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Register a new userAn Approach to Modeling and Scaling of Hysteresis in Soft Magnetic Materials. I Magnetization Curve
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Krzysztof Sokalski prof
Publication date
2014
fields
Physics
and research's language is
English
Authors
Krzysztof Z. Sokalski
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A new mathematical model of hysteresis loop has been derived. Model consists in an extansion of tanh($cdot$) by extanding the base of exp function into an arbitrary positive number. The presented model is self-similar and invariant with respect to scaling. Scaling of magnetic hysteresis loop has been done using the notion of homogenous function in general sense.
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Assuming that core loss data of Soft Magnetic Materials obey scaling relations, models describing the power losses in materials exposed to non-sinusoidal flux waveforms and DC Bias conditions have been derived. In order to test these models, the measurement data for two materials have been collected and the core losses calculated. Agreement between the experimental data and the model predictions is satisfactory.
Data collapse enables comparison of measurement data measured in different laboratories on different samples. In the case of energy losses in Soft Magnetic Materials (SMM) the data collapse is possible to achieved only if the measurement data can be described by the two components formula. For more complicated cases we propose to perform data collapses sequence in the two-dimensional subspaces $L_{i,i+1}$ spanned by the appropriate powers of frequency ${f^{i},f^{i+1}}$. Such approach enables the data comparison in the different two-dimensional subspaces. This idea has been tested with measurement data of the four SMM-s: amorphous alloy textrm{Fe}_{78}textrm{Si}_{13}textrm{B}_{9}$, amorphous alloy $textrm{Co}_{71.5} textrm{Fe}_{2.5} textrm{Mn}_{2} textrm{Mo}_{1} textrm{Si}_{9} textrm{B}_{14}$, crystalline material -- oriented electrotechnical steel sheets 3% Si--Fe, iron--nickel alloy $79% textrm{Ni}-textrm{Fe}$. Intermediate calculations revealed interesting property of the energy losses in the cristalline and amorphous SMM-s which lead to the following hypothesis. Let $P_{tot,1,2}=f_{1,2}(1+f_{1,2})$ be scaled two-components formula for the energy loss in SMM, where $f_{1,2}$ is the corresponding scaled frequency. Then the scaled energy losses values in amorphous SMM are below the second order universal curve $P_{tot,1,2}=f_{1,2}(1+f_{1,2})$, whereas the scaled energy losses values in crystalline SMM are above that universal curve.
In published papers, the Gibbs free energy of ferroelectric materials has usually been quantified by the retention of 6th or 8th order polarization terms. In this paper, a newly analytical model of Gibbs free energy, thereout, a new model of polarization-electric field hysteresis loops in ferroelectric materials has been derived mathematically. As a model validation, four patterns of polarization-electric field hysteresis loops of ferroelectric materials have been depicted by using the model. The calculated results indicated that the self-similar model can characterize the various patterns of hysteresis loops in ferroelectric materials through adjusting the external excitation or the synthetically parameter (e.g., electric, temperature, and stress, etc.) employed in the model.
In order to understand the physical hysteresis loops clearly, we constructed a novel model, which is combined with the electric field, the temperature, and the stress as one synthetically parameter. This model revealed the shape of hysteresis loop was determined by few variables in ferroelectric materials: the saturation of polarization, the coercive field, the electric susceptibility and the equivalent field. Comparison with experimental results revealed the model can retrace polarization versus electric field and temperature. As a applications of this model, the calculate formula of energy storage efficiency, the electrocaloric effect, and the P(E,T) function have also been included in this article.
Magnetic junction is considered which consists of two ferromagnetic metal layers, a thin nonmagnetic spacer in between, and nonmagnetic lead. Theory is developed of a magnetization reversal due to spin injection in the junction. Spin-polarized current is perpendicular to the interfaces. One of the ferromagnetic layers has pinned spins and the other has free spins. The current breaks spin equilibrium in the free spin layer due to spin injection or extraction. The nonequilibrium spins interact with the lattice magnetic moment via the effective s-d exchange field, which is current dependent. Above a certain current density threshold, the interaction leads to a magnetization reversal. Two threshold currents are found, which are reached as the current increases or decreases, respectively, so that a current hysteresis takes place. The theoretical results are in accordance with the experiments on magnetization reversal by current in three-layer junctions Co/Cu/Co prepared in a pillar form.
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