Analytical way of transforming the hyperbolic hysteresis model into the T(x) model

Abstract

PurposeThis paper aims to present an analytical way of formulating the vital parameters of an equivalent hysteresis loop of a composite, multi-component magnetic substance. By using the hyperbolic model, the only model, which separates the constituent parts of the composite magnetic materials, an equivalent loop can be composed analytically. So far, it was only possible to superimpose the tanh functions by numerical method. With this transformation, all multi-component composite substances can be treated mathematically as a single-phase material, as in the T(x) model, and include it in mathematical operations. The transformation works with good accuracy for major and minor loops and provides an easy analytical way to arrive to the vital parameters. This also shows an analytical way to the easy solution of some of the difficult problems in magnetism for multi-component ferrous materials, such as Fourier and Laplace transforms, accommodation and energy loss, already solved for the T(x) model.Design/methodology/approachThe mathematical single loop formulation of hysteresis loop of a multi-phase substance shows the way in good approximation of the sum of constituent loops, described by tanh functions. That was so far only possible by numerical methods. By doing so, it becomes equivalent to the T(x) model for mathematical operations.FindingsThe described method gives an analytical formulation [identical to the T(x) model] of multi-component hysteresis loops described by hyperbolic model, leading to simple solution of difficult problems in magnetism such as loop reversal.Research limitations/implicationsAlthough the method is an approximation, its accuracy is good enough for use in magnetic research and practical applications in industries engaged in application of magnetic materials.Practical implicationsThe hyperbolic model is the only one which separates the magnetic substance, used in practice, to constituent components by describing its multi-component state. Superimposing the components was only possible so far by numerical means. The transformation shown is an analytical approximation applicable in mathematical calculations. The transformation described here enables the user to apply all rules applicable to the T(x) model.Social implicationsThis study equally helps researchers and practical users of the hyperbolic model.Originality/valueThis novel analytical approach to the problem provides an acceptable mathematical solution for practical problems in research and manufacturing. It shows a way to solutions of many difficult problems in magnetism.