On methods for the construction of networks dual to non-planar networks

Abstract

Two networks are said to be dual to each other if the following correspondence exists between them: - Each element of one network has a counterpart in the other one. The currents through the elements in one network and the voltages across their counterparts in the other network are proportional to each other. A network dual to a given one is usually constructed by connecting elements of the appropriate kind in a circuit the diagram of which forms the topological counterpart to the diagram of the original circuit. If this original circuit is a non-planar one (i.e. if it cannot be drawn without crossings between some of its branches), then there exists no topological dual, and for this reason it has been said that there exists no dual to a non-planar electrical network. However, if we are interested in an electrical network not because its circuit diagram has certain geometrical properties, but because it constitutes an assembly of a number of passive and active elements which perform in a certain manner (on account of their impedance properties and of certain relations of mutual constraint imposed by Kirchhoff's mesh and junction relations), then there is really no restriction of this kind. We need only convert the original network into another with the same elements and the same performance, but with a planar circuit diagram and with additional constraints implemented by ideal transformers. This equivalent network can be then converted into its dual counterpart, which is another planar network subject to the dual counterparts of the transformer constraints of the first case. In this way we arrive at an assembly which performs in the required dual manner. Introducing thus the concept of physical duality as opposed to geometrical duality, the paper explains in an elementary manner a number of ways in which this conversion into equivalent planar networks can be achieved. These methods are: - A method originally published by Julia. The dual counterpart of this method. The method of fictitious junction points (an electrical analogue to F. Schur's method of fictitious junction points in pin-jointed frameworks). The method of the transferred terminal. The general method of separation. Some of these methods require a smaller number of ideal transformers than Julia's original method. The paper ends with a short discussion of the errors introduced when the dual network is set up on a network analyser and when the ideal transformers have to be replaced by real ones; the numerous methods available for the realization of an equivalent network enable a choice to be made so as to minimize these errors, or possibly eliminate them altogether. The existence of a dual to a non-planar network is of some interest in connection with electro-mechanical analogies as it removes a restriction to which the direct analogy would otherwise be subjected.