# The Induction Machine: Modeling and Control

## C.1 Space Vector Notion

The space vector $\overline{x}$ is defined by (Fig. C.1):

$\overline{x}=\frac{2}{3}\left({x}_{1}+a{x}_{2}+{a}^{2}{x}_{3}\right)$

where

$a={e}^{j\frac{2\pi}{3}}$

and

$\{\begin{array}{c}{x}_{1}={X}_{1}\mathrm{cos}(wt+{\phi}_{1})\\ {x}_{2}={X}_{2}\mathrm{cos}(wt+{\phi}_{2})\\ {x}_{3}={X}_{3}\mathrm{cos}(wt+{\phi}_{3})\end{array}$ (C.1)

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