SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

j0 j0 j0
aˆ
1 0 1
bˆ
j e e e
1
3
2 2
j120 j2 /3j j e e e a
(3.35)
1 3
j120 j2 /3 j
2
cˆ
j e e e a
2 2
Similar to the CRT example presented earlier, the MMF produced due to the combined effect of
current in the three windings could be represented by an MMF space vector. In the CRT example
the relationship between voltage and the force it produces is a constant of proportionality and in a
machine the relationship between current and the MMF it produces is also a constant of
proportionality—the same as that in Equation (3.1). Using this constant, the concept of Equations
(3.32)-(3.34) from the CRT example, and the directionality of the magnetic axes of windings
shown in Figure 3.1, the MMF produced by current space vector could be described as Equation
(3.36). 22
Ne Ne
() ˆ () ˆ Ne
f i () ˆ
A t i B t iC t
2
a
2
b
2
c
(3.36)
Further, the CRT example showed that the constant of proportionality could be pulled out front
and the force space vector could be described by defining a voltage space vector. The similar case
here is to define the MMF space vector by defining a current space vector as shown in Equation
(3.37), where the current space vector is given by Equation (3.38).
Ne
f () ˆ () ˆ
i () ˆ
A t i B t iC
t
2
a b c
(3.37)
i i () ˆ () ˆ () ˆ
A t i B t iC
t
a b c
(3.38)
The basis vectors in Equation (3.36)-(3.38) are typically represented by one of the rectangular or
polar notations in Equation (3.35). Therefore the current space vector, for example, is typically
written in the form of Equation (3.39) or Equation (3.40); this report will favor the form of
Equation (3.40). In addition a very important transformation constant (k) has been added to scale
the result to account for the fact that the basis vectors are not linearly independent; it will be
discussed in detail later but for simplicity its importance will be minimized for now by selecting
k=1 until further notice.
22
Until the section titled “The SV as a Distribution,” the reader is asked to ignore the idea of sinusoidally
distributed quantities in a machine.
97