## 2 Three-Phase Single-Stage AC-DC Converter in Inductive-Power-Transfer-Based Wind Power Systems
### Operation Modes of Inductive-Power-Transfer-Based Three-Phase Single-Stage AC-DC Converter
\[\langle\text{I}_{\text{LBa1}}\rangle=\frac{1}{\text{T}}\int_{0}^{(\text{D}_{1} \text{T}+\text{D}_{2}\text{T})}\text{I}_{\text{LBa1}}\text{d}(\text{out})= \frac{\text{D}_{1}^{2}\text{V}_{\text{Link}}\sin(\text{out})}{2\text{f}_{\text {s}}\text{L}_{\text{B}}\left(\frac{\text{V}_{\text{Link}}}{\text{V}_{\text{pk} }}-2\sin(\text{out})\right)}, \tag{5}\]  
where fs denotes the switching frequency (=1/T). In the interval \(\Phi_{\text{cr,a}}\) \(<\) out \(\leq\)\(\pi\)/2, where the boost inductor current (I\({}_{\text{LBa2}}\)) operates with three rising and reset slopes, the increased a-phase voltage (V\({}_{\text{AN}}\) = V\({}_{\text{pk}}\)sin(\(\text{out}\))) results in the boost inductor current (I\({}_{\text{LBa2}}\)) having a gentler slope during the reset period D\({}_{2}\)T. This is due to the reduced differential voltage (V\({}_{\text{AN}}\) \(-\) V\({}_{\text{Link}}\)/2) between the a-phase voltage (V\({}_{\text{AN}}\) = V\({}_{\text{pk}}\)sin(\(\text{out}\))) and the link capacitor (C\({}_{1}\)) voltage (V\({}_{\text{Link}}\)/2). Consequently, the boost inductor current (I\({}_{\text{LBa2}}\)) is not fully reset within the half-switching period (T/2), resulting in three slopes. The D\({}_{3}\)T interval occurs when switching devices Q\({}_{1}\) and Q\({}_{2}\) are turned off, and the boost inductor current (I\({}_{\text{LBa2}}\)) resets to zero. During this interval, the large  
Figure 4: Operating waveforms of phase voltage (V\({}_{\text{AN}}\), V\({}_{\text{BN}}\), V\({}_{\text{CN}}\))/boost inductor current (I\({}_{\text{B1}}\), I\({}_{\text{B2}}\), I\({}_{\text{B3}}\))/boost inductor voltage (V\({}_{\text{LBI}}\), V\({}_{\text{LB2}}\), V\({}_{\text{LB3}}\)) and link voltage (V\({}_{\text{Link}}\)) in the 3-phase single-stage AC-DC converter for the small wind power system. differential voltage between the a-phase voltage (V\({}_{\rm{AN}}\) = V\({}_{\rm{pk}}\)sin(\(\omega\)t)) and the combined voltage of link capacitors C\({}_{1}\) and C\({}_{2}\) (V\({}_{\rm{Link}}\)) causes the boost inductor current (I\({}_{\rm{LBa2}}\)) to reset quickly, allowing it to operate in Discontinuous Conduction Mode (DCM). In the D\({}_{2}\)T period, where three slopes appear, the relationship between the boost inductor voltage (V\({}_{\rm{LB1}}\) = V\({}_{\rm{pk}}\)sin(\(\omega\)t) \(-\) V\({}_{\rm{Link}}\)/2)  
\[\Phi_{\text{cr\_b1}}= \begin{cases}\begin{array}{c}0\text{ if }0\geq-\frac{\pi}{3}+\Phi_{\text{ cra}}\\ -\frac{\pi}{3}+\Phi_{\text{ cra}}\text{ if }0<-\frac{\pi}{3}+\Phi_{\text{ cra}}\leq\frac{\pi}{6}\\ \frac{\pi}{6}\text{ if }\frac{\pi}{6}<-\frac{\pi}{3}+\Phi_{\text{ cra}}\end{array}\\ \Phi_{\text{cr\_b2}}=\begin{cases}\frac{\pi}{3}\text{ if }\frac{2\pi}{3}-\Phi_{\text{ cra}}\geq\frac{\pi}{3}\\ \frac{2\pi}{3}-\Phi_{\text{ cra}}\text{ if }\frac{\pi}{6}>\frac{2\pi}{3}-\Phi_{\text{ cra}}\leq\frac{\pi}{3}\\ \frac{\pi}{6}\text{ if }\frac{\pi}{6}>\frac{2\pi}{3}-\Phi_{\text{ cra}}\end{cases}\\ \Phi_{\text{cr\_c},c}=\begin{cases}\frac{\pi}{3}-\Phi_{\text{ cra}}\text{ if }0<\frac{\pi}{3}-\Phi_{\text{ cra}}\leq\frac{\pi}{3}\\ 0\text{ if }0\geq\frac{\pi}{3}-\Phi_{\text{ cra}}\end{cases}\end{cases}  \tag{14}\]  
As shown in Figure 4, the instantaneous reset currents (I\({}_{\text{Link\_a},x}\) I\({}_{\text{Link\_b},y}\) I\({}_{\text{Link\_c},c}\)) of the boost inductors (L\({}_{\text{BI}}\), L\({}_{\text{B2}}\), L\({}_{\text{B3}}\)) are determined according to the critical angle \(\Phi_{\text{cr\_x}}\) for each phase. During the phase control (D\({}_{1}\)T) switching operation, where the switching devices Q\({}_{1}\)/Q\({}_{2}\) or Q\({}_{3}\)/Q\({}_{4}\) are turned on, the energy stored in the boost inductors (L\({}_{\text{BI}}\), L\({}_{\text{B2}}\), L\({}_{\text{B3}}\)) is transferred to the voltage divider link capacitors (C\({}_{1}\), C\({}_{2}\)) at the output of the input rectifier diode over the D\({}_{2}\)T or D\({}_{2}\)T + D\({}_{3}\)T time period.