Impedance Transformations of Cascaded Active or Passive Identical Twoports

The number of iterative impedances for any active or passive twoport is one, two or infinite, the first and last cases being of only limited significance and application. For the second case, a previous article determined which of the two impedances (called stable) was ultimately approached by the sequence of input impedances to cascades of identical networks (active or passive) with ever-increasing numbers of sections. It was also noted that this convergence was in general nonmonotonic. This article provides a solution to the problem of determining the impedance values which, when used as terminations for such a cascade, will generate a sequence of input impedances which steadily approaches the stable iterative impedance. To facilitate the proofs involved, it also presents a resume of the relatively obscure geometric interpretation, the isometric circle method of construction, and the classification of the underlying bilinear transformations. The theorems presented could have important application in the estimation of error in approximation problems involving artificial lines and filters working into varying terminating impedances and even, perhaps, over wide frequency ranges. Secondary results include 1) the passive iterative impedance for a cascade of passive networks is the stable one when the other is active, and 2) conditions on, and examples of, networks which generate periodic sequences of input impedances.