SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Equation (2.44) is the component of stator flux linkage due to the rotor (the rotor-stator flux
linkage) and R is the peak value.7 It is easy to verify that the peak value given in terms of the
parameters in Equation (2.43). The area of the pole face of the coil is given by Equation (2.45).
When the rotor is aligned to zero degrees the total flux is the average flux given by Equation
(2.46). Then, the total flux linkage (which is the peak value at zero degrees) is given by Equation
(2.47).
A ( D/2) Y
(2.45)
2
Bpsin( ) d Bp
(2.46)
0
R Np NDY Bp
(2.47)
The rotor-stator flux linkage in the trapezoidal motor can be found graphically by considering the
flux density as a function of the angle around the rotor, α, as shown in Figure 2.21. The plots of
rotor-stator flux linkage as a function of the rotor position θr are given in Figure 2.26. The
sinusoidal case is simple but the trapezoidal case is more complicated. If the airgap flux density
of the trapezoidal motor in Figure 2.21 was a rectangular wave (with zero segments
corresponding to the gaps between rotor magnets), the rotor-stator flux linkage would be the
lighter trapezoid shown below. Due to fringing effects and the hysteretic B-H curve of the iron
the actual flux density deviates from the ideal, as shown in Figure 2.21, thus the actual rotorstator
flux linkage differs, as shown in Figure 2.26. Note that the angle β (which defines the ideal
case) has correspondence between Figure 2.20, Figure 2.21, and Figure 2.26.
7 Notice that the peak value is denoted by the uppercase . This should perhaps be denoted only the
subscript ‘p’ as with other variables, but the notation adopted here is used to better align with space vector
notation in the rest of the report.
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