## 2 Materials and Methods
### WPT System: Experimental and Simulated Data
#### 2.2.1 WPT NFC Impedance Matching  
The proposed method was initially applied to a Wireless Power Transfer (WPT) system that employs Near-Field Communication (NFC) technology, operating at a frequency of 13.56 MHz [41]. This WPT system is composed of two magnetically coupled resonators, in a series-series configuration, that enable the transfer of power from the transmitter to the receiver, represented in a circuit in Figure 1. The magnetically coupled resonator coils used are planar of class 1 with standard ISO/ICE 14443 [42]. The circuit has an LC\(-\)series resonance obtained using the capacitors \(C_{res}\). In addition to the symbolic regression (SR) approach, a neural network (NN) embedded within a microcontroller was implemented to optimize the circuit's performance by achieving optimal impedance matching conditions. This adjustment is crucial for improving the overall efficiency of the WPT system. The NN controls an Adaptive Impedance Matching Network (AIMN), which is designed using an LC circuit consisting of three cascaded L\(-\)type low-pass networks arranged in a series-series configuration as shown in Figure 2.  
Figure 1: Electrical circuit schematic of the WPT system proposed in [41]. This circuit is represented here with SIMULINK. \(V_{in}\) is the input source of power, and \(R_{s}\) is the series impedance of the generator, with a resistance of 50 \(\Omega\). The parameters used to simulate the transmitter coil are \(C_{res}=22.6\) pF, \(C_{p1}=5.7\) pF and \(R_{rf1}=4.48\)\(\Omega\). The receiver is simulated with \(R_{rf2}=4.48\)\(\Omega\), \(C_{p2}=5.7\) pF, \(C_{res2}=22.6\) pF and with a purely resistive load of 50 \(\Omega\). The coils are simulated with a coupled inductor with a mutual inductance of 4.87 \(\upmu\)H.  
The capacitances within the AIMN are variable ceramic capacitors, allowing for the dynamic control of capacitors C\({}_{1}\), C\({}_{2}\), and C\({}_{3}\) to achieve effective impedance matching. The entire system is managed by a microcontroller that continuously monitors the scattering parameters, represented by the S-parameter complex matrix. From these measurements, the input impedance (\(Z_{in}\)) of the system can be precisely calculated. This integrated approach not only enhances power transfer efficiency but also ensures that the system operates under optimal conditions for maximum performance. The WPT circuit comprises two circular coils separated by a distance \(d\) as shown in the scheme of Figure 3. The coils are not perfectly aligned, with misalignments measured by \(\Delta x\) and \(\Delta y\); see Figure 3. Additionally, the receiver coil is rotated by an angle \(\theta\) to simulate a realistic scenario. The receiver is connected to a fixed load, while the transmitter is linked to the microcontroller and the AIMN. A simplified version of this system is shown in Figure 3.  
As previously mentioned, maximum power transfer is achieved by adjusting the AIMN to reach the working point of perfect impedance matching. The control variables for the AIMN are the capacitance values \(C_{1}\), \(C_{2}\), and \(C_{3}\). The NN described in [41] is trained  
Figure 3: Simplified scheme of the main blocks of the circuit proposed in the experiment with the AIMN. For a more complete scheme and more details of the system used, see [41]. Here, the two coils are represented with Tx for the transmitter ones and Rx for the receiver. They are separated by a distance d, along the z direction, and their centers are misaligned by \(\Delta y\) (Dy) and \(\Delta x\) (Dx). The Rx coil is connected to a purely resistive load as shown also in Figure 1. The Tx coil is connected through the AIMN to the microcontroller and the signal block, that represents a more complex system with signal generator, measurement instruments, and directional couplers. Figure 2: Circuit of the AIMN proposed in [41]. This circuit uses three variable capacitances to adjust the impedance of the circuit for obtaining the perfect impedance matching.