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Chapter 7 Induction motors 193

R s ¼ 0.43U, X s ¼ 0.51U, R m ¼ 150 U, X s ¼ 31U, R r ¼ 0:38U and X r ¼ 0:98U: The supply
voltage is 380 V line-to-line.
Starting torque
At zero rotor speed, hence using Eq. (7.6)
2
3I R r p
r
T e ¼
su e
where the slip at standstill is given by
u e u r
s ¼ ¼ 1
u r
u e ¼ 2pf e ¼ 158 rad s 1
Hence
T s ¼ 121:5 Nm

Breakdown slip and torque
The value of the breakdown slip can be determined by using Eq. (7.9),
R r
R r
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:245
s b ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ q
2 2 2 2 2
R þ u ðL s þ L r Þ R þðX s þ X r Þ
s
e
s
A slip of 0.245 equates to a speed of 1132 rev min 1 and 1868 rev min 1 as shown in
Fig. (7.4).
From the slip values, the torque can be calculated; giving 232.0 Nm at a slip of þ0.245 and
-410.1 Nm at a slip of -0.245.
nnn
7.2 Scalar control
A wide range of induction-motor-speed-control strategies exist, including voltage
control, voltage and current-fed variable-frequency inverters, cycloconverters, and
slip-energy recovery. However, within the application areas being considered, the use of
voltage- and current-fed inverters predominates; additional speed-control systems are
widely discussed in the literature (Bose, 2006; Sen, 1989).
The torque-speed curve of an induction motor can be modified by using a variable-
voltage supply, where the motor’s supply voltage is controlled either by a variable
transformer or by a phase-controlled anti-parallel converter in each supply line
(Crowder and Smith, 1979). By examination of Eq. (7.9), it can be seen that the slip at
which breakdown occurs is not dependent on the supply voltage. Only the magnitude of
the torque is affected, and this results in the family of curves which are shown in Fig. 7.5.
When the load’s torque-speed characteristic is also plotted on the same axes, the
characteristics of speed control under voltage control can be seen. This form of control is