SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

current does not influence the value of the rotor flux linkage (which is true when rare-earth
magnets are used).
T cR
i sin r
T c
i sin
(3.158)
R
If this same procedure is followed in the rotor reference frame a great simplification results.
Beginning again with Equation (3.156) and using the same technique yields Equation (3.159).
R R
T ci R R R
cLi s R i
R R R R
cLi s i cR i
R R
T cRi
(3.159)
R
From the figure, i ˆ j
Ie
R
and
j0
e . Substituting these into the cross product
R R R
yields Equation (3.160), where c (3 / 2) for a two-pole machine [87, p.68]. It is important to
note that the derivation assumes a magnitude-invariant SV transform.
R R
T c R i sin( 0)
R
c i sin( )
R
3
T ˆ
RIsin(
)
(3.160)
2
It is seen that Equation (3.160) reduces to Equation (3.21) (which was developed in Part I for the
SPE model) for the case of balanced sinusoidal currents. Continuing, if the angle between the q
axis and the current SV is defined as Γ, sin( ) sin( / 2) cos( ) . Then Equation (3.160)
can be written as Equation (3.161).
3
T R iq
(3.161)
2
Ignoring temperature and saturation effects, Equation (3.161) shows that for a given machine
torque is directly proportional to the q-axis component of stator current. (This is evident in
Equation (3.158) as well.) By controlling the current SV to be completely along the q axis the
maximum torque per current is produced. Another way to look at it is to realize that only the qcomponent
of current generates torque. The rotor-oriented FOC discussed in Chapter 5 exploits
this concept.
153