*4.3. Loss Separation Models*

The loss separation method (LSM) is well-known, accurate and straightforward in many applications [94]. It defines the core loss (*PC*) under a dynamic magnetic excitation as the sum of three components: the hysteresis loss (*P<sup>h</sup>* ), the Eddy current loss (*Peddy*) and the residual or anomalous loss (*Panom*) [82] is given by:

$$P\_{\mathcal{L}} = P\_{\mathcal{h}} + P\_{\text{eddy}} + P\_{\text{anom}}.\tag{8}$$

According to Equation (8), the power loss per unit volume is the sum of a hysteresis and dynamic contribution; Eddy currents and residual loss are part of the latest one [100].

The loss separation method was proposed in 1924 by Jordan, which describes the core losses as the sum of the hysteresis loss (or static loss) and the Eddy loss (or dynamic loss) [98]

$$P\_{\mathbb{C}} = k\_{\mathbb{H}} f B\_{\mathbb{S}} + k\_{\text{eddy}} f^2 B\_{\mathbb{S}}^2. \tag{9}$$

Years later, Bertotti extended Jordan's proposal, adding an extra term to calculate residual losses. G. Bertotti is the maximum exponent in the loss separation methods; his theory provides a solid physical background. The total power loss can be calculated at any magnetizing frequency as the sum of three components [44]. This is show in the following equation

$$P\_{\mathbb{C}} = k\_{\hbar} f B\_{\mathbb{s}}^{\mathbb{X}} + k\_{\text{eddy}} f^2 B\_{\mathbb{s}}^2 + k\_{\text{anom}} f^{1.5} B\_{\mathbb{s}}^{1.5} \,\,\,\,\,\tag{10}$$

where *k<sup>h</sup>* , *kc*, *kanom* are the hysteresis, Eddy currents, and anomalous coefficients, respectively. The hysteresis coefficient decreases if the magnetic permeability increase. The frequency is represented by *f* , while *x* is the Steinmetz coefficient and has values between 1.5 and 2.5, according to the permeability of the material. Finally, *B<sup>s</sup>* is the peak value of the flux density amplitude [23,100].

From Equation (10), Bertotti defined eddy currents and anomalous losses in Equations (11) and (12), respectively, for lamination materials.

$$P\_{cdy} = \frac{\pi^2 d^2 f^2 B\_s^2}{\rho \beta} \tag{11}$$

$$P\_{anom} = 8\sqrt{\frac{GAV\_0}{\rho}}B\_s^{1.5}\sqrt{f} \,\tag{12}$$

where *G* is about 0.2, *ρ* is the electrical resistivity, *d* is the lamination thickness, *A* is the cross sectional area of the lamination, *β* is the magnetic induction exponent, *G* is a dimensionless coefficient of Eddy current, and *V*<sup>0</sup> is a parameter that characterizes the statistical distribution of the magnetic objects responsible for the anomalous eddy currents [101]. However, this method only provides average information, so it is not able to calculate core loss under harmonic excitation [94]. Still, it can calculate core loss under square waveforms considering DC bias [102].
