SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

can always be written directly in any reference frame. Once again, the fact that the rotor-stator
flux linkage in the rotor frame is simply R could have been recognized from the beginning.
R jr R j r R jr
e Li s e
R e
R R R
Li s
R
R R
Li
(3.148)
s R
Thus, Equations (3.147) and (3.148) are simply Equations (3.138) and (3.139) expressed in the
rotor frame. There is some similarity between Equations (3.138) and (3.147) in that each contains
both a resistance multiplied by a current and the derivative of stator flux linkage. However,
R
transformation into the rotor frame introduced a new term, j R
. Although it is not obvious in
the given form, this term represents a coupling between the d- and q- components of voltage. To
see this, resolve Equations (3.147) and (3.148) into their dq components (in accordance with
Equation 3.106) to yield Equations (3.149) and (3.150).
d
vd Rid d q
dt
d
vq Riq q d
dt
L i
q
Lsiq d s d R
(3.149)
(3.150)
In Equation (3.149) it is clear that the d-component voltage has a q-component of flux linkage
and vice versa. These are speed voltages and they are the coupling mechanism. However,
Equation (3.149) does not follow obviously from Equation (3.147). To see how it was derived,
examine how the imaginary unit is represented in dq form, Equation (3.151), from which
Equation (3.152) follows.
j /2 cos( / 2) sin( / 2) 0 1
j e
sin( / 2) cos( / 2) 1 0
0 j
0
This concept is illustrated in Figure 3.42.
(3.151)
(3.152)
148