AIR-GAP FLUX DISTRIBUTION 295 



If r be the length of the vector, and the angular distance from 

 the reference axis, we may write 



Area of triangle OSP = Y^r X rd8 



and the area included between any given angular limits and 

 a is 



Area OAB (shaded) = 2 



y*. (average value of r 2 ) X (ft a) 



twice area of curve 

 whence average value of r 2 = - -jr 



Applying this rule to the case of a periodically varying e.m.f. 

 or current, we have in Fig. 120 a representation of an e.m.f. 

 wave plotted to polar coordinates. This may be thought of as the 

 actual e.m.f. wave obtained by the graphical method previously 

 outlined, but transferred from rectangular to polar coordinates. 

 The radius vector (moving in a counter-clockwise direction, 

 covers the complete area of one lobe when it has moved through 

 an angle of 180 degrees; because, in this diagram, the electrical 

 degrees are correctly represented by the actual space degrees. 

 The angle moved through during the half period is TT radians, and 

 the virtual value of the alternating e.m.f. is therefore 



(area of one lobe) 



E '-'- 



The area of the curve is easily measured with a planimeten 

 and the value of E thus obtained has merely to be divided by the 

 previously calculated average value in order to obtain the form 

 factor of the irregular wave. 



102. Equivalent Sine-waves. Equivalent sine-waves are a 

 great convenience in power calculations because they permit the 

 use of vectors, and enable us to express the power factor as the 

 cosine of a definite angle. Whenever vector diagrams are used, 

 the alternating quantities must be sine functions of time; and 

 when applied to practical calculations involving irregular (i.e.; 

 non-sine) wave shapes, they must be thought of as representing 

 "equivalent" sine functions. It will of course be understood 

 that, in many cases, the substitution of a sine curve for the actual 

 wave form is not permissible; the effects, for instance, of the 

 higher harmonics on a condensive load cannot be annulled by 



