# A Multi-Domain Model for Variable Gap Iron-Cored Wireless Power Transmission System
## 3 Mutual Induction Model: The Dynamic Aspects
### Copper Losses  
High-frequency excitation induces various loss phenomena in copper; the most critical of these are direct current (DC) and alternating current (AC) resistance, and the more dangerous skin and proximity effects.  
According to [PERSON] et al. in [23], one of the most common ways to counteract this phenomenon is to design the device wound through Litz wires, due to the reduced dimensions of the device under analysis and the necessity to maintain cheap production costs. The authors' work intended to formulate a numerical model that considers a generic type of standard round wire.  
In his work [24], Professor [PERSON] proposes a series of models for circular and rectangular conductors, which allow a reliable solution to be obtained for any considered frequency without making use of the complex Bessel functions.  
Another simple alternative to the Bessel functions is reported by [PERSON] et al. in [25]. The proposed work introduces a complete characterization of the Faraday shield losses of a transformer, giving a detailed mathematical modeling of [PERSON]'s derived equations of an equivalent resistor to introduce in circuit simulation.  
The proposed Dowell formulation tries to summarize all the AC, DC, proximity and skin effects that characterize the loss of the conductor into a single expression for the equivalent ohmic resistance.  
As described in the above paragraph for the core losses, the following reports a set of equations for the equivalent mathematical formulation used in the implementation of a behavioral Spice-based resistance.  
The equivalent AC resistance expression \(R_{ACi}\) for the \(i\)-th winding is formulated as a combination of different hyperbolic functions \(\zeta_{i}\) and \(\zeta_{i}\).  
\[R_{ACi}=R_{DCi}\;F_{ri}=R_{DCi}\;\Delta_{i}\left[\zeta_{i}+\frac{2}{3}\left(p_ {i}^{2}-1\right)\zeta_{i}\right],\;i=1,2\ldots \tag{12}\]  
The DC resistance of the windings \(R_{DCi}\) is directly experimentally measured. The proximity \(\zeta_{i}\) and skin \(\zeta_{i}\) effect instead need to be numerically calculated through Equations (13) and (14) through a combination of harmonic functions of the penetration ratio \(\Delta_{i}\).  
\[\zeta_{i}=\frac{\sinh\Delta_{i}-\sin\Delta_{i}}{\cosh\Delta_{i}+\cos\Delta_{i}} \tag{13}\]  
\[\zeta_{i}=\frac{\sinh(2\Delta_{i})+\sin(2\Delta_{i})}{\cosh(2\Delta_{i})-\cos (2\Delta_{i})} \tag{14}\]According to Equations (15)-(17) the penetration depth \(\Delta_{i}\) is essentially a function of the physical and geometric parameters of the material.  
\[\Delta_{i}=\frac{\sqrt{\eta_{i}}d_{wi}}{\delta_{i}} \tag{15}\]  
\[\eta_{i}=\frac{t_{i}\sqrt{k_{i}}d_{wi}}{h_{c}} \tag{16}\]  
\[\delta_{i}=\sqrt{\frac{\rho_{i}}{\pi\mu f}} \tag{17}\]  
where:  
* \(\eta_{i}\) is the porosity factor of the material;
* \(d_{wi}\) is the equivalent thickness of the material and for a round wire is equal to \(\frac{\sqrt{\pi}d_{i}}{2}\), where \(d_{i}\) is the diameter of the wire;
* \(\delta_{i}\) is the skin depth in the wire;
* \(t_{i}\) is the number of turns per layer;
* \(k_{i}\) is the amount of Litz wire that is 1 for round wire;
* \(h_{c}\) is the height of the core window;
* \(\rho_{i}\) is the resistivity of the wire material;
* \(\mu\) is the permeability constant equal to \(\mu_{r}\mu_{0}\);
* \(f\) is the principal operating frequency;
* \(p_{i}\) is the equivalent number of layers.  
A more in-depth description and identification of the parameters as well as the whole mathematical demonstration can be found in [25].