SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

in relative time like an oscillograph. Each sweep in an oscilloscope will the trigger on the same
point in the periodic waveform and for the phasor diagram that trigger point is the peak of the
reference signal. In contrast, SV diagrams are drawn with reference to the phase axes (or
real/imaginary axes). For illustration two SVs are defined by Equations (3.121) and (3.122).
These are shown on a SV diagram (Figure 3.40) for arbitrary values ωt=30° and δ=15°.
ˆ j1 x1 Xe 1
jt X p1e
(3.121)
x ˆ j2 X e
jt X e
(3.122)
2 2 p1
Figure 3.40 – Space vector diagram.
It is useful and appropriate to show the instantaneous SV relative to the phase axes for some
value of ωt (such Figure 3.26) when showing the position of the SV at a particular instant. But
often the steady-state SV diagram will be drawn at t 0 and the SV diagram will look like
exactly like a phasor diagram. Comparing Equations (3.119)-(3.120) with Equations (3.121)-
(3.122) it is seen that at t 0 the phasor is equal to the SV (with the exception of scaling). For
sinusoidal quantities in steady state the phasors and SVs trace a circle and any imbalance
transient condition causes the phasor trajectory to deviate from the circle. In contrast,
nonsinusoidal quantities have a SV trajectory that is not circular in the steady state and these
cannot be represented by phasors.
Returning to Equations (3.116)-(3.118) it is seen that the phasor transform takes a cosinusoidal
quantity to a stationary vector in the complex plane; this is the phasor domain. It is common
practice to say that the phasor rotates, but the phasor itself ( X ) does not rotate—rotation is
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