SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

2
i k iA()1 t ai B() t a iC()
t
jj i k iA()1 t i B() t e iC() t e
Using this notation the MMF space vector is written as Equation (3.41).
Ne
jj f kiA()1 t i B() t e iC() t e
2
(3.39)
(3.40)
(3.41)
Recognizing that Equation (3.40) and Equation (3.41) differ by only a constant, the MMF space
vector can be written directly in terms of the current space vector as shown in Equation (3.42).
This corresponds to Equation (3.34) of the CRT example and should help solidify the
understanding of mechanism by which a space vector in α (with units corresponding to υ) can be
assigned units that correspond to α by means of a simple constant of proportionality.
N e
f i
2
(3.42)
2
but that the space
It was mentioned earlier that in the CRT example the vector space α was
vector uses the vector space . From the definition of the complex basis vectors in Equation
(3.35) and the definition of the space vector it can be seen that the three phase variables
3
(belonging to ) are mapped onto the complex plane, hence the space vector is in . The
important consequences of this mapping will be examined throughout this chapter.
Finally, it is clear that the magnitude and angle of the current SV will depend on the nature of the
three phase currents in Equation (3.40). This concept is true for any SV and will be elaborated for
the SV of the general quantity x whose SV is defined by Equation (3.43).
jj x k xA1xBe xC e
(3.43)
If the three phase quantities ( xA, xB, x C ) form a balanced sinusoidal set (Equation 3.44) the SV
will have a constant magnitude and its angle will be equal to ωt. This space vector is represented
as Equation (3.46); this result is found by using the identity in Equation (3.45) to represent each
phase quantity as its complex equivalent before substitution into Equation (3.43).
xA() t X pcos(
t)
x xB() t X pcos(
t)
xC()
t X pcos(
t)
(3.44)
98