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This 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.

The 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.

The 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.

Although 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.

The 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.

This study equally helps researchers and practical users of the hyperbolic model.

This 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.

The purpose of this paper is to model one of the unsolved problems of magnetism, the reversal of hysteresis loops, in an analytical way. The mathematical models, describing the multiphase steel used in engineering practice, without any exception, are unsuited to provide a way to reverse the hysteretic process. In this paper, a proposal is put forward to model it by using analytical expressions, applying the reversal of the Langevin function. This model works with a high accuracy, giving useful answers to a long unsolved magnetic problem, the lack of reversibility of the hysteresis loop. The use of the proposal is shown by applying the reversal of Langevin function to a sinusoidal and a triangular waveform, the two most frequently used waveforms in research, test and industrial applications. Schematic representations are given for the wave reconstruction by using the proposed method.

The unsolved reversibility of the hysteresis loop is approached by a simple analytical formula, providing close approximation for most applications.

The proposed solution, applying the reversal of Langevin function, to the problem provides a good practical solution.

The simple analytical formula has been applied to a number of loops of widely different shapes and sizes with excellent results.

The proposed solution provides a missing mathematical tool to an unsolved problem for practical applications.

The solution proposed will reduce the work required and provide replacement for expensive complex test instrumentation.

To the best of the authors’ knowledge, this approach used in this study is the first successful approach in this field, irrespective of the required waveform, and is completely independent of the model used by the user.

The mathematical complexity of the B_J(x) Brillouin function makes it unsuitable for most calculations and its application difficult for computer programming in magnetism. Here, its approximation with the tanh function is proposed to ease the mathematical operations for most cases. The approximation works with good accuracy, acceptable in practical calculations. This approximation has already formed the foundation of the “hyperbolic model” in magnetism for the study of hysteretic phenomena. The reversal of the Brillouin function is an important but difficult mathematical problem for practical purposes. Here, a proposal has been put forward for an easy approximation using an analytical expression. This provides a good workable solution for the B_J(x)⁻¹ function dependent on J, the angular momentum quantum number of the material used. The proposed approximation is applicable within the working range of practical applications. The paper aims to discuss these issues.

The multi-variant Brillouin function is closely approximated by the tanh function to ease calculations. Its mathematically unsolved reversed function is approximated by a simple analytical expression with a good working accuracy.

The Brillouin function and its reversal can be approximated for practical users mostly for professionals working in Magnetism.

Most if not all practical problems in Magnetism can be solved within the limitations of the two approximations.

Both proposed functions can ease the mathematical problems faced by researchers and other users in Magnetism.

Ease the frustration of most users working in the field of Magnetism.

The application of the tanh function for replacing the Brillouin function led to the creation of the hyperbolic model of hysteresis. To the author's knowledge, the reverse function was mathematically only solved in 2015 with a vastly complicated mathematics, and is hardly suitable for practical calculations in Magnetism. The proposed simple expression can be very useful for theorists and experimental scientists.

A brief account of the exponential model introduces the reader to one of the mathematical descriptions of the double non‐linearity of the hysteretic phenomena. The model described here satisfies the requirement for calculating the Laplace transforms in closed form for excitation waveforms constructed of straight lines. The method is demonstrated by applying it to a triangular excitation in the hysteretic process. It is shown that the Laplace transform of the induction waveform can also be calculated when the same excitation waveform is being applied in an anhysteretic process. It is also shown that when the excitation is small and falls within the limits of the Rayleigh region the calculation becomes simpler. This is demonstrated by formulating the Laplace transform of the induction waveform that resulted from triangular excitation in the Rayleigh region for both the hysteretic and anhysteretic cases.

This paper sets out to develop analytical solution to the hysteresis, eddy current and excess losses using the T(x) model. Based on Steinmetz' postulation, the losses, represented by the area enclosed by the hysteresis loop, are individually formulated in analytical form. The model is applied to sinusoidal and triangular excitation wave forms.

The equivalent interaction fields introduced into the model represent the losses individually by applying the separation and superposition principle.

Contrary to the presently used models, this model describes the hysteresis loop with its natural sigmoid shape and describes the losses individually in simpler mathematical formulation.

Experimental verification will still be needed as to the accuracy of the model and the applicability to the various magnetic materials.

The model presented here gives a more realistic presentation of the hysteresis loop and by using simpler mathematics than other models it is more accessible to the practical user. At the same time with the easy mathematics and its visual presentation it is a great value to people engaged in theoretical research in the field of magnetics.

In contrast with present magnetic loss models, using almost exclusively MSPM with “flat power” loop or the elliptical equivalent loop approximations, these calculations based on the T(x) model of hysteresis and uses realistic shape for the hysteresis loop, resulting in a simpler mathematical formulation.

The paper presents the derivation of the Fourier coefficients of the magnetisation wave form in closed form for periodic triangular field excitation. The mathematical model used here to approximate the hysteresis loop applies exponential functions and it is presented briefly in the first part of the paper. This method of calculation is applicable to a group of excitation wave forms, constructed of straight lines, such as square, triangular and trapezoid wave forms. The criteria for the Rayleigh region is given and an appropriate formulation of the Fourier coefficients in closed form for the Rayleigh region is also described. It is shown that the calculation is also applicable to anhysteretic processes both in the saturation and in the “small” signal (Rayleigh) region. The Fourier components of the periodic magnetisation wave form resulting from an anhysteretic magnetisation process are also calculated in closed form for triangular field excitation.

The paper sets out to develop the T(x) model, based on well known principles and using the free energy of the ferromagnetic binary system as a starting point.

With the inclusion of the coercive and the intermolecular forces the model fully describes the hysteretic process for both major and minor loops, with exchange field between adjacent magnetic moments.

The paper formulates the dependence of the hysteretic system's normalised free energy on parameters such as the temperature, coercivity, other magnetic and materials properties of the ferromagnetic medium.

Experimental verification will still be needed as to the accuracy of the model and the applicability to the various magnetic materials.

The paper provides an easy mathematical and visual method to present the energy state and its variation of the magnetic materials during magnetisation, including non‐saturation conditions. It has a great value to people engaged in theoretical research in magnetism.

So far free energy calculations were only possible for major hysteresis loops. The T(x) model, as presented here, is applicable to the calculation of the free energy flow of any symmetrical minor loops as well.