A Justification of the Characteristic-Impedance Method for Lumped Periodic Cascades

A semi-infinite cascade of identical lumped two-ports has in general an infinity of solutions, each solution being a set of branch currents and voltages that satisfy Kirchhoff's laws and Ohm's law. The characteristic-impedance method picks out one of those solutions. The question is "which one?" This has been a long-standing lacuna in the arguments justifying that method. Our paper is aimed at this lacuna. It shows that the characteristic-impedance method picks out that solution for which the branch voltages comprise a vector in Hilbert's coordinate space
l_{2}
and that all other solutions will not satisfy this condition. This result holds for purely resistive cascades. For
RLC
cascades a similar theorem holds in the time domain, but the "
l_{2}
condition" must be imposed on the real axis of the complex frequency domain.