Evaluation of the minimum radiation Q for electrically small antennas in spheroidal coordinates

We have calculated the quality factor or Q of an electrically small antenna enclosed by a spheroidal surface as a function of both the wave number and aspect ratio. We considered cases in which the height to radius ratio (aspect) was greater than 1 (prolate) or less than 1 (oblate). In a previous paper we had evaluated the Q by including only the lowest mode. To understand the effect of the addition of higher modes we calculated the inclusion of modes 2, 4, and 6. The Q depends in a non-linear manner on the mode structure. We found a simple way to isolate the effects of mode and to minimize the Q with respect to the addition of higher modes. We found that the addition of mode 2 that minimizes Q is independent of the aspect ratio. The reduction in the value of Q is significant if the aspect ratio is very different from 1. The coefficients for modes 4 and 6 to minimize Q are effectively zero. We show the implications of these results for the design of electrically small antennas.