Elliptical Loop Antennas of Arbitrary Size and Ellipticity: Generalized Exact Expressions for the Radiation Resistance and Directivity

There has been renewed interest in understanding the properties and radiation characteristics of loop antennas due to their wide applicability, especially in infrared and optical regimes. Despite this interest, little work in either the RF or in higher frequency regimes has been devoted to investigating full radiation solutions for a generalized elliptical loop with an arbitrary current distribution due to the additional complexity required to solve the perturbed (from the conventional circular case) radiation integrals. In this work, we present the full mathematically exact solution for an elliptical loop's radiated power, directivity, and other far-field quantities assuming the current is represented as a Fourier cosine series. Further, we demonstrate that these solutions are accurate and in agreement with full-wave solutions for ellipses with axial ratios of five and show the convergence and accuracy of the provided analytical solutions up to extreme axial ratios of ten. Most importantly, these mathematically exact solutions provide a fundamental step toward a complete theoretical description for the radiation properties of elliptical loops, thereby enabling rapid analysis and optimization in comparison with computationally expensive full-wave solution methods. This is especially true for dispersive nanoloops, which typically require many thousands of mesh elements to accurately model.