Power Electronics Handbook, 4th Edition

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 ...