Abstract :
The solution of some advanced problems on alternating currents leads to rather complicated trigonometric transformations. Let now, in addition to the real sinusoidal currents and voltages, certain imaginary currents and voltages be assumed to exist in the same circuit. These imaginary quantities may be selected of such a magnitude and phase that together with the real quantities they will give simpler mathematical expressions than the real quantities alone. In the final results the real and the imaginary terms can be readily separated, because an imaginary voltage cannot produce a real current, and vice versa. This method is based on some remarkably simple properties of certain mathematical functions which contain a real and an imaginary term, as compared to the properties of similar functions containing real variables only. The following two analogs may make this method clearer. A. In order to make extremely fine platinum wire, a piece of heavier platinum wire is coated with silver and then passed successively through several dies until it is reduced to the smallest practicable size. Then the tubular coating of silver is dissolved in nitric acid. It would not be possible to draw platinum alone to the same size. Here the use of silver is analogous to that of imaginary quantities in alternating currents. Silver is carried along in the operations and separated in the end. B. In the manufacture of common ether, sulphuric acid is combined with alcohol and carried through certain operations. In the end this sulphuric acid is separated and used over and over again. Sulphuric acid in this case may be likened to imaginary currents and voltages which are added to real quantities at the beginning of the problem and separated in the end. The immediate occasion for the writing of this article was a paper by Mr. Gilman referred to under (2) in the article. Had Mr. Gilman used superimposed imaginary currents, several pages of tedious mathematical transformations could be saved. - more general reason for writing this article was a desire to bring this method to the attention of American engineers. It is hardly mentioned in American text-books, while it is quite well-known in England and on the Continent of Europe. As is stated in the introductory paragraph to the article, the method is not new, and the examples quoted are too self-evident to claim any originality. A brief introduction to the method will be found in the author´s “Electric Circuit,” p. 97. The importance of the method lies in the possibility of solving certain electrical problems in a shorter and more direct way, and possibly in making the solution of certain problems feasible which in the ordinary way lead to too complicated expressions. It is a mathematical tool, and as such should find its place among other useful mathematical methods used in the solution of physical and engineering problems.
