## 1. Introduction

[2] A significant number of military and commercial applications require slot/aperture type antennas that are conformal to the surface of a perfect electrically conducting (PEC) sphere coated or partially coated with a lossy thin dielectric/magnetic material [*Tomasic et al.*, 2002; *Sipus et al.*, 2008]. Thus, the electromagnetic compatibility (EMC) and the electromagnetic interference (EMI) between these antennas become important, and their prediction requires an accurate, and if possible efficient, analysis of mutual coupling between the antennas and hence, surface fields excited by these antennas. However, such an analysis becomes a challenging task when the radius of the sphere and the distance between the antennas along the geodesic path are large in terms of the wavelength. A possible remedy for this challenging task is to approximate the boundary conditions on aforementioned spherical surfaces by an impedance boundary condition [*Penney et al.*, 1996; *Rojas and Al-hekail*, 1989; *Senior and Volakis*, 1995], and to perform the analysis using a high-frequency based asymptotic solution that, in general, contain a Fock type integral representation [*Fock*, 1965].

[3] Several high-frequency based asymptotic solutions for the radio wave propagation around the earth that model the earth by a spherical impedance surface have been presented [*Fock*, 1945, 1965; *Wait*, 1960, 1962, 1965, 1967; *Spies and Wait*, 1966, 1967; *Hill and Wait*, 1980, 1981; *King and Wait*, 1976], and attracted significant attention. Among them, *Wait* [1960] discusses the surface waves excited by a vertical dipole and their propagation on a sphere where the spherical surface exhibits an inductive reactance. In his solution, the electric field is expressed as the radiation field of the dipole if it were placed on the surface of a PEC plane multiplied by an attenuation factor (ground wave attenuation factor) that takes the curvature effects into account and possess a Fock type integral representation. *Spies and Wait* [1966] discuss the calculation of this ground wave attenuation factor at low frequencies. They use both residue series and power series based on the distance of the observation point from the source. In the work of *Spies and Wait* [1967], analytical and numerical procedures are described for the evaluation of some Fock type integral functions that appear in a method presented by *Wait* [1967] to compute the tangential magnetic field on the surface of a smooth inhomogeneous earth excited by a plane wave. Then, *Hill and Wait* [1980] generalize the computation of the ground wave attenuation function for a spherical earth with an arbitrary surface impedance, where ground waves are excited by a vertical electric dipole located at the surface of the earth. Their attenuation function is represented in terms of a Fock type integral, and is in general computed using a residue series approach. However, when the argument of the attenuation function is small (i.e., small curvature case), the attenuation function is computed preferably using either its power series representation given by *Bremmer* [1958] and *Wait* [1956, 1958], or its small curvature expansion [*Wait*, 1956; *Bremmer*, 1958] based on the complementary error function. More references on the subject of ground wave propagation, including the early work, can be found in the work of *Wait* [1998].

[4] However, the aforementioned solutions are in general valid far from the source location. Therefore, a different high-frequency based asymptotic analysis from that used traditionally in the ground wave propagation problems is developed in this paper. Our solution is a uniform geometrical theory of diffraction (UTD) [*Kouyoumjian and Pathak*, 1974] based representation of the surface fields excited by a magnetic current located on the surface of a sphere that has a uniform surface impedance, *Z*_{s} with a positive real part. The radius of the sphere and the length of the geodesic path between the source and observation points, when both are located on the surface of the sphere, are assumed to be large compared to the wavelength. Unlike the UTD-based solution for a PEC sphere developed by *Pathak and Wang* [1978], some higher-order terms and derivatives of Fock type integrals are included as they may become important for certain impedance values. It is shown that when *Z*_{s} → 0, our UTD-based solution recovers to that of PEC case developed by *Pathak and Wang* [1978] with higher-order terms and derivatives of the corresponding Fock type integrals. Furthermore, the methodology developed by *Pathak and Wang* [1978] to correct the surface fields at the caustic of the PEC sphere is extended to the impedance sphere case. It should be noted that, together with the UTD-based solution for a circular cylinder with impedance boundary condition (IBC) [*Tokgöz and Marhefka*, 2006; *Alisan et al.*, 2006], the solution presented in this paper may form a basis toward the development of UTD-based asymptotic solutions valid for arbitrary smooth convex surfaces with an IBC that can model thin material coated/partially material coated surfaces [*Pathak and Wang*, 1978, 1981].

[5] The organization of this paper is as follows: Section 2 presents the formulation of the UTD-based solution for the surface fields on an impedance sphere excited by a magnetic current located on the surface of the sphere. In the course of obtaining the high-frequency representations for the surface fields, first a method similar to that developed by *Fock* [1965] is followed to obtain the necessary potentials without any assumption or approximation, and then UTD-based high-frequency solution is obtained in a similar manner to that developed by *Pathak and Wang* [1978]. Caustic corrections, limiting situations (i.e., *Z*_{s} → 0) and numerical evaluation of Fock type integrals are also provided in this section. Numerical results are presented in section 3, followed by a brief conclusion. An *e*^{jωt} time convention is assumed and suppressed through out this paper, where *ω* = 2*πf* with *f* being the operating frequency.