## 2 Dead-Time Effect of the Auxiliary Resonant Snubber Inverter (ARSI)
### Principle
#### 2.1.2 Light Load Condition  
When the load current is low, the resonant capacitors cannot be discharged to zero-voltage during the dead-time. NZVS of S\({}_{2}\) and S\({}_{3}\) fails [22]. Therefore, the auxiliary circuit is operated to achieve AZVS of S\({}_{2}\) and S\({}_{3}\). One switching cycle of the operating waveforms in the LLM is shown in Figure 3.  
Figure 3: Key waveforms of the ARSI in the light load condition. Before \(t_{L2}\), S\({}_{72}\) is turned on to charge the resonant inductor \(L_{r}\), thus increasing the switch current. After S\({}_{1}\) and S\({}_{4}\) are turned off at \(t_{L2}\), the resonant capacitors resonate with the resonant inductor. Due to the higher switch current, C\({}_{72}\) and C\({}_{73}\) can be discharged to zero-voltage. Thus, S\({}_{2}\) and S\({}_{3}\) can be turned on at the ZVS condition. The voltage error also only occurs in the resonant stage \(t_{L2}\)-\(t_{L3}\). This resonant stage is similar to the stage \(t_{H6}\)-\(t_{H7}\) in Figure 2. The same equations can be obtained as Equations (5)-(9). However, the initial resonant conditions are different as follows:  
\[v_{ds1}\left(t_{L2}\right)=v_{ds4}\left(t_{L2}\right)=0 \tag{22}\]  
\[v_{ds2}\left(t_{L2}\right)=v_{ds3}\left(t_{L2}\right)=V_{s} \tag{23}\]  
\[i_{Lr}(t_{L2})=-I_{Lrm2} \tag{24}\]  
Therefore, the inductor current and drain-source voltages of the main MOSFETs can be obtained as follows according to Equations (5)-(9) and (22)-(24):  
\[v_{ds1}\left(t\right)=v_{ds4}\left(t\right)=\frac{1}{2}V_{s}-\frac{1}{2}V_{s} \text{cos}\omega_{A}\left(t-t_{L2}\right)+\frac{1}{2}Z_{A}I_{boost2}\text{sin} \omega_{A}\left(t-t_{L2}\right) \tag{25}\]  
\[v_{ds2}\left(t\right)=v_{ds3}\left(t\right)=\frac{1}{2}V_{s}+\frac{1}{2}V_{s} \text{cos}\omega_{A}\left(t-t_{L2}\right)-\frac{1}{2}Z_{A}I_{boost2}\text{sin} \omega_{A}\left(t-t_{L2}\right) \tag{26}\]  
\[i_{Lr}\left(t\right)=-I_{boost2}\text{cos}\omega_{A}\left(t-t_{L2}\right)- \frac{V_{s}}{Z_{A}}\text{sin}\omega_{A}\left(t-t_{L2}\right)+i_{o} \tag{27}\]  
where \(I_{boost2}\) is the switch current at the initial resonant time \(I_{boost2}=I_{Lrm2}+i_{o}\)  
Thus, the actual pole voltage can be calculated as follows,  
\[v_{ab}(t)=v_{ds3}\left(t\right)-v_{ds4}\left(t\right)=V_{s}\text{cos}\omega_{ A}\left(t-t_{L2}\right)-Z_{A}I_{boost2}\text{sin}\omega_{A}\left(t-t_{L2}\right) \tag{28}\]  
whereas the ideal pole voltage is:  
Figure 3:0  
The voltage error caused by the resonant stage can be calculated as follows:  
Figure 3:1  
For Equation (28) \(v_{ab}(t)=-V_{s}\), the resonant time can be obtained:  
Figure 3:2  
Regarding the dead-time \(t_{L8}\)-\(t_{L10}\), the principle is the same as \(t_{H6}\)-\(t_{HS}\) in the heavy load condition, which is caused by the rise- and fall-times of the pole voltage. Therefore, the voltage error and the resonant time can be obtained:  
Figure 3:3  
Figure 3:4  
where \(I_{boost3}\) is the switch current at the initial resonant time \(I_{boost3}=I_{Lrm3}-i_{o}\)  
The voltage error in a switching cycle can be obtained from Equations (30) and (32) in the light load condition:\[v_{err}=\left\{ \begin{array}{ccc}&V_{s}\text{cos}\omega_{A}\left(t-t_{L2}\right)- Z_{A}I_{boost2}\text{sin}\omega_{A}\left(t-t_{L2}\right)+V_{s}&t_{L2}\leq t\leq t _{L3}\\ &-V_{s}-V_{s}\text{cos}\omega_{A}\left(t-t_{L8}\right)+Z_{A}I_{boost3}\text{sin} \omega_{A}\left(t-t_{L8}\right)&t_{L8}\leq t\leq t_{L9}\\ 0&t_{L0}\leq t<t_{L2}\text{ \ or \ \ }t_{L3}<t<t_{L8}&\text{ \ or \ \ }t_{L9}<t<t_{L12}\end{array} \right. \tag{34}\]  
Thus, the average voltage error in a switching cycle can be calculated as follows:  
\[V_{err}=\frac{1}{T_{s}}\int_{t_{L0}}^{t_{L12}}v_{err}dt=\frac{\Delta t_{L23}- \Delta t_{L80}}{T_{s}}V_{s}=\frac{V_{s}}{T_{s}}\left(\left|\frac{2}{\omega_{A }}\text{arcsin}\frac{V_{s}}{\sqrt{V_{s}^{2}+Z_{A}^{2}I_{boost2}^{2}}}\right| -\left|\frac{2}{\omega_{A}}\text{arcsin}\frac{V_{s}}{\sqrt{V_{s}^{2}+Z_{A}^{ 2}I_{boost2}^{2}}}\right|\right) \tag{35}\]