SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

the components are defined relative to the rotor—if the stator current SV has constant magnitude
but the instantaneous rotor velocity changes, so will the dq components). Therefore the voltages
defined by the second terms in Equation (3.154) are transient components. On the other hand, the
third terms in Equation (3.154) will be nonzero even in steady state operation. They are the dq
embodiment of the inductive voltage drop (which is always 90° ahead of the current SV). Each
component of that voltage drop will thus be 90° ahead of that component of current. A
component that is 90° ahead of the d axis will lie along the q axis; a component that is 90° ahead
of the q axis will lie along the –d axis. Hence, these third terms are responsible for the crosscoupling
and the polarities (these polarities should be verified between Figure 3.42 and Equation
3.154). Finally, the q-component of voltage contains a fourth term, which is recognized to be the
bEMF. By definition the d axis is along the rotor flux (which is cophasal with the rotor-stator flux
linkage) so the bEMF must always lie along the q axis, and the equation confirms this. Further,
the bEMF scales with velocity, as expected.
Now the SV diagram of the motor in the rotor frame can be drawn. A major difference from that
in the stationary frame is that we do not have to pick a reference SV because the rotor flux (hence
rotor-flux linkage and bEMF) are always along the said axes. As before, we will choose to
enforce the steady-state condition j
, causing the second terms in the voltage equation
above to vanish, thereby simplifying the diagram. The steady-state SV diagram is shown in
Figure 3.43 and Figure 3.44, where the various SVs have been split between two diagrams to
improve clarity. 27 (Usually BPMS motors are operated with the current being cophasal or leading
the bEMF, as shown.)
For comparison with the stationary and SPE models, Equation (3.154) can be put into SV form as
Equation (3.155), where only the steady-state terms have been kept in the final line.
27 The direction (additivity) of SVs and phasors depends on how the polarity conventions are selected in the
KVL equations. This report always assumes all circuit terms have the same polarity, defined to be the
opposite of the applied voltage. The SVs in the figure still satisfy this rule. However, to avoid the pesky
problem of distinguishing between a negative value and a negative projection direction, the components are
always shown for positive values of current. In the figures, id is negative. Accordingly, reversing the id
component directions will show that the components add to give the SVs, using the prior convention.
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