An efficient trans-impedance Green's function that describes the electromagnetic behavior of a ring circulator is presented. A rigorous derivation composed of an infinite summation of modified Bessel functions of the first and second kinds is included. As with more traditional circulator descriptions, the formulation herein contains a weak singularity when the measurement point is located near the impressed source point on the same radius. To accelerate convergence of the series, this singularity is extracted from the formulation and integrated analytically. To complete the formulation, two circulators are presented; the first with ports that emanate at equal angles from the outer radius, and the second with two ports associated with the outer radius and one port that connects to the inner radius. The computation time associated with the proposed analysis lasted approximately 0.25 s, whereas an identical structure simulated via a common full-wave solver lasted approximately 10 hours. Comparison of impedance data between the proposed analysis and full-wave simulation is presented.
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