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Fundamental properties of certain popular integral equations for thin-wire circular-loop antennas with nonsingular kernels are studied. The cornerstone of our study is a large-asymptotic formula for the n'th Fourier coefficient of the nonsingular kernel. Four different methods of driving the circular loop are considered; namely, the delta-function generator, the finite-gap generator, a certain type of frill generator, and the case of plane-wave incidence. The excitation model is crucial to our discussions, since it determines the behavior of the solution convergence. Also discussed are associated difficulties that may arise when moment methods are applied to the aforementioned equations, as well as a simple improvement to the Fourier-series method for determining the current in the case of the frill generator. The main results herein closely parallel recent results for the case of the straight antenna, and similarities and differences between the straight and circular cases are discussed.