Fundamental directivity limitations of dense array antennas: a numerical study using Hannan´s embedded element efficiency

Hannan introduced in 1964 an embedded element efficiency concept that explains the so-called element -gain paradox in antenna arrays, i.e., that the array gain always is smaller than the sum of the element gains. In the present letter, we show for the first time the usefulness of his approach by evaluating directivities and aperture efficiencies of an array of open-ended waveguides by commercial full -wave electromagnetic (EM) solvers for a large range of element spacing. The results show also that by using embedded element analysis, the realized gain of regular arrays actually becomes equal to the sum of the realized gains of the embedded elements. Thus, the embedded element efficiency is more practical to use in design and numerical analysis than the more commonly used active element pattern approach. We also show that the embedded element efficiency can be approximated by a simple formula when the element spacing is smaller than half -wavelength.