Drive system control - Propulsion Systems for Hybrid Vehicles - page 374

machine currents) to the controller synchronous frame where sinusoidal variables
become dc quantities is [ R ] and its inverse for the backward transformation:

cos ð q Þ sin ð q Þ

½ R ¼

sin ð q Þ cos ð q Þ

ð 7 : 9 Þ

cos ð q Þ sin ð q Þ
sin ð q Þ

½ R 1 ¼

cos ð q Þ

where it is apparent from the trigonometric identity that [ R ][ R ] 1 = I . In an FOC
system the transformation of sensed machine currents in the stationary reference
frame to synchronous reference frame d-q variables is described by (7.10):

I qds ¼½ R ½ T I abc
I abc ¼½ T 1

ð 7 : 10 Þ

½ R 1 I qds

In a closed loop system the transformation pair given by (7.10) forces the sensed
currents I abc to equal the reference currents I abc . The controller manipulates the d - q
variables according to the hybrid propulsion system control law such that the
operator commanded torque is delivered by the M/G. So, if the transformations
discussed were applied to a balanced set of 3-phase sinusoidal currents, the result of
(7.10) would be that same set of currents. Even so, it is important to an understanding
of FOC that a brief illustration of the transformation given by the top expression in
(7.10) be carried out. Suppose that the induction machine stator currents, I abc ,are
sinusoidal with magnitude I m =10A pk at an electrical frequency w e ,thentheappli-
cation of the transformation to the synchronous frame is as shown in Figure 7.6:

I as ¼ I m cos ð w e t f Þ
I bs ¼ I m cos ½ w e t ð 2p = 3 Þ f
I cs ¼ I m cos ½ w e t ð 4p = 3 Þ f

ð 7 : 11 Þ

Had the sinusoidal 3-phase currents used in this example been given a phase
shift f > 0 then the synchronous frame currents, I qde , would have a non-zero d -axis
component. When the inverse transforms are applied to the synchronous frame
currents, the original I abc currents are restored at the proper magnitude and fre-
quency. A common error is to swap the vector rotator matrices during the forward
and backward processes. If this is done the synchronous frame currents, instead of
being at some dc value relative to the phase shift of the I abc currents, would be at
twice the electrical frequency and offset.
Next, to illustrate the result of a controller action in the synchronous frame,
suppose that the torque component of current, I qe , is given a step change as illu-
strated in Figure 7.7 that first commands a 50% increase in M/G torque for some
dwell time and then commands a torque reduction to 25% of the original value.
When the torque command given in Figure 7.7 is applied to the FOC controller
in the synchronous frame, it results in step changes in the stationary frame d-q