360 THE INDUCTION MOTOR. 



exactly the same way as the average value of the power 

 in a circuit (c e cos 0) was determined (seepage 95). 



It is thus seen that the effect of the rotor leakage is two- 

 fold, ( 1 ) the value of the current in the rotor conductors 



AC BC 



instead of being-^ (see Fig. 1 79) is only ^ , i.e. the current is 



/ 1 /i 



D/^ 



reduced by the factor -r^ = cos </>. (2) Owing to the 



phase of the current not being the same as that of the 

 main field, the effectiveness of the current is further 

 diminished by the factor cos <f>. 



Hence the torque is reduced in amount to cos 2 </> x 

 (value of torque without leakage). 



The calculation previously given for the value of the 

 torque on the shaft may be employed for a motor with 

 leakage, the final expression being multiplied by the 

 factor cos' 2 <. 



Thus for a motor with leakage we must write the 

 equations (3) and (6) for current and torque given pre- 

 viously in the following form : 



_ 7T S p Z .Q. 



' VTr^ 

 T = -" |. Z cos' * - (9) 



Referring to Fig. 179, we may write, 



1 



cos 2 (f> ^= 1 = \ /leakage volts V* 



1 + tan 2 useful volts ' 



The value of the reactance voltage due to leakage is 

 2irsL 2 C, where L 2 is the coefficient of self-induction of 

 the rotor circuit and s is the slip per second, which is 

 also the frequency of the currents induced in the rotor. 



.. : f - Thus the value of tan <f> is s ^-? 



and we may write this in the form * 



tan </> = s k, where k = - 



JK 



and cos - < = T -7^ 



Thus in the actual motor- with leakage the torque is 

 T = cn Z 



