The Use of Superimposed Imaginary E. M. Fs. Currents, and Fluxes in the Solution of Alternating-Current Problems

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 contaning 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 tthhede manufacture of common ether, sulphuric acid is combined with alcohol and carried through certain operations.