A general synthesis approach is proposed for reflectarrays using second order Phoenix cells. It relies on an original spherical representation that transforms the optimization domain in a continuous and unbounded space with reduced dimension. This makes the synthesis problem simpler and automatically guarantees smooth variations in the optimized layout. The proposed mapping is combined with an Artificial Neural Network (ANN) based behavioral model of the cell and integrated in a min/max optimization process. Bi-cubic spline expansions are used to decrease the number of variables. As an application, a contoured beam for space communication in the [3.6-4.2] GHz band is considered. The gain improvement compared to an initial Phase Only synthesis (POS) is up to 1.62 dB at the upper frequency. Full wave simulation of the final array is provided as a validation.
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