.2019.2896187

Issue Date 2019-06

Doc URL http://hdl.handle.net/2115/74696

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Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP > 0163 < 1

Identification of Magnetization Characteristics of Material from Measured Inductance Data

Akito Maruo1, Hajime Igarashi1, Yuki Sato2, Kenji Kawano2, IEEE Member

1 Graduate School of Information Science and Technology, Hokkaido University, Sapporo 060-0814, Japan 2 Texas Instruments Japan Limited, Tokyo 160-8366, Japan

This paper introduces a novel identification method of the magnetization properties of the core material for an from measured inductance data. The proposed method allows us to obtain the magnetic characteristics without a special measurement instrument for material BH properties. The proposed method determines the parameters included in the distribution function for the Preisach model from the measured L- characteristics. It is shown that the identified initial magnetization curve and the minor loops are in good agreement with the original BH curves. The uniqueness for the identification problem is numerically verified. 𝑰𝑰 Index Terms— Genetic Algorithm, Inductor, Inverse Problem, , Preisach Model.

I. INTRODUCTION II. NUMERICAL METHOD INITE element (FE) analysis of power and reactors A. Preisach model F is based on the initial BH curves or hysteresis characteristics [e.g., 1]. It is, however, not always possible to obtain such data The Preisach model [5-7] is one of the hysteresis models because a special measurement instrument for BH which are widely used in FE analysis. In the Preisach model, characteristics of magnetic material is necessary [2-4]. the magnetic characteristic is represented by the Moreover, it is sometimes hard to generate sufficiently strong superimposition of the basic hysteresis loops as shown in Fig. to measure the magnetic properties. 1, where and are the upper and lower thresholds of the On the other hand, it is not difficult to measure the macroscopic basic hysteresis loop. The magnetization becomes positive 𝐻𝐻𝑢𝑢 𝐻𝐻𝑣𝑣 properties of power inductors; the dependence of inductance on when increases to become greater than . Similarly, 𝑀𝑀 the DC-bias current and amplitude of the imposed AC current becomes negative when decreases to become smaller than . 𝐻𝐻 𝐻𝐻𝑢𝑢 𝑀𝑀 can be measured using, e.g., a network analyzer. It is The Preisach model expresses in the following form: 𝐻𝐻 𝐻𝐻𝑣𝑣 remarkable that magnetic saturation can occur near the coil 𝑀𝑀 edges in an inductor core even when a weak current is imposed. = ( , ) + (1) Moreover, the measured inductance is attributed to the 𝑢𝑢 𝑣𝑣 𝑢𝑢 𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚 distributed magnetic field that changes in time to draw major or where𝑀𝑀 �𝐷𝐷 and𝐾𝐾 𝐻𝐻 𝐻𝐻 denote𝑑𝑑𝐻𝐻 𝑑𝑑 𝐻𝐻the distribution𝑀𝑀 function and the minor loops in the inductor core. From these observations, a minimum value of , respectively. Moreover, is a triangular 𝑚𝑚𝑚𝑚𝑚𝑚 question arises; is it possible to identify the magnetization domain𝐾𝐾 in the𝑀𝑀 Preisach plane shown in Fig.2 which characteristics of the core material used in an inductor from the satisfies 𝑀𝑀 . When the interval𝐷𝐷 [ , ] is measured inductance data? [ , ] 0 < < 1 divided into 𝑠𝑠 subintervals𝑣𝑣 𝑢𝑢 𝑠𝑠 𝑠𝑠 ,𝑠𝑠 the In this paper, we propose a method to identify the BH integral in− 𝐻𝐻(1) ≤is 𝐻𝐻partitioned≤ 𝐻𝐻 ≤ into𝐻𝐻 −𝐻𝐻 𝐻𝐻 characteristics including magnetic hysteresis from the 𝑁𝑁𝑝𝑝 𝐻𝐻𝑛𝑛 𝐻𝐻𝑛𝑛−1 � 𝑛𝑛 𝑁𝑁𝑝𝑝 − � measured dependence of the inductance of an inductor on the ( , ) = ( , ) amplitude and bias of the imposed current. The hysteresis 𝐻𝐻𝑗𝑗+1 𝐻𝐻𝑖𝑖+1 (2) property is assumed to be described by the Preisach model with 𝐾𝐾′ 𝑖𝑖 𝑗𝑗 � � 𝐾𝐾 𝐻𝐻𝑢𝑢 𝐻𝐻𝑣𝑣 𝑑𝑑𝐻𝐻𝑢𝑢𝑑𝑑𝐻𝐻𝑣𝑣 a distribution function while the proposed method can be Then, (1) is rewritten𝐻𝐻𝑗𝑗 𝐻𝐻𝑖𝑖 for numerical evaluation as extended to other hysteresis models such as Play model. The parameters included in the distribution function are determined = 𝑁𝑁𝑝𝑝−1 𝑖𝑖 ( , ) ( , ) + (3) by the genetic algorithm (GA) so that the measured inductance ′ properties are reproduced. We discuss the accuracy of the 𝑀𝑀 � � 𝐾𝐾 𝑖𝑖 𝑗𝑗 𝐷𝐷 𝑖𝑖 𝑗𝑗 𝑀𝑀𝑚𝑚𝑚𝑚𝑚𝑚 identified BH characteristics and uniqueness for the where 𝑖𝑖=(0, 𝑗𝑗)= is0 defined by identification. 1 positive magnetization ( , 𝐷𝐷) =𝑖𝑖 𝑗𝑗 0 negative magnetization (4) ∶ 𝐷𝐷 𝑖𝑖 𝑗𝑗 � The initial ∶magnetization curve as well as the major and Manuscript received April 1, 2015; revised May 15, 2015 and June 1, minor BH loops can be expressed using the Preisach model. 2015; accepted July 1, 2015. Date of publication July 10, 2015; date of current version July 31, 2015. (Dates will be inserted by IEEE; “published” is the date The AC magnetic property, shown in Fig. 3, of the material the accepted preprint is posted on IEEE Xplore®; “current version” is the date used for an inductor is computed from the slope of the minor the typeset version is posted on Xplore®). Corresponding author: A. Maruo loops governed by the Preisach model for the biased magnetic (e-mail: [email protected]). Digital Object Identifier (inserted by IEEE). field as shown in Fig. 4.

0018-9464 © 2015 IEEE. Personal use is permitted, but republication/redistribution𝐻𝐻 requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. (Inserted by IEEE.) > 0163 < 2

(ii) For the magnitude of the density at each M Hv H finite element, the magnetic permeability is computed s from Fig. 3. 𝑩𝑩 (iii) Assuming a small input current, the linear FE analysis is -H s performed using the magnetic permeability evaluated in Hv Hu H H Hu s step (ii), and is computed from (6).

-Hs 𝑑𝑑𝑑𝑑 𝐿𝐿 Fig. 1. Basic hysteresis loop Fig. 2. Preisach plane III. IDENTIFICATION METHOD 2000 In this study, the BH characteristic of the material used by an

1600 inductor is identified from the inductance values, and defined in II, using GA, the detail of which is described in 𝑎𝑎𝑎𝑎 1200 [7]. The distribution function ( , ) in (1) 𝐿𝐿is here 𝑑𝑑𝑑𝑑 expressed𝐿𝐿 by the Gaussian function with an elliptic exponent 800 𝑢𝑢 𝑣𝑣 , 𝐾𝐾 𝐻𝐻 𝐻𝐻 relative permeability that includes three parameters and as follows: 400 𝐴𝐴 𝜎𝜎1 𝜎𝜎+2 0 2 2 2 2 0 20 40 60 80 100 120 140 = exp (7) 2 𝐻𝐻𝑢𝑢2− 𝐻𝐻𝑣𝑣 𝐻𝐻𝑢𝑢2 𝐻𝐻𝑣𝑣 𝐴𝐴 � � � � Magnetic field H (A/m) 𝐾𝐾 �− 2 − 2 � Fig. 3. AC magnetic property for biased magnetic field 𝜋𝜋𝜎𝜎1𝜎𝜎2 𝜎𝜎1 𝜎𝜎2 The magnetic flux density is expressed by 0.6 = + (8) 0.5 𝑩𝑩

(T) out

B 0 𝑟𝑟 0.4 where𝑩𝑩 𝜇𝜇 𝜇𝜇 is𝑯𝑯 the 𝑴𝑴relative permeability in the fully saturated region andout is obtained by the Preisach model based on (7). 0.3 𝑟𝑟 We determine𝜇𝜇 the unknown parameters ( , , , ) in (7) 0.2 𝑴𝑴 Flux density density Flux and (8) by solving the optimization problem defined outby Minor loops at biased magnetic 1 2 𝑟𝑟 0.1 field 𝐴𝐴 𝜎𝜎 𝜎𝜎 𝜇𝜇 1 ( ) ( ) 0 min = 𝑁𝑁𝑖𝑖 -100 0 100 200 300 400 ( )0 2 -0.1 𝐿𝐿𝑑𝑑𝑑𝑑 𝑖𝑖 − 𝐿𝐿𝑑𝑑𝑑𝑑 𝑖𝑖 Magnetic field H (A/m) 0 (9) 𝐹𝐹 𝑖𝑖 � � � 𝑁𝑁 𝑖𝑖 1𝐿𝐿𝑑𝑑𝑑𝑑 𝑖𝑖 ( ) ( ) Fig. 4. Initial magnetization curve and minor loops + 𝑁𝑁𝑗𝑗 ( )0 2 𝐿𝐿𝑎𝑎𝑎𝑎 𝑗𝑗 − 𝐿𝐿𝑎𝑎𝑎𝑎 𝑗𝑗 0 𝑗𝑗 � � 𝑎𝑎𝑎𝑎 � B. Analysis of inductance where , run over the𝑁𝑁 number𝑗𝑗 of 𝐿𝐿sampling𝑗𝑗 points , , and We evaluate the inductance of an inductor by FE analysis in and represent the original and computed DC , 𝑖𝑖 𝑗𝑗 𝑁𝑁𝑖𝑖 𝑁𝑁𝑗𝑗 respectively,0 and the same convention is used for the AC which the following magnetostatic equation is discretized: 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝐿𝐿inductance.𝐿𝐿 The cost function defined in (9) is minimized by rot( rot ) = (5) GA for a set of optimization variables , , , , in which , the fitness is evaluated as follows: out where𝜈𝜈 𝑨𝑨 and𝑱𝑱 denote the reciprocal of magnetic 1 2 𝑟𝑟 permeability, vector potential and , respectively. (i) The initial magnetization curve and𝐴𝐴 AC𝜎𝜎 magnetic𝜎𝜎 𝜇𝜇 property After the𝜈𝜈 solution𝑨𝑨 𝑱𝑱of the FE equation, the inductance is shown in Fig.3 for the current parameters ( , , , ) are computed using the Preisach model based on (7) outand computed. When considering magnetic saturation, we may 1 2 𝑟𝑟 define inductance from the derivative or 𝐿𝐿the (8). 𝐴𝐴 𝜎𝜎 𝜎𝜎 𝜇𝜇 average ( ) . The measured inductance depends on (ii) The and characteristics are the definition of inductance which differs for each𝑑𝑑Φ measurement⁄𝑑𝑑𝑑𝑑 computed using the method mentioned in II.B. 𝐿𝐿𝑎𝑎𝑎𝑎 − 𝐼𝐼𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝐿𝐿𝑑𝑑𝑑𝑑 − 𝐼𝐼𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 instrument∫ .𝛷𝛷 Here,𝐼𝐼 𝐼𝐼⁄ for𝐼𝐼 𝑑𝑑𝑑𝑑 simplicity, we compute the AC inductance (iii) The cost function that measures the difference between using the peak value of an AC current from the computed and original inductance values, and , is evaluated𝐹𝐹 by (9). 0 𝑎𝑎𝑎𝑎 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝐿𝐿 𝐼𝐼 In GA, the individuals0 have a different set of , , 𝐿𝐿, − that𝐿𝐿 = (6) 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝐿𝐿 − 𝐿𝐿 out Ω correspond to the and characteristics. ∫ 𝑨𝑨 ∙ 𝑱𝑱 𝑑𝑑Ω 𝐴𝐴 𝜎𝜎1 𝜎𝜎2 𝜇𝜇𝑟𝑟 On𝐿𝐿 the other2 hand, the DC inductance is computed by the They evolve to reproduce the original inductance characteristics. 𝐼𝐼 𝐿𝐿𝑎𝑎𝑎𝑎 − 𝐼𝐼𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝐿𝐿𝑑𝑑𝑑𝑑 − 𝐼𝐼𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 following procedure: 𝑑𝑑𝑑𝑑 (i) The magnetic flux density is computed𝐿𝐿 at each finite element for a bias current using the nonlinear FE analysis based on the initial 𝑩𝑩magnetization curve. 𝐼𝐼𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

> 0163 < 3

IV. NUMERICAL RESULT z TABLE I A. Preparation of Inductance Characteristics INDUCTOR PARAMETERS [mm] 0.45 core [mm] 0.65 We consider here the ferrite inductor shown in Fig. 5 for a 1 𝑟𝑟 [mm] 1.00 2 numerical example. Table I summarizes the specific parameters 𝑟𝑟 [mm] 0.35 3 of the inductor. The measured major BH curve of ferrite used Coil 𝑟𝑟 [mm] 0.75 1 N𝑧𝑧 umber of coil turns 8 r for the is shown in Fig. 6. The initial 𝑧𝑧2 magnetization curve obtained from Fig. 6 is used to compute characteristic. Moreover, we determine the , , , parameters𝑎𝑎𝑎𝑎 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 ( ) in (7) and (8) from the measured 𝐿𝐿curve− in𝐼𝐼 Fig. 6 by the curveout fitting that solves 1 2 𝑟𝑟 𝐴𝐴 𝜎𝜎 𝜎𝜎 𝜇𝜇 Fig. 5. Analysis model

1 ( ) ( ) 0.6 min = 𝑁𝑁𝑖𝑖 (10) ( ) 2 𝐵𝐵 𝑖𝑖 − 𝐵𝐵0 𝑖𝑖 0.4 𝐸𝐸 � � � (T) 0.2 𝑖𝑖 B using GA, 𝑁𝑁where,𝑖𝑖 and𝐵𝐵 𝑖𝑖 denotes the magnetic flux density shown in Fig. 6 and that computed by the Preisach model. The 0 0 -600 -400 -200 0 200 400 600 resultant parameters𝐵𝐵 are summarized𝐵𝐵 in the left column of Table -0.2 Flux density Flux Ⅲ. Based on this result, characteristic is computed -0.4 Preisach using the method mentioned in II.B. Measured value 𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 -0.6 𝐿𝐿 − 𝐼𝐼 Magnetic field H (A/m) B. Identified Results Fig. 6. Magnetic property of the ferrite core

The and characteristics computed TABLE Ⅱ ANALYSIS CONDITIONS from the𝑎𝑎𝑎𝑎 measured𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 data shown𝑑𝑑𝑑𝑑 𝑏𝑏 𝑏𝑏𝑏𝑏𝑏𝑏in Fig. 6 and fitted by the proposed𝐿𝐿 method− 𝐼𝐼 mentioned𝐿𝐿 in− III𝐼𝐼 are plotted in Fig. 7 and Fig. [A/m] 400 [A/m] 4 𝑠𝑠 8. The parameters identified from the inductance characteristics 𝐻𝐻 200 Δ𝐻𝐻 are summarized in the right column of Table III. Although there Population𝑝𝑝 of GA 400 Generation𝑁𝑁 of GA 1000 are small discrepancies for small input currents in TABLE Ⅲ characteristic, satisfactory coincidence is obtained.𝑎𝑎𝑎𝑎 𝑝𝑝𝑝𝑝The𝑝𝑝𝑝𝑝 convergence history of GA for solving problem (9) 𝐿𝐿is shown− 𝐼𝐼 in PREISACH PARAMETERS identified from L identified from Fig.6 Fig. 9. We can find in Fig. 9 that the cost function becomes characteristics sufficiently small after 1000 generations. The initial 0.81 0.82 𝐹𝐹 23.44 29.52 magnetization curve and the BH loops identified from the 𝐴𝐴 24.50 21.60 1 𝜎𝜎 170.21 173.06 inductance characteristics shown in Figs. 7 and 8 are plotted 2 𝜎𝜎out with the original curves obtained from the measured data in Fig. 𝑟𝑟 𝜇𝜇 45 6 and the Preisach model in Fig. 10 and 11. It can be seen that 40 Proposed method both are in good agreement. Figure 12 shows the hysteresis loss ) 35 Original data uH ( 30 with respect to the amplitude of magnetic flux density. The L 25 hysteresis loss is overestimated by the proposed method in 20 comparison with the measured value. This discrepancy comes 15 from the errors in the BH loops shown in Fig. 11; the ratio Inductance 10 5 , where and denote the area of the measured and 0 identified minor loops, respectively, is almost the same as the 0 0.01 0.02 0.03 0.04 0.05 1 2 1 2 ratio𝑆𝑆 ⁄𝑆𝑆 of the measured𝑆𝑆 𝑆𝑆and computed hysteresis losses shown in DC-bias current I (A) dc Fig. 12. Fig. 7. characteristic 120 𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 C. Discussions 𝐿𝐿 − 𝐼𝐼 Proposed method 100 The above-mentioned method is based on the assumption ) Original data μH

( 80

that the BH characteristics can be described by the Preisach L model with the distribution function (7). If the core material 60 does not obey this model, the present identification would be 40 unsuccessful. If the distribution function of the material can be Inductance 20 described by an explicit function of parameters, the proposed 0 method can be extended to identify the parameters. The 0 0.02 0.04 0.06 0.08 0.1 extension of the proposed method to identification of implicit AC current Iac (A) distribution functions is remained as an open question. Fig. 8. characteristic

𝐿𝐿𝑎𝑎𝑎𝑎 − 𝐼𝐼𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

> 0163 < 4

0.6 D. Uniqueness of Solution

F 0.5 We consider here if the BH characteristics can be uniquely 0.4 determined from the inductance characteristics. When

0.3 assuming the distribution function of (7), the BH characteristics are described by the parameters ( , , , ). To verify if 0.2 these parameters are uniquely determined fromout the inductance Objective function Objective 1 2 𝑟𝑟 0.1 characteristics, we performed additional𝐴𝐴 𝜎𝜎 𝜎𝜎 five𝜇𝜇 identifications 0 starting from different initial populations generated randomly. 1 10 100 1000 Generations The identified parameters and the final value of are summarized in Table Ⅳ. It can be concluded from these results Fig. 9. Convergence history of GA that there exists uniqueness to the solution at least for the𝐹𝐹 data 0.6 considered here. The theoretical proof of the uniqueness is 0.5 remained for a future work.

(T)

B 0.4

0.3 V. CONCLUSION 0.2

Flux density density Flux Proposed method We have proposed a new method to identify the 0.1 Measured value magnetization characteristics of ferrite used for the magnetic 0 core of an inductor from the inductance characteristics. This 0 100 200 300 400 approach is useful because special measurement instruments for Magnetic field H (A/m) Fig. 10. Identified initial magnetization curve BH loops are not needed. The BH characteristics are computed from a measured major loop. The inductance characteristics are 0.3 0.6 obtained from those BH data. Then, the BH characteristics have 0.2 0.4 (T) (T) B B 0.1 0.2 been identified from the inductance characteristics. The 0 0 identification result is satisfactory. The uniqueness for the -50 0 50 -100 -50 0 50 100 -0.1 -0.2 identification result is numerically verified. The extension of

Proposed mehod Fluxdensity Fluxdensity -0.2 -0.4 Proposed method the proposed method to implicit distribution functions and Measured value Measured value -0.3 -0.6 theoretical proof of the uniqueness have been remained as open Magnetic field H (A/m) Magnetic field H (A/m) (a) = 40 (b) = 80 questions. 0.6 0.6 𝑚𝑚 𝑚𝑚 𝐻𝐻 0.4 𝐻𝐻 0.4 (T) (T) B 0.2 B 0.2 REFERENCES 0 0 -200 -100 0 100 200 -400 -200 0 200 400 [1] K. Watanabe, et al., “Optimization of Inductors Using Evolutionary -0.2 -0.2 Algorithms and Its Experimental Validation,” IEEE Transactions on Fluxdensity -0.4 Proposed method Fluxdensity -0.4 Proposed method Magnetics, vol. 46 no. 8, pp. 3393-3396, 2010. -0.6 Measured value -0.6 Measured value [2] T. Matsuo and M. Shimasaki, “An identification method of play model Magnetic field H (A/m) Magnetic field H (A/m) with input-dependent shape function,” IEEE Transactions on Magnetics, (c) = 160 (d) = 240 vol. 41, no. 10, pp. 3112-3114, 2005. [3] Y. Takeda, et al., “Iron Loss Estimation Method for Rotating Machines Fig. 11. Identified BH loops, where denotes the amplitude of 𝐻𝐻𝑚𝑚 𝐻𝐻𝑚𝑚 Taking Account of Hysteretic Property,” IEEE Transactions on 7 𝑚𝑚 Magnetics, vol. 51, no. 3, pp. 1-4, 2015. Proposed method𝐻𝐻 𝐻𝐻

) 6 [4] IWATSU, “B-H Analyzer SY-8218 / SY-8219”, Measured value https://www.iti.iwatsu.co.jp/en/products/sy/sy8218_top_e.html. mW

( 5

c [5] F. Preisach, “Über die Magnetische Nachwirkung,” Zeitschrift für Physik, P 4 vol. 94, pp. 277-302, 1935. 3 [6] G. Bertotti, “Hysteresis in ,” Academic Press, 1998 [7] E. Della Torre., “Magnetic Hysteresis,” IEEE Press, 1999. 2 1 Hysteresis loss 0 0 0.1 0.2 0.3 0.4 0.5 0.6

Amplitude of flux density Bm (T) Fig. 12. Hysteresis loss with respect to amplitude of magnetic flux density

TABLE Ⅳ IDENTIFIED PARAMETERS AND RESIDUAL #1 #2 #3 #4 #5 #6 0.82 0.82 0.81 0.83 0.82 0.81 29.52 29.66 28.98 30.06 29.43 29.42 𝐴𝐴 21.60 21.61 21.56 21.65 21.45 21.53 1 𝜎𝜎 173.06 173.94 173.27 173.83 173.95 174.30 2 F𝜎𝜎inalout 3.99×10-3 3.99×10-3 3.72×10-3 4.39×10-3 3.89×10-3 3.86×10-3 𝜇𝜇𝑟𝑟 𝐹𝐹