ANALYSIS OF CURVES. 421 



The instantaneous value of the harmonic which has a 

 frequency of 2n will be 



e, 2 = E 2 sin (farnt + 2 ) 



Avhere E* is the maximum value of this harmonic, and # 2 is 

 its phase at the same moment as 6 is the phase of the 

 fundamental. 



The instantaneous value of the voltage composed of 

 fundamental and harmonics will consequently be 

 e = E, sin (2* + t ) + 2 sin (krnt + 0,) + E 3 sin (6 irnt + 3 ) 

 +Et sin (%Trnl + 6~} + . . . 



In this equation the constants E 1} E 2 , &c., give the 

 maximum value of the corresponding harmonic. 



We may write sin (%irnt +0) = sin %rnt cos + cos ^nt 

 sin 6, and since Q is a constant term, each of the sine func- 

 tions in the equation may be written as the sum of a sine 

 and a cosine term, and the equation then takes the form 

 e - a x sin 27rwf + a.> sin ^nt + a 3 sin Qirnt + . . . 

 + 6j cos %-irnt + 6 2 cos 4:Trnt + b 3 cos font + . . . 

 where a,, a 2 , a s , b l} b. 2 , b 3 are constants. 



This form of the equation is sometimes more con- 

 venient. v 



Harmonics of a Symmetrical Curve. The wave forms pro- 

 duced by the electromotive force of an alternator ars 



FIG. 200. THIRD HARMONIC IN OPPOSITION, WAVE PEAKED. 



always symmetrical about the horizontal axis, i.e., the 

 negative half waves are exactly similar to the positive 

 half waves. This follows from the fact that the north 

 and south poles of an alternator are similar, and therefore 

 generate similar electromotive forces. From this follows 

 the important fact that the wave forms met with in an 

 alternating circuit contain only " odd harmonics " that 

 is to say, that the periodicity of any harmonic is 1, 3, 



