130 THE MAGNETIC CIRCUIT [ART. 43 



trical degrees, or, what is the same, by 120 electrical degrees; 

 hence, their m.m.fs. mutually cancel at each point along the 

 air-gap. This can be proved by drawing three sine waves dis- 

 placed by 120 degrees and adding their ordinates point by point; 

 or else one can replace each wave by a vector, and show that the 

 sum of the three vectors is zero because they form an equilateral 

 triangle. 



The reasoning given for the two- and three-phase windings can 

 1)0 extended to any number of symmetrical phases, say m, pro- 

 vided that the windings are displaced in space by 360/m electrical 

 degrees, and also provided that the currents in these windings are 

 displaced in time by I/rath of a cycle. The gliding fundamental 

 waves due to each phase which go in one direction are in phase 

 with each other, and, when added, give a wave m times larger 

 than that expressed by eq. (63) ; while the fundamental waves 

 going in the opposite direction are displaced in space by 720/m 

 electrical degrees, and their combined m.m.f. is zero. The direc- 

 tion hi which the resultant m.m.f. travels is from the leading to 

 the lagging phases of the winding. Thus, for any symmetrical 

 77i-phase winding 



....... (64) 



w*here M denotes the amplitude of the fundamental sine wave of 

 the resultant gliding m.m.f., n is the number of turns per pole per 

 phase, and i is the effective value of the current in each phase. 



Prob. 4. It is desired to build a 60 horse-power, 550-volt, 4-pole, 

 Y-connected induction motor, using a stator punching with 4 slots per 

 pole per phase, and a winding pitch of one hundred per cent. The required 

 maximum m.m.f. per pole is estimated at 1550 ampere-turns. What is 

 the total required number of stator turns (for all the phases) if the mag- 

 netizing current must not exceed 25 per cent of the full-load current? 

 The estimated full-load efficiency is 92 per cent, the power factor at 

 full load is about 90 per cent? Ans. Not less than 504. 



Prob. 6. What is the required number of turns in the preceding 

 problem, if the stator winding is to be delta-connected and to have a 

 winding pitch of about 75 per cent? Ans. Not less than 936. 



Prob. 6. What is the amplitude of the first harmonic of the total 

 armaturere action in a 100-kva, 440- volt, 6-pole, two-phase alternator 

 with non-salient poles? The stator has 72 slots; the coils lie in slots 1 

 and 9 ; the number of conductors per slot is C s . In practice, the arma- 

 ture reaction must not exceed a certain limit, and this helps to determine 

 the permissible value of C 8 Ans. 4800C S amp.-turns. 



