## 2 Methodology  
Electromagnetic and electric simulations were performed using COMSOL 6.2 and LTspice software version 24.1.4. COMSOL is a finite element simulation program that divides a geometric model into a mesh of spatial elements and solves partial differential equations (PDEs) in the electromagnetic field [15]. In our COMSOL model, we imposed magnetic insulation on the sphere (world) and initialized Ampere's Law on the world, while also applying Ampere's Law with the B-H curve formulation to the ferrite core. To each coil, we generated a ramp-up wave separately at each measured location. The mesh consists of 775,046 domain elements, 15,602 boundary elements, and 3286 edge elements using the Free Tetrahedral option. In the study, we employed a sweep on the location along the x-axis, combined with a Coil Geometry Analysis and a Time-Dependent solver. LTspice is a software tool used to simulate electrical circuits [16]. Using COMSOL and the method described in [17], we were able to extract the coupling coefficients between the transmitting coils and each of the receiving coils. These coupling values were then incorporated into LTspice to simulate the electrical circuit. Using the results, we calculated both the input and output power to estimate the efficiency of the system. We then changed the output values based on the location of each receiving coil, allowing us to calculate the efficiency relative to the position of each coil. Measurements of the lab-scaled models were performed utilizing an HP 8904A synthesizer to excite the transmitting coils. An amplifier was added to allow sufficient voltage to activate the rectifying circuit. Measurements of the resonance frequency of the system were conducted using Keysight Vector Network  
Figure 4: VC magnetic field density logarithmic distribution. Colors represent magnetic field intensity [T]. Analyzer (VNA) [18]. The VNA applies power at varying frequencies to the system and measures the output across the frequency spectrum. By analyzing the power level as a function of frequency, we can determine the efficiency of the power transfer at each frequency. The resonance frequency is identified as the point at which the system achieves maximum power transfer, indicated by the peak in the measured spectrum. Ferrite materials are widely used in wireless power transfer systems due to their high magnetic permeability and low magnetic losses. In this work, we used a stack of 56 \(\times\) 100 \(\times\) 1.1 mm\({}^{3}\) ferrite plates with magnetic characteristics described in ref. [19]. To facilitate the relative movement of the transmitting and receiving coils with respect to each other, we printed the 3D rail system displayed in Figure 5[20]. This setup allows position adjustments of the transmitter coils relative to the receiver coils, which remain position-fixed. The electronic box depicted in Figure 5 contains the simple elements used to form the circuit on the receiving side displayed in Figure 6. For each vertical coil, a capacitor of 150 nF is connected to the coil in series. Then, a rectifying bridge continues to carry a resistor load of 0.1 \(\Omega\). The BNC ports shown in Figure 5 enable for each VC circuit measurements of the coil, capacitor, and load voltage waveforms. Of course, the same values of inductance, capacitance, and load resistor were used both for the simulations and experiments. For the large VC models, capacitor values of 13.3 nF were used in the simulations to maintain the same resonance frequency with coils of higher inductance. We began by simulating full-scale models of the DDQ and VC receiver layout. We then continued to simulate and experimentally measure small, lab-scale models of both. In verifying that for the reduced-size models, simulations and experiments are in accordance, we can deduce that one may rely on simulation results for the full-scale designs.