preface
\[ \theta = \omega t + \phi \]
- \(\theta\) angular position
- \(\omega\) angular frequency
- \(\phi\) initial phase
three phase machine
\(abc \longrightarrow dq0\)
stationary \(ABC\) frame into synchronous \(dq\) frame
\[ \begin{equation} \begin{bmatrix} d\\ q\\ 0 \end{bmatrix} =\frac{2}{3} \begin{bmatrix} \cos(\theta) & \cos(\theta-\frac{2\pi}{3}) & \cos(\theta+\frac{2\pi}{3})\\ -\sin(\theta) & -\sin(\theta-\frac{2\pi}{3}) & -\sin(\theta+\frac{2\pi}{3})\\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \begin{bmatrix} a\\ b\\ c \end{bmatrix} \end{equation} \]
\(dq0 \longrightarrow abc\)
synchronous \(dq\) frame into stationary \(ABC\) frame
\[ \begin{equation} \begin{bmatrix} a\\ b\\ c \end{bmatrix} = \begin{bmatrix} \cos(\theta) & -\sin(\theta) & 1\\ \cos(\theta-\frac{2\pi}{3}) & -\sin(\theta-\frac{2\pi}{3}) & 1\\ \cos(\theta+\frac{2\pi}{3}) & -\sin(\theta+\frac{2\pi}{3}) & 1 \end{bmatrix} \begin{bmatrix} d\\ q\\ 0 \end{bmatrix} \end{equation} \]
\(abc \longrightarrow \alpha \beta 0\)
stationary \(ABC\) frame into stationary \(\alpha\beta\) frame
\[ \begin{equation} \begin{bmatrix} \alpha\\ \beta\\ 0 \end{bmatrix} =\frac{2}{3} \begin{bmatrix} \cos(0) & \cos(-\frac{2\pi}{3}) & \cos(\frac{2\pi}{3})\\ -\sin(0) & -\sin(-\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3})\\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \begin{bmatrix} a\\ b\\ c \end{bmatrix} =\frac{2}{3} \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2}\\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2}\\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \begin{bmatrix} a\\ b\\ c \end{bmatrix} \end{equation} \]
\(\alpha \beta 0 \longrightarrow abc\)
stationary \(\alpha\beta\) frame into stationary \(ABC\) frame
\[ \begin{equation} \begin{bmatrix} a\\ b\\ c \end{bmatrix} = \begin{bmatrix} 1 & 0 & 1\\ -\frac{1}{2} & \frac{\sqrt{3}}{2} & 1\\ -\frac{1}{2} & -\frac{\sqrt{3}}{2} & 1 \end{bmatrix} \begin{bmatrix} \alpha\\ \beta\\ 0 \end{bmatrix} \end{equation} \]
\(\alpha \beta \longrightarrow dq\)
stationary \(\alpha\beta\) frame into synchronous \(dq\) frame
\[ \begin{equation} \begin{bmatrix} d\\ q \end{bmatrix} = \begin{bmatrix} \cos(\theta) & \sin(\theta)\\ -\sin(\theta) & \cos(\theta) \end{bmatrix} \begin{bmatrix} \alpha\\ \beta \end{bmatrix} \end{equation} \]
\(dq \longrightarrow \alpha \beta\)
synchronous \(dq\) frame into stationary \(\alpha\beta\) frame
\[ \begin{equation} \begin{bmatrix} \alpha\\ \beta \end{bmatrix} = \begin{bmatrix} \cos(\theta) & -\sin(\theta)\\ \sin(\theta) & \cos(\theta) \end{bmatrix} \begin{bmatrix} d\\ q \end{bmatrix} \end{equation} \]
\(ABC\longrightarrow NP0\)
- \(N\) (negative sequence)
- \(P\) (positive sequence)
- \(0\) (zero sequence)
\[ \begin{equation} \begin{bmatrix} P\\ N\\ 0 \end{bmatrix} = \frac{1}{3} \begin{bmatrix} 1 & \alpha & \alpha^2\\ 1 & \alpha^2 & \alpha\\ 1 & 1 & 1\\ \end{bmatrix} \begin{bmatrix} A\\ B\\ C\\ \end{bmatrix} \end{equation} \]
\[\alpha = e^{j\frac{2\pi}{3}}\]
\(NP0 \longrightarrow ABC\)
\[ \begin{equation} \begin{bmatrix} A\\ B\\ C\\ \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1\\ \alpha^2 & \alpha & 1\\ \alpha & \alpha^2 & 1\\ \end{bmatrix} \begin{bmatrix} P\\ N\\ 0 \end{bmatrix} \end{equation} \]
dual three-phase machine
\(ADBCDEF \longrightarrow \alpha \beta x y o_1 o_2\)
\[ \begin{equation} \label{eq:VSD Matrix} \begin{bmatrix} \alpha\\ \beta\\ x\\ y\\ o_1\\ o_2 \end{bmatrix} =\mathbf{T}_{VSD} \begin{bmatrix} A\\ B\\ C\\ D\\ E\\ F \end{bmatrix} =\frac{1}{6} \begin{bmatrix} 2 & -1 & -1 & \sqrt{3} & -\sqrt{3} & 0\\ 0 & \sqrt{3} & -\sqrt{3} & 1 & 1 & -2\\ 2 & -1 & -1 & -\sqrt{3} & \sqrt{3} & 0\\ 0 & -\sqrt{3} & \sqrt{3} & 1 & 1 & -2\\ 2 & 2 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 2 & 2 \end{bmatrix} \begin{bmatrix} A\\ B\\ C\\ D\\ E\\ F \end{bmatrix} \end{equation} \]
\(ABCDEF \longrightarrow \alpha \beta x y o_1 o_2\)
\[ \begin{equation} \label{eq: inverse VSD Matrix} \begin{bmatrix} A\\ B\\ C\\ D\\ E\\ F \end{bmatrix} = \begin{bmatrix} 1 & 0 & 1 & 0 & 1 & 0\\ -\frac{1}{2} & \frac{\sqrt{3}}{2} & -\frac{1}{2} & -\frac{\sqrt{3}}{2} & 1 & 0\\ -\frac{1}{2} & -\frac{\sqrt{3}}{2} & -\frac{1}{2} & \frac{\sqrt{3}}{2} & 1 & 0\\ \frac{\sqrt{3}}{2} & \frac{1}{2} & -\frac{\sqrt{3}}{2} & \frac{1}{2} & 0 & 1\\ -\frac{\sqrt{3}}{2}& \frac{1}{2} & \frac{\sqrt{3}}{2} & \frac{1}{2} & 0 & 1\\ 0 & -1 & 0 & -1 & 0 & 1 \end{bmatrix} \begin{bmatrix} \alpha\\ \beta\\ x\\ y\\ o_1\\ o_2 \end{bmatrix} \end{equation} \]
备注
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