On the dynamics of electrical networks

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Brayton and Moser [I] and Brayton [2] initiated a method of analyzing nonlinear electrical networks using the mixed potential functions. Their approach is interesting in that it is geometric in nature. Smale [3] observed that the dynamics of a class of electrical networks can be viewed, in a natural manner, as flows on nontrivial manifolds. His approach is ineteresting not only from mathematical point of view but from the circuit theoretic point of view, also. Desoer and Wu [4] obtained a result along this direction. In this paper we will discuss a fairly general class of electrical networks including transistors, vacuum tubes, and various other electronic devices. These devices are very important in electrical networks. The mathematical models of these electronic devices contain elements called dependent sources. In order to discuss networks containing dependent sources, one has to relax some of the conditions assumed in [l-3]. W e will regard dependent sources as coupled resistors and give a formula describing the dynamics of the class of networks. We will, then, give conditions under which the dynamics is the gradient of a real valued function with respect to a certain 2-tensor field which comes naturally from capacitors and inductors in a network. It turns out that these conditions are precisely what circuit theorists call reciprocity conditions. Finally, we will discuss a property called forced degeneracy of networks which is left unsolved in [3]. We will give one way of resolving such degeneracy. Preliminary results are reported in [5].

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