## 2 Materials and Methods
### WPT System: Experimental and Simulated Data
#### 2.2.2 WPT Phase Control  
The same SR technique is also utilized in a different WPT system to address a distinct issue, demonstrating the method's versatility and applicability across various problems. This system, described in [44], employs inductive WPT for charging the receiver coil. The two coils used were manufactured by Wurth Elektronik, specific for wireless charging applications and compliant with the Qi standard, characterized by a winding inLitz wire and a core in high magnetic permeability material (ZnNi). The system has a series LCresonance in each antenna, obtained by connecting in series a compensation capacitor (\(C_{1}\) and \(C_{2}\) in Figure 4 with the coil inductor. In this case, the frequency is variable, necessitating a control mechanism to adjust it according to varying load conditions and distances between the coils. To evaluate this, the phase difference between the voltage and current of the transmitter coil is measured. A dedicated module is implemented to measure the phase shift between voltage and current in the transmitter antenna using high-speed operational amplifiers and an XOR port. Subsequently, a proportional-integral controller (PI control) is implemented on a microcontroller, Teensy 4.1, to fine-tune the WPT system's frequency, aiming to enhance the power factor \(\cos\phi\). The feedback control aims to achieve a phase angle \(\phi\) below \(5^{\circ}\). The operating frequency ranges from 50 kHz to 90 kHz. The PI controller is executed on a microcontroller with fixed values for the proportional and integral constants set at 1 and 10, respectively. The phase difference serves directly as the error signal since the target phase is 0 degrees and the control loop ceases operation when the phase exceeds 5 degrees. The variable manipulated by the control loop is the WPT system's frequency. As proposed in [44], the PI control can be enhanced by using an adaptive neuro-fuzzy inference system (ANFIS) that dynamically adjusts the PI constants for faster control. During the experiment, extensive data are collected using an oscilloscope, voltmeter, and ammeter, and subsequently processed using LabView 2020 and MATLAB 2021. This study focuses on four parameters: frequency, phase, load condition, and the distance between the two coils. The experiment is conducted 18 times under varying load conditions, \(R\), and distances between the transmitter and receiver, \(z_{0}\). The circuit used in this experiment is represented in Figure 4.  
Specifically, tests are performed with six load values, \(R\) = 100, 200, 500, 1000, 1500, and 2000 \(\Omega\), and three distance values, \(z_{0}\) = 0, 4, and 8 mm. This dataset is open-source and accessible at [45]. From this dataset, it is noteworthy to observe the phase trends concerning frequency, with respect to \(R\), as shown in Figure 5, and with respect to \(z_{0}\)  
Figure 4: Electric circuit of the WPT system used in the experimental setup for the phase control system. This circuit was created by [44]. The main part is the mutual inductance that represents the two coils separated by a physical distance \(z_{0}\). The value of the mutual inductance changes with \(z_{0}\). The MOSFET transistors are controlled by the microcontroller to adjust the frequency of the WPT system in order to minimize the phase angle. In this simplified scheme, the microcontroller is omitted to reduce circuit complexity, resulting in open MOSFET gates, as indicated by the red circle. We report the parameters of the circuit: input voltage, \(V_{DC}\) = 12 V, transmitter and receiver coil self inductance, \(L_{1}=L_{2}\) = 24 \(\mu\)H, transmitter and receiver coil internal resistance, \(R_{1}=R_{2}=0.07\)\(\Omega\), transmitting and receiving antenna compensation capacitor, \(C_{1}=C_{2}=200\) nF, shunt resistance \(R_{s}=3\)\(\Omega\), load resistor \(R\) = 100, 200, 500, 1000, 1500, 2000 \(\Omega\), and operating frequency \(f\) = 50\(-\)90 kHz.