Radiation quality factor analysis of planar phased arrays

The energy storage and radiation quality factor of infinite planar phased arrays are studied for several cases of practical interest, including dipole arrays in free space and over a ground plane, as well as more general infinite phased arrays modeled with planar currents that excite only a small number of Floquet modes. The radiation Q is computed in a manner analogous to the classic Chu result for finite sized antenna elements [1], by calculating the stored electric and magnetic energies contained in a unit cell volume of the array (minus the energy density associated with the radiation field), and by calculating the radiated power from each unit cell. This is done using a full-wave moment method approach, with the appropriate Green's function for the geometry under consideration [2] (i.e., free-space, or over a ground plane). For lossless arrays in free space, in the absence of grating lobes, it is known that radiated power is contained only in the (0,0) Floquet mode, and that stored energy is contained only in higher order Floquet modes. This is in contrast to the Chu results for finite antennas, where each spherical wave mode contains some radiated power and some stored energy. The result is slightly different for arrays over a ground plane, where the (0,0) Floquet mode may contribute to the stored energies in the volume between the ground plane and the current elements. Both the radiated power and stored energies are evaluated in terms of the two-dimensional spectral domain Floquet mode series representation and the Fourier transform of the element current. Mode-by-mode correspondence of each Floquet mode contribution to the element input impedance is demonstrated, as is satisfaction of the complex Poynting theorem. The exact quality factor results are compared with approximate values based on the input impedance vs frequency from an independent calculation.