EQUIVALENT SINE CURVES. 81 



F" 2 and /, To give a geometrical interpretation, ^ is the phase- 

 difference between two simple periodic functions whose E.M.S. 

 values are F" 2 and /. 



We call an equivalent sine curve one which has the 

 same E.M.S. value as a given periodic curve which is not itself 

 necessarily a simple sine curve. 



54. It follows that, so long as the E.M.F.s and currents are 

 such that each half-wave is identical with the preceding one, 

 except in sign, as is usually the case, their K.M.S. values can be 

 compounded in a vector fashion. 



If we are dealing with a circuit whose self-induction and 

 resistance are constant, the diagram of E.M.F.s is shown in Fig. 32. 



'L P I 



FU 



FIG. 32. 



E is the E.M.S. value of the impressed P.D. The E.M.F. neces- 

 sary to overcome the resistance is RI, and that required to balance 

 the self-induction is pLI, where p = 2?r times the frequency, L is 

 the self-induction, and / the E.M.S. value of the current. 



If there is iron in the immediate neighbourhood, the per- 

 meability depends upon the current, and the self-induction of the 

 circuit is no longer a constant quantity. 



In consequence of this the E.M.F. curve becomes distorted, 

 and differs in shape from the current curve. 



It should be noticed that Fig. 32 holds good either for maxima 

 or E.M.S. values of E.M.F. and current, when the self-induction of 

 the circuit is constant, and the impressed P.D. is a sine function of 

 the time. 



We have seen that the power given to an inductive circuit may 

 be written in the form of 



P = VI cos 



where V and / are E.M.S. values, and cos is a multiplying 

 factor less than unity. This shows that in cases where only E.M.S. 

 values are concerned, the effects are just the same as if the actual 

 P.D. and current were replaced by a sine P.D. and current having 



G 



