288 



ELEMENTS OF ELECTRICAL ENGINEERING. 



end ring from the rods of one band. Let one side of the square, Fig. 248, represent 

 the current entering an end-ring from one band of rotor rods, and let each side of 

 the square be divided into n equal parts ( Fig. 248 is constructed for n = 6), then the 

 lines a, b, c, d, e, etc., drawn from the center of the square to the equidistant points 

 on the sides of the square represent the values of current in the portions of the end- 

 rings between the points of attachment of the ends of the rods, and the average value 

 of the square of the currents , b, c y d, <?, etc., is represented by the square of the 

 radius of a circle (= (2/ // ) 2 /7r) having the same area as the square in Fig. 248. Let 

 r be the resistance of one of the portions of the end-rings from center to center of 

 adjacent rods, then rX (2/") 2 /7r is the average power lost in each portion, and, 



Fig. 248. 



Fig. 249. 



since there are 2Z // portions, counting both end-rings, the total power lost in both 

 end-rings is 2Z"r. (2nf") 2 /n. 



For the case of a three-phase stator -winding. As before let n be the number of 

 rods in each of the above mentioned bands (= Z"/3/), then nl" is the total current 

 entering an end-ring from the rods of one band. Let one side of a regular hexagon, 

 Fig. 249, represent the current nP f entering an end-ring from a band of rotor rods, 

 and let each side of the hexagon be divided into n equal parts (Fig. 249 is constructed 

 forw=4), then the lines, a, b, c, d, etc., drawn from the center of the hexa- 

 gon to the equidistant points on the periphery represent the values of current in the 

 portions of the end-rings between points of attachment of the ends of the rods, and 

 the average value of the square of the currents, a, b, c t d, etc., is represented by 

 the square of the radius of a circle [ 3V / 3(/ // ) 2 /(27r)] having the same area as the 

 hexagon. Therefore the total power lost in both end-rings is 2.Z"r sV^/2Tr (/")*. 



138, The algebraic formulation of the problem of the induction 

 motor. Figure 250 represents a combination of simple circuits 

 which is equivalent to a given induction motor running at slip s, 

 and Fig. 251 represents a combination of simple circuits which is 

 approximately equivalent (magnetizing current assumed to be in- 



