678 IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 2, FEBRUARY 2009 Modeling the Electromagnetic Behavior of Nanocrystalline Soft Materials Peter Sergeant and Luc Dupré Faculty of Engineering, Department of Electrical Energy, Systems and Automation, Ghent University, B-9000 Ghent, Belgium Department of Electrotechnology, Faculty of Applied Engineering Sciences, University College Ghent, B-9000 Ghent, Belgium A model that describes the magnetic behavior of nanocrystalline ring cores is useful for simulations of electronic circuits that contain inductors or transformers using these cores. A general but computationally demanding model combines a macroscopic model of the ribbon with a dynamic Preisach hysteresis model. In this paper, we present two models that—taking into account the principle of loss separation—make it possible to avoid the use of the CPU time consuming dynamic Preisach model. Both models compute the waveform of the magnetic flux density for an arbitrary waveform of the magnetic field, or vice versa. The first model uses a macroscopic model based on the plane wave theory and the classical rate-independent Preisach formalism. The macroscopic model operates in the frequency do- main and applies the harmonic balance principle. Because of nonlinearity, the model is solved iteratively by a Newton-Raphson scheme. The second model starts from a single evaluation of the classical Preisach model. Additionally, it uses a lookup table that is a function of the flux density and its time derivative to evaluate the classical and excess field to be added. The models are validated by measurements between 2 and 100 kHz on Vitroperm nanocrystalline ring cores. Index Terms—Hysteresis, losses, nanocrystalline material. I. INTRODUCTION T HE high frequency behavior of passive magnetic compo- nents is important in power electronic circuits. These high frequencies originate from the fundamental frequency and har- monics of modulated power signals. By fast switching between two or more voltage levels, the desired voltages and currents can be synthesized. The faster the switching, the more accurate the desired waveform can be generated by, e.g., pulse width mod- ulation. For electrical machines, the frequency of pulse width modulation is typically 20–25 kHz, with “extremes” between 2 and 100 kHz. Here, nanocrystalline materials are useful thanks to a rela- tively high induction (compared to ferrites), a high permeability and low electromagnetic losses up to rather high frequencies. They are widely employed in power electronic circuits to build inductors or transformers that create filters of high harmonics, galvanic isolation, or energy storage. For several types of soft magnetic materials, manufacturers provide data about the geometry, the magnetic permeability, the saturation and the losses as a function of the frequency [1]–[4]. However, the losses are usually given for sinusoidal waveforms, while many applications in power electronics use other wave- forms such as square waves. Based on the statistical loss theory (Section II-B), the losses can be computed to whatever supply waveform, provided that all the material coefficients are evalu- ated starting from a limited number of measurements under si- nusoidal excitation. Recently, the losses in nanocrystalline cores under square waveforms have been studied in [5] by a model based on the one-dimensional transmission line theory in com- bination with an impedance function to represent the hysteresis and excess losses [6]. In this paper, we do not only want to compute the losses, we also want to reproduce the waveforms by using fast models so that circuit simulations of a network including nonlinear and Manuscript received August 07, 2008; revised October 16, 2008. Current ver- sion published February 11, 2009. Corresponding author: P. Sergeant (e-mail: peter.sergeant@ugent.be). Digital Object Identifier 10.1109/TMAG.2008.2008109 hysteretic magnetic components can be elaborated. The consid- ered frequency range in the simulations is 2–100 kHz, which is sufficient for applications of pulse width modulation in elec- trical machines. Nevertheless, the presented models are valid also outside this frequency range. II. HOW TO MODEL ELECTROMAGNETIC BEHAVIOR IN A LAMINATION? A numerical model that describes the magnetic behavior in a lamination of a material, has to be a combination of a macro- scopic model that implements the Maxwell equations, and a constitutive law. One such model has been developed in [7] where the macroscopic model is a 1-D time domain finite ele- ment model (FEM) and the constitutive law is a rate-dependent (dynamic) Preisach model [8]. In this paper, two other methods are presented, based on the statistical loss theory of Bertotti [6]. In method 1—Section III-A—the macroscopic model uses the one-dimensional plane wave theory in the frequency domain and the constitutive law is a rate-independent classical Preisach model (CPM) [9], [10]. In method 2—Section III-B—the macroscopic model is a lookup table and the constitutive law is also the CPM. Before discussing the two methods, Section II-A explains the macroscopic model that will be used in method 1: the plane wave model (WM) based on the plane wave theory. Section II-B describes how the macroscopic model and the constitutive law together calculate the three loss terms—hysteresis loss, classical loss, and excess loss—in the statistical loss theory. A. Plane Wave Model (WM) The plane wave theory is explained in [11] to find the field in magnetic shields in the frequency domain for one frequency component (no additional harmonics), but we extend the theory towards harmonics. In this one-dimensional approach, all electromagnetic vector quantities are represented by scalars 0018-9464/$25.00 © 2009 IEEE