SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

3 N e
(3.6): f ( , t)
I
p cos( t
)
2 2
In the foregoing equations FS represented the peak amplitude of the stator MMF cosine wave,
which is given by the terms that multiply the cosine in Equation (3.6). Substituting this amplitude
into Equation (3.16) gives Equation (3.17).
3 Ne
T c2Bp Ipsin( )
2
2
(3.17)
The angle δ was defined as the difference between the peaks of cosine waves, measured from the
rotor wave to the stator wave (stator minus rotor). The angle of stator MMF in Equation (3.6) is
equal to ωt when referenced to the magnetic axis of phase-A (θ=0°) and the rotor position is
defined from the same reference. Thus the stator angle minus the rotor angle is equal to δ (in
other words, θ=θr in Equation 3.6). The stator MMF defined by Equation (3.6) and the rotor flux
density defined by Figure 3.15 are cosine waves, thus the sine of δ in Equation (3.17) is still
valid. The torque may then be written as Equation (3.18), where the constant c3 is defined by
Equation (3.19).
3
T c3BpIpsin( )
(3.18)
2
Ne
c3DY 2
2
(3.19)
For a given motor, Bp is fixed by the magnetic design thus Equation (3.18) shows that torque is
proportional to the amplitude I p of the sinusoidal three-phase currents and is scaled by the sine
of the angle δ between the rotor and stator fields. For a given current it is obvious that the
maximum torque is achieved when the fields are at δ=90°; this concept will form the basis of
FOC discussed in Chapter 5. It is important to understand that torque is produced by the
interaction of these two quantities as they maintain a fixed phase-relationship relative to one
another while they both rotate at the synchronous frequency, ω. Equation (3.18) describes only
the component of torque produced by fundamental stator and rotor quantities (the number of
effective stator turns describes the fundamental MMF and Bp represents the amplitude of rotor
flux density).
Note that if all else is held constant in Equation (3.12), torque is proportional to the magnitude of
the stator MMF. Thus the factor of 3/2 from the stator MMF expression comes through in the
torque expression, as expected. This shows that a sinusoidal rotor flux will interact with the
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