Plane-wave expansion for arrays of arbitrarily oriented piecewise linear elements and its application in determining the impedance of a single linear antenna in a lossy half-space

An infinite array of arbitrarily oriented identical elements with arbitrary identical currents is considered. The field from this array is expanded into plane inhomogeneous waves, and the mutual impedance between the array elements and an exterior arbitrarily oriented element is derived. The formulation is particularly useful when the array is located adjacent to a dielectric interface. Numerical examples are given and the relationship to earlier formulations pointed out. It is further shown that the impedance of a single element can be obtained as the average of the scan impedance taken over the entire hemisphere (called the array scanning method (ASM)). This technique has a clear physical interpretation which greatly facilitates its uses, which include the moment method solutions of wire antennas as applied to the Sommerfeld integral. Numerical evaluation is straightforward when the dipole is in the lossy half-space, and the utility of the method is demonstrated by the presentation of results for the input impedance of dipoles in a variety of half-space environments. Solution is by Galerkin's method with a piecewise sinusoidal expansion for the current. Computer time is proportional to
d^{-1}
, where
d
is the distance of the dipole to the interface. For conducting media and low frequencies an approximation is made to reduce computation time. The moment method solution of a dipole buried at a depth as small as 1/150000 wavelength in the earth is presented.