Empirical Core Losses Proposals

The primary purpose of this section is to show a few empirical core losses models proposals to emphasize the diversity of parameters that are taken into account: voltage waveforms, temperature effect, and the versatility of using them to calculate core losses.

Generally, square waveforms are used in power electronic applications resulting in triangular waveform induction and flux density; therefore, core losses models must contemplate this waveform. However, a sinusoidal approximation for a duty rate of 50% is valid [121].

Curve fitting is used to approximate and characterize the model's core losses parameters. Nonetheless, curve fitting by polynomials are unstable with minor changes at entry, resulting in a big variation on coefficients, so the designer must be meticulous in selecting the model as the curve fitting method and logarithm curve [121].

The Composite Waveform Hypothesis (CHW) was proposed in 2010; it is based on a hypothesis that establishes that if a rectangular waveform given is decomposed in two pulses, total core losses are the sum of the losses generated by each one of the two pulses [26,122]. This method is described by

$$P\_{\rm CHW} = \frac{1}{T} \left[ P\_{\rm sqr} \left( \frac{V\_1}{N}, t \right) t\_1 + P\_{\rm sqr} \left( \frac{V\_2}{N}, t \right) t\_2 \right], \tag{21}$$

where *Psqr* is the power losses of a rectangular waveform with the parameters given, *V*1/*N* and *V*2/*N* are the voltage by turn in time *t*<sup>1</sup> and *t*2, respectively.

Villar's proposal [120,123] calculates core losses for three-level voltage profiles through a lineal model by parts. Villar takes the models based on Steinmetz equations and modifies them by a linear model by parts (22)–(24); this includes a duty ratio parameter in the original equations, getting expressions to calculate core losses for three-level voltage profiles.

$$P\_{\text{VillarOSE}} = k f^{\alpha} B\_{\text{s}}^{\beta} D^{\beta} \tag{22}$$

$$P\_{VillarIGSE} = 2^{a+\beta} k\_i f^\alpha B\_s^\beta D^{\beta-a+1} \tag{23}$$

$$P\_{Villar\text{WcSE}} = \frac{\pi}{4} \left( 1 + \frac{\omega}{\pi} \right) kf^{\alpha} B\_{\text{s}}^{\beta} D^{\beta}.\tag{24}$$

In Equation (24) the parameter *ω* is the duration of zero-voltage period this is, power electronics converters' switching devices take a short time to start their conduction mode; therefore, Villar includes this effect in its model by the flux density's equation as follows:

$$B\_{\rm s} = \frac{1}{2} \frac{\mathcal{U}}{N A\_{eff}} \left(\frac{T}{2} - T\_{\omega}\right), T\_{\omega} = \omega \frac{T}{2\pi}. \tag{25}$$

where *U* is a DC constant voltage, *Ae f f* is the core's area effective, and *T<sup>ω</sup>* is the length of zero-voltage period.

Villar's proposal also is applied to the equivalent elliptical loop (EEL); however, the proposal does not include thermal parameters.

Another alternative is the model proposed by Gorécki given by Equation (26) in [124,125], which includes 20 parameters. Those are divided into three groups: electrical parameters, magnetic parameters, and thermal parameters. At the same time, the magnetic parameters' are subdivided into three groups: core material parameters, geometrical parameters, and ferromagnetic material parameters corresponding to power losses. The model parameters can be obtained by the datasheets of the cores and experimental way when the inductor is tested under specific conditions, which is called local estimation.

$$P\_{\text{Goreci}} = P\_{v0} f^a B\_s^\theta (2\pi)^a \left[ 1 + a\_p (T\_R - T\_M)^2 \right] \left[ 0.6336 - 0.1892 \ln(a) \right] \tag{26}$$

$$P\_{v0} = a \exp\left( -\frac{f + f\_0}{f\_3} \right) + a\_1 (T\_R - T\_M) + a\_2 \exp\left( \frac{f - f\_2}{f\_1} \right)$$

$$\beta \begin{cases} 2 \left[ 1 - \exp\left( -\frac{T\_R}{a\_T} \right) \right] + 1.5 & \text{if } \exp\left( \frac{-T\_R}{a\_T} \right) > 0\\ 1.5 & \text{if } \exp\left( \frac{-T\_R}{a\_T} \right) < 0 \end{cases}$$

In these equations, *T<sup>R</sup>* is the core temperature, *T<sup>M</sup>* is the core temperature at which the core has a minor loss, *α<sup>p</sup>* is the losses' temperature coefficient in ferromagnetic material, *a*, *a*1, *a*2, *f*0, *f*1, *f*2, *f*3, and *α<sup>T</sup>* are material parameters.

Gorecki's model is also valid for the triangular waveform following Equation (27), where *D* is the duty ratio of the waveform.

$$P\_{\rm Greenki} = P\_{\rm \upsilon 0} f^{\rm \upsilon} B\_{\rm s}^{\beta} (2\pi)^{a} \left[ 1 + \mathfrak{a}\_{p} (T\_{\rm R} - T\_{M})^{2} \right] \left[ D^{(1-a)} + (1+D)^{(1-a)} \right]. \tag{27}$$

The main drawback of Gorecki's model is to find the overall parameters mentioned before, and solving the equations related to the power core losses.
