The mutual impedance between concentric curved dipole antennas above a spherical conducting surface

Abstract

In this paper an analytical expression is derived for the mutual impedance between concentric curved dipole antennas located in close proximity to a conducting sphere. The boundary value problem is solved using the spherical vector mode function and the solution is formulated in an infinite series of spherical Bessel functions, spherical Hankel functions, associated Legendre polynomials and the derivatives of these functions. The mutual impedance expression was found to be a function of the operating frequency, the height above the sphere and the orientation of the two antennas. The analytical expression for the mutual impedance is valid for any antenna current distribution but, for the purpose of numerical calculations, the current distribution was assumed to be sinusoidal. Calculations were made for four different cases of antenna orientation. Case A was defined for two curved antennas oriented such that the planes containing the antennas intersected along the y-axis of the rectangular coordinate system of one of the antennas. Case B was defined for antennas in a staggered orientation. Case C was defined such that the planes of the two antennas were perpendicular and the plane of one antenna passed through the feedpoint of the other. Case D was defined for curved antennas oriented such that both antennas were in the x-y plane of the rectangular coordinate system of one of the antennas. For these four cases , numerical values were calculated as the antennas were maintained at a constant height of 20 cm. above the sphere while the frequency was varied from 250 MHz. to 400 MHz. in 50 MHz. steps, as the antennas were varied simultaneously from a height of 5 cm. above the sphere to a height of 35 cm. in 5 cm. steps, while the frequency was maintained at a constant 300 MHz. and as one antenna was varied from a height of 5 cm. above the sphere to a height of 35 cm. in 5 cm. steps, while the other antenna was maintained at a constant height of 35 cm. above the sphere and the frequency was maintained at 300 MHz. Laboratory measurements were made to verify the numerical results for the four cases of orientation. A detailed analysis of errors discovered in the measuring system was made and computed values of meter readings were compared with the meter readings obtained in the laboratory.