Page 205 - Electric Drives and Electromechanical Systems

200 Electric Drives and Electromechanical Systems

s
s
balanced; secondly, the q and d axes are considered to be coincident, then q can be set
to zero. The net effect of this is to simplify the mathematical relationships; therefore, the
speed of any computation is increased.
s
The second transformation required is the transformation from the stationary d q s
axes to the corresponding rotational d q axes. If the d q reference frame is rotating at
the induction motor’s synchronous speed, u_e, relative to the fixed frame (see
Fig. 7.11B), the transformations are given by
s
s
v q ¼ v cos u e t v sin u e t (7.15a)
q
d
s
s
v d ¼ v sin u e t v cos u e t (7.15b)
q d
and the inverse relationship is given by,
s
v ¼ v q cos u e t v d sin u e t (7.16a)
q
s
v ¼ v q sin u e t v d cos u e t (7.16b)
d
s
If v s ¼ V m sin u e t, and v s , v , and v s form a balanced three-phase supply,
a a b c
substituted into Eq. (7.16), will result in v q ¼ V m and V d ¼ 0; hence the supply voltage
within the stationary frame is transformed to a d.c. voltage within the synchronous
rotating reference frame. This approach can be extended to all time-dependent variables
within the motor’s model; this results in a simplified mathematical model for an
induction motor.
7.3.2 Implementation of vector control
The theory of vector control discussed above shows that torque control of an induction
motor can be performed by the effective decoupling of the flux- and torque-producing
components of the stator current (Bose, 2006). It should be noted that, within an in-
duction motor, the rotor currents and the flux cannot normally be directly measured.
The separation of the stator current into flux and torque-producing components can,
however, be undertaken by the use of the transformations and the relationships already
discussed. This decoupling of the orthogonal field and the armature axes permits high-
performance dynamic control, in a similar fashion to the control of d.c. brushed motors.
In order to implement the vector control of an induction motor, information, either
measured or derived, about the position and magnitude of the currents and the fluxes
within the motor is required.
The overview of a vector-controller scheme given in Fig. 7.12 allows the various
transformations that need to be considered to be located. Those to the right of the motor
terminals are within the motor, while those to the left are within the controller and they
must be implemented in real time. The inverter is omitted from this block diagram, but it
can be assumed to give an ideal motor supply current. It follows, therefore, that for an
induction motor to operate under vector control, the values of sin u e t and cos u e t need to
be determined as part of the overall control strategy.