Stored energies in electric and magnetic current densities for small antennas

Electric and magnetic currents are essential to describe electromagnetic-stored energy, and the associated antenna Q and the partial directivity to antenna Q-ratio, D/Q, for arbitrarily shaped structures. The upper bound of previous D/Q results for antennas modelled by electric currents is accurate enough to be predictive. This motivates us to develop the analysis required to determine upper and/or lower bounds for electromagnetic problems that include magnetic model currents. Here we derive new expressions for the stored energies, which are used to determine antenna Q bounds and D/Q bounds for the combination of electric and magnetic currents, in the limit of electrically small antennas. In this investigation, we show both new analytical results and we illustrate numerical realizations of them. We show that the lower bound of antenna Q is inversely proportional to the largest eigenvalue of certain combinations of the electric and magnetic polarizability tensors. These results are an extension of the electric only currents, which come as a special case. The proposed method to determine the minimum Q-value which is based on the new expressions for the stored energies, also yields a family of current-density minimizers for optimal electric and magnetic currents that can lend insight into antenna designs