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 sin
^u
(θ) cos
^u
(θ), with u > -(1/2) and u > -1, and u, u ∈ ℤ. Variations of this expression for different values of u and u are analyzed from the analytical and numerical perspectives. The parameters found in the closed-form expression are related to the order u 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 (θ
₀
, φ
₀
). The expression found in this communication has been validated by numerical results, considering distinct configuration scenarios.