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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 4, JULY 2002 453 A Static Hysteresis Model for Power Ferrites Paiboon Nakmahachalasint, Student Member, IEEE, Khai D. T. Ngo, Senior Member, IEEE, and Loc Vu-Quoc Abstract—Basso and Bertotti’s physics-based, yet simple, static hysteresis model is brought to the power electronic community as an alternative for simulation of magnetic components embedded in a power electronic converter. The model is reviewed and its equations cast in application/simulation-oriented forms. It is then revised to better characterize the very soft saturation behavior of commercial power manganese–zinc (MnZn) ferrites. The procedures to extract the model parameters from voltage and current measurements are described. The improved models have been verified against experimental data for major and minor hysteresis loops of three commercial power ferrites. resnorm Index Terms—Domain-wall, hysteresis modeling, power MnZn ferrites, preisach model, soft magnetic materials. NOMENCLATURE flops Magnetic flux density. Magnetic flux density pertaining to a turning point. Maximum flux density of the major – loop. Remanent flux density. Saturation flux density. Magnetic flux density at which is zero. Coefficient of the reversible process used in (1). Number of floating-point operations. Magnetic field intensity. Magnetic field intensity pertaining to a turning point in Fig. 1. Coercive force. Magnetic field intensity at which . Half the width of the elemental hysteresis loop in Fig. 9(b). Bias field of the elemental hysteresis loop in Fig. 9. Magnetization; . Saturation magnetization. Normalized magnetization. Normalized magnetization pertaining to a turning point. Normalized magnetization at which is zero. Manuscript received February 19, 2002; revised March 5, 2002. This work was supported by the National Science Foundation under Grant ECS-9906254, the State of Florida Integrated Electronics Center, and the Royal Thai Government. Recommended by Associate Editor C. R. Sullivan. The authors are with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611-6200 USA. Publisher Item Identifier 10.1109/TPEL.2002.801000. Positive integer of the irreversible process used in (3). Parameter used in (7b) for an alternate form of . Domain-wall surface function given in (4), (7a), and (7b). Parameter used in (4) for . Squared 2-norm of the residual from least-square-fitting. Mean domain-wall position. Domain-wall position pertaining to a turning point in Fig. 1. Mean domain-wall position of the initial magnetization curve. Mean domain-wall position of any return branch. Maximum susceptibility used in (1a) and (1b) for . Permeability, . I. INTRODUCTION H YSTERESIS models for power ferrites are an integral part of a computer-aided design (CAD) system for power electronic converters. The models need to capture the dependence of hysteresis on such physical phenomena as major loop, minor loops, irreversible magnetization, reversible magnetization, dynamic effects, shape effects, and temperature effects. This paper deals primarily with static hysteresis at room temperature. The methods described in [1] and [2] could be used to incorporate the dependence of the hysteresis phenomena on rate, temperature, and core shape. The Stoner–Wolhfarth, Jiles–Atherton, Globus, and Preisach models are four well-known physics-based macroscopic models of static hysteresis. Their characteristics and applicability were discussed and compared in [3]. These models generally do a better job at describing the major loop than the minor loops although power electronic transformers and inductors are normally designed to operate in minor loops. Static hysteresis is better characterized than dynamic hysteresis although the later is more pertinent in power converters. Recently, Basso and Bertotti describes a model that is physics-based and quite simple to use [4], and that could be another choice for core models in circuit simulators for power electronic engineers. This model was experimentally verified for amorphous and nanocrystalline alloys [5], which exhibited hard saturation. The applicability of the model to the soft manganese–zinc (MnZn) power ferrites (e.g., MN8CX ferrite from Ceramic Magnetics [6]) used in high-frequency power transformers and inductors was demonstrated in [7]. 0885-8993/02$17.00 © 2002 IEEE Authorized licensed use limited to: University of Florida. Downloaded on January 22, 2009 at 13:53 from IEEE Xplore. Restrictions apply. 454 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 4, JULY 2002 Fig. 1. Mean domain-wall position and B –H loop for H = 50 A/m, = , c = 0. : 0 01 (A/m) One objective of the paper is to introduce the Basso–Bertotti static hysteresis model to the power electronic readers. This is accomplished in Section II-A and the Appendix, which review the model described in [4] and [5] from the simulation/application standpoint. The second objective of the paper is to describe a refined Basso-Bertotti model for soft ferrites. This is accomplished in Section II-B, which identifies the need to improve the original model and suggests the corresponding modifications. While rigorous implementation of the refined model in a commercial circuit simulator remains a future topic, a numerical algorithm is suggested in Section II-C for the simulation of a static hysteresis loop using the model. Parameter extraction and experimental verification based on voltage and current measurements of sample toroids are discussed in Section III. The main results are summarized in Section IV. Fig. 2. Comparison of theoretical and measured R(m). netization), (maximum susceptibility, ), (associated (coercive force), and the adwith reversible magnetization), ditional parameters in (1a) (1b) (2a) or (2b) where II. MODEL DESCRIPTION AND IMPLEMENTATION A. Review of Basso–Bertotti Model for Static Hysteresis Basso–Bertotti’s model for static hysteresis accepts as an input the applied magnetic field intensity , and outputs the (related to the magnetic flux density by magnetization ) via an intermediate parameter (the mean and , the domain-wall position). The model also requires and of the most recent turning point, which is the tip of a -loop/branch, to generate the minor and major hysteresis loops. From the application standpoint, Basso-Bertotti static hysteresis model can be summarized by (1)–(3). (These equations are derived in the Appendix for those readers interested in the physics of magnetism.) Let the magnetization process start at . The “initial magthe de-magnetized state with netization curve” is generated by (1a), which computes from , and (2b), which computes from . For any -branch , is found from via (1b) and (2b) in a simwith ilar fashion. Equation (1) is illustrated in Fig. 1. As explained is associated with the irrein the Appendix, the function (saturation magversible process. The model parameters are (3) is the inverse function of , i.e., . and (called “domain-wall surface function” in The function [4]) is critical in shaping the saturation regions of the hysteresis is given in [4] loop. One choice of (4) by the curve labeled which is illustrated in Fig. 2 for . When is employed in (2b), as plotted in Fig. 3. The corresponding loop generated by (1)–(4) saturates rather abruptly as shown in Fig. 1. Although (4) has been found to be satisfactory for the two sample materials in [5], a good fit between modeling and measurement has been found to be difficult for the very soft saturation characteristic of power MnZn ferrites, as is evident in Fig. 4. Authorized licensed use limited to: University of Florida. Downloaded on January 22, 2009 at 13:53 from IEEE Xplore. Restrictions apply. NAKMAHACHALASINT et al.: STATIC HYSTERESIS MODEL 455 Fig. 5. Fig. 3. (2b). Comparison of theoretical and measured dR(m)=dm. Normalized magnetization vs mean domain-wall position obtained by Thus, the suitability of for a given magnetic material can with the measured be assessed by comparing . This is done in Fig. 5 to evaluate the appropriateness of (4) for MN8CX soft ferrite. The need for alternate is evident. forms of B. Static Hysteresis Model for Power Ferrites can be obtained from measureAs suggested by (5), from the major loop data. ments by computing These data points are plotted in Fig. 2 along the suggested in [5], which is (4) with . Much better fit, however, is achieved by the other two curves in Fig. 2 that correspond to and (7a) (7b) and for MN8CX ferrite. Note that the in (7a) was suggested in [7]. The given by (7b) may be considered to be the generalized from of the original BassoBertotti’s function in (4). The two parameters ( and ) that (7b) has more than (7a) are expected to provide more flexibility in fitting a variety of hysteresis shapes. On the other hand, the in (7a) is more mathematics below will show that the efficient numerically. where Fig. 4. Measured and fit major loops at room temperature and 10 kHz for MN8CX ferrite. The fit major loop is generated by the original Basso & Bertotti model [4], using the parameters in Table I for R(m) = 1 m . 0 Two arguments are now given to explain the difficulty in fitting is shown in Appendix B and in [5] near saturation. First, of the saturation loop. Thus, to be proportional to ought to approach zero very slowly as approaches unity for of power ferrites approaches zero power ferrites, for the very slowly in the saturation region. This is not seen in Fig. 2. increases monoIn fact, the (magnitude of the) slope of decreases from tonically as increases from 0 to 1, or as 1 to 0. Secondly, the softness of the static hysteresis loop can be with respect to quantified to be the rate of change of , which can be obtained from the upward branch of the major loop using (A.11) as or C. Model Implementation (5) To recap, the improved domain-wall model for static hysteresis of soft ferrites consists of the following equations: (1)–(3), and (7a) or (7b). The pseudo-code for the implementation of the preceding equations is outlined in Algorithm 1, which calls Algorithm 2 for a given according to (2a) to numerically solve for if a closed-form solution or a lookup table for in term of is given in (7a), Algorithm 2 unavailable via (2b). For the is not needed since a closed-form solution for is available (6) (8) Authorized licensed use limited to: University of Florida. Downloaded on January 22, 2009 at 13:53 from IEEE Xplore. Restrictions apply. 456 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 4, JULY 2002 For the given in (7b), Algorithm 2 is needed during parameter extraction since a closed-form solution for in term of is generally unavailable. Once all model parameters have been extracted, a lookup table can be generated and used to determine for a given , instead of Algorithm 2, to reduce computation time. In steps 3 and 13 of Algorithm 1, is assumed to be negli, gible relative to , permitting one to write etc. The derivation of step 10 in Algorithm 2 is shown in Appendix C. The Matlab [8] codes for Algorithms 1 and 2 are listed should not be allowed to equal 1 in Appendix D. Note that unless the solution algorithm for (2) can handle the singularity . at Algorithm 1. Generation of any hysteresis branch . 1) Data: model parameters and . 2) Input: . 3) Calculate 4) Goal: compute , , and 5) if , (initial magnetization curve) . 6) set via (1a); 7) Calculate 8) else (any return branch) via (1a). 9) Calculate via (1b). 10) Calculate 11) endif 12) Calculate via, e.g., (8), or via is unavailable. Algorithm 2 if . 13) Find Algorithm 2. Numerical solution of (2) for by Newton’s method 1) Data: Tolerance (tol), Iterative-limit (iter), and -limit (mlim) , , , 2) Input: , 3) Let . (initial guess). and . 4) Set and , 5) while 6) Set 7) if , . 8) Set 9) endif 10) Calculate . . 11) Update 12) endwhile III. PARAMETER EXTRACTION AND EXPERIMENTAL VERIFICATION The model using (7a) for has one integer parameter and four real parameters ( , , , and ) to be extracted. In addition to these parameters, (integer) and (real) need to be . extracted if (7b) is used for Fig. 6. Experimental setup for B –H loop measurement. Fig. 7. Measured and fit major loops at room temperature and 10 kHz for MN8CX ferrite. The fit major loop is generated by the improved model, using the extracted parameters in Table I for R(m) = (1 m ) . 0 The first step in parameter extraction is the acquisition of the quasistatic hysteresis loops via voltage and current measurements [1], [9]. In this paper, the hysteresis loops were measured with a 10 kHz sinusoidal excitation on an MN8CX test toroid at room temperature. The test core has the following dimensions: outer diameter 12.70 mm, inner diameter 6.35 mm, height 3.18 mm, mean path length 29.92 mm, and cross-sectional area 10.1 mm . A primary winding and a secondary winding were wound bifilar on the core, both having 25 turns of gauge AWG #28. LabVIEW [10] was used to automatically acquire hysteresis loop data by controlling the measuring instruments via IEEE 488 interface as shown in Fig. 6. The test core was demagnetized before obtaining each set of the measured data by slowly reducing the magnitude of the ac excitation voltage from its saturation value to zero [9]. To prevent self-heating, the test core was excited for only few seconds before each set of data was measured and stored for post-processing. The first quadrant of the measured major static hysteresis loop is shown in Fig. 7. The measured major hysteresis loop is shown together with several measured minor loops in Fig. 8. describes the reduction of the magnetization near Since the saturation region, its parameters are extracted from the major Authorized licensed use limited to: University of Florida. Downloaded on January 22, 2009 at 13:53 from IEEE Xplore. Restrictions apply. NAKMAHACHALASINT et al.: STATIC HYSTERESIS MODEL 457 TABLE I EXTRACTED PARAMETERS FOR THE HYSTERESIS MODEL FOR MN8CX FERRITE loop data. Nevertheless, the minor loop data are also included to check the accuracy of the improved model with respect to minor loop behavior. From the versus data for the major hysteresis loop, and, then, are computed. Appendix B suggests the following algorithm to estimate the initial values , and : for the five real parameters , , , (9) (10) (11) Fig. 8. Measured and fit major and minor loops at room temperature and 10 kHz for MN8CX ferrite. The fit B –H loops are generated by the improved model, using the extracted parameters in Table I for R(m) = cos (m=2). the model can be attributed to the physics base of the model. The model has also been verified for 3D3 ferrite [11] (resnorm 0.002 016) [7] and for ferrite [12] (resnorm 0.006 720). (12) IV. CONCLUSION is the magnetic flux density at which is where zero. Using the preceding equations and unity as the initial guess for , the initial guesses for MN8CX ferrite are found to be T, (A/m) , , A/m, . and Matlab [8] was then used to extract the model parameters. Since and are integers, they could not be extracted using the least-square-fitting function, lsqcurvefit, in a straightforward manner. Thus, was swept between 1 and 5, and between 1 pair, the five real parameters , , , and 4. For each , and were extracted using lsqcurvefit, and the squared 2-norm error (called resnorm in Matlab) recorded. The sets of parameters with the minimum squared 2-norm errors are listed in Table I. The numbers of floating-point operations (flops) of is about the the improved model using same as the flops of the original model, whereas the flops of is about the improved model using two hundred times of the flops of the original model (without retable-lookup). The model using quires more computation because of the integration of (2) via Algorithm 2. The resnorms of the improved models are at least twenty times less than the resnorm of the original model, indicating that the improved models provide a better fit to the measured data. Fig. 7 compares the measured major loop with the fit major . The close agreeloop calculated with ment between theory and experiment justifies the use of the improved models for power MnZn ferrites. Fig. 8 compares the measured and predicted minor loops . Interestingly, the fit between using theory and experiment is still good although the minor loop data were not used to extract the parameters. The robustness of To better characterize the very soft saturation characteristic functions with a particular shape have of power ferrites, been proposed for Basso-Bertotti model of static hysteresis so approaches zero asymptotically as the that the slope of functions have resulted core saturates. These improved in good agreement between measured and fit data for the major and minor loops of three commercial power ferrites. The model parameters for the other commercial power ferrites will be extracted to make this work complete. In addition, other physical phenomena that are important to power electronics, such as temperature, frequency, and core shape, ought to be incorporated into the model. The complete model should be implemented in a commercial circuit simulator, offering power electronic designers another choice for magnetic core models. APPENDIX A DERIVATION OF THE MEAN DOMAIN-WALL POSITION Similar to the classical Preisach model [13], the simplified Preisach model in [4] employs a collection of noninteracting, statistically distributed elemental hysteresis loops like those shown in Fig. 9 to describe the macroscopic hysteresis phenomena. Each elemental hysteresis loop is associated with an “idealized” one-dimensional domain-wall that is allowed to is move freely when an external magnetic field intensity applied. Domain-wall motion could be reversible (no loss) or irreversible (with loss). The reversible domain-wall motion is modeled by the zero-width elemental hysteresis loop shown in Fig. 9(a), and the irreversible domain-wall motion by the elemental hysteresis loop in Fig. 9(b). The elemental hysteresis loops are distributed statistically ac. In [5], cording to the Preisach distribution function Authorized licensed use limited to: University of Florida. Downloaded on January 22, 2009 at 13:53 from IEEE Xplore. Restrictions apply. 458 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 4, JULY 2002 Fig. 9. Elemental hysteresis loops for (a) reversible and (b) irreversible domain-wall motions. Fig. 11. Preisach diagrams for the derivation of the mean domain-wall motion: (a) for the initial magnetization, (b) for the downward branch, and (c) for the upward branch. In the classical Preisach model, the magnetization under a certain field history is found by proper integration of the distri. In the Basso-Bertotti domain-wall bution function model, however, magnetization is calculated from the domain-wall position that, in turn, is found by proper integration . of described in With the Preisach distribution function (A.1)–(A.3) and the Preisach diagram in Fig. 11, the mean domain-wall position for the initial magnetization curve, the downward branch, and the upward branch are derived, respectively, as shown in (A.4)–(A.6). The generalized equation (A.7) for any return branches can then be obtained by combining (A.5) and (A.6) Fig. 10. Irreversible Preisach distribution function 50 A/m, = 0. c is uniform with respect to cording to p (h ) in (A.2) for H and depends on = (A.4) ac- (A.1) (A.5) (A.2) (A.3) consists of two terms, and The function describing irreversible and reversible processes, , , and are defined respectively. Since , , and such that , the coefficient is used to weigh the contribution of the reversible process; the coefficient then represents the contribution of the irreversible process. in (A.2) is plotted in Fig. 10 for various The function is the Dirac delta function values of . The function enables the initial susin (A.3). As explained in [5], ceptibility and the susceptibility at a turning point (e.g., point in Fig. 1) to be non-zero. (A.6) (A.7) versus is exemplified in Fig. 1, where it is A plot of could reach infinity as approaches infinity. noted that versus are exemplified in Fig. 1 for a The plots of downward branch originating from a large positive value of , Authorized licensed use limited to: University of Florida. Downloaded on January 22, 2009 at 13:53 from IEEE Xplore. Restrictions apply. NAKMAHACHALASINT et al.: STATIC HYSTERESIS MODEL 459 and for an upward branch originating from a large negative value of . Thus APPENDIX B , AND DERIVATIONS OF , , or By chain rule If the test core is driven back from saturation to as (A.17) at (A.8) as defined by (A.6) With (A.18) as (A.19) Thus If FIND (A.9) (A.20) APPENDIX C NEWTON’S METHOD BY Using (2b), define Substitution of (A.9) in (A.8) yields (A.10) Thus as or (A.11) (A.21) as (A.12) If the test core is driven to saturation from the demagnetized state (A.13) By definition (A.14) Thus as If the test core is driven back from saturation to (A.15) at (A.16) APPENDIX D MATLAB CODES FOR ALGORITHMS 1 AND 2 The Matlab codes for Algorithms 1 and 2 are listed below along with comments, signified by the % sign. function B = findB(H,H0,B0,chi,c,Hc,Bs,n,r,q) %Data: model parameters {chi, c, Hc, Bs, n, r, q} %Input: H and {H0, B0} m0 = B0/Bs; %Goal: compute delx = x-x0, m, B if (H0 == 0) & (B0 == 0), %initial magnetization curve x0 = 0; x = chi*sign(H)*((1c)*Pirr(abs(H),Hc,c,n)+c*abs(H)); else %any return branch x0 = chi*sign(H0)*((1c)*Pirr(abs(H0),Hc,c,n)+c*abs(H0)); Authorized licensed use limited to: University of Florida. Downloaded on January 22, 2009 at 13:53 from IEEE Xplore. Restrictions apply. 460 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 4, JULY 2002 x = x0+2*chi*sign(H-H0)*… ((1-c)*Pirr(abs(H-H0)/2,Hc,c,n)+c*abs(HH0)/2); end m = G(x,x0,m0,r,q); % or m = 2/pi*atan(pi/2*x); B = m*Bs; % subfunction Pirr(delH) function output = Pirr(delH,Hc,c,n) sum = 0; for k = 1:n, sum = sum+k/n/prod (1:n-k)*(n*delH/Hc*(1c))^(n-k); end output =(delH-Hc/(1-c))+Hc/(1-c)*exp(n*delH/Hc*(1-c))*sum; function m = G(x,x0,m0,r,q) %Data: Tolerance (tol), Iterative-limit (iter), and m-limit (mlim). %Input: x, x0, m0, r, q. iRm = inline(’1./(1m.^(2*r)).^q’,’m’,’r’,’q’); % 1/R(m) tol = 0.0001; iter = 500; mlim = 0.99; m = 0; %initial guess delm = tol; i = 0; while (abs(delm) >= tol) & (i < iter), i = i+1; if abs(m) > mlim, m = sign(m)*mlim; end delm = (quad8(iRm,m0,m,[ ],[ ],r,q)-x+x0) /iRm(m,r,q); m = m-delm; end [6] Engineered Ferrites Catalog, Ceramic Magnetics, 1999. [7] P. Nakmahachalasint and K. D. T. Ngo, “An improved domain-wall model of static hysteresis for power ferrites,” Electron. Lett., vol. 36, no. 24, pp. 2020–2022, Nov. 2000. [8] Optimization Toolbox User’s Guide, Version 2, MathWorks, Natick, MA, 1999. [9] IEEE Standard for Test Procedures for Magnetic Cores, IEEE Std 3931991. [10] LabVIEW User Manual for Windows, Version 5.1, National Instruments, Austin, TX, 1999. [11] Soft Ferrites, Data Handbook MA01, Phillips Components, 2000. [12] Ferrite Cores Design Manual, Magnetics, Inc., 1999. [13] G. Bertotti, Hysteresis in Magnetism. Boston, MA: Academic, 1998. Paiboon Nakmahachalasint (S’99) received the B.Eng. degree in industrial instrumentation from King Mongkut’s Institute of Technology Ladkrabang, Thailand, in 1991, and the M.S. degree in electrical engineering from the University of Florida, Gainesville, in 1994, where he is currently pursuing the Ph.D. degree. Since 1992, he has been with Thammasat University, Thailand, where he is currently an Assistant Professor. His current research interests include magnetic materials and components. Khai D. T. Ngo (S’82–M’84–SM’02) received the B.S. degree from California State Polytechnic University, Pomona, in 1979, and the M.S. and Ph.D. degrees from the California Institute of Technology, Pasadena, in 1980 and 1984, respectively, all in electrical and electronics engineering. He was a Member of Technical Staff, General Electric Corporate Research and Development Center, Schenectady, NY, from 1984 to 1988. He has been an Associate Professor in the Department of Electrical and Computer Engineering, University of Florida, since 1988. His current research interests are magnetic materials and components, power converters, and power integrated circuits. ACKNOWLEDGMENT The authors wish to thank Dr. V. Basso for his valuable comments during the course of this work. REFERENCES [1] J. T. Hsu and K. D. T. Ngo, “A Hammerstein-based dynamic model for hysteresis phenomenon,” IEEE Trans. Power Electron., vol. 2, pp. 406–413, May 1997. [2] , “Application of field-based circuits to the modeling of magnetic components with hysteresis,” IEEE Trans. Power Electron., vol. 2, pp. 422–428, May 1997. [3] F. Liorzou, B. Phelps, and D. L. Atherton, “Macroscopic models of magnetization,” IEEE Trans. Magn., vol. 36, pp. 418–428, Mar. 2000. [4] V. Basso and G. Bertotti, “Hysteresis models for the description of domain wall motion,” IEEE Trans. Magn., vol. 32, pp. 4210–4212, Sept. 1996. [5] V. Basso, “Hysteresis models for magnetization by domain wall motion,” IEEE Trans. Magn., vol. 34, pp. 2207–2212, July 1998. Loc Vu-Quoc received the Dip.Ing. degree in structural engineering (with highest honors) from the Institut National des Science Appliquées, Lyon, France, in 1979, the M.S. degree in structural mechanics from the Illinois Institute of Technology, Chicago, in 1981, and the the M.S. degree in electrical engineering and computer science and the Ph.D. degree in structural engineering and structural mechanics from the University of California, Berkeley, in 1985 and 1986, respectively. He worked for two years (1979–1981) developing finite element codes at the Centre Technique des Industries Mécaniques, Senlis, France. He joined the University of Florida in 1988, after two years of postDoctoral work at Stanford University, Stanford, CA, and Berkeley, CA, he is currently Professor of aerospace engineering, mechanics and engineering science. His current research interests are in applied/computational electromagnetics/mechanics, and in power electronics simulation. Dr. Vu-Quoc received the NSF Presidential Young Investigator award in 1990. Authorized licensed use limited to: University of Florida. Downloaded on January 22, 2009 at 13:53 from IEEE Xplore. Restrictions apply.