Cascaded Active Twoports

Abstract

If a number of identical twoport networks (active or passive) are cascaded, the input impedance to the cascade will usually approach one of the iterative impedances of the twoport no matter what termination is used. In the passive-termination passivetwoport case, this will be the passive iterative impedance which is unique provided the other iterative impedance is active. In the passive twoport cases, where there are two passive iterative impedances or an active termination, and in the active case, simple conditions are established for the existence and nonexistence of a limit and the limit identified. It is shown that the unique limiting value of the input-impedance sequence is the iterative impedance at which the derivative of the twoport impedance transformation assumes a magnitude less than unity. Secondary results include a closed form for the input impedance of the general twoport cascade, the proof of nonexistence (except in a single special case) of periodic fluctuations in input impedance as sections are added, and the proof that the derivatives of the transformation evaluated at the two fixed points are mutually inverse. Examples of both passive and active networks are presented and the rate of convergence indicated.