SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

The phasor transform, the phasor, and the inverse phasor transform are given by Equations
(3.116), (3.117), and (3.118), respectively [39, p.418].
() ; () cos
X P xt xt X t
(3.116)
j
p p
p
X X e X
(3.117)
jt j jt Re Re p
(3.118)
1
P X X e X e e
In phasor analysis we are not concerned with the absolute position of signals in time—only the
position relative to the reference signal—thus the reference point can be chosen at will. Since the
cosine is used in the definition, the reference quantity is that which peaks at t 0 . As an
example consider the two signals given by Equations (3.119) and (3.120). The reference phasor is
defined in Equation (3.119) and all other phasors would be defined with respect to this reference,
such as the phasor in Equation (3.120). 2 () x t leads 1 () x t by / seconds; 2 X leads 1 X by φ
degrees.
x () t ReX e ReX e X cos t
(3.119)
p
jtjtp
p
j jt jtp
1 1 1 1
x () t ReX e e ReX e X cos t
(3.120)
2 2 2 2
The phasor diagram of these two phasors is shown in Figure 3.39 along with their time-domain
waveforms (plotted against ωt to make the angular displacement easier to see). This demonstrates
how any phasor is defined with respect to the reference phasor.
Figure 3.39 – Time-domain waveform and corresponding phasor diagram.
By convention, the phasor diagram is drawn for the condition ωt=0, which forces the reference
phasor to be aligned with the real axis as shown in Figure 3.18. The phasor diagram is a snapshot
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