Minimum Q for Lossy and Lossless Electrically Small Dipole Antennas

General expressions for the quality factor (Q) of antennas are minimized to obtain lower-bound formulas for the Q of electrically small, lossy or lossless, combined electric and magnetic dipole antennas confined to an arbitrarily shaped volume. The lower-bound formulas for Q are derived for the dipole antennas excited by both electric and magnetic surface currents as well as by electric surface currents alone. With either excitation, separate formulas are found for the dipole antennas containing only lossless or nondispersive-conductivity material and for the dipole antennas containing highly dispersiveconductivity material. The formulas involve the quasi-static electric and magnetic polarizabilities of the associated perfectly conducting volume of the antenna, the ratio of the powers radiated by the electric and magnetic dipoles, and the efficiency of the antenna