SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

peak value is represented as an uppercase letter. Power system engineers use RMS-valued
phasors exclusively but both peak- and RMS-valued space vectors are found in the motor control
literature. Because peak space vectors are used in this report, peak phasors will also be used.
Further, this report uses the cosine as reference; the sine reference is used in a portion of the
literature but seems to complicate the math. The phasor is seen to represent a scalar quantity as
the real part of a complex vector that is made to rotate at angular velocity ω.
jt jt
et () E cos( t)
ReEeReE e
(3.24)
p p
~
j0
E E pe
E p0
(3.25)
The phasor of Equation (3.25) is defined as the reference phasor because the scalar signal it
represents has its peak value whent 0 . By definition then, the reference line must be the real
axis and the phasor diagram therefore represents the snapshot in time when t 0 . The phasor
diagram “captures” all phasors when at the reference signal peaks just like a digital storage
oscilloscope set to trigger upon detection of a peak. Phasors for other sinusoids are defined with a
phase measured relative to the reference phasor; traces on an oscillogram have a phase defined
from the trigger point of the reference waveform. For example, the applied terminal voltage,
Equation (3.26), is represented in phasor form as in Equation (3.27). Figure 3.17 shows clearly
how the phase ( ) of a V is defined relative to a E .
j jt jt
vt () Vcos( t) ReVeeReV e
(3.26)
p p
V V e V
(3.27)
j
p p
According to the discussion in the previous chapter, the bEMF is the time derivative of the rotor-
stator flux linkage, which is Ra ,
for phase-A. In the phasor domain the time derivative
corresponds to multiplication by jω. Therefore the bEMF a
E~ will always lead Ra ,
in phase by
90° and its amplitude will be proportional to rotor speed; this agrees with the development of the
previous chapter and the relationship is shown in Figure 3.19. Following a convention used in
[27], phasors of electromagnetic quantities are represented with a hollow diamond head.
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