Electromagnetic fields in nonuniform lossless transmission lines

Abstract

Methods are developed for electromagnetic field calculations in nonuniform lossless transmission lines which support quasi-one dimensional propagation of a single baseband wave species. Approximate solutions are obtained in perturbation series for smoothly tapered lines by expanding Maxwell's equations and boundary conditions in the dimensionless parameter ƞ, given by the ratio of typical cross-section dimension to the length of the tapered section. The method emphasizes construction of a single ''warped field description rather than the local modal expansions of Schelkunoff's generalized telegraphist's equations. Expansions in Cartesian coordinates yield the traditional distributed circuit parameter equations in the lowest approximation. Correction terms appear in even powers of ƞ, and their effects are shown most clearly by calculating waveform aberrations introduced by line transitions of nominally constant characteristic impedance. Improved field descriptions in nonuniform regions are obtained by reformulating the exact equations in special nonorthogonal coordinate systems more closely related to the essential structure of the field problem. The lowest term of the ordered expansion is now exact for a uniform finite angle taper. New circuit level nonuniform line equations are obtained which reduce to the well-known forms for gradual tapers. These techniques are extended to treat tapered plate lines with curved center lines and then to give a description of coaxial lines in which the field pattern is locally dominated by the boundaries, and the electrical center line is located in the propagation region. Odd-sequence field distortion terms now appear in third and higher orders. In all the systems investigated, distributed circuit equations give results, outside the nonuniform region, that are valid to within second order terms in the taper scale parameter.