A Closed-Form Analytical Evaluation of the Directivity of Planar Array Antennas with Different Types of Radiating Elements

This work presents a fast and general formulation for the directivity computation of planar antenna arrays composed of radiating elements modeled as cosqθ transmitters. The main novelty in the proposed formulation as compared to the literature is the fact that an arbitrary number of different radiating elements, each one characterized by a different q-factor, can be considered. The proposed framework enables the evaluation and validation of array architectures previously introduced in the literature, including concentric-ring configurations with tapered element sizes. A unified expression is derived in which the directivity depends solely on two inputs: the array factor and the class-dependent radiating element size. This reduced representation allows real-time computation of directivity, even for electrically large planar arrays comprising multiple rings and heterogeneous element classes. The resulting computational complexity is significantly lower than that of classical integral-based approaches traditionally used for directivity evaluation. Numerical results confirm that the proposed formulation yields accurate directivity values that are fully consistent with existing analytical and numerical results reported in the literature.