652 ELECTRICAL MEASUREMENTS 



is prohibitive, but machines called harmonic analyzers have IHMMI 

 devised, some of which practically effect the determination of 

 Ak and B k in this manner. 



The various coefficients may be determined arithmetically as 

 follows: A complete cycle of the curve to be analyzed is plotted. 

 The base, (27r), is then divided into 2n equal spaces, and 

 ordinates erected. Let the measured values of the 2n ordinates 

 t> e 2/0, 2/1, 2/2, ... 2/2n-i. The distance between consecutive 



ordinates is A0 = ~- = - 

 2n n 



By (10a), 



2 



A k = ~ [2/o sin k A0 + yi sin 1&A0 + 1/2 sin 2A0 + . . . + 



2/2n-z sin (2n - 2) fcA0 + 2/2n-i sin (2?i - 1) 



(13) 



If the data are furnished by the contact method, the measure- 

 ments being made at equal intervals along the wave, the values 

 of i/o, 7/1, etc., are given directly and the curve need not be plotted. 

 Equation (13) may be written 



m-2n 1 



> sin km M 



where m is the number of the ordinate concerned in the multiplica- 

 tion; similarly 



B k = - [t/o cos Qkde + 2/1 cos fcA0 + ?y 2 cos 2A-A0 + . . . + 



,_ 2 cos (2n - 2)fcA0 + y Zn -i cos (2n - 1) fcA0] (15) 



- m = 2n-l 



y m cos fcmA0 (16) 



Runge Method of Grouping Terms. Theoretically it is easy 

 to calculate the coefficients. The practical difficulty lies in the 

 great expenditure of time which is necessary for carrying out 

 the process as indicated. For this reason Runge 9 has intro- 

 duced an abridged method of calculation which is carried out by 

 aid of a systematically arranged schedule. 



In the majority of cases, the two halves of an alternating-cur- 

 rent wave are the same except for the algebraic sign; that is, 



