Discussion on “A practical method of harmonic analysis”

Abstract

communicated) : The study of the wave-shapes occurring in alternating-current work is of such great importance, that any attempt to improve the methods of the analysis of such wave-forms is to be welcomed. Mr. Kemp's paper is, in addition, of interest as it emphasizes the fact that any mathematical method of harmonic analysis based upon a limited number of selected ordinates, such as the original Perry method, or its simplification by Runge, or the adaptation of Runge's method to the" case of odd-harmonics only, as set forward by Thompson, does not give correct results under certain conditions. In some respects, however, the paper is a little disappointing. To obtain the various multiplying factors, two sets of nine simultaneous equations, containing the various factors, are used involving, as one may well imagine,-" considerable initial labour." This labour would appear to be unnecessary, as the constants are exactly those which would be used in the ordinary Perry method or any of its modifications, namely, one-ninth of sin 5 0 , of sin 15 0 , of sin 25 0 , etc., the ordinates being erected at 5 0 , 15 0 , 25 0 , etc. In fact, the chief point of difference that is apparent between the method of analysis put forward by the author and the Thompson method, would appear to lie in the fact that the ordinates are erected at these angles, instead of at io°, 20 0 , 30 0 , etc. In certain cases it is true that by basing calculations upon the mid-ordinates more accurate results are obtained than by using the normal method ; but it is equally true that in other cases the use of mid-ordinates will give rise to a greater error than the normal method. It is not possible to enter fully here f into the magnitude of the errors that are likely to arise in practice by the use of Thompson's method of analysis. It may be stated, however, that for the wave-shapes normally met with, the theoretical accuracy.of the method is very much greater than the accuracy of the necessary measurements is likely to be, and that the average error, i.e. averaged over a number of waves of different form, will be much the same whether the mid-ordinate or the normal method is used. Taking as examples the three simple waveforms represented respectively by:(1) fi x = 100 sin p t + 5 sin 35 p t -5 sin 37 p i ;(2) « a = 100 sin p t + 5 sin 35 p t -5 sin 71 p t;(3) e 3 = 100 sin p t + 5 sin 35 p t + 5 sin 71 p t.