Closed-Form Directivity Expression for Arbitrary Volumetric Antenna Arrays

TL;DR

This paper derives a closed-form formula for the directivity of arbitrary volumetric antenna arrays using a generalized element pattern, validated through numerical analysis across various configurations.

Contribution

It introduces a novel closed-form expression for antenna array directivity based on a general element pattern, linking parameters to array geometry and excitation.

Findings

Validated the formula with numerical results for different array configurations

Analyzed variations of the expression for different pattern parameters

Established relationships between parameters and array characteristics

Abstract

It is proposed a closed-form expression of directivity for an arbitrary volumetric antenna arrays using a general element pattern expression of type $\sin^u{(\theta)}\cos^v{(\theta)}$, with $v > -\frac{1}{2}$ and $u > -1$, and $u, v \in \mathbb{Z}$. 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 $(\theta_0, \phi_0)$. The expression found in this work has been validated by numerical results, considering distinct configuration scenarios.