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#### 15.4.3.3 Model Transformation using Decoupling Matrix

Since the system under discussion is a seven-phase one, the complete model can only be elaborated in seven-dimensional space. The first two-dimensional spaces are d-q, the second one is x1-y1, the third one is x2-y2, and the last is zero-sequence component that is absent due to the assumption of isolated neutral. On the basis of the general decoupling transformation matrix for an n-phase system, inverter voltage space vectors in the second two-dimensional subspace (x1-y1) and third two-dimensional subspace (x2-y2) are determined as

$\begin{array}{l}{\underset{¯}{v}}_{x1y1}^{INV}=\frac{2}{7}\left({v}_{a}+{\underset{¯}{a}}^{2}{v}_{b}+{\underset{¯}{a}}^{4}{v}_{c}+{\underset{¯}{a}}^{6}{v}_{d}+\underset{¯}{a}{v}_{e}+{\underset{¯}{a}}^{3}{v}_{f}+{\underset{¯}{a}}^{5}{v}_{g}\right)\\ {\underset{¯}{v}}_{x2y2}^{INV}=\frac{2}{7}\left({v}_{a}+{\underset{¯}{a}}^{3}{v}_{b}+{\underset{¯}{a}}^{6}{v}_{c}+{\underset{¯}{a}}^{2}{v}_{d}+\underset{¯}{{a}^{5}}{v}_{e}+\underset{¯}{a}{v}_{f}+{\underset{¯}{a}}^{4}{v}_{g}\right)\end{array}$

(15.134)

Thus, 128 space vectors of phase-to-neutral voltage ...

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