Closed-Form Directivity Expression for Arbitrary Volumetric Antenna Arrays

It is proposed a closed-form expression of directivity for an arbitrary volumetric antenna arrays using a general element pattern expression of type sinu(θ)cosv(θ), with v > -frac12 and u > -1, and u, v ∈ mathbbZ. Variations of this expression for different values of v and u are analyzed from the analytical and numerical perspectives. The parameters found in the closed-form expression are related to the order v and u of the element patterns, the rectangular spatial coordinate of each antenna element, the magnitude and phase excitation coefficients (complex excitation) of all elements, and the desired angle in spherical coordinates (θ0, φ0). The expression found in this work has been validated by numerical results, considering distinct configuration scenarios.