## 5 Analysis of Gain Characteristics in DC-DC Stage with Series-Series (S-S) Compensation Circuit  
Figure 10 shows the AC equivalent model of the series-series (S-S) compensation circuit applied to the proposed three-phase single-stage AC-DC converter. The model consists of the leakage inductances (L\({}_{\rm II}\), L\({}_{\rm II}\)) on the primary and secondary sides, the magnetizing inductance (L\({}_{\rm m}\)), the resonant capacitors (C\({}_{\rm r1}\), C\({}_{\rm r2}\)) on the primary and secondary sides, and the AC equivalent resistance (R\({}_{\rm ac}\)). The internal resistance components of the primary and secondary windings have been neglected. C\({}_{\rm r1}\) and C\({}_{\rm r2}\) are the resonant capacitors on the primary and secondary sides, respectively, and R\({}_{\rm ac}\) was derived by converting the DC load resistance (R\({}_{\rm DC}\)) into the AC equivalent resistance (R\({}_{\rm ac}\) = 8R\({}_{\rm DC}\)/\(\pi^{2}\)). The voltage gain characteristics of the series-series (S-S) compensation circuit in the DC-DC stage of the IPT-based three-phase single-stage AC-DC converter were analyzed using the Fundamental Harmonic Approximation (FHA). In this study, only the fundamental frequency was considered, ignoring the third, fifth, and seventh harmonics. Equation (26) represents the input-output gain G\({}_{\rm V}\) of the S-S compensation circuit in terms of the coupling coefficient k, the magnetizing inductances (L\({}_{\rm I}\), L\({}_{\rm 2}\)), and the leakage inductances (L\({}_{\rm II}\), L\({}_{\rm 2}\)) of the contactless coupler. The voltage gain curve as a function of frequency, derived from the AC equivalent model, is shown in Figure 11b. Figure 8: Equivalent circuit operating at the resonant frequency (f\({}_{\rm r}\)) viewed from the secondary side. Figure 7: Single-turn FEA model of the contactless coupler for an IPT design. Figure 9: Fabricated IPT-based contactless coupler. The resonant frequencies of the series-series (S-S) compensation circuit include three frequencies: \(\mathrm{f_{m}}\), \(\mathrm{f_{r}}\), and \(\mathrm{f_{p}}\). The first resonant frequency, \(\mathrm{f_{m}}\), is the resonant frequency associated with the primary and secondary resonant capacitors (\(\mathrm{C_{r1}}\), \(\mathrm{C_{r2}}\)), the secondary-side magnetic inductance (\(\mathrm{L_{2}}\)), and the magnetizing inductance (\(\mathrm{L_{m}}\)), as shown in Equation (27). At the resonant frequency (\(\mathrm{f_{m}}\)), the switching operation can always occur with a leading current, making it suitable for hard switching. The second resonant frequency, \(\mathrm{f_{r}}\), is the resonant frequency associated with the leakage inductances (\(\mathrm{L_{11}}\), \(\mathrm{L_{2}}\)) of the primary and secondary sides and the resonant capacitors (\(\mathrm{C_{r1}}\), \(\mathrm{C_{r2}}\)), as depicted in Equation (28). At the frequency \(\mathrm{f_{r}}\), the AC equivalent resistance \(\mathrm{R_{ac}}\) does not affect the input-output voltage gain characteristics, maintaining a consistent gain characteristic despite changes in \(\mathrm{R_{ac}}\). The third resonant frequency, \(\mathrm{f_{p}}\), is the resonant frequency between the primary-side magnetic inductance (\(\mathrm{L_{1}}\)) and the resonant capacitor (\(\mathrm{C_{r1}}\)), as shown in Equation (29). This frequency (\(\mathrm{f_{p}}\)) is where the gain characteristics are most significantly affected by changes in the equivalent AC resistance (\(\mathrm{R_{ac}}\)). Particularly, as the load increases, a bifurcation phenomenon occurs where the slope of the gain characteristics decreases. Thus, if a fixed alternating switching operation is performed at the third resonant frequency (\(\mathrm{f_{p}}\)) under rated-load conditions, the primary-side resonant current (\(\mathrm{f_{p}}\)) may lead the phase of the primary-side square wave voltage (\(\mathrm{V_{P}}\)), leading to hard switching.