Page 124 - Handbook of Electrical Engineering

106 HANDBOOK OF ELECTRICAL ENGINEERING

Hence the torque-slip curve has a steep straight-line section near to the region of zero
slip. Small changes in slip cause large changes in torque. If the value of R 2 is increased to raise
the starting torque then the slope of the full-load straight-line section is reduced, and the speed
regulation for changes in load torque becomes poor. The efficiency at and near full-load also falls
with increasing values of R 2 .
c) The condition for maximum torque.

The maximum torque T max can be found by differentiating the torque with respect to the slip and
equating the derivative to zero. The torque occurs at a particular slip s max , which is found to be:-

R 2
(5.4)
s max =
2 2
R 1 + X 12
The torque T max is found by substituting s max into (5.1):-
2
s max R 2 V s
T max = (5.5)
2
(s max R 1 + R 2 ) + (s max X 12 ) 2
For actual motors chosen for oil company applications the value of s max is very small
when compared with unity. Therefore some approximations can be made. In the denominator the
resistive term can be simplified as:-
2 2
(s max R 1 + R 2 ) R 2
The reactive component approaches zero in value for small values of slip. Therefore the
maximum torque can be expressed as:-
2 2
V s s max V s
or (5.6)
T max ≈
2 2
R 1 + X 12 R 2
In practice R 1 is much smaller than X 12 and so the maximum torque is very dependent upon
the value of the leakage reactances, especially the rotor leakage reactance X 2 . The maximum torque
is also called the ‘breakdown’ torque.
Motors are usually started ‘direct-on-line’ with no series impedance added or starting trans-
formers inserted between the supply and the stator terminals of the motor. The starting current is
therefore high and an associated volt-drop occurs in the feeder cable to the motor. It is normally a
requirement in motor specifications that the motor should start and run up to speed whilst the terminal
voltage is reduced to 80% of its rated value. This is an allowance for the volt-drop in the feeder
cable. The torque produced by the motor varies with the square of the terminal voltage. Consequently
at 80% voltage the torque is reduced to 64% of its value at any slip in the range of zero to unity.
This is shown in Figure 5.4 for a 22 kW motor and in Figure 5.5 for a 200 kW low voltage motor.
For most designs of motors the ability to start at a voltage of 80% is assured if the motor drives
a centrifugal machine. If the voltage falls much lower e.g. 70% or less, then the motor may not
develop enough torque to accelerate the load to its full speed. In practice the motor would accelerate
the load up to some intermediate speed and then remain at that speed. It would draw a high current
and eventually fail from overheating, or be shut down by the protective devices in the switchgear. It
is also important that the motor develops a sufficient minimum torque during the run-up period. This
torque is often called the ‘pull-up’ torque and it must not fall below the load torque at the associated