SINE WAVES 3 



2. Simple Sine Waves. Vector Diagrams 



In some cases (as, e.g., in incandescent lighting, or electric furnace 

 work) the results obtained by using an alternating current of given 

 r.m.s. value are entirely independent of its wave-form; so long as 

 the r.ra.s. value is unaltered, we may use a current of any wave-form 

 we please without in any way affecting the results. In other cases, 

 however, the effects produced will, for a given r.m.s. value, depend, 

 to a greater or less extent, on the wave-form (as, e.g., in connection 

 with motors, transformers, arc lights, and especially in cases where 

 capacity is present, as in concentric cables). 



Now in order to simplify the theoretical treatment of the subject 

 as much as possible, it is an obvious advantage to select a wave-form 

 which shall lend itself readily to mathematical treatment; and of 

 all possible wave-forms, the simplest and easiest to deal with is that 

 known as a simple harmonic or sine wave. The equation to such a 

 wave is 



y = Y sin pt * 

 and its graph is shown in Fig. 2. 



In the classical theory of alternating currents, it is usual to 

 assume that the waves dealt with are simple sine waves. Since 

 there are, as mentioned above, cases in which the results obtained 

 are independent of the wave-form, the assumption of the simplest 

 wave-form in such cases is perfectly justifiable, and leads to correct 

 results while considerably simplifying the treatment. Unfortunately, 

 there are other cases in which the assumption of sine waves is no 

 longer admissible, and leads to results more or less erroneous, and 

 at times entirely misleading. 



For this reason, the use of sine waves has been severely criticized 

 by some writers. But there is nothing else that can be usefully 

 substituted for them, and since in a large number of problems they 

 yield results sufficiently accurate for practical purposes, their use is 

 certainly justifiable. It must be remembered, however, that results 

 deduced on the sine-wave hypothesis must be used with due caution 

 when applied to certain problems. 



There is, however, a still further justification for the assumption 

 of sine waves in alternating- current theory, and this is due to the 

 gradual recognition of the fact that for most practical purposes also 

 the sine wave is a desideratum. Especially is this the case in con- 

 nection with long-distance power transmission. Successful attempts 

 have recently been made to construct alternators capable of giving 

 pure sine waves of e.m.f. ; and although many of the alternators at 

 present in use give somewhat irregular wave-forms, it seems highly 



* The angle pt, where t is the time, is expressed in radiant. 



