## 3 Proposed Control Scheme of 3-Phase AC/DC PWM Converter Using SOGI-FLL
### Positive Sequence Voltage Extraction and Phase Detection Using 3-Phase SOGI-FLL  
In 1937, [PERSON] proposed a method for decomposing balanced 3-phase voltages \(v_{a}\), \(v_{b}\), and \(v_{c}\) into positive, negative, and zero sequence voltage in the time domain, respectively, while in 1918, [PERSON] proposed a method in the frequency domain [23]. Using these methods, the balanced 3-phase voltage is decomposed into positive, negative, and zero sequence voltages, as shown in Equations (9)-(11), respectively.  
\[\left[\begin{array}{c}v_{a}^{+}\\ v_{b}^{+}\\ v_{c}^{+}\end{array}\right]=\frac{1}{3}\cdot\begin{bmatrix}1&\alpha&\alpha^{2} \\ \alpha^{2}&1&\alpha\\ \alpha&\alpha^{2}&1\end{bmatrix}\cdot\left[\begin{array}{c}v_{a}\\ v_{b}\\ v_{c}\end{array}\right]\;(v_{abc}^{+}=[T_{+}]\cdot v_{abc}) \tag{9}\]  
\[\left[\begin{array}{c}v_{a}^{-}\\ v_{b}^{-}\\ v_{c}^{-}\end{array}\right]=\frac{1}{3}\cdot\begin{bmatrix}1&\alpha^{2}&\alpha \\ \alpha&1&\alpha^{2}\\ \alpha^{2}&\alpha&1\end{bmatrix}\cdot\left[\begin{array}{c}v_{a}\\ v_{b}\\ v_{c}\end{array}\right]\;(v_{abc}^{-}=[T_{-}]\cdot v_{abc}) \tag{10}\]  
\[\left[\begin{array}{c}v_{a}^{0}\\ v_{b}^{0}\\ v_{c}^{0}\end{array}\right]=\frac{1}{3}\cdot\begin{bmatrix}1&1&1\\ 1&1&1\\ 1&1&1\end{bmatrix}\cdot\left[\begin{array}{c}v_{a}\\ v_{b}\\ v_{c}\end{array}\right]\;(v_{abc}^{0}=[T_{0}]\cdot v_{abc}) \tag{11}\]  
where \(\alpha=e^{j\frac{2\pi}{3}}=-\frac{1}{2}+j\frac{\sqrt{3}}{2}\) and \(\alpha^{2}=e^{j\frac{4\pi}{3}}=-\frac{1}{2}-j\frac{\sqrt{3}}{2}\).  
3 \(\times\) 3 matrices in each equation can be expressed as \([T_{+}]\), \([T_{-}]\), and \([T_{0}]\). If the balanced 3-phase voltage is transformed to the voltage in the stationary reference frame, it can be defined as Equation (12). Equation (12) is then simplified to Equation (13), with the assumption that no zero sequence components exist in the balanced 3-phase voltage.  
\[\left[\begin{array}{c}v_{\alpha}\\ v_{\beta}\\ v_{0}\end{array}\right]=\frac{2}{3}\cdot\begin{bmatrix}1&-\frac{1}{2}&-\frac {1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\end{bmatrix}\cdot\left[\begin{array}[] {c}v_{a}\\ v_{b}\\ v_{c}\end{array}\right]\;\;\;\;\;v_{a\beta 0}=[T_{a\beta 0}]\cdot v_{abc} \tag{12}\]  
\[\left[\begin{array}{c}v_{\alpha}\\ v_{\beta}\end{array}\right]=\frac{2}{3}\cdot\begin{bmatrix}1&-\frac{1}{2}&- \frac{1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\end{bmatrix}\cdot\left[\begin{array}[] {c}v_{a}\\ v_{b}\\ v_{c}\end{array}\right]\;\;\;\;\;v_{a\beta}=[T_{a\beta}]\cdot v_{abc} \tag{13}\]  
The transpose matrix \(\left[T_{a\beta}\right]\) is given by the following Equation (14).  
\[\left[T_{a\beta}\right]^{T}=\frac{3}{2}\cdot\begin{bmatrix}\frac{2}{3}&0\\ -\frac{1}{3}&\frac{\sqrt{3}}{3}\\ -\frac{1}{3}&-\frac{\sqrt{3}}{3}\end{bmatrix} \tag{14}\]  
Using Equations (9)-(14), the positive and negative sequence voltages in the stationary reference frame can be summarized, as shown in Equations (15) and (16).  
\[\begin{array}{c}v_{a\beta}^{+}\\ \\ \\ \\ \\  
In Equations (15) and (16), \(q=e^{-j\frac{q}{2}}\), which indicates that the phase is delayed by 90 degrees. In this case, the positive and negative sequence voltage extractors are added. Figure 12 is then modified, as shown in Figure 13.