IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 12, DECEMBER 2010 4001 Effect of Anisotropy and Direction of Magnetization on Complex Permeability of Ferromagnetic Rectangular Thin Slabs Behzad Ahmadi , Hervé Chazal , Thierry Waeckerlé , and James Roudet Grenoble Electrical Engineering Laboratory (G2ELab), UMR CNRS 5269, 38402 St Martin d’heres, France Imphy research centre, ArcelorMittal, 58160 Imphy, France Ultrasoft magnetic materials, such as amorphous, nanocrystalline, and polycrystalline alloys, have been successfully used for power electronic applications during recent years. However, enhancements are needed for the integration of power electronic features, which involves high power densities and operating frequencies up to a few megahertz. Complex permeability spectra, which are used to describe material behavior in these applications, depend mainly on domain-wall motions and coherent magnetization rotation mechanisms. In this paper, we present a model describing these mechanisms, in terms of magnetic anisotropies and domain structure of the material. To validate the model, we measured permeability spectra of polycrystalline Ni-Fe alloys under mechanical stress using a specific setup. These measurements are suitable for comparing the physical model according to different magnetoelastic anisotropies. Results are useful for correlating high-frequency magnetic behavior and the annealing process of materials. Index Terms—Magnetic anisotropy, magnetic domains, magnetization processes, permeability measurement. I. INTRODUCTION P OWER electronic converters use switching techniques to convert the electrical energy. In these converters, the magnetic components are unavoidable to associate input and output voltage sources. For medium switching frequencies from 10 kHz to 1 MHz, spinel ferrites are often used for these components. These ferrites exhibit very low conductivity and their characterization methods are well known [1], [2]. In recent years, some conductor magnetic materials, such as amorphous, nanocrystalline alloys (FeSi- CuNbB), and polycrystalline Ni-Fe alloys, were introduced in these applications [3]. These alloys are cast in form of very thin ribbons, of about 20 m thick, using planar flow casting or lamination techniques. The magnetic properties of these rib- bons are adjusted by specific heat treatments under mechanical stress or magnetic field, to develop ultrasoft magnetic materials. Due to eddy-current effects, in medium frequencies, magnetic properties of these materials are downgraded. First materials encountered this problem were polycrystalline Si-Fe sheets used in electrical applications, such as power transformers, for operating frequencies up to 400 Hz. Analytical models com- bined with numerical methods are employed to analyze this problem and to optimize the design of these transformers [4], [5]. It should be highlighted that magnetic skin depth to ribbon thickness ratio are the same in 0.35 mm Si-Fe thin sheets, used in 50 Hz applications, and in 20 m thin ribbons at 15 kHz, which are used in power electronic applications. Analyzing the effects of different annealing processes, different chemical compositions and intrinsic physical properties on high-fre- quency behavior of these ribbons is of the main interest to the research on this topic nowadays; either from a metallurgical [6], [7] or power electronic point of view [8]. Manuscript received March 13, 2010; revised May 14, 2010 and June 28, 2010; accepted August 06, 2010. Date of publication September 13, 2010; date of current version November 30, 2010. Corresponding author: B. Ahmadi (e-mail: behzad.ahmadi@g2elab.grenoble-inp.fr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2076339 At medium frequencies, where the domain-wall movements are dumped by eddy currents, magnetization rotation inside the magnetic domains should also be considered to describe the be- havior of material. These two mechanisms constitute the overall magnetization process. The importance of each one depends on physical char- acteristic and the microstructure of the material. To study the frequency behavior of these phenomena, com- plex permeability spectra of material are measured and anal- ysed, which are also significant parameters for power electronic designers. In Section II-A, the physical model describing domain-wall motions is presented and related works are recalled. Mesoscopic behavior of material is highlighted and pointed out through statistical and scale hypothesis in Section II-B. Then in Section II-D, according to a depicted scenario, variation of magnetic domain structure is investigated depending on mag- netic anisotropy induced by mechanical stress. A unified model considering both magnetization rotation and domain-wall motions in a wide frequency range is then pro- posed. An experimental setup is described in Section III and permeability spectra of thin Ni-Fe polycrystalline ribbons under mechanical static stress are measured. The theoretical results are compared to the experiment in Section IV. The complex permeability spectra and so specific eddy-current signatures of both magnetization mechanisms are then used to check the consistency of the model. II. PHYSICAL MODEL A. Review of Existing Magnetic Domain-Wall Models (DW) The conductor ferromagnetic materials are depicted in the frame of magnetic domain structure. The basic theory of mag- netic domains has been developed by many writers; a good review of pioneer works is given in [9]. Different domain geometries are first introduced by Landau and then have been developed by Néel [10]. For laminated ferromagnetic conduc- tors, a domain structure with 180 saturated domains separated with Bloch walls is generally accepted. For frequencies where skin depth to material thickness ratio is almost one, a good 0018-9464/$26.00 © 2010 IEEE