# A Multi-Domain Model for Variable Gap Iron-Cored Wireless Power Transmission System
## 4 Multi-Physics Numerical Model
### The 3D Finite Volume Model of the Coupled Inductor  
Figure 4 shows the implemented numerical model of the coupled coils developed through the ESRF Radia [27] package and solved through the 3D finite volume method. Figure 4: Three-dimensional magneto-static finite volume model. Figure 3: (a) PyS price–MATLAB/Simulink co-simulation logic; (b) ESRF Radia–MATLAB/Simulink co-simulation logic. The necessity of implementing such a model arose from the lack of information provided by the manufacturer about the technical operating characteristics of the individual coils. The system, as shown in Figure 1, consists of two coils (\(n=2\)) separated by an air gap, whose variation can only occur in the direction that is normal to the coupled faces. To obtain a detailed description of the parameters constituting the static inductive coupling phenomena, it is necessary to reconstruct, for each air gap configuration reached by the system, the entire inductance matrix reported in Equation (4). The air gap range equal to \(X_{air\ gap}=[0;0.7]\) mm is discretized with a step interval of 0.02 mm. For each of these air gap configurations, the numerical 3 DFVM model is solved iteratively for \(n\) times, where \(n\) is equal to the total number of coils evaluated in the inductive coupling, individually energizing a different coil for each \(n\)-th iteration. In detail, for the single air gap value and the single resolutive iteration of the numerical model, the \(n\)-th winding is energized through a constant current, with a value that generates an induced electromagnetic field sufficiently far from the material saturation. The Finite Volume Model is solved numerically, and the inductance parameters' values \(L_{ij}\) are solved according to Equations (1)-(3) through the average magnetic flux density \(B_{j}\), which is measured across the core cross section delimited by the winding. Figure 5a,b depict, respectively, the inductance and coupling coefficient trend along a gap variation from 0 to 0.7 mm. It is necessary to note that for the mutual inductance trend \(L_{12}\), two slightly different curves are reported, which are obtained from two different iterations of the model. This discrepancy is due to an intrinsic error of the resolution of [PERSON]'s equations through the finite volume method. For the sake of simplicity, the final value adopted during further calculations is obtained as the average of the two curves. The coupling coefficient trend instead highlights the close dependence between the performances and the air gap; the graph depicts a 19.5% reduction in transmission properties only after a tenth of a mm. In detail, to obtain a stable and reliable resolution of the magnetic field presented in Figure 4 as well as high fidelity of the reported results, according to the FVM modelling approach, the core material is partitioned as follows:  
Figure 5: (**a**) Auto and mutual-inductance parameters vs. air gap variation; (**b**) coupling factor vs. air gap variation. * Along X-axis: 51 subdivisions;
* Along Y-axis: 13 subdivisions;
* Along Z-axis: 5 subdivisions. This deviation in the parameters is due to the fact that the magnetic field lines inside the core material are mainly arranged parallel to the X-Y plane. This aspect introduces the necessity to refine the volumes' subdivision along these main axes to obtain a better solution, even where the field lines tend to curve, such as at the ends of the ferrites where the core material is not wound by the coil. Instead, the copper windings are modeled as a racetrack coil of 28 turns for the emitter side and 53 turns for the receiver, with a wire diameter of 0.1 mm for both inductors. From the point of view of the simulation, the resulting inductance values as well as the coupling factor obtained from a specific air gap configuration can be processed through two main different strategies: the real time co-simulation or the pre-simulation, and look-up table interpolation.