## 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
It also enlarges the AC phase voltages (VAN, VBN, VCN) and the boost inductor currents (I\({}_{\rm B1}\), I\({}_{\rm B2}\), I\({}_{\rm B3}\)) in the interval 0 \(<\) out \(\leq\)\(\pi\)/3. In the enlarged waveform, the negative voltages of the three-phase AC phase voltages (V\({}_{\rm AN}\), V\({}_{\rm BN}\), V\({}_{\rm CN}\)) are represented in absolute values. The rising and reset slopes of the boost inductor currents (I\({}_{\rm B1}\), I\({}_{\rm B2}\), I\({}_{\rm B3}\)) are divided into two or three segments depending on the magnitudes of the three-phase AC phase voltages (V\({}_{\rm AN}\), V\({}_{\rm BN}\), V\({}_{\rm CN}\)) in the interval 0 \(<\) out \(\leq\)\(\pi\)/3, the link voltage (V\({}_{\rm Link}\)), and the load [R\({}_{\rm Link}\) = (4N)\({}^{2}\)R\({}_{\rm DC}\)]. The critical angle that differentiates the two or three slopes of the boost inductor currents (I\({}_{\rm B1}\), I\({}_{\rm B2}\), I\({}_{\rm B3}\)) is defined as \(\Phi_{\rm cr,x}\) (x = a, b, c), and this is represented in Equation (1) by applying the flux balance condition. Here, taking the a-phase of the PMSG small wind turbine three-phase AC phase voltages (V\({}_{\rm AN}\), V\({}_{\rm BN}\), V\({}_{\rm CN}\)) as the reference, \(\Phi_{\rm cr,a,t}\), the critical angle, is symmetric about \(\pi\)/2 during the half-cycle (0 \(<\) out \(\leq\)\(\pi\)). Using this, the condition for \(\Phi_{\rm cr,a}\) during the interval 0 \(<\) out \(\leq\)\(\pi\)/2 for each phase is represented as shown in Equation (1), where \(\Phi_{\rm cr}\) is defined as shown in Equation (2). \[\Phi_{\rm cr,a}=\left\{ \begin{array}{ll}\frac{\pi}{2},&\mbox{if $\[PERSON]_{\rm cr }>\frac{\pi}{2}$}\\ \[PERSON]_{\rm cr},&\mbox{if $\[PERSON]_{\rm cr}\leq\frac{\pi}{2}$}\end{array} \right. \tag{1}\]  
\[\Phi_{\rm cra} =\sin^{-1}\left(\frac{V_{\rm Link}}{2V_{\rm pk}}(1-2\rm D_{1})\right) \tag{2}\]  
As depicted in Figure 4, the boost inductor current (I\({}_{\rm B1}\)) of the a-phase operates in Discontinuous Conduction Mode (DCM), resulting in two slopes (rise and reset) within the interval 0 \(<\) out \(\leq\)\(\Phi_{\rm cr,a}\). The period during which the main switching devices (Q\({}_{1}\)/Q\({}_{2}\) or Q\({}_{3}\)/Q\({}_{4}\)) are simultaneously turned on is defined as D\({}_{1}\)T, where D_i, i = 1,2,3, is the duty ratio and T is the switching period. During D\({}_{1}\)T, the a-phase voltage across the filter capacitor C\({}_{\rm FA}\) (V\({}_{\rm AN}\) = V\({}_{\rm pk}\)sin(\(\omega\)t)) is applied to the boost inductor L\({}_{\rm B}\). This causes the boost inductor current I\({}_{\rm B1}\) to increase, storing energy in the inductor. The D\({}_{2}\)T interval is defined as the period when switching device Q\({}_{1}\) turns off, allowing the reset current to flow into the voltage divider capacitor C\({}_{1}\) and transferring energy. During the D\({}_{1}\)T period, when the a-phase voltage (V\({}_{\rm AN}\) = V\({}_{\rm pk}\)sin(\(\omega\)t)) is applied to the boost inductor (L\({}_{\rm B1}\)), causing the boost inductor current (I\({}_{\rm B1}\)) to rise and store energy, the highest peak point is denoted as I\({}_{\rm pk,a}\), as shown in Equation (3). The reset period D\({}_{2}\)T can be expressed using the flux balance condition in Equation (4). During the D\({}_{2}\)T period, the energy stored in the boost inductor is quickly reset by the differential voltage (V\({}_{\rm AN}\) \(-\) V\({}_{\rm Link}\)/2) between the lower a-phase voltage (V\({}_{\rm AN}\) = V\({}_{\rm pk}\)sin(\(\omega\)t)) and the voltage of the link capacitors (C\({}_{1}\), C\({}_{2}\)) (V\({}_{\rm Link}\)/2), causing the boost inductor current (I\({}_{\rm B1}\)) to reset completely to zero. \[\rm I_{pk,a}=\frac{V_{pk}\sin(\omega\)t}{L_{\rm B1}}D_{1}T \tag{3}\]Therefore, during the period 0 \(<\) out \(\leq\) D\({}_{2}\)T of the switching period (T), the instantaneous average current (<I\({}_{\text{LBa1}}\)>) of the boost inductor current (I\({}_{\text{LBa1}}\)) with two slopes is expressed using I\({}_{\text{pk,a}}\), D\({}_{1}\)T, and D\({}_{2}\)T, as shown in Equation (5).