Nonlinear Networks. Iv

It may be well to point out explicitly that the networks considered in this paper and in the preceding papers are definitely not arbitrary nonlinear networks. Rather, it has been the aim to define nonlinear networks which share the uniqueness properties of passive linear networks. Physical considerations suggest that such networks may be aptly termed reliable networks. The networks considered here consist of transformers with ferromagnetic cores (assumed to have negligible hysteresis) and Ohmic resistors, arbitrarily interconnected to a set of generators. It is found that as in the case of linear networks, under given electromotive forces from the generators, the currents flowing are uniquely determined after sufficient time has elapsed. Moreover, if the electromotive force is periodic, a unique, periodic current flow can exist. The proof begins by integrating the network equations with respect to time. It turns out that the integrated equations have the same form as the network equations of a network containing linear capacitors and nonlinear resistors. The differential permeability of a ferromagnetic lies between positive limits; in the integrated equations this is imaged by the condition that the differential resistance lies between positive limits. The results stated are then easily deduced from theorems developed in preceding papers for networks containing such quasi-linear resistors [l, 2].1 (These papers will hereinafter be referred to as I and III.) In most applications of transformers the nonlinearity of the magnetization curve is an undesirable feature; however, the operation of flux gate magnetometers, magnetic voltage regulators, frequency multipliers, and magnetic amplifiers depends essentially on the nonlinearity. Considerable space is devoted to the concepts of a transformer and of a transformer network. In §1 the transformers are restricted to be of a symmetrical type so that their properties can be determined fairly rigorously from elementary considerations. The general transformer is treated in §4 by means of the hypothesis of the magnetic network. The concept of topologically coupled networks employed there will be found quite interesting from an abstract point of view.