An exact integration procedure for vector potentials of thin circular loop antennas

A direct integration procedure for far-zone vector potentials of thin circular loop antennas has been known for many years. This method is general in the sense that it leads to simple integrals which have closed form solutions for most commonly assumed loop current distributions. However, a comparable integration technique has not been available for evaluating the more complicated near-zone vector potentials. This paper introduces a systematic approach for the exact integration of general near-zone vector potentials associated with current-carrying circular loop antennas. A particular example is considered where this new integration technique is used to find exact solutions to the vector potential and electromagnetic field integrals for loops with a Fourier cosine-series expansion of the current. The observation is made that degenerate forms of these exact representations lead to simplified expressions for the important special cases of a uniform and cosinusoidal current loop. Two equivalent forms of exact series expansions are derived for the uniform current vector potential and field integrals. It is shown that the familiar small-loop approximations, as well as the classical far-field expressions, may be obtained as limiting cases of the more general exact series representations for the uniform current loop obtained in this paper. Convenient asymptotic far-field expansions are derived for the loop with a cosinusoidal current distribution. Finally, the far-field analysis for the cosinusoidal loop is generalized to loops having an arbitrary current represented by a Fourier cosine series.