Handbook of Electrical Engineering

104 HANDBOOK OF ELECTRICAL ENGINEERING

A reasonable and practical approximation can be made forZ 2 m, which is that the magnitudes
ofRcandXmare each much greater than the magnitude ofR 2 andX 2. (For a more precise analysis
see Reference 1, Chapter 12.) HenceZ 2 mreduces to:-

Z 2 m=

R 2

s

+jX 2

And soVmbecomes:-

Vm=

Vs(R 2 +jsX 2 )

sR 1 +R 2 +jsX 12

whereX 12 =X 1 +X 2

And soVm^2 becomes:-

Vm^2 =

Vs^2 (R 22 +js^2 X 22 )

(sR 1 +R 2 )^2 +s^2 X 122

Hence the torque becomes:-

Te=

sR 2 Vs^2

(sR 1 +R 2 )^2 +s^2 X 122

( 5. 1 )

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 torqueTmax.

a) The starting condition.

When the slipsequals unity the starting torqueT 1 can be found from equation (5.1) as:

T 1 =

R 2 Vs^2

R 122 +X 122

( 5. 2 )

Where,

R 12 =R 1 +R 2

The starting torque is very dependent uponR 2 because for typical parameters the total

reactanceX 12 is significantly larger than the total resistanceR 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 torqueT 0 can be approximated as:-

T 0 ≈

sVs^2

R 2

( 5. 3 )