On Electrical Circuits and Switching Circuits

From an abstract point of view, both electric circuits and switching circuits may be considered as weighted nonoriented graphs. There are two main differences between electric and switching circuits from this point of view. The first of these is the algebra to which the weights belong. In electrical network theory the weights belong to the complex number field or to the field of rational functions of a complex variable since they are impedances or admittances. In switching theory they are Boolean functions. The second difference is that one is interested, in electrical network theory, in the circuits or the "loops" in the system, whereas in switching theory one is interested in the paths. This paper seeks to relate the two theories by means of topological considerations. Formulas are derived relating the switching function and the driving point admittance function of a two-terminal network. Certain relations between dual networks are also established. The paper concludes with a synthesis procedure for a type of switching circuit of academic interest, the single contact switching circuit, and statements of some important unsolved problems.