SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

The SV as a Vector
The first facet of the SV to study is its simple vector nature. Both the SV and the phasor are
complex-valued quantities but in the simplest sense it is their vector nature that makes them
useful. The vector properties of the SV are similar to the familiar properties of regular vectors
thus we begin by comparing the SV with linear algebra. This development will then be extended
by introducing complex basis vectors. Finally, the manner in which MMF and current SVs add is
examined.
Comparison with Linear Algebra
The space vector is like any other vector in that it describes a resultant vector quantity (in a vector
space α) that is produced by a linear combination of products of a scalar quantity (belonging to a
scalar field υ) multiplied by a basis vector in α. Consider the horizontal and vertical deflection
plates of a CRT to which two scalar variables (voltages) are applied. This potential creates an
electric field which imparts force on the electrons composing the beam. Looking into the electron
gun, the Cartesian plane is the vector space α and can be described by the orthonormal unit
vectors xˆ and yˆ
. The two scalar voltages applied to the plates belong to the field υ and have no
meaning in the vector space α. But the voltage applied to each pair of plates produces an electric
field that imparts a force whose direction can be described by some linear combination of the
basis vectors ( xˆ and yˆ
) and this result obviously does have meaning in α; this is the coupling
mechanism found in space vector theory. The force on an electron is given by Equation (3.32),
where c1 is a constant of proportionality that relates electric potential v to electric field strength E
and then to force F on an electron. 20 Since F is defined in the vector space α it could be called a
space vector. 21
F(c v xˆ c v yˆ) c ( v xˆ v
y ˆ)
(3.32)
1 x 1 y 1 x y
The net effect of the two voltages could be combined into a vector in the space α and this vector
may be called the voltage space vector defined in Equation (3.33).
v ˆ ˆ
x vy
v x y (3.33)
The force could be written as Equation (3.34).
20
The electric field would have a polarity opposite that of the applied voltage but the charge of the electron
is negative.
21 The simple vectors in space in this example (v, F) carry a slightly different meaning than the complex
space vectors developed next. Hence the boldface notation is used in this example but all space vectors in
the remainder of the report use the superscribed arrow notation listed in the beginning of the report.
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