6 ALTERNATING CURRENTS 



referred to as the angle of lag and lead respectively in the two 

 cases. 



The ordinary graphs of the two waves y and y\ are shown in 

 Kg. 4, the full-line curve representing y and the dotted one y\. 



It is, of course, obvious that two sine waves whose equations are 

 7/3 = Y 3 sin (pt + 63) and y = Y 4 sin (pt -f 4 ) have a phase 

 difference 3 4 ; this being the fixed angle between the two cor- 

 responding rotating vectors (of lengths Y 3 and Y 4 ) in the clock 

 diagram. 



Instead of assuming an axis fixed in space, and considering the 

 projection on it of a rotating vector, the direction of rotation of which 

 is counter-clockwise, we may suppose the vector to remain fixed, and 

 consider its projection on a rotating axis, the angular velocity being 

 clockwise and numerically equal to p. It is evident that both these 

 methods yield identical results. 



What we are concerned with in practice when dealing with a 

 number of sine waves of the same frequency is to know (1) the 

 r.m.s. value of each simple sine wave ; and (2) their phase dif- 

 ferences. Hence in constructing a vector diagram it is generally 

 convenient to make the length of each vector correspond to the 

 r.m.s. value of the sine wave, instead of, as in the original mode of 

 representation, to its maximum value. 



3. Relations connecting Amplitude, r.m.s., and 

 Arithmetic Mean Values of Simple Sine 

 Wave 



In order to enable us to pass from the maximum value or ampli- 

 tude of a sine wave to its r.m.s. value, we have to find the relation 

 connecting these two quantities. The r.m.s. value is the square root 

 of the mean value of y* = Y 2 sin 2 pt over a period. Now the mean 

 value of 2/ 2 = Y 2 sin 2 pt may be found by determining the area of the 

 graph of this function over a period, and dividing this area by the 



length of the base-line, i.e. by T. The area is, however, represented 

 n 



by the definite integral] Q Y 2 sin 2 pt . dt, the value of which may be 

 obtained as follows : 



T T /" T 



Y 2 sin 2 pt . dt = ? ? Y 2 (1 - cos 2pt)dt 

 > o J o 



= 1Y 2 | I dt- I cos 2pt . dt\ 



(J ^ 



