Modal decomposition in prolate spheroidal coordinates to minimize the quality factor of an antenna

In a technical report published in 2004 we presented results for the evaluation of the quality factor (or Q) of an antenna surrounded by a surface in the shape of a prolate spheroid. There was assumed to be no energy within the surface. In a report published at a conference of the IEEE we extended this result to oblate spheroids. For the lowest mode and extremely long wavelengths we presented analytic formulas that supported numerical results that compared Chu's result for a sphere to those of spheroids with either the same height or same volume. In this paper we extend the results to prolate spheroids in which the current exciting the antenna has higher modes. There is no coupling between odd and even modes. There is coupling between the lowest order even modes. We present a simple result for the values of the degree of mixing of modes 2, 4 and 6 that minimize the Q. We find that the degree of coupling for long wavelength excitations should be proportional to the square of the frequency for mode 2 and to the fourth power of the frequency for mode 4. The coefficient that multiplies the mode 2 coupling is independent of the aspect ratio (height to diameter) of the surface. The coefficient for mode 4 that minimizes the Q is zero. The current density distribution for electrically small antennas that produces the minimum Q is presented.