Abstract
Two sufficient conditions for the stability of one-dimensional active transmission lines with arbitrary imperfections (i.e., discrete or continuous reflections) are derived. The first stability condition guarantees stability for any arbitrary distribution of reflection. The second stability condition is restricted to a special case of interest that includes discrete reflectors with nominally equal magnitude and spacing; the stability condition for this restricted class is greatly improved over the general stability condition described above. These results, aside from their own interest, provide rigorous justification for previous calculations for the gain statistics of such a device with random discrete reflectors. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> They may also be used to find an upper bound on the probability of instability of such a device with random reflectors. Certain types of optical maser amplifiers and traveling-wave tubes provide examples of practical devices with distributed gain to which these results, or similar ones, might be applied.
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