On the Exact Mutual Reactance of a Line Source Array: A Hilbert Transform Methodology

The mutual reactance between two elements of a lineal array is formulated using Fourier transform, autocorrelation, convolution, and Hilbert transform concepts, with the invocation of the Hilbert transform being the novel contribution of this paper. It is shown herein that the mutual reactance of an array can be determined from the radiation pattern, if one considers the entire time-averaged power delivered to both the visible and the nonvisible far-field regions. Power to the nonvisible region corresponds to stored energy in the Poynting volume; stored energy corresponds to reactive power, which is in quadrature with the nonvisible, time-averaged power; and quadrature suggests the invocation of the Hilbert transform. The end result of the Hilbert transform process is a straightforward, closed-form analytical expression for the mutual reactance. Since the closed-form mutual resistance has been determined previously, both closed-form expressions can be combined to provide an elegant, but easy-to-use formulation for the mutual impedance. For a given current distribution, the results are exact and validated against data obtained numerically. Cases associated with the half-wave dipole, generalized dipole distribution, and the uniform distribution are provided herein as examples of the robustness of the presented theory.