$Q$ Bounds for Planar and Ellipsoidal Antennas
The procedure for numerically deriving equivalent circuits for the six lowest-order modes of arbitrarily shaped, electrically small antennas is reviewed. Element values and the effective radius are tabulated for thin prisms, spheroids, and ellipsoids with a wide range of aspect ratios. A self-consistency test for the elements is defined and applied. These circuits allow calculation of the internal and external electric and magnetic reactive energies and thus establish lower bounds for Q of antennas with these shapes as a function of their electrical radius, ka. Electromagnetic simulations on a family of meander dipoles and a planar loop provide Q values that are compared with the bounds over a range of ka up to one.