SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Ne
1
jj f , t i e i * e
2 2
(3.54)
Defining the first of the term in the square brackets as the complex number z, the identity in
Equation (3.55) can be used to rewrite (3.54) as Equation (3.56).
Re
1
(3.55)
2
z z z
N
f t i e
2
e
j
, Re
(3.56)
According to Equation (3.46), the current SV (k=1) of three-phase balanced sinusoidal currents of
amplitude Ip is given by Equation (3.57).
pe I i
3
2
jt
Substituting this into Equation (3.56) yields Equation (3.58).
Ne
3jt j
f , t Re
Ipe e
2 2
3 N
f t Ipt
2 2
e
, cos
(3.57)
(3.58)
Equation (3.58) is identically Equation (3.6), the real-valued expression of stator MMF due to
balanced sinusoidal currents developed in Part I of this chapter. To be clear, Equation (3.6) was
developed without any knowledge of the space vector and it is written in terms of scalar currents.
With some manipulation (by transforming it into complex form, but without making any changes)
it was rewritten as the real part of a complex-valued expression in which the current space vector
appeared along with a term that contained the angle about the stator periphery, θ. The space
vector corresponding to balanced sinusoidal currents was substituted in and this resulted in the
real-valued expression for the MMF that one would expect from those balanced sinusoidal
currents. This roughly illustrates what this subsection set out to show: that the MMF SV
represents a cosinusoidal MMF distribution.
Perhaps the result is questionable since we made the substitution of the current SV that represents
balanced sinusoidal currents. That substitution was made in order to show the clear link to the
result obtained earlier in the chapter but now we can treat the arbitrary current case. The current
SV in Equation (3.56) can be expressed in terms of its magnitude and phase using the notation
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