138 THE MAGNETIC CIRCUIT [ART. 45 



other. Thus, in a two-phase machine, the 3d, 7th, llth, etc., 

 harmonics travel against the direction of the main m.m.f., while the 

 5th, 9th, 13th, etc., harmonics travel in the same direction as the 

 fundamental m.m.f., though at lower peripheral speeds. 



Applying a similar reasoning to a three-phase winding (Fig. 

 356) we find that the three L n waves travel at a relative distance of 

 $7r(n 1), while the relative distance between the three R n waves is 

 7r(n+ 1) electrical degrees. We thus obtain the following table of 

 the angular distances between the waves due to the three phases : 



Order of the harmonic 1 3 5 7 9 11 13 15 



Distance between the three Ln waves. .. TT |?r -^ V^ K 

 Distance between the three Rn waves. . . |TT x ^n - 2 /7r *n ^n 



The component waves, of any harmonic, which travel at a distance 

 zero from each other, are simply added together, and give a resul- 

 tant wave of three times the amplitude of the component. The 

 three waves which travel at an angular distance of TT or one of 

 its multiples from each other give a sum equal to zero. Thus, in a 

 three-phase machine, the 1st, 7th, 13th, etc., harmonics travel in 

 one direction, while the 5th, llth, 17th, etc., harmonics travel 

 against the direction of the fundamental m.m.f. The higher the 

 order of a harmonic the lower its peripheral speed. The harmonics 

 of the order 3, 9, 15, etc., are entirely absent. 



Prob. 16. What are the amplitudes of the fifth and the seventh 

 harmonics, in percentage of that of the fundamental wave, for a three- 

 phase winding placed in 2 slots per pole per phase, when the winding- 

 pitch is 5/6? Ans. 1.4 and 1.0 per cent respectively. 



Prob. 17. Show that, in order to eliminate the nth harmonic in 

 the m.m.f. wave, the winding-pitch must satisfy this condition ; namely, 

 Y/- = (2q + l)/n, where f is defined in Fig. 16, and q is equal to either 0, 

 1, 2, 3, etc. Hint: Cos %rn must be = 0. 



Prob. 18. Investigate the direction of motion of the various har- 

 monics of the m.m.f. in a symmetrical w-phase system. 



Prob. 19. Show that only the nth harmonic in the m.m.f. wave, 

 due to the nth harmonic in the exciting current, moves synchronously 

 with the fundamental gliding m.m.f., and therefore distorts it perma- 

 nently. 



Prob. 20. A poorly designed 2-phase, 60-cycle induction motor has 

 4 poles, 1 slot per phase per pole, and a winding pitch of 100 per cent. 

 At what sub-synchronous speed is it most likely to stick? Hint: The 

 torque due to any harmonic reverses as the motor passes through the 

 corresponding sub-synchronous speed. Ans. 360 r.p.m. 



