## 3 Compensation Method  
In the HLM, the ZVS realization of the main switches is \"AZVS + NZVS.\" In the LLM, the ZVS realization is \"AZVS + AZVS.\" To distinguish the HLM and LLM, the threshold current \(I_{th}\) is used, which is given as follows. \[I_{th}=\frac{2C_{r}V_{s}}{t_{dead}} \tag{36}\]  
When the magnitude of the load current is lower than \(I_{th}\), the load current cannot discharge the resonant capacitors to zero-voltage. Thus, the ARSI operates in LLM. When the magnitude of the load current is higher than \(I_{th}\), the HLM is adopted. Table 1 shows the realization type of ZVS from zero load to full load. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & \(i_{o}<-I_{th}\) & \(-I_{th}\leq i_{o}\leq I_{th}\) & \(i_{o}>I_{th}\) \\ \hline S\({}_{2}\) and S\({}_{3}\) & AZVS (S\({}_{2}\)) & AZVS (S\({}_{2}\)) & NZVS \\ S\({}_{1}\) and S\({}_{4}\) & NZVS & AZVS (S\({}_{x1}\)) & AZVS (S\({}_{x1}\)) \\ Load Condition & Heavy Load & Light Load & Heavy Load \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Realization type of ZVS from zero load to full load. Figure 5: Dead-time effect of hard-switching inverter with single pole (**a**) the operating stages when \(i_{o}>0\); (**b**) the key waveforms when \(i_{o}>0\); (**c**) the key waveforms when \(i_{o}<0\). The ARSI can reduce the switching loss, whereas the conduction loss is increased due to the auxiliary current. To maintain a low conduction loss, the auxiliary current should be as low as possible. Thus, the initial resonant current \(I_{boost}\) is controlled to be constant from zero load to full load as follows. \[I_{boost1}=I_{boost2}=I_{boost3}=I_{boost} \tag{37}\]  
The dead-time effect in the case of positive output current is introduced in Section 2.1. When the output current is negative, the dead-time effect is similar. Therefore, the voltage error caused by the dead-time can be obtained based on Equations (21), (35) and (37):  
\[V_{err}=\left\{ \begin{array}{c}\frac{V_{s}}{T_{s}}\left(\left|\frac{2C_{r}V _{s}}{l_{o}}\right|-\left|\frac{2}{\omega_{A}}\text{arcsin}\frac{V_{s}}{\sqrt {V_{s}^{2}+Z_{A}^{2}I_{boost}^{2}}}\right|\right)\quad i_{o}>I_{th}\\ 0\\ \frac{V_{s}}{T_{s}}\left(\left|\frac{2}{\omega_{A}}\text{arcsin}\frac{V_{s}}{ \sqrt{V_{s}^{2}+Z_{A}^{2}I_{boost}^{2}}}\right|-\left|\frac{2C_{r}V_{s}}{l_{o}} \right|\right)\quad i_{o}<-I_{th}\end{array} \right. \tag{38}\]  
The voltage error is related to the resonant time. Due to the same \(I_{boost}\), the voltage error is zero in the LLM. As for the HLM, the voltage error occurs because the resonant time is adaptively related to the output current to achieve NZVS, whereas the commutation time to achieve AZVS is constant. Due to the dead-time effect, a voltage error occurs between the actual pole voltage and ideal pole voltage. The ARSI can be modeled as a proportional gain \(K_{pvm}\) without consideration of delays. \(K_{pvm}\) is the ratio between the DC voltage and peak value of the carrier in the PWM modulator. The reference voltage \(v_{c}\) is amplified \(K_{pvm}\), thus obtaining the ideal pole voltage \(v_{ab,id}\). The actual pole voltage can be obtained by adding the voltage error \(v_{err}\). Figure 6 shows the transfer function of the ARSI. The output current can be calculated as follows. \[i_{o}=\frac{1}{L_{o}s+R_{o}}v_{ab,id}+\frac{1}{L_{o}s+R_{o}}v_{err} \tag{39}\]  
where the actual voltage \(v_{ab,id}=K_{pvm}v_{c}\). The voltage error caused by the dead-time effect is considered as a disturbance in the ARSI. To compensate the voltage error, the feedforward method can be utilized, which is shown in Figure 7.  
Figure 6: Transfer function of the ARSI. Figure 7: Transfer function of the ARSI with feedforward compensation. After the feedforward compensation is utilized, the output current can be calculated as follows.