# A Multi-Domain Model for Variable Gap Iron-Cored Wireless Power Transmission System
## 3 Mutual Induction Model: The Dynamic Aspects  
### Core Losses  
Power losses in electro-magnetic components are important design parameters, which limit many high-frequency designs. The high-frequency magnetic field variation induces eddy currents in the core loops, according to [PERSON]'s law of induction. As summarized by [PERSON] et al. in [14] and by [PERSON] et al. in [15], different geometric and material parameters can influence the induced core losses in ferrites, as well as the related physical non-linear phenomena, such as the permeability of the material, the magnetic saturation point, the core masses and the geometric properties. On the other hand, losses also depend on the type of application, such as the frequency operating range, temperature, waveform and amplitude of the magnetic flux density excitation, as well as its DC offset [16]. The literature contains several standard-based Steinmetz models for studying the power losses in ferro-ceramic or iron-based cores [17; 18]; however, the majority of them have been developed to work in a limited frequency range and temperature, and low magnetic flux [19]. Thanks to its ease of use and standardization, manufacturers provide core-losses documentation under the form of an experimentally obtained set of curves, which is based on the reported Steinmetz equation (SE) (6). \[\overline{P}_{v}=kf^{\alpha}\,\beta^{\beta} \tag{6}\]  
where \(\overline{P}_{v}\) is the time-averaged power loss per unit volume, \(f\) is the excitation frequency, \(\hat{B}\) is the amplitude of the magnetic flux density and, finally, \(\alpha\), \(\beta\) and \(k\) are the so-called Steinmetz parameters (SP). Due to its simplicity, the equation has two main drawbacks that limit its usage for general applications: its ineffectiveness when handling non-sinusoidal excitations, and its non-sensitivity with respect to applications with the presence of DC bias. Further improvements to the SE model are proposed in [20; 21] up to the last formulation of a revised generalized Steinmetz equation (RGSE) model proposed by [PERSON] and [PERSON] in [22]. In their work, the authors provide a mathematically based approach for core loss estimation implemented through the PSpice engine for the equivalent circuit simulation. The proposed generalized RGSE mathematical model is based on the standard Steinmetz equation and some further geometrical data. The reported set of Equations (7)-(9) allows a behavioral Spice-based current source to be implemented for the equivalent circuit estimation. More detailed information along with the full mathematical demonstration can be found in [22]. As previously introduced, the equivalent Spice-based current source power loss \(P_{c}\) presents a similar formulation with respect to the standard SE, but with the introduction of new terms that can handle a general exciting waveform and DC bias. \(k_{1}\), described in (8), maintains its independence with respect to the operating frequency and is simply related with the standard SP; \(B_{AC}\), whose formulation is reported in (9), presents an amplitude voltage \(v(t)\) and frequency \(f=\frac{1}{t}\) dependency, as expected by the standard magnetic flux density formulation, and it also introduces the DC bias dependency through the term \(B_{DC}\); finally, \(A_{e}\), \(V_{e}\) and \(N\) represent, respectively, the effective cross section and volume of the core, and the number of turns of the exciting coil.