## 3 Principle of the Proposed Eios-TPS
### Modulation Strategy in the Low-Power Range When k > 1
It is noteworthy that the optimal operating trajectory precisely resides at the boundary between mode E and mode F. Based on (5), the optimal trajectories for modes E and F are plotted. As shown in Figure 4, the cyan plane represents the interface between mode E and mode F, and the red line illustrates the optimal path. Under optimal operating conditions, the transmitted power range is [0, \(2(k-1)/k^{2}\)]. According to (4) and (5), the waveform under low-power operation with \(k>1\) is shown in Figure 5a, where only two switches achieve ZVS, and the remaining six switches operate under ZCS. Although ZCS mitigates turn-off losses, the lack of ZVS leads to the dissipation of energy stored in the output capacitance of the switches as heat. Consequently, it is essential to optimize the duty-cycle combinations to ensure ZVS for all switches while simultaneously minimizing the peak inductor current. While ZCS reduces turn-off losses, the absence of ZVS results in energy stored in the switch's output capacitance being dissipated as heat. Therefore, it is necessary to adjust the duty-cycle combinations to achieve ZVS for all switches while maintaining a low inductor current peak. As shown in Figure 5a, at \(t_{1}\), the switch current is zero, leading to a ZVS failure for \(S_{5}\). At \(t_{2}\), the rising edges of \(v_{1}\) and \(v_{2}\) overlap, causing a ZVS conflict for switches \(S_{4}\) and \(S_{8}\). By modifying the relationship among \(D_{1}\), \(D_{2}\), and \(\Phi\) to enter mode F, as shown in Figure 5b, and ensuring \(i_{L}(t_{2})>0\) and \(i_{L}(t_{3})<0\), ZVS can be achieved for all switches. Specifically, by imposing the condition \(i_{L}(t_{2})+i_{L}(t_{3})=0\), both current polarities and an appropriate current magnitude are ensured, enabling full ZVS operation. From this condition, the intrinsic relationship among \(D_{1}\), \(D_{2}\), and \(\Phi\) can be derived as:  
\[\Phi=(k-\frac{1}{2})D_{1}-\frac{1}{2}D_{2} \tag{6}\]  
According to mode F constraints, it is known that \(1-kD_{1}>0\). Introducing a scaling factor \(\lambda\) (\(0<\lambda<1\)) and setting \(1-D_{2}=\lambda(1-kD_{1})\), the normalized inductor current \(i_{Ln}(t_{2})\) can be expressed as:  
\[i_{Ln}(t_{2})=(1-\lambda)(1-kD_{1}) \tag{7}\]  
Therefore, it can be ensured that \(i_{L}(t_{2})>0\) and \(i_{L}(t_{3})<0\), enabling full ZVS for all switches. For mode F, when the transmitted power is positive, \(i_{L}(t_{0})<0\) is always satisfied. As shown in Figure 6a, the ZVS region of mode F is defined by the planes \(i_{L}(t_{1})=i_{L}(t_{2})=0\) and \(i_{L}(t_{3})=0\), as well as the coordinate planes, forming a closed volume. Different values of \(\lambda\) correspond to different optimization paths, all of which lie on the green surface, which itself is located within the ZVS region of mode F.  
Figure 4: The optimized operating path in the low-power region for \(k>1\). (**a**) A three-dimensional visualization of the optimized path. (**b**) A two-dimensional projection of the optimized path. Figure 5: (**a**) Operating waveforms of mode E and mode F under minimum current-stress conditions. (**b**) Further optimized waveforms under low-power operation for \(k>1\). Figure 7 illustrates the relationships between the normalized transmitted power \(p\), the normalized current stress \(i_{m}\), and the normalized inductor current \(i_{Ln}(t_{2})\) for different values of \(\lambda\) (0.4, 0.6, 0.8, and 0.9). As shown in Figure 7a, for different \(\lambda\) values, the maximum transmitted power of mode F remains constant at \([0,2(k-1)/k^{2}]\). As \(\lambda\) increases, the current stress decreases. Conversely, as depicted in Figure 7b, for the same transmitted power, the inductor current \(i_{Ln}(t_{2})\) decreases with increasing \(\lambda\). To ensure sufficient discharge of the parasitic capacitance of SiC MOSFETs, the inductor current \(i_{Ln}(t_{2})\) should retain a relatively high magnitude. Therefore, a slightly smaller \(\lambda\) value should be selected to maintain the required current magnitude.