Kron’s method of subspaces
Introduction. Gabriel Kron has introduced new and powerful methods of applying tensor analysis to complicated engineering problems, presenting his major contributions in the field of electrical engineering. The manner of presentation, and the rarity of a simultaneous knowledge of the hitherto almost unrelated subjects of electrical engineering and tensor analysis, have unfortunately served to limit his audience. Recent experimental confirmation of some of his investigations dealing with equivalent circuits, however, has attracted the serious attention of a wider engineering following. In view of the growing importance of the whole subject, and of the controversy which has surrounded it, it has seemed desirable to present some particular aspect of Kron's work in a form which may appeal to a less highly specialized audience. To avoid complications as far as possible, the present paper must ignore such important topics as electrical networks, electrical machines, and equivalent circuits. It confines itself to purely dynamical problems, and to that particular idea of Kron's which may be called the method of subs paces.] Much discussion has arisen over Kron's claim that he uses tensor analysis. It is the considered opinion of the present writer that Kron does indeed make a full and proper use of tensor analysis. Possibly the belief that Kron employs only matrices may have arisen from the fact that, in order to present his actual mathematical procedure in a form that may be understood and used by those not familiar with the intricacies of the tensor calculus, he often presents this procedure in matrix form. However, he is always careful to point out that the underlying concepts are wholly tensorial in character. In the present paper the method of subspaces will first be presented in terms of a simple example, and in purely matrix form, merely as a set of rules of procedure, the essentially tensorial significance of the procedure being discussed only after the actual procedure has been brought before the reader. The theoretical discussion will then be followed by three simple, related examples illustrative of various aspects of the method. Scope of the method of subspaces. Let us consider a system, which may be dynamical, electrodynamical, or otherwise, containing several standard parts, such as a fly-wheel, a governor, a pair of synchronous machines, a system of levers, etc. The equations of performance of the individual parts are usually well known, but the equations of performance of the complex whole will depend on the manner in which they are interconnected. Usually it is extremely difficult to trace out the full influence of each interconnection in setting up the equations of performance. The method