SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Torque and Back-EMF
An expanded version of Figure C.1 is shown in Figure C.20. It is clear that torque is produced via
the interaction of the stator and rotor fields and an expression could be found in terms of these.
But the stator field is directly related to the current by the winding function thus an expression for
torque could also be found in terms of the rotor field and the stator current; this is desirable since
it is the stator current that we can control.
Figure C.20 – ‘Natural’ torque and bEMF relationships.
As discussed in Chapter 2, the torque produced by a winding in a nonsalient brushless permanent
magnet motor (found via coenergy) is given by Equation (C.21). Also, bEMF can be found (using
Faraday’s law and some manipulation) to be given by Equation (C.22).
d
Tt () R( r)
it ()
(C.21)
d
r
d
et () R( r) ()
t
(C.22)
d r
Using this information the relationships can be redrawn as Figure C.21.
Figure C.21 – Torque and bEMF in terms of rotor-stator flux linkage.
It was also found in Chapter 2 that the torque and bEMF functions were the same as the position
derivative of rotor-stator flux linkage, Equation (C.23).
d
R( r) ke( r) kt(
r)
(C.23)
d
r
Torque and bEMF are determined by the derivative of rotor-stator flux linkage. Since all three
functions in Equation (C.23) are easily visualized and are equivalent it is easy to see how the
motor construction affects torque production and bEMF generation. The previous discussion
showed how the fundamental and harmonic winding factors affect MMF and rotor-stator flux
linkage. The same affects apply to torque and bEMF.
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