650 ELECTRICAL MEASUREMENTS 



WAVE ANALYSIS 



Curves taken by the foregoing methods show that in practice 

 both the potential difference and the current waves may depart 

 widely from the sinusoidal form. In any particular case after 

 having obtained the graph of the wave, it is possible to write 

 its equation as the sum of a series of sinusoidal terms of multiple 

 frequencies which have the proper time-phase relations. 



To effect such an harmonic analysis, recourse is had to the 

 work of Fourier who, in 1812, first explicitly showed that a 

 function which is subject to certain mathematical conditions 

 can be represented by a constant term plus the sum of a sine 

 and a cosine series. 8 This result he published in his "Thorie 

 Analytique de Chaleur," 1822. 

 Accordingly 



B (9) 



+ -y + #1 cos + j cos 20 + 3 cos 30 + . . . 



The coefficients are given by the following equations: 8 



A k = - f/(0) sin k6d9 (10) 



Vo 



coskede (ii) 



The expression (9) is called a Fourier series. 



The sine and cosine terms may be combined, for 



A sin + B cos = \A 2 + B 2 sin (0 + tan- 1 T ) *= Csin (0+a'). 



\ A. I 



This gives 



f(0) = ^ + Ci sin (0 + a',) + C z sin (20 + a' 2 ) + 



C 3 sin (30 + a',) + . . , (12) 



If the origin is taken at the zero of the fundamental, which 

 is convenient if the waves are to be plotted, 



f(0) = - + Ci sin + C 2 sin 2(0 + 2 ) + 



C 3 sin 3(0 + a,) + . . . (12a) 

 where 



= 0^2 _ , = aj _ af . etc 



ELECTRICAL LAC03ATORY, 



OF APPLIED SCILNCE. 



