Binary Relations between Magnitudes of Different Dimensions Used in Material Science Optimization Problems Pseudo-State Equation of Soft Magnetic Composites

Abstract

New algoritm for optimizing technological parameters of soft magnetic compozites has been derived on the base of topological structure of the power loss characteristics. In optimization processes of magnitudes obeying scaling it happen binary relations of magnitudes having different dimensions. From mathematical point of view in general case such a procedure is not permissible. However, in a case of the system obeying the scaling law it is so. It has been shown that in such systems binary relations of magnitudes of different dimensions is correct and has mathematical meaning which is important for practical use of scaling in optimization processes. Derived here structure of the set of all power loss characteristics in soft magnetic composite enables us to derive a formal pseudo-state equation of SMC. This equation constitutes a realation of the hardening temperature, the compaction pressure and a parameter characterizing the power loss characteristic. Finally, the pseudo-state equation improves the algoritm for designing the best values of technological parameters.

Scaling in Magnetic Materials

“…This concept is based on the assumption that power losses in SMC obey the scaling law. The efficiency of this approach has been confirmed in [9]. The scaling is very useful tool due of the three reasons:…”

“…By introducing new independent variables, we introduce new scales and corresponding new scaling parameters [8]. We have shown in [8,9] that it is always possible to introduce a new characteristic scale; however, one must investigate whether the considered system possesses corresponding symmetry. This can be known only from investigations of the measurement data.…”

“…Γ n , α and β parameters have been estimated for different values of pressure and temperature [9]. For the purpose of this paper, we take into account only one family of power losses characteristics which are presented in Figures 15 and 16.…”

“…The corresponding estimated values of the model parameters are presented in Table 7. For all other details concerning SMC material and measurement data, we refer to [9]. Now we are ready to formulate the goals of this section.…”

An Approach to Modeling and Scaling of Hysteresis in Magnetic Materials. Magnetization Curve

“…Whereas in designing the magnetic materials we need analytic formulas returning real values of physical magnitudes which are relevant in this process. Just the mathematical models together with an experimental data constitute powerful tool for designers [18], [19]. In this paper we have derived the simple model which is promising for the mentioned above applications.…”