112 



ELECTRICAL ENGINEERING 



The radius of the circle is the maximum value of the sine 

 function and is called its amplitude. If the radius vector is 

 rotated at uniform angular velocity w in the counter-clockwise 

 direction, its vertical projection at any time t is 



y = R sin co = R sin 6, 



where time is measured from the horizontal. Plotting the values 

 of y on a base of angle 9 for a complete revolution of the radius 

 vector gives the cycle shown. 

 To represent the sine function 



e = E sin 0, 



a circle of radius E Q is taken and the vertical projections of the 

 revolving vector are plotted on base of angle 0. (Fig. 80 (a).) 



EC, 



lo 



(6) 

 FIG. 80. Vector representation of harmonic functions. 



To represent a sine function 



i = 7 sin (0 <), 



a circle of radius 7 is taken, but since i does not pass through 

 zero until angle = <j>, the sine wave is displaced to the right by 

 angle <f> as shown and the e.m.f. wave e = E Q sin leads the cur- 

 rent wave i = 7 sin (0 <) by angle <f>. Instead of using com- 

 plete circles to represent sine waves, their radii E and 7 can be 

 used as in Fig. 80 (6) or since alternating quantities are repre- 

 sented by their effective values instead of their maximum values 

 the two vectors OE and 07, Fig. 80 (c), are used to represent the 



two sine waves in Fig. 80 (a). OE 



pi 



E = ^ is the effective value 



V2 



