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246 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996 Dynamic art I: Theor 0 etic Cores for Low- and of t ivalent Core Loss Jian. G. Zhu, Member, IEEE, S. Y. R. Hui, Abstruct- This paper describes the theoretical calculation of the equivalent core loss resistance for a dynamic magnetic core loss model. The equivalent core loss resistance incorporates the effects of both the classical eddy current and anomalous losses. Derivation of a generalized nonlinear core loss resistance expres- sion is presented. This new equivalent core loss resistance can be incorporated into a generalized dynamic magnetic core circuit model suitable for low and high frequency applications (Part I1 of this paper). I. INTRODUCTION ITH the continuous improvements in switching speed and power capability of modem power electronics, the study of the magnetic core behavior becomes increasingly important owing to the facts that magnetic cores are often operated in the nonlinear characteristic regions and the core loss increases with the operating frequency. A dynamic circuit model for magnetic cores that can accurately predict such nonlinear behavior would be very useful and practical to engineers who want to assess the core loss and optimize the switching frequency and efficiency of the overall system. The model must be able to account for hysteresis, eddy current, and also anomalous losses. It should be based on parameters that can be obtained easily either from manufacturers’ data sheets or simple measurements, and it should be easy to implement. Two major aspects of modeling magnetic cores by an equiv- alent circuit are the representation of 1) the nonlinear dynamic behavior of the hysteresis effects and 2) the equivalent core loss resistance. Several papers have investigated various meth- ods for modeling hysteresis effects [ 11-[4]. Among different hysteresis modeling methods, the Preisach model appears to be a practical and easy-to-use technique. It is strongly related to the mechanisms of magnetic hysteresis and can describe various macroscopic hysteretic phenomena. Although the description of the classical Preisach theory requires compli- cated mathematics, the resultant formula obtained by a simple graphical approach is rather simple and easy to implement. Details of such graphical approach can be found in [5]. The Manuscript received February 14, 1995; revised October 20, 1995. J. G. Zhu and V. S. Ramsden are with the School of Electrical Engineering, S . Y. R. Hui is with the Department of Electrical Engineering, University Publisher Item Identifier S 0885-8993(96)09195-1. University of Technology, Sydney, NSW 2007, Australia. of Sydney, NSW 2006, Australia. Senior Member, IEEE, and V. S. Ramsden results of Preisach modeling fit the experimental behavior of different magnetic materials [ l]-[.5]. The modeling of core losses using an equivalent resistance, however, needs more investigation. The total eddy current loss includes the classical eddy current loss and anomalous loss. In the past, most circuit models proposed included the hysteresis and classical eddy current losses only. The anomalous loss, which is due to the effects of magnetic domain wall movement on the distribution of eddy currents [6]-[13], has not been included. Fig. l(a) shows a typical circuit model for magnetic core together with the winding resistance R, of the coil wound in the core. R, and L, represent the core loss and inductance in the equivalent circuit, respectively. If anomalous loss is included, it can be shown [14] that R, is composed of two parallel resistances, namely the classical eddy current loss resistance Re and the anomalous loss resistance R,, as shown in a dynamic circuit model [Fig. l(b)], in which L( i ) represents the differential inductance. At constant temperature, R, and Re can be regarded as constants. R,, however, is a nonlinear resistor. If hysteresis is involved, L( i ) is nonlinear and should be determined by the magnetization history of the core using a mathematical model of hysteresis. It was pointed out in [14] that the circuit models shown in Fig. l(a) and (b) were suitable only for low-frequency applica- tions since the reaction of eddy currents on the distribution of excitation magnetic field, which is known as the skin effect, was not properly accounted for. In order to generalize the model so that it can be used for high-frequency applications as well, it is necessary to extend the circuit to a ladder network as shown in Fig. 2. (Details of the ladder network will be given in Part I1 of this paper.) In this approach, the cross section of the magnetic core is divided into a number of assumed eddy current paths. Within each path, the variation of eddy current density with the depth can be approximately fit by a straight line, and the distribution of flux density can then be considered as uniform. Applying the circuit model shown in Fig. l(b) to each assumed eddy current path results in the ladder network model. In the ladder network, d@k/dzk ( k = 1 , 2 , . . . , n , where n is the number of stages of the ladder network) is a differential inductance related to the kth eddy current loop, and RI, ( k = 1 , 2 , . . . , n) is the equivalent core loss resistance of 0885-8993/96$05.00 0 1996 IEEE Authorized
licensed use limited to: University of Sydney. Downloaded on September
02,2020 at 08:32:26 UTC from IEEE Xplore. Restrictions apply. ZHU et al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART I 241 t (b) Fig. 1. model of magnetic core. (a) Traditional circuit model of magnetic core. (h) Dynamic circuit a d i’1 Fig. 2. A generalized equivalent circuit model (ladder network) for magnetic cores. the kth eddy current loop, which consists of both the classical eddy current and anomalous losses represented by Re and R, in Fig. l(b), respectively. Part I of this paper deals with the calculation of the equivalent core loss resistance in the generalized dynamic circuit model for magnetic cores shown above. This work is based on a low-frequency core loss model [14] and a high- frequency model [ 151 previously published by the authors. The new core loss resistance expression derived here includes both the classical eddy current loss and the anomalous loss and is developed in such a form that it not only suits both low- and high-frequency applications, but it also simplifies the formation of a generalized ladder network for the core loss model. Details of the generalized core model and its implementation are presented in Part I1 of this paper. 11. APPROXIMATE EDDY CURRENT DISTRIBUTION I A MAGNETIC CORE Fig. 3 illustrates the cross section of a magnetic core of length I,, width a, and thickness b where 1, >> a >> b. When a I Fig. 3. due to a changing magnetic field in the 2 direction. Cross section of a magnetic core showing induced eddy current paths time-varying magnetic field is applied in the 2-direction, eddy currents are induced in the X and Y direction. For a sinusoidal excitation, and since 1, >> a >> b, the phasor expressions of flux density and eddy current density are found to be cosh Q y cosh B, 1 B,- and QB, sinh ay p cosh (2) where B, is the peak value of flux density at the core surface, frequency, f the excitation frequency, and a and p are the conductivity and permeability of the core material, respectively [16]. The Y axis component of the eddy current density Jy, and the variation of the eddy current density with x, can be ignored since a >, b. If 6 >> b, the magnetic core is regarded as a thin sheet and approximate expressions of flux density and eddy current density can be obtained from (1) and (2) as J -__- x – , 6 = ,E the skin depth, w = 2 ~ f the angular B, N” B, (3) and J, s jwaB ,y . (4) Therefore, in a thin sheet, the reaction of eddy currents on the magnetic field distribution can be ignored. The magnetic field can be considered as uniformly distributed over the cross section of the core, and the amplitude of eddy current density is proportional to the distance from the center of the core. If b >> 6, however, the reaction of eddy currents on the magnetic field distribution cannot be neglected. For an approximate solution with an excitation of arbitrary waveform, we divide the cross section of the core into a number of slice pairs as shown in Fig. 4. Each slice pair forms an assumed eddy current loop (J , is ignored since a >> b). The thickness of a slice Ay is chosen such that within the slice the variation of eddy current density against y is almost linear. Inside the slice, therefore, the reaction of eddy currents on the magnetic field distribution can be ignored, and the flux density can be considered as uniformly distributed, or independent of y. In the kth slice pair, let the flux density be B = B,k ( 5 ) Authorized
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02,2020 at 08:32:26 UTC from IEEE Xplore. Restrictions apply. 248 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996 per unit volume, the instantaneous power loss in the kth slice pair is (12) 2a1, Y k p e k = 7 Li, J,” dy . From (9), we have (13) 1 dy = ~ d J z . a ( % ) , Hence Fig. 4. core. An assumed eddy current path formed by a slice pair in a magnetic where the bold face symbols are vectors, k is the unity vector in the 2-direction in Fig. 4, and subscript z refers to the 2-direction component. Since a >> b, the eddy current density in the kth

slice pair is J = J,i (4 ) where i is the unity vector in the X-direction in Fig. 4, and subscript z denotes the X-direction component. Substituting (5) and (6) into J = aE and (7) where E is the electric field strength in the kth slice pair, we have since B, is independent of y and Jz only varies in Y direction in the slice pair. The eddy current density in the kth slice pair can then be obtained by performing integration on both sides of (9) as where J k and J k – 1 are the eddy current densities at surfaces y = yk and y = yk-1, respectively. Because B, is considered as uniformly distributed inside the kth slice pair, as discussed earlier, (%)k can also be treated as independent of y. Therefore Resistance in the current paths along the Y axis is ignored because the thickness b is very small when compared with width a and length 1,. From (1 l), one obtains Therefore, the classical eddy current loss in the kth slice pair becomes where is a correction factor that takes into account the variation of eddy current density J , with y. If the distribution of current density is uniform, i.e., J k = J k – 1 , T J k becomes one. w. CONSIDERATION OF ANOMALOUS LOSS It has been pointed out in [14] that the eddy current loss P, and the anomalous loss Pa expressions (in Wkg) in a thin sheet are given by 2 P, = .($) and where where Ce and C, are the coefficients of the classical eddy current loss and the anomalous loss, respectively, and B is assumed uniform within the depth. The sum of the classical eddy current loss and the anomalous loss is AY = Yk – Yk-1. 111. CLASSICAL EDDY CURRENT Loss The classical eddy current loss in the kth slice pair can be expressed as the resistive loss of the eddy current path in the kth slice pair. Since the classical eddy current loss P, = $ J 2 Pea = Pe + Pa d B , 3’2 -ce __ + C __ – (d2r d t 1 Authorized
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02,2020 at 08:32:26 UTC from IEEE Xplore. Restrictions apply. ZHU et al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART I 249 where is a correction factor if the anomalous loss (the effects of domain wall movement) is included in the calculation of the eddy current loss. Therefore, the sum of the eddy current and anomalous losses in the kth slice pair P e a k from (16) is where is the correction factor for the kth slice pair. For the time instant at which I%Ik = 0, p e a k = 0, and Tck approaches infinity. Numerically, a large number, for example 1020, is assigned to Tck in this case. The coefficients of the classical eddy current and anomalous losses used in (21) for correction factor rc can be obtained by the alternating core loss separation procedure described in [ 141. V. EQUIVALENT CORE Loss RESISTANCE Since a >> b, the induced emf in the kth slice pair at y = yk is VI, = 2 a E k . (23) Since JI, = oEI,, then for the kth slice pair. From the ladder network shown in Fig. 2, it can be seen that vk is also the voltage across the resistance R k ( k = 1 , 2 , . . . , n) since where GJ = 2aAyB,, ( j = 1 , 2 , . . . , n) is the magnetic flux through the cross section of the jth slice pair. Therefore, the equivalent core loss resistance for the kth slice pair can be calculated by (26) vi R k = -. P e a k Thus, from (22) and (24), one obtains ( F J k ) 2 ( 2 a z / ) J ~ ~ J I , ~ ~ I , RI, = where R D C k = 2% is the dc resistance of the kth slice pair. if the variation of eddy current dlensity in the kth path corresponding to the skin effect in the magnetic core is considered. The resistance expression changes to the form in (27) if the anomalous loss is also included. Equation (27) provides a simple general expression for the equivalent eddy current and anomalous loss resistance for each eddy current loop in the magnetic core. This equation can be used in the generialized magnetic core model as explained in Part I1 of this pape:r. It is important to note that RI, is expressed in terms of the dc resistance, which can be easily determined. This enables (27) to be applied to magnetic cores of different cross-sectional shapes. The dc resistance R D c k changes to VI. CONCLUSION This paper has presented the theoretical calculation of the magnetic core loss in a generalized core model suitable for both low and high-frequency applications. A new expression for the effective core loss resistance, which includes both the classical eddy current and anomalous loss, has been derived and explained. This core loss resistor, together with the hysteresis model, can be used to develop a generalized dynamic magnetic core circuit model. The validity of this new expression has been confirmed by measurements and simulations. Implementation of and results obtained from such models are presented in Part I1 of this paper. REFERENCES I. D. Mayergoyz, Mathematical Models of Hysteresis. New York Springer Verlag,, 1991. F. Ossart and G. Meunier, “Comparison between various hysteresis models and experimental data,” ZEEE MAG-26, no. 5, pp. 2837-2839, 1990. S. R. Naidu, “Simulation of the hysteresis phenomenon using Preisach’s theory,” ZEE Proc.-A, vol. 137, no. 2, pp. 73-79, 1990 F. Preisach, “1Jber die magnetische Nachwirkung,” Zeifschhriji fur Physik, vol. 94, pp. 277-302, 1935. S. Y. R. Hui and J. G . Zhu, “Numerical modeling and simulation of hysteresis effects in magnetic cores using the transmission line modeling and Preisach theory,” ZEE Proc.-Sci. Meas. Technol., vol. 142, no. 1, R. H. Pry and IC. P. Bean, “Calculation of the energy loss in magnetic sheet materials using a domain model,” J. Appl. Phys., vol. 29, no. 3, F. Brailsford arnd R. Fogg, “Anomalous iron losses in cold reduced grain-oriented transformer steel at very low frequencies,” in Proc. IEE, vol. 113, pp. 1562-1564, 1966. J. A. Robey, “Domain wall motion and eddy current loss in grain oriented silicon iron sheet,” J. Sci. & Tech., vol. 36, no. 1, pp. 3-10, 1969. C. D. Graham, “Physical origin of losses in conducting ferro-magnetic materials,” J. Appl. Phys., vol. 53, no. 11, pp. 8276-8280, 1982. pp. 57-62, 1995. pp. 532-533, 1958. Authorized
licensed use limited to: University of Sydney. Downloaded on September
02,2020 at 08:32:26 UTC from IEEE Xplore. Restrictions apply. 250 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11. NO. 2, MARCH 1996 [lo] K. Foster, F. E. Werner, and R. M. Del Vecchio, “Loss separation measurements for several electrical steels,’’ J. Appl. Phys., vol. 53, no. 11, pp. 8308-8310, 1982. [ I l l T. Sat0 and Y. Sakaki, “Physical meaning of equivalent loss resistance of magnetic cores,” IEEE MAG-26, no. 5, pp. 2894-2897, 1990. [12] F. Fiorillo and A. Novikov, “An improved approach to power losses in magnetic laminations under nonsinusoidal induction waveform,” IEEE MAG-26, no. 5, pp. 29042910, 1990. [I31 H. Pfutzner, B. Erbil, 6. Harasko, and T. Klinger, “Problems of loss separation for crystalline and consolidated amorphous soft magnetic materials,” IEEE MAG-27, no. 3, pp. 34263432, 1991. [14] J. G. Zhu, S. Y. R. Hui, and V. S . Ramsden, “Discrete modeling of magnetic cores including hysteresis, eddy current, and anomalous losses,” IEE Proc.-A, vol. 140, no. 4, pp. 317-322, 1993. [15] J. G. Zhu, S. Y. R. Hui, and V. S. Ramsden, “A dynamic circuit model for solid magnetic cores,” in IEEE Power Electronics Specialists Cons (PESC’93) Rec., Seattle, WA, June 20-24, 1993, pp. 11 11-1 115. [16] R. L. Stoll, The Analysis of Eddy Currents. Oxford, U.K.: Clarendon, ch. 1, sec. 2.7, pp. 21-23, 1974. Jian 6. Zhu (M’92) was bom in Shanghai, China, in 1958 He received the B.Eng. degree from Jiangsu Institute of Technology (JIT), China, in 1982, the M Eng degree from Shanghai University of Tech- nology, China, in 1987, and the Ph.D. degree from the University of Technology, Sydney (UTS), Aus- tralia, in 1995. From 1987-1990, he worked as a Lecturer at JIT, China. He joined the School of Electrical Engineer- ing, UTS, as a Lecturer in 1994. His research areas include numerical analysis of electromagnetic fields, measurement and modeling of magnetic properties of materials, and electrical machines and drive systems. S. Y . R. (Ron) Hui (SM’94) was bom in Hong Kong in 1961. He obtained the B Sc. (Hons.) at the University of Birmingham, U K., in 1984, and the D.I.C. and Ph.D. degrees at the Impenal College of Science and Technology, London, U K , in 1987. He was appointed Lecturer in power electronics at the University of Nottmgbam, U.K., in 1987. In 1990, he went to the University of Technology, Australia, where he became a Senior Lecturer in 1991. In January 1993, he joined the University of Sydney, Australia, where he is now a Reader of Electrical Engineering and Coordinator of Power Electronics Research. He has pubhshed over 30 refereed joumal papers and over 40 conference papers on various aspects of power electronics and simulation techniques His current research interests include soft-switching techniques for converters and inverters, power electronic clrcuit and magnetic core simulation, power factor correction circuits, PWM methods, new topologies for power conversion, and electronic ballast. Dr. Hui is a Fellow of the IEAust. V. S. Ramden, publication. photograph and biography not railable at the time of Authorized
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