SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

To analyze a salient machine one must employ the “full” dq theory (the two-reaction theory used
by Park and those before him). In addition to separating vectors into their dq components (as we
have seen), the full dq theory assumes the effects of rotor saliency can be represented by different
inductances in the d- and q- directions ( d L and q L ) instead of the synchronous inductance L s . In
that case the simple SV analysis must be modified from what was shown in Chapter 3 to use these
separate inductances. The only result of concern here is that Equation (5.8) must be modified to
be Equation (5.10).
L i
q
Lqiq d d d R
(5.10)
Substituting these flux linkages into Equation (3.160) yields the torque for a salient machine,
Equation (5.11).
3
T d q q d
2 i
i
3
Lii d d qRiqLii q q d
2
3
T RiqLdLqidi q
2 (5.11)
In Equation (5.11) the first term describes the mutual torque produced via interaction of the stator
field with the magnets and the second term describes the reluctance torque produced as the salient
rotor attempts to align with the stator field in order to minimize the reluctance of the magnetic
circuit. The definitions of the inductances are not given here, but in a nonsalient machine they are
equal to one another and to the synchronous inductance: ( L L L ) . Thus, for a nonsalient
d q s
machine, Equation (5.11) simplifies to Equation (5.9) as expected. To investigate the meaning of
the d- and q-axis inductances, examine the three types of rotors shown in Figure 5.15.
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