126 THE MAGNETIC CIRCUIT [ART. 42 



28.) For the same reason, the value of the winding-pitch factor, 

 k w , deduced in Art. 29, holds for the m.m.fs. as well as for the 

 induced e.m.fs. 



When adding the waves of the higher harmonics due to several 

 coils, one must remember that an angle of a electrical degrees for 

 the fundamental wave is equivalent to 3a electrical degrees for the 

 third harmonic, 5a for the fifth harmonic, etc. Therefore, when 

 using the formula (29) and Fig. 19, different values of a and of per 

 cent pitch must be used for each harmonic, and in this connection 

 the reader is advised to review Art. 30. In the practical problems 

 given below the higher harmonics of the armature m.m.f. are dis- 

 regarded altogether. The results so obtained are in a sufficient 

 agreement with the results of experiments to warrant the great 

 simplification so achieved. For the completeness of the treatment, 

 and as an application of the general method, an analysis of the 

 effect of the higher harmonics of an m.m.f. is given in Art. 45 

 below. However, this article may be omitted, if desired, without 

 impairing the continuity of the treatment in the rest of the book. 



Resolution of a Pulsating m.m.f. into Two Gliding m.m.fs. The 

 reader is aware from elementary study that the pulsating m.m.fs. 

 produced by two or three phases combine into one gliding (revolv- 

 ing) m.m.f. in the air-gap. It is therefore convenient to consider 

 even a single-phase pulsating m.m.f. as a combination of m.m.fs. 

 gliding along the air-gap in opposite directions. In this wise, the 

 m.m.fs. due to different phases are later combined in a simple 

 manner. This method of treatment is similar to that used in 

 mechanics, when an oscillatory motion is resolved into two rotary 

 motions in opposite directions. Also in the analysis of polarized 

 light a similar method of treatment is used. 



Take the first harmonic of the m.m.f. (Fig. 34) and assume the 

 current in the exciting coil to vary with the time according to the 

 sine law; then the amplitude of the m.m.f. wave also varies with 

 the time according to the sine law. Imagine two m.m.f. waves, of 

 half the maximum amplitude of the pulsating wave, gliding uni- 

 formly along the air-gap in opposite directions; the superposition 

 of these waves gives the original pulsating wave. One can see 

 this by drawing such waves on two pieces of transparent paper and 

 placing them in various positions over a sketch showing the pul- 

 sating wave. It will be found that the sum of the corresponding 

 ordinates of the revolving waves gives the ordinate of the pulsating 



