CHAPTER IX. 



THE USE OF SINE CURVES IN ALTERNATING-CURRENT PROBLEMS. 

 EFFECT OF HIGHER HARMONICS. 



5O. The representation of alternating currents and E.M.F.s by 

 sine, or simple periodic, curves is frequently objected to on the 

 ground that they do not accurately represent the actual variations 

 of the current or E.M.F., as the case may be, and consequently 

 cannot lead to accurate results. 



Let us examine carefully the value of this objection, and 

 ascertain whether we may expect to obtain true results when we 

 assume that alternating currents and E.M.F.s may be expressed 

 as sine functions of the time. 



It will readily be granted that all alternating currents and 

 E.M.F.S are periodic ; that is, that they are all of such a nature 

 that there is a certain time, T, called the periodic time, in which 

 their values go through a complete cycle of changes, and that in 

 each succeeding time T this cycle is repeated. It is quite correct, 

 then, to represent any alternating current or E.M.F. whatever by 

 some periodic function of the time. 



Now, by a theorem due to Fourier, any periodic function of 

 the time of frequency n may be represented by an expression of 

 the form 



ai sin (pt 0j) + 2 sin (2pt - 2 ) + a 3 sin (3pt - 3 ) -f . . . etc. 



where p = 2-n-n, and a\ t a%, etc., are the amplitudes, and 61, 0%, etc., 

 the phases. 



The frequency of the first term is n t and those of the other 

 terms are respectively 271, 3n, etc. These terms are called the 

 first, second, etc., harmonies of the first term, which is itself 

 called the fundamental term. 



