Lower Bounds on Substructure Antenna Q -Factor

Calculation of the lower bound on $Q$ -factor for currents within a fixed region (equivalently on a perfectly conducting structure) is used to derive corresponding bounds for currents located within any subregion (substructure). Both tuned and untuned $Q$ -factors are studied using eigenvalue techniques and the Poincaré separation theorem. We show that the tuned $Q$ -factor of a substructure is bounded from below by the minimum $Q$ -factor of the full structure. Furthermore, we derive that all modal untuned $Q$ -factors of a substructure are bounded from below by the corresponding modal untuned $Q$ -factors of the full structure. Mathematical results are demonstrated through numerical examples involving substructures produced at random, by heuristic design and by genetic optimization.