Empirical Formulas for Superdirectivities of Broadside Linear and Square Arrays

Empirical formulas are proposed for the superdirectivities of broadside linear and square arrays. The formulas are derived based on a large amount of numerical results of optimum directivity, for the superdirectivity of a broadside linear array of isotropic elements with an electrical element spacing varying from 0 to 0.5. The formulas for the linear array can also be developed for the square array. Besides the isotropic elements, the superdirectivity of the square array of different unidirectional elements are also numerically analyzed. It is found that the square array of unidirectional elements offers higher directivity than that of isotropic elements for a small number of elements, but tend to consistency as the element number increases. The numerical results are validated with the full-wave simulation of a square array of planar inverted-F antennas, considering coupling effects between elements.