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 <inline-formula> <tex-math notation="LaTeX">$\sin ^{u}{(\theta)}\cos ^{v}{(\theta)}$ </tex-math></inline-formula>, with <inline-formula> <tex-math notation="LaTeX">$v > -({1}/{2})$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$u > -1$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$u$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$v \in \mathbb {Z}$ </tex-math></inline-formula>. Variations of this expression for different values of <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$u$ </tex-math></inline-formula> are analyzed from the analytical and numerical perspectives. The parameters found in the closed-form expression are related to the order <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$u$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$(\theta _{0}, \phi _{0})$ </tex-math></inline-formula>. The expression found in this communication has been validated by numerical results, considering distinct configuration scenarios.