*4.4. Empirical Models*

A big group of empirical models is based on a simple power-law equation, which was proposed in 1982 by Charles Steinmetz to calculate hysteresis loss without including the frequency relation [103]. This equation is known as the Steinmetz equation or original Steinmetz equation (OSE),

$$P\_{\rm OSE} = k f^{\alpha} B\_{\rm s}^{\beta} \, \tag{13}$$

where *P*OSE is the time-average core loss per unit volume, *B<sup>s</sup>* the peak induction of the sinusoidal excitation, *k* is a material parameter, *α*, and *β* are the frequency and magnetic induction exponents, respectively, often referred to as Steinmetz parameters [76]. Typically *α* is a number between 1 and 2, and *β* is typically between 1.5 and 3 [104].

The Steinmetz parameters can be determined from a double logarithm plot by linear curve fitting of the measured core loss data. Therefore Equation (13) assumes only sinusoidal flux densities with no DC bias [76,77,98,105].

In modern power electronics applications, sinusoidal-wave voltage excitation is not practical because many of them, like power converters, require square-wave voltage excitation [106]. Therefore, a square waveform's core loss can be lower than the sinusoidal-wave's losses for the same peak flux density and the same frequency [76,107].

To overcome the aforementioned situation, modifications to OSE have been made. The result was the developing Modified Steinmetz Equation(MSE) [108], General Steinmetz Equation (GSE) [109], Doubly Improved Steinmetz Equation (Improved-Improved Steinmetz Equation, *i* <sup>2</sup>GSE) [110], Natural Steinmetz Equation (NSE) [85], and Waveform-Coefficient Steinmetz Equation (WcSE) [111]. It is important to mention that the OSEs's constants *α*, *β*, and *k* remain in those expressions.

Modified Steinmetz Equation was the first modification proposed to calculate core loss with non-sinusoidal excitation, incorporating into OSE the influence of a magnetization rate. MSE is proposed in [112] as follows

$$P\_{\rm MSE} = (kf\_{eq}^{\alpha -1} B\_s^{\beta}) f\_r$$

$$f\_{eq} = \frac{2}{\Delta B^2 \pi^2} \int\_0^T \left(\frac{dB(t)}{dt}\right)^2 dt,$$

where *f<sup>r</sup>* is the periodic waveform fundamental frequency, *feq* is the frequency equivalent, ∆*B* is the magnetic induction peak to peak, and *dB*(*t*)/*dt* is the core loss magnetization rate.

A drawback of Equation (14) is that its accuracy decreases with the increasing of waveform harmonics, and for waveforms with a small fundamental frequency part [77,85,113].

Another modification of OSE is the Generalized Steinmetz Equation to overcome the drawbacks of OSE and MSE for sinusoidal excitations, whose expression is:

$$P\_{GSE} = \frac{k\_1}{T} \int\_0^T \left| \frac{dB(t)}{dt} \right|^\alpha |\mathcal{B}(t)|^{\beta - \alpha} dt \tag{15}$$

$$k\_1 = \frac{k}{2^{\beta - \alpha} (2\pi)^{\alpha - 1} \int\_0^{2\pi} |\cos \theta|^\alpha |\sin \theta|^{(\beta - \alpha)} d\theta}.$$

In this equation, the current value of flux is considered additional to the instantaneous value of *dB*(*t*)/*dt*, *B*(*t*), *T* is the waveform period, and *θ* is the sinusoidal waveform phase angle. The main GSE's advantage is it DC-bias sensitivity. However, its accuracy is limited if a higher harmonic part of the flux density becomes significant [85,98].

The Improved Generalized Steinmetz Equation is considered one of the best methods because it is a practical and accurate [114]. It is defined as follows:

$$P\_{\rm iGSE} = \frac{k\_i}{T} \int\_0^T \left| \frac{dB(t)}{dt} \right|^\alpha |\Delta B|^{\beta - \alpha} dt \tag{16}$$

$$k\_i = \frac{k}{2^{\beta - \alpha} (2\pi)^{\alpha - 1} \int\_0^{2\pi} |\cos \theta|^\alpha d\theta}.$$

Unlike GSE, which uses the instantaneous value *B*(*t*), *i*GSE considers its peak-to-peak value ∆*B*. Any excitation waveform can be calculated by it; therefore, it has better accuracy with waveforms that contained strong harmonics [114–117].

The Natural Steinmetz Extension is similar approach to the *i*GSE; this means that ∆*B* was also taken into account [85,118]. It is written as:

$$P\_{NSE} = \left(\frac{\Delta B}{2}\right)^{\beta - a} \frac{k\_N}{T} \int\_0^T \left|\frac{dB(t)}{dt}\right|^a dt\tag{17}$$

$$k\_N = \frac{k}{(2\pi)^{a-1} \int\_0^{2\pi} |\cos\theta|^a d\theta}.$$

The NSE focuses on the impact of rectangular switching waveform like PWM, so Equation (17) can be modelled for a square waveform with duty ratio *D* by:

$$P\_{NSE} = k\_N (2f)^a B\_s \left[ D^{1-a} + (1-D)^{1-a} \right]. \tag{18}$$

The Improved–Improved Generalized Steinmetz Equation (*i* <sup>2</sup>GSE) considers the relaxation phenomena effect in the magnetic material used due to a transition to zero voltage [115]. The *i* <sup>2</sup>GSE was developed to work with any waveform but its main application is with trapezoidal magnetic flux waveform [110]. For a trapezoidal waveform, *i* <sup>2</sup>GSE is described as follows:

$$P\_{i^2 \text{GSE}} = \frac{1}{T} \int\_0^T k\_i \left| \frac{d\mathcal{B}(t)}{dt} \right|^a (\Delta B)^{(\beta - a)} dt + \sum\_{l=1}^n Q\_{rl} P\_{rl} \tag{19}$$

$$P\_{rl} = \frac{k\_r}{T} \left| \frac{d}{dt} \mathcal{B}(t-) \right|^{a\_r} (\Delta B)^{\beta\_r} (1 - e^{-\frac{t\_1}{\tau}})$$

$$Q\_{rl} = e^{-q\_r \left| \frac{dR(t+)/dt}{dB(t-)/dt} \right|}.$$

Note that *Prl* and *Qrl* calculate the variation of each voltage change and the voltage change, respectively. However, to use the *i* <sup>2</sup>GSE requires additional coefficients as can be seen in Equation (19) where *k<sup>r</sup>* , *α<sup>r</sup>* , *β<sup>r</sup>* , *τ* and *q<sup>r</sup>* are material parameters and they have to be measured experimentally; given that, they are not provided by manufacturers and the steps to extract the model parameters are detailed in [119]. The expression given by Equation (19) can also be rewritten by a triangular waveform.

Finally, the Waveform-Coefficient Steinmetz Equation (WcSE) was proposed in [111] to include the resonant phenomena , and it applies only at situations with certain loss characteristics. This empirical equation is used in high-power, and high-frequency applications, where resonant operations are adopted to reduce switching losses. The WcSE is a simple method that correlates a non-sinusoidal wave with a sinusoidal one with the same peak flux density; the waveform coefficient (*FWC*) is the ratio between the average value of both types of signals [119,120]. WcSE can be written as follows:

$$P\_{\rm WC} = F \mathsf{WC} k f^{\mathbb{A}} B\_{\rm s}^{\beta} \,. \tag{20}$$

The most important characteristics of each approximation based on the Steinmetz equation are listed in Table 1.


**Table 1.** Details of Steinmetz's equations.

In addition to the methods listed before, in [110] the authors provided several graphs for different materials at different operating temperatures making use of the Steinmetz Premagnetization Graph (SPG), which is a simple form to show the dependency of Steinmetz's parameters on premagnetization to calculate the core losses under bias conditions.

Using the SPG, the changes of Steinmetz in *i*GSE can be considered; it is also possible to calculate the core loss under any density flux waveform [19,74].
