## 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
\[\text{V}_{\text{Link\_x1}}\!=\text{V}_{\text{pk}}\sin(\omega\text{t}+\text{ \theta}_{\text{x}})\left(1\!+\!\frac{\text{R}_{\text{Link}}\sqrt{\frac{\text{ f}_{\text{L}}\text{B}_{\text{I}}}{\text{R}_{\text{Link}}}\Big{(}\frac{\text{ f}_{\text{L}}\text{B}_{\text{I}}}{\text{R}_{\text{Link}}}\!+\!\text{D}_{1}^{2} \Big{)}}{\text{f}_{\text{s}}\text{L}_{\text{BI}}}\right) \tag{20}\]  
\[\text{V}_{\text{Link\_x2}}=\frac{\text{V}_{\text{pk}}\sin(\omega\text{t}+\text{ \theta}_{\text{x}})\left\{\Big{(}\!-\!8\text{D}_{1}^{2}\!+\!2\!+\!\frac{32\text{f}_{ \text{L}}\text{B}_{\text{I}}}{\text{R}_{\text{Link}}}\Big{)}\pm\sqrt{\Big{(}\!-\!8 \text{D}_{1}^{2}\!+\!2\!+\!\!\frac{32\text{f}_{\text{L}}\text{L}_{\text{BI}}}{ \text{R}_{\text{Link}}}\Big{)}^{2}\!+\!64\text{D}_{1}^{2}\Big{(}\!\frac{32 \text{f}_{\text{L}}\text{L}_{\text{BI}}}{\text{R}_{\text{Link}}}+\left(2\text{D}_{1} \!-\!1\right)^{2}\!\Big{)}\right\}}}{2\Big{\{}\frac{32\text{f}_{\text{L}}\text{L}_{ \text{BI}}}{\text{R}_{\text{Link}}}+\left(2\text{D}_{1}\!-\!1\right)^{2} \!\Big{\}}} \tag{21}\]  
As a result, the average link voltages <V\({}_{\text{Link\_x2}}\)> can be determined according to the slopes of the boost inductor currents (I\({}_{\text{BI}}\), I\({}_{\text{B2}}\), I\({}_{\text{B3}}\)) using Equations (20) and (21), and are given by Equations (22)-(24). Therefore, within the interval \(0<\omega\text{t}\leq\pi/3\), the final three phase link voltage V\({}_{\text{Link}}\) is represented as the sum of the average link voltages for each phase, <V\({}_{\text{Link},\text{a}}\)>, <V\({}_{\text{Link},\text{b}}\)>, and <V\({}_{\text{Link},\text{c}}\)>, as shown in Equation (25). \[\left\langle\text{V}_{\text{Link},\text{a}}\right\rangle=\frac{3}{\pi}\bigg{\{} \int_{0}^{\Phi_{\text{cr},\text{a}}}\text{V}_{\text{Link},\text{a}}\text{d}( \omega\text{t})+\int_{\Phi_{\text{cr},\text{a}}}^{\pi/3}\text{V}_{\text{Link}, \text{a}}\text{d}(\omega\text{t})\bigg{\}} \tag{22}\]  
\[\left\langle\text{V}_{\text{Link},\text{b}}\right\rangle=\frac{3}{\pi}\bigg{\{} \int_{0}^{\Phi_{\text{cr},\text{b}}}\text{V}_{\text{Link},\text{b}}\text{d}( \omega\text{t})+\int_{\Phi_{\text{cr},\text{b}}}^{\pi/6}\text{V}_{\text{Link}, \text{b}}\text{d}(\omega\text{t})+ \tag{23}\]  
\[\left\langle\text{V}_{\text{Link},\text{c}}\right\rangle=\frac{3}{\pi}\bigg{\{} \int_{0}^{\Phi_{\text{cr},\text{c}}}\text{V}_{\text{Link},\text{c}}\text{d}( \omega\text{t})+\int_{\Phi_{\text{cr},\text{a}}}^{\pi/3}\text{V}_{\text{Link}, \text{c}}\text{d}(\omega\text{t})\bigg{\}} \tag{24}\]  
\[\text{V}_{\text{Link}}=\frac{1}{\sqrt{3}}(\text{V}_{\text{Link},\text{a}}+ \text{V}_{\text{Link},\text{b}}+\text{V}_{\text{Link},\text{c}}) \tag{25}\]  
Figure 5 shows the simulation results of the boost inductor values (L\({}_{\text{B}}\) = L\({}_{\text{B1}}\) = L\({}_{\text{B2}}\) = L\({}_{\text{B3}}\)) and the PFC stage link voltage (V\({}_{\text{Link}}\)) variations according to phase control (D\({}_{\text{1}}\)T) obtained using Equation (25) at a 1 kW link equivalent load resistance (R\({}_{\text{Link}}\)). As can be seen from the simulation results, a smaller boost inductance (L\({}_{\text{B1}}\), L\({}_{\text{B2}}\), L\({}_{\text{B3}}\)) results in a wider boost control range, but a higher V\({}_{\text{Link}}\) increases the boost inductor currents (I\({}_{\text{B1}}\), I\({}_{\text{B2}}\), I\({}_{\text{B3}}\)), which reduces efficiency. Considering that one-quarter of the link voltage (V\({}_{\text{Link}}\)) is applied to the primary side of the contactless coupler and the remainder is applied to the secondary side according to the turn ratio, the maximum range of the link voltage is set to 600 V to control an output voltage of 400 V at the rated load (1 kW). Consequently, the boost inductor is selected as 60 \(\mu\)H.