There are three mathematical conditions that must be solved simultaneously for the analysis of a fully-symmetric radio-frequency (RF) crossover. When additional reciprocal two-port networks - which might be of an arbitrarily high complexity - are appended at each port of a crossover, analysis of the modified crossover becomes very tedious. Therefore, this paper examines the requirement of the three conditions in such scenario. We show that two of the three conditions can be invoked without considering the additional two-port networks altogether. This is a remarkable simplification considering that the additional two-port networks, in general, would necessitate dealing with more involved algebraic calculations. To demonstrate the usefulness of the presented theory, for the first time, analysis and design of a dual-frequency port-extended crossover is included. A prototype of the dual-frequency crossover operating concurrently at 1 GHz and 2 GHz is manufactured on a Rogers RO4350B laminate having 30 mil substrate height and 3.66 dielectric constant. The close resemblance between the EM simulated and measured results validates the analytical equations.
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