## 2 System Design and Analysis
### Verification via Simulation  
Figure 3 illustrates the system configuration for the proposed method, namely the simultaneous identification of multiple parameters in WPT systems. The signal generator works together with the power amplifier as a high-frequency voltage source \(u_{p}\). The primary variable capacitor, namely the SCC, consists of two constant capacitors: one is connected with a relay switch \(S_{r}\) in parallel and another is connected with two MOSFETs switches in parallel. Thus, the SCC can operate in three modes:  
1. When the relay \(S_{r}\) is open, and both MOSFETs paralleled with \(C_{2}\) are conducting, the equivalent capacitance overall is \(C_{1}\). 2. When the relay \(S_{r}\) is closed, and both MOSFETs paralleled with \(C_{2}\) are turned off, the equivalent capacitance overall is \(C_{2}\). 3. When the relay \(S_{r}\) is open, \(C_{2}\) and the two MOSFETs paralleled to form an SCC, which can achieve continuously adjustable capacitance from \(C_{1}\) in series with \(C_{2}\) to \(C_{1}\). The critical parameters in this paper are listed in Table 1. The variation of primary equivalent capacitance corresponds to the change of primary resonant frequency from 75 to 130 kHz, which can cover the frequency range required in the SAEJ2954 standard. In this paper, three predetermined frequencies, \(f_{p1,2,3}\) of 74.76, 105.73, and 129.49 kHz, are utilized to achieve the parameter identification. The values of such frequencies are designated to accommodate the resonant frequency range (75 kHz\(\sim\)130 kHz) on the secondary side. Moreover, regardless of the variation of the parameters to be identified, the above three  
Figure 3: System configuration for parameter identification with multi-frequency operating capability. Figure 2: Flowchart of gradient descent algorithm used for parameter identification. frequencies are used in each identification case, the range of which is defined by the lower and upper bounds of the space vector. It should be noted that the proposed system can easily identify and accommodate unknown receivers as long as their actual resonant frequencies are within the aforementioned range. To identify unknown receivers with different resonant frequencies and load conditions, the secondary capacitance \(C_{s}\) and load resistance \(R_{L}\) were intentionally adjusted from 22 to 45 nF and from 2 to 50 \(\Omega\) in solution space, respectively. For simplifying the verification, the \(C_{s}\) are deliberately set to 26.7, 33.2, and 37.8 nF when the secondary coil inductance \(L_{s}\) is constant at 101 uH, which in turn corresponds to the \(f_{s}\) of 81.45, 86.9, and 96.9 kHz, respectively. In any case, mutual inductance varies from 10 to 50 uH in solution space. As shown in Figure 4, twelve cases of simulation were conducted with variations of secondary resonant frequency \(f_{s}\) and load resistance \(R_{L}\). Cases i, i + 1, i + 2, and i + 3 (i = 1,5,9) share the same \(f_{s}\) value, while Cases j, j + 4, j + 8 (j = 1, 2, 3, 4) share the value of \(R_{L}\) that is shown in Figure 4b. The dotted line in the figure is the set value in the simulation. By comparing the identified values of \(M_{ps}\), \(R_{L}\), and fs derived from (12) with the set value, it can be seen that the parameter identification basically functions as predicted but is not precise enough. The maximum identified errors of mutual inductance, load resistance, and secondary resonant frequency are about 2.6 uH, 4.78 \(\Omega\), and 5.24 kHz, respectively. To further explore how to increase the accuracy of the algorithm, the worst case (Case 4), which shows the maximum identified error in all parameters, is studied. Its loss surface is demonstrated in Figure 5a to see the location of the optimum point delivered by the algorithm. The loss surface is flat near the optimum point, which means that the algorithm is not able to distinguish the differences in the dark-colored area.