184 ALTERNATING CURRENTS 



106. Induction Motor replaced by Equivalent 

 Transformer 



Let us now suppose that the motor is running with a slip s, 

 exerting a definite torque, and that while the torque remains unaltered, 

 resistance is introduced into the rotor windings, the slip increasing 

 until ultimately the rotor comes to rest still exerting its original 

 torque. Let r 2 be the resistance of each phase of the rotor winding, 

 and let r be the additional resistance which must be introduced into 

 each phase in order to reduce the rotor to rest. Then the total resist- 

 ance 11 2 of each phase, when the rotor has been reduced to rest, is 

 E 2 = r 2 + r. If n 2 denote the original frequency of the rotor currents, 

 then it is clear that by reducing the rotor to rest we have increased 

 the e.m.f. induced in its windings (by the hypothetical field due to 



M 



the stator currents) in the ratio ; the frequency and reactance of the 



rotor windings have each been increased in the same ratio. It is, 

 therefore, clear that in order that the secondary current and its phase 

 relatively to the induced secondary e.m.f. may remain unaltered, the 

 total resistance of the rotor windings must also be increased in the 



same ratio , i.e. we must have 

 n 2 



P _% - T * 



1*2 - '2 



n% s 

 or 



I - s 

 r = ^-r. 2 (1) 



The primary and secondary currents, their phase relations, and 

 the total power supplied to the primary, remain unaltered. But the 

 rotor is now at rest, and although still exerting its original torque, 

 does not develop any mechanical power. It is evident that the power 

 which formerly corresponded to mechanical power is now employed 

 in producing heat in the additional or external resistance r, and is 

 represented by rI 2 2 per phase, I 2 being the secondary current in each 

 phase. 



We have thus reduced our induction motor to an equivalent three- 

 phase transformer, the total mechanical power developed by the motor 

 corresponding to the power absorbed by the resistances r external to 

 the rotor windings, and the slip s of the motor corresponding, according 

 to equation (1), to 



.->- . (2) 



