The Driving-Point Immittances of Infinite Cascades of Nonlinear Resistive Two Ports

Abstract

Infinite cascades of nonlinear resistive two ports can have characteristic immittances in the uniform-cascade case and driving-point immittances in the nonuniform-cascade case at each of their ports. By definition any such immittance is a curve specifying the voltage-current pairs at a given port corresponding to a polarity convention for power flowing to the right (or left) under the condition that the power injected into the cascade from infinity at the right (respectively, left) is zero. Our result is shown by imposing certain monotonicity, continuity, Lipschitz, and boundedness conditions on the lattice parameters of the two ports. They ensure that the equivalent lattice networks of the two ports do not balance as wheatstone bridges. In this case, the forms of the their trajectories of consecutive port voltage-current pairs can be quite completely described. However, when balancing does occur, those trajectories can behave in peculiar ways. For example, an infinite uniform cascade can be totally unenergized in a section extending infinitely to the left and totally energized in its other part. Also, trajectories can intersect; in fact, an infinity of different trajectories can meet at a nonequilibrium point. Furthermore, there can be certain portions of the voltage-current plane that cannot be penetrated by any trajectory.