SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

otor MMF wave to the peak of the stator MMF wave. 18 The direction in which the angle δ is
measured is a result of the convention that developed electromagnetic torque in a motor is taken
to be positive. Equation (3.12) is recognized as the scalar equivalent of the vector cross product
the vector quantities shown in Figure 3.15.
T c1FRF
S sin( )
(3.12)
In Equation (3.12) the constant c1 is given by Equation (3.13), where D is the inside diameter of
the stator steel laminations (or rotor diameter, assuming a negligible airgap), Y is the length of the
rotor and stator lamination stack, and G is the airgap length.
c DY
1
0
2
G
(3.13)
Since the rotor field has been introduced using flux density, the torque expression can be
reformulated in terms of BR as Equation (3.14) simply by rearranging the constant terms.
FR
T DY 0FSsin( )
2 G
T DY 0HRFS
sin( )
2
T DY BRFS
sin( )
2
(3.14)
Since the traditional symbol for flux density is always uppercase it is difficult to distinguish
between peak and instantaneous values. For this reason BR will now be replaced by Bp. Given that
we typically think of rotor flux density and stator MMF, it will be understood that Bp refers to
rotor flux density and Equation (3.14) is rewritten as Equation (3.15).
T DY BpFS
sin( )
2
(3.15)
Now the constants can be lumped together and torque is expressed by Equation (3.16) in terms of
stator MMF and rotor flux density, as was desired.
T c2BpFS sin( )
(3.16)
Since the stator MMF is created by the stator current it is useful to expand the FS term to show
this. Equation (3.6) gave the expression for stator MMF due to three sinusoidal balanced currents
of amplitude I p .
18 The torque angle and load angle can refer to two different angles in the same machine but this distinction
is not necessary for this report.
80