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Electromagnetic fields are analyzed in complementary and self-complementary structures. As is well known, Maxwell's equations exhibit a complementary (duality) symmetry between the electric and magnetic fields. Self-complementary field solutions (when these exist) on self-complementary structures have particular properties of constant impedance (resistance) associated with certain pure traveling wave fields. The contribution of this paper is that it rigorously treats the electromagnetic field in a self-complementary waveguide. It shows a self-complementary field solution is a pure traveling wave. A particular waveguide structure is selected that illustrates circumstances when a self-complementary field solution fails to exist in a self-complementary structure. A computer model was developed to analyze a family of waveguide structures that differ only in an aspect ratio parameter, an aspect ratio of 1.0 corresponding to the self-complementary case. The original structures and the corresponding complementary structures are analyzed independently. Results are presented for a wide range of frequencies and aspect ratios. At low frequencies, when only one TEM mode propagates, field solutions for the input TEM mode exhibit discontinuous behavior as the aspect ratio approaches one from above and below. In this frequency range, no time harmonic self-complementary field solution exists for the precisely self-complementary waveguide structure.