SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

F c 1 v
(3.34)
The situation is shown graphically in Figure 3.24.
Figure 3.24 – Equivalent force produced by plates aligned with resultant force.
Per the above, the composition of the space vector is no different than a standard vector in linear
algebra: any vector quantity is composed of a summation of the products of a basis vector with a
scalar. In the simplest cases, such as the Cartesian plane, the two basis vectors ( xˆ and yˆ
) are
2
, as it is in
orthonormal and therefore linearly independent; in this case, the vector space α is
the CRT example. In contrast, the space vector method uses complex exponentials as basis
vectors and therefore the vector space is . Further, there is no restriction that the spanning set
must be orthonormal. In fact the basis vectors are not even linearly independent (for three-phase
systems without a grounded neutral) and this is part of the reason that that the factor 3/2 arises
in the analysis of three-phase machines, irregardless of whether traditional or space vector
analysis is used.
Complex Basis Vectors
In three-phase systems there are three basis vectors, one aligned to each of the three electrical
axes (A, B, C) which are 120 electrical degrees apart according to the convention established in
Figure 3.1. Regardless of what type of quantity (voltage, current, flux, flux linkage) the space
vector describes in a three-phase system, the basis vectors are the axes are given by several
equivalent rectangular and polar forms in Equation (3.35). Note that 120 2 / 3,
j240 j120
e e
degrees.
, and that by definition the phase-A axis is the real axis which corresponds to zero
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