Dqo Transformation - Definition

Definition

The dqo transform applied to three-phase currents is shown below in matrix form:

I_{dqo} = TI_{abc} = \sqrt{\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{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2} \end{bmatrix}\begin{bmatrix}I_a\\I_b\\I_c\end{bmatrix}

The inverse transform is:

I_{abc} = T^{-1}I_{dqo} = \sqrt{\frac{2}{3}}\begin{bmatrix}\cos(\theta)& - \sin(\theta)&\frac{\sqrt{2}}{2}\\ \cos(\theta - \frac{2\pi}{3})& - \sin(\theta - \frac{2\pi}{3})&\frac{\sqrt{2}}{2}\\ \cos(\theta + \frac{2\pi}{3})& - \sin(\theta + \frac{2\pi}{3})&\frac{\sqrt{2}}{2}\end{bmatrix} \begin{bmatrix}I_d\\I_q\\I_o\end{bmatrix}

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