Steinmetz law in iron–phenolformaldehyde resin soft magnetic composites Peter Kollár a,n , Vladimír Vojtek a , Zuzana Birčáková a , Ján Füzer a , Mária Fáberová b , Radovan Bureš b a Institute of Physics, Faculty of Science, Pavol Jozef Šafárik University, Park Angelinum 9, 04154 Košice, Slovakia b Institute of Materials Research, Slovak Academy of Sciences, Watsonova 47, 04001 Košice, Slovakia article info Article history: Received 24 July 2013 Received in revised form 9 October 2013 Available online 23 October 2013 Keywords: Soft magnetic composite Steinmetz law Magnetic property Energy loss Hysteresis loop abstract The validity of Steinmetz law describing the dc energy losses as a function of maximum induction has been investigated for iron based soft magnetic composites (SMCs) up to 1.4 T with the effort to find a physical meaning of the coefficients in Steinmetz law. In the Rayleigh region the coefficients were expressed mathematically using the Rayleigh law. Further the “range of validity of Steinmetz law” was found to be from 0.3 T to 1.2 T. The typical “straight” shape of hysteresis loops of SMCs at lower maximum induction was approximated by linear functions in order to express the dc losses in form of Steinmetz law. & 2013 Elsevier B.V. All rights reserved. 1. Introduction One of the most important characteristics of magnetic materials is the total energy dissipated during the magnetization process which causes heating of devices. For that reason the effort is to minimize these energy losses as much as possible. In practice it is useful to be able to express the energy losses mathematically, therefore for more than a hundred years scientists and engineers are trying to find the most appropriate relations. The energy losses come from the irreversible mechanisms of magnetization process: the irreversible domain wall displacement and magnetization vector rotation. According to a general picture of energy losses [1] the eddy currents induced by the magnetic induction changes accompanying the mentioned mechanisms are responsible for all the energy losses in magnetic materials, but the traditional concept separating the total losses W t to be a sum of three components: the dc losses W dc , the classical losses W c and the domain wall eddy current losses W dw [2–4], is still being used today, primarily by materials engineers. The dc losses W dc are related mainly to the structural imperfec- tions in magnetic material, the stress regions and impurities [3] as well as the surface roughness [5], becoming the sources of pinning sites hindering the domain wall motion and the local eddy currents are dissipated as the Barkhausen jumps occur. Various empirical relations for W dc have been proposed up to the present time, e.g. by Richter (1910): W dc ¼ aB m þ bB m 2 , where B m is the maximum magnetic induction and a and b denote the coefficients; by Anderson and Lance (1922): W dc ¼ aB m H C , where H C is the coercive field [6]; but the most popular and convenient is the oldest law stated by Steinmetz in 1892 [7] which describes the relation between W dc and B m as following: W dc ¼ K dc B m 1.6 (J/m 3 ), or more general W dc ¼ K dc B x m ; ð1Þ where K dc and x are parameters depending on the material and including the structural aspects, the domain wall pinning and magnetization reversal [3]. Coefficient x is sometimes called the “Steinmetz coefficient”. Since Steinmetz formulated the relation, many studies have dealt with it in an effort to improve it or to extend its validity. Many kinds of magnetic materials were investigated and many modifications of the law were proposed, e.g. the “ac Steinmetz law”:P t ¼ K ac B x m f α (W/m 3 ), where f is the magnetizing frequency [8], or the equivalent description introducing quantity “hysteresis field” H hyst ¼ W dc =4B m [9], and several “Steinmetz-like” relations formulated for other magnetic quantities as the magnetization, the coercive field, the remanence or the remanence work, obtained by an analysis of minor hysteresis loops in studies of Kobayashi et al. [10–12]. For the dc losses they write the Steinmetz law in the form W dc ¼ W 0 dc ðB m =B S Þ n , where B S is the saturation magnetic induction and W 0 dc and n are the coefficients. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jmmm Journal of Magnetism and Magnetic Materials 0304-8853/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2013.10.017 n Corresponding author. Tel.: þ421 55 2342529; fax: þ421 55 6222124. E-mail address: peter.kollar@upjs.sk (P. Kollár). Journal of Magnetism and Magnetic Materials 353 (2014) 65–70