THE INDUCTION MOTOR. 355 



Hence Z = -wal x average value of 6 



17 



and we may substitute for B the expression ^ , 



Thus c = = Tr 3 



Power developed in rotor 



= W = Nc-r= 



But from equation (1) JF = 2 TT s T 



7T 2 S 2 Z 2 2V 



Consequently, 2 TT s T = a - 



W TT <! N Z 2 



.-. rotor torque = J = " ... (4) 



2 TT * 4r 



This expression applies to one N and $ pole only. 

 If the motor has p pairs of poles we must write Z p 

 instead of Z and the torque becomes 



(5) 

 4 r 



By substituting the value for s given in equation (3) 



we have since s = ** " _ 

 p Z 



T - w ^~^ crN p- Z' 2 _ c N p Z ,,,, 



" 



The equations (3) and (6) above show the fundamental 

 relations of the quantities affecting the performance of 

 an induction motor having no magnetic leakage. From 

 (3) we see that the current in the rotor having a given 

 field is directly proportional to the slip. From equation 

 (6) it appears that the torque depends only on the pro- 

 duct of the current and field strength in a given 

 motor, and is directly proportional to the current in the 

 case of a motor having a constant field. 



Both of these relations are exactly analogous to the 

 case of a direct-current motor. In the first case, the 



current in a direct-current motor = ^ * when E 



It 



= voltage applied to the armature and e is the back 



* See the Author's " Practical Dynamo and Motor Testing," p. 106. 



