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A detailed analysis of the circular, homogeneous ferrite microwave circulator is provided. Particular emphasis is on the circulator's Green's function and the impact of the asymptotic term within the Green's function on convergence, data quality, and design methodology. The asymptotic term is shown to be logarithmic, which suggests that the Green's function is weakly singular when the source and observation points occupy the same location. With the Green's function properly understood, two techniques - one analytical and one numerical - are then offered to integrate that function in order to obtain Z-parameter data and, subsequently, S-parameter data. Data are provided to show rapid convergence of all parameters of interest. A small coupling angle approximation is then given for the Z-parameters and, from that approximation, a first-order design equation is obtained that relates the coupling angle to circulator radius. A circulator design example is presented and compared to a design associated with the Wu and Rosenbaum method; the comparison substantiates the small coupling angle approximation and design formula.