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It is proposed a closed-form expression of directivity for an arbitrary volumetric antenna arrays using a general element pattern expression of type sinu (&#x03B8;) cosu (&#x03B8;), with u &gt; -(1/2) and u &gt; -1, and u, u &#x2208; &#x2124;. 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 (&#x03B8;0, &#x03C6;0). The expression found in this communication has been validated by numerical results, considering distinct configuration scenarios.

Closed-Form Directivity Expression for Arbitrary Volumetric Antenna Arrays