## 3 Results
The values of the self-inductance, mutual-inductance, and coupling coefficients are influenced by the geometric properties of the coils and their position in space. This means that at each specific location, the values between the transmitting coil and each of the receiving coils will vary, reflecting changes in their relative positions and distances. To generate the matrix of values at each position, we applied the method suggested in [17]. For each coil, we simulate a constant current ramp rate and then the voltage at each of the other coils. We then calculate the mutual-inductance between the coils according to Equation (3):  
\[M_{1,2}=\frac{v_{2}}{(\frac{dI_{1}}{dt})} \tag{3}\]  
The coupling coefficient is then calculated using Equation (4):  
\[k=\frac{M_{1,2}}{\sqrt{L_{1}\cdot L_{2}}} \tag{4}\]  
In Tables 2 and 3, columns and rows 2-6 represent the five receiving coils in the full-scale VC system, respectively, while column and row 7 represent the transmitting coils. The cell values in Table 2 give the inductance in uH units. The diagonal represents the self-inductance of each VC coil and the rest of the cells exhibit the mutual-inductance values between pairs of VC coils. Similarly, Table 3 displays the coupling coefficients for the set of VC coils. As expected, VC 3, which is right on top of the center of the transmitting coils, presents the highest coupling between a receiver coil and the transmitter, 0.105. VC 1 and 2 are far apart from the axis of symmetry and, hence, have lower k values. Since the magnetic field drops sharply as one moves away from the center, the k values drop dramatically and it is clearly seen that the reduction in k of VC 1 to VC 2 is larger than that of VC 2 to VC 3. VC 4 and VC 5 show a symmetric behavior with symmetric k-values. In the same way, Tables 4 and 5 display the inductance and coupling values of the DDQ layout in the same way. Columns and rows 2-3 correspond to the DD and Q coils, respectively, and column and row 4 represent the transmitting side in the DDQ system. Here, since coil Q is positioned at the center of symmetry, its coupling with the transmitter coils is negligible. At this position, the DD coils only contribute to energy transfer with a maximum value of k of 0.081. Tables 3 and 4 demonstrate that for the \"0\" position (when the centers of transmitting and receiving coils overlap), the VC design exhibits higher k values than the DDQ. Not only is the center coil VC 1 k higher by almost 30% than the DD k, but all other VC coils also participate and contribute while the Q coil in the DDQ configuration does not. This situation may change with position and therefore, for each position, a new table was generated, and these values were then incorporated into the LIspice calculations. This approach allowed us to account for the variations in the coupling coefficients between the transmitting and receiving coils as their relative positions shifted. Although this method provides a discrete solution for the dynamic transition of the EV above the transmission coils, we use it as an approximation for a continuous dynamic pass. One might initially assume that if we know the coupling coefficient at the 0 position for one vertical coil, the same value will apply to any other coil when it is located at the 0 position. However, the non-linear nature of the ferrite complicates this situation. Due to demagnetization effects, the local permeability of the ferrite plate varies with the distance from its edges; hence, the magnetic field intensity is inhomogeneous throughout its cross-section. A vertical coil wound around the central cross-section would experience a higher field intensity than a coil wound near the edge. As a result, k values at the position 0 differ between the VCs, however symmetric around the center coil. The non-linear nature of the B-H curve makes it even more complex since the permeability, \(\mu\), and therefore k, become dependent on the amplitude of the current of the transmitter coils.