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A circuit model is used to present the basis forthe existence of complex propagation constants and backward wave modes in lossless coupled dispersive transmission lines. In particular, for the 2-line lossless case, conditions for realizing complex modes, as well as the possible forms for the diagram are given. Methods for approximating the mode behavior of lossless waveguide systems, uniform in the propagation direction, but with transverse plane inhomogeneities, are presented using coupled-line models. These methods are derived from the Schelkunoff representation of such a waveguide structure by an infinite set of coupled TE and TM dispersive transmission lines. The techniques used in this paper depend on an approximation which reduces the infinite number of lines to a finite number. One approximation method interpolates the parameters of a realizable 2-line system to approximate the behavior of the infinite line system up to a frequency in the vicinity of the secondlowest eigenmode. A second method is a systematic network procedure for eliminating coupling ports of the infinite structure so as to obtain a finite number of coupled lines to approximate the mode behavior. These two methods are illustrated by 2-line models for round waveguide loaded with a dielectric rod. The mode behavior of the loaded guide is surprisingly well approximated by both the 2-line models.