Page 678 - Mechatronics with Experiments

664 MECHATRONICS
where N is the number of turns in the primary coil. The rotor sees a magnetic flux frequency
1
of w = w syn − w . The induced output voltage as a result of the Faraday’s law of induction
rm
s
is
dΦ
v (t) = N (8.243)
r 2
dt
N V d
s
2
=− [cos(w − w )t] (8.244)
rm
e
N w dt
1
e
N (w − w )
2
rm
e
= V sin(w t) (8.245)
s
s
N w
1 e
The operating principle of an AC induction motor in terms of cause and effect
relationship is illustrated below. The stator current is a result of the applied stator voltage
(v (t)), and induced rotor current is a result of the induced rotor voltage (v (t)).
s r
V (w syn ) ⟹ i (w syn ) ⟹ B (w syn ) (8.246)
s
s
s
w ⟹ V (t) induced ⟹ i (t) induced ⟹ B (w syn ) (8.247)
s
r
r
r
B (w syn ) & B (w syn ) ⟹ T (torque) (8.248)
r
m
s
Notice that B and B rotate with the synchronous speed, w syn . The rotor mechanically rotates
s
r
close to the synchronous speed with the difference being the slip speed, w = w syn − w .
rm
s
When the mechanical speed of the rotor is smaller than the synchronous speed, torque is
generated by the motor (motoring action). Whereas if the mechanical speed of the rotor
is larger than the synchronous speed, torque is consumed by the motor (it is in generating
mode). In the vicinity of the rotor mechanical speed close to synchronous speed, the
torque is proportional to the slip speed. When the slip is zero, the generated torque is zero
(Figure 8.41).
The steady-state torque-speed characteristics of an AC induction motor can be sum-
marized as follows:
1. In the vicinity of a small slip, the torque is proportional to the slip frequency for
a given stator excitation. If the rotor speed is smaller than the synchronous speed
(slip is positive), then the torque is positive. The motor is in motoring mode.Ifthe
rotor speed is larger than the synchronous speed (slip is negative), then the torque is
negative. The motor is in generator mode.
2. At a certain value of slip, the torque reaches its maximum value. For slip frequencies
larger than that, the inductance of the rotor becomes significant and the current is
limited at the higher slip. As a result, torque drops after a certain magnitude of slip
frequency.
3. The shape of the steady-state torque-speed curve can be modified for different appli-
cations by modifying the rotor conductor shape and stator winding distribution.
The torque generation in an AC induction motor can be viewed as the result of the
interaction between the magnetic flux distributions of the stator and the rotor. From basic
reasoning regarding the interaction of magnets and magnetic fields (Figure 8.40), it can be
concluded that
T = K ⋅ B ⋅ B ⋅ sin( ) (8.249)
rs
m
m
r
s
which states that the torque is proportional to the size and design parameters of the motor
(K ), the flux density in rotor and stator, and the angle between the two flux vectors. The
m
so-called vector control algorithm discussed later in this chapter attempts to maintain a