Stability of Active Transmission Lines with Arbitrary Imperfections

The preceding paper1 has considered the theory of active transmission lines with discrete imperfections. First, lines with equally-spaced identical reflectors were studied; in particular, gain-frequency curves were determined as functions of the various parameters, and the stability of the device was studied under these special conditions. It was pointed out that the mathematical expression for gain would yield a perfectly definite result for any values of the parameters, but that this mathematical result would have physical significance only if the device is stable, i.e., does not oscillate. Next, the case of random imperfections was studied.1 Here the statistics of the transmission were determined in terms of the statistics of the discrete reflectors, which were assumed to have random position and 203 294 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 19G4 magnitude. Again, these results have physical significance only if the device is stable (or if the probability of instability is negligible). However, in the random case no precise information about stability was given; the computed statistics of the transmission were felt to be valid if the rms magnitude of the discrete reflectors was sufficiently small, but only intuitive feelings of what was "small enough" were available. In the present paper we derive a sufficient condition for stability of an active transmission line with arbitrary reflectors; we further show (by one example) that this sufficient condition cannot be greatly improved (if at all) in the general case.