424 ANALYSIS OF CURVES. 



consider the third and fifth harmonic in addition to the 

 fundamental. This will, however, be largely governed by 

 the number of slots per phase in which the armature is 

 wound, the shape of the poles, &c. 



Example of Curve Analysis. The following examples of 

 the analyses of a curve obtained experimentally will 

 enable the student to analyse for himself any curve which 

 he obtains, even if he is not able to follow the mathematics 

 on which the methods are based. He may omit the 

 mathematical proof if he is not able to follow it. 



Outline of Mathematical Proof of Method. Denoting the 

 variable quantity by e, 



Let e = A + A l sin + A^ sin 2 6 + A 3 sin 3 + . . . 

 + B, cos + B 2 cos 2 8 + B 3 cos 3 6 + . (1) 

 Multiply both sides of the equation by sin 0, then 

 e sin 6 = A sin + A l sin 2 6 + A., sin 2 Q sin 



+ A 3 sin 3 sin 6 + ... 



+ BI cos sin + 2 cos 2 sin + B 3 cos 3 sin + . (2) 

 Integrating both sides of this expression between the 

 limits and 2?r , we have : 



2ir 2ir 



| e sin d d = j A, sin 2 6 d . . . (3) 

 o o 



the other terms all vanishing with the limits chosen. 

 Hence, mean value of e sin 6 mean value of A l sin 2 6 



i.e., A! = 2 x mean value of e sin . . . (4) 



Similarly by multiplying equation (1) by 



sin 2 0, sin 3 6 . . .1 , , f/i 2 , A 3 



cos 6, cos 2 0, cos 3 6] tlie values of \ B* B*, B 3 . . 



may be determined. 



Thus A. i = 2 x mean value of e sin 2 0, 

 B l = 2 x mean value of e cos 0. 

 By = 2 x mean value of e cos 2 0, 

 And so on. 



Since all symmetrical wave forms will consist of odd 

 or even terms only, only half of the constants need to be 

 determined. In any ordinary alternating circuit the wave 

 form will contain odd harmonics only. 



