348 ELECTRICAL ENGINEERING 



tion motor is sf, where / is the frequency of the supply and s is the 

 slip. Thus the frequency impressed on the stator of the second 

 motor is the frequency of slip of the first motor. The speed of 

 the two motors is always the same and thus at no load (1 s)f = 

 sf and s = 0.5. 



Therefore, two similar motors connected in cascade tend to 

 approach a speed of half synchronous speed at no load and fall 

 below this speed under load. 



Speeds below half synchronous speed are obtained by inserting 

 resistance in the rotor windings of the second motor. 



For speeds above half synchronous speed the stator of the second 

 motor must be connected to the line and the rotor of the first 

 motor closed through resistances. 



This method of control is used for some three-phase traction 

 systems and is very similar to the series-parallel control of direct- 

 current series motors. The induction motor, however, does not 

 tend to increase its speed indefinitely and if it operates above 

 synchronous speed it acts as a brake and pumps back power into 

 the lines. 



221. Analysis by Rectangular Coordinates. Using rectan- 

 gular coordinates the performance characteristics of an induc- 

 tion motor can be determined if the constants of the motor are 

 known. 



Let 



y g jb = stator exciting admittance per phase, meas- 

 ured with the rotor circuits open so that the friction losses are 

 not included. The rotor must be driven at synchronous speed. 



?i = FI H~ j%i = stator impedance per phase. 



?2 = ?*2 + jsx 2 = rotor impedance per phase at slip s. 



Assume that the ratio of turns is HI : HZ = 1 : 1 and take as real 

 axis of coordinates the e.m.f. generated in the stator by the flux 

 of mutual inductance. The quantities used refer to one phase of 

 the stator and the corresponding phase of the rotor. 



E' = e.m.f. generated in the stator. 

 E 2 = E f = e.m.f. generated in the rotor at standstill. 

 sE 2 = e.m.f. generated in the rotor at slip s. 



These three e.m.f. 's are written as absolute values without the 

 dot since they lie along the axis. 



