The driving-point immittances of infinite cascades of nonlinear three-terminal networks: I-Uniform cascades and characteristic immittances

The characteristic impedance or admittance of a uniform infinite cascade of resistive networks is a concept that has been extant for a long time, but up to now it has only been available for linear networks. This work develops that idea for nonlinear networks. It shows that uniform infinite cascades of certain nonlinear three-terminal networks have characteristic immittances that are in general nonlinear curves in the voltage-current plane. The prototype of such cascades is the uniform ladder network whose shunting conductances and series resistances are nonlinear, continuous, positive for positive arguments, negative for negative arguments, zero at the origin, and monotone in a neighborhood of the origin. Monotonicity elsewhere is not required. The resulting characteristic immittances can vary quite radically and can even be multivalued functions of both voltage and current. On the other hand, an example is also given of a nonlinear ladder network whose characteristic immittance is perfectly linear.