*4.1. Mathematical Models*

Mathematical hysteresis modelling is divided into approaches based on the theory of micro-magnetics and the methods based on curve-fitting. The hysteresis models require complex computation to calculate the model parameters, the material parameter that manufacturers do not provide, or mathematical approximations where the accuracy depends on the number of data points to fit hysteresis loops [86,87]. These drawbacks remain if they are analyzed from a purely mathematical or a physical point of view [78].

The Preisach model and the Jiles-Atherton (J-A) model have been widely used in practical problems to calculate core loss, they are in continuous improvement, and they are considered a valuable and convenient tool to the hysteresis modelling [87–89].

The Preisach is a scalar-static model that considers several quantities of basic's domainwalls [72]. It is an accurate-phenomenological model, such that it could describe any system that shows a hysterical behavior [87,90]. This model can be a link between theory and experimentation to describe a microscopical system by measuring macroscopic behavior [78].

Given a typical hysteresis loop for separate domains as shown in Figure 4, *H<sup>d</sup>* and *H<sup>u</sup>* are the switching magnetic fields "down" and "up", respectively. The magnetization *M*(*t*) of a particle having the hysteresis loop *m*ˆ(*Hu*, *H<sup>d</sup>* ) is described as the magnetic moment *m<sup>s</sup>* while the particle is switched up *m*ˆ(*Hu*, *H<sup>d</sup>* )*H*(*t*) = +*m<sup>s</sup>* or down *m*ˆ(*Hu*, *H<sup>d</sup>* )*H*(*t*) = −*m<sup>s</sup>* [87]. It is assumed that all domains have a distribution of reversal fields *H<sup>u</sup>* and *H<sup>d</sup>* that can be characterized by a distribution function *φ*(*Hu*, *H<sup>d</sup>* ), so the Preisach model is usually defined as:

$$M(t) = \iint\_{H\_{\rm u} \ge H\_{\rm d}} \phi(H\_{\rm u}, H\_{\rm d}) \, \* \, \mathfrak{m}(H\_{\rm u}, H\_{\rm d}) \, \* H(t) dH\_{\rm u} dH\_{\rm d}.\tag{1}$$

**Figure 4.** Hysteresis loop of (**a**) an oriented magnetic domain, (**b**) a multidomain particle of a 3C90 Epcos ferrite core at 500 Hz and 28.5 ◦C. Source: Adapted from [91].

A drawback of the Preisach model is the several measurement data to adapt it to the B-H environment's changes [72]. Nonetheless, it can track complex magnetization processes and minor loops [78].

On the other hand, the J-A model requires solving a strongly non-linear system of equations, with five scalar parameters, which are determined in an experimental way; this is, measuring the hysteresis loop [92]. This model predicts the major hysteresis loops under quasi-static conditions, besides there is a dynamic conditions version in which eddy current loss, DC-bias fields, anisotropy, and minor loops are included [86,93].

The J-A model is described as the sum of reversible *Mrev* and irreversible *Mirr* magnetizations, hence, the total magnetization *M* is:

$$M = M\_{irr} + M\_{rev} \tag{2}$$

$$\frac{dM\_{rev}}{dH} = c \left( \frac{dM\_{an}}{dH} - \frac{dM\_{irr}}{dH} \right) \tag{3}$$

$$\frac{dM\_{irr}}{dH} = \frac{M\_{an} - M\_{irr}}{\delta \mathbf{x}/\mu\_0 - \alpha\_{ic}(M\_{an} - M\_{irr})} \tag{4}$$

$$M\_{an} = M\_s \left( \coth \frac{H\_{eff}}{a\_s} - \frac{a\_s}{H\_{eff}} \right) \tag{5}$$

$$H\_{eff} = H + \mathfrak{a}\_{ic} M.\tag{6}$$

In these equations, *c* is the reversibility coefficient, *αic* is the inter-domain coupling, *δ* indicates the direction for magnetizing field (*δ* = 1 for the increasing field, and *δ* = −1 for decreasing field), *x* is the Steinmetz loss coefficient, *M<sup>s</sup>* is the saturation magnetization, *Man* is anhysteretic magnetization, *He f f* is the effective field, *H* is the magnetic field, *µ*<sup>0</sup> is the permeability in free-space, and *a<sup>s</sup>* is the shape parameter for anhysteretic magnetization [73]. A drawback of the J-A model is the computing time and the computing resources to solve a strongly non-linear system.

On the other hand, Eddy's losses are produced due to changing magnetic fields inside the analyzed core. These fast changes generate circulating peak currents into it, in the form of loops, generating losses measured in joules [94].

These losses depend on the actual signal shape instead of only the maximum value of the density flux. The power loss is proportional to the area of the measured loops and inversely proportional to the resistivity of the core material [50,94].

Eddy currents are perpendicular to the magnetic field axis and domain at highfrequency, as well, they are proportional to the frequency. This phenomenon can be

treated as a three-dimensional character or in a simplified way [95]. In a magnetic circuit, Eddy currents cause flux density changes in specific points of its cross-section [96].

One way to describe the Eddy currents distribution is based on Maxwell's equations [95,97]. In practice, especially on higher frequencies, there is a difference in the measured total loss between the sum of the hysteresis loss and the Eddy current loss; this difference is known as anomalous or residual losses [13]. The effect of temperature and the relaxation phenomena are enclosed in this category [82].
