SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

sinusoidal windings with a sinusoidal rotor and then study the same windings with the
squarewave rotor. 49 Example rotor flux distributions were given in Chapter 2 for the sinusoidal
rotor and an arbitrary trapezoidal rotor. The sinusoidal rotor flux varied as the cosine away from
the rotor axis and the trapezoidal rotor flux had the positive portion centered on the rotor axis; the
only change here is that the magnets span a full pitch, as shown later by Figure C.14.
In order to study both windings with a sinusoidal rotor it is useful to derive an expression that
gives the flux linked by a single-turn coil whose axis is at an arbitrary position. The full-pitch
coil’s axis and the rotor position are defined as shown in Figure C.12.
Figure C.12 – Full-pitch coil axis at θc and sinusoidal rotor at position θr.
The rotor’s flux density profile is a cosine about the rotor’s axis. When the rotor’s axis is at θr (as
measured from zero) the value of flux density at some arbitrary angle θ around the stator given by
Equation (C.12).
B( , ) B cos( )
(C.12)
r p r
The flux through the single-turn coil is found by integrating the flux density over a differential
area defined in Figure C.13.
49 A rotor can never produce airgap flux that is a squarewave. The rotor would need to have a continuous
magnet and if this was manufactured as a cylinder and then magnetized, it would seem impossible to
perfectly magnetize the poles where they met; a dead spot would be expected. If the magnets were made
separately and attached to the rotor, one would expect flux leakage that would produce a transition from
North to South that was not sharp but somewhat rounded. Even with this limitation studying an ideal
squarewave profile will illuminate the concepts this appendix is to demonstrate.
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