SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

phase variables and the SV trajectory are difficult to describe thus a ratio is impossible to define.
The simples explanation is to examine the projection onto the α axis by using Equation (3.84) (or
any 0 variant such as Equation 3.88).
3 3
A B C
2 2
(3.84): x k x j x x
(3.88):
3 3
x k xA j ( xA 2
xB)
2 2
In these particular forms it is clear that the projection onto α is affected by phase-A only.
Therefore it does not matter what the instantaneous value of xA() t is, the projection onto the α
axis will be 3 k /2 larger than that instantaneous value. This demonstrates that the scaling is the
same although the interpretation is not as clear. Or one could observe the results of Equation
(3.95).
T 3
CC I
2
T 2
CC m m I
3
T
CC I
p p
(3.95)
Inverse Clarke Transform
The inverse of the Clarke transformation has already been presented but will be elaborated
briefly. Either Equation (3.79)/(3.85) or Equation (3.89) can be used to find phase variables
corresponding to a SV represented by its α- and β- components.
(3.85):
2 0
x 3
A
1 1 1 x
x
B
k
3
3 x
x
C
1 1
3 3
In Equation (3.85) the phase variables in x abc are the result of the inverse of the Clarke
transform. They are NOT the same values that would be obtained if the SV was simply projected
onto the phase axes in Figure 3.29 because of the inherent scaling in the transformation.
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