Page 122 - Handbook of Electrical Engineering

104 HANDBOOK OF ELECTRICAL ENGINEERING

A reasonable and practical approximation can be made for Z 2m , which is that the magnitudes
of R c and X m are each much greater than the magnitude of R 2 and X 2 . (For a more precise analysis
see Reference 1, Chapter 12.) Hence Z 2m reduces to:-

R 2
Z 2m = + jX 2
s
And so V m becomes:-

V s (R 2 + jsX 2 )
V m = where X 12 = X 1 + X 2
sR 1 + R 2 + jsX 12
2
And so V becomes:-
m
2
2
2
2
V s (R 2 + js X 2 )
2
V m =
2 2 2
(sR 1 + R 2 ) + s X 12
Hence the torque becomes:-
2
sR 2 V s
T e = (5.1)
2 2 2
(sR 1 + R 2 ) + s X 12
There are three important conditions to consider from the torque equation:
a) The starting condition in which the slip is unity.
b) The full-load condition in which the slip is small, i.e. 0.005 to 0.05 per-unit.
c) The value and location of the maximum torque T max .
a) The starting condition.
When the slip s equals unity the starting torque T 1 can be found from equation (5.1) as:

2
R 2 V s
T 1 = (5.2)
2 2
R 12 + X 12
Where,
R 12 = R 1 + R 2
The starting torque is very dependent upon R 2 because for typical parameters the total
reactance X 12 is significantly larger than the total resistance R 12 . During the starting process the
denominator remains fairly constant until the slip approaches a value that creates the maximum
torque, which is typically a value between 0.05 and 0.2 per-unit, as seen in Figures 5.4 and 5.5
for two ratings of low voltage motors. The higher value of slip generally applies to the lower kW
rated motors.
b) The full-load condition
Full-load is obtained when the slip is typically in the range 0.005 to 0.05 per-unit. The higher
values apply to the lower kW rated motors. The full-load torque T 0 can be approximated as:-

2
sV s
T 0 ≈ (5.3)
R 2