A generalized expression of directivity for planar antenna arrays

Abstract

A single expression of directivity for arbitrary planar antenna array is derived in closed form. Element Pattern in the array is taken as sin <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">μ</sup> θcos <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">υ</sup> θ, {μ > -1, θ > - 1/2;}. Other parameters of this expression are location, excitation amplitude and phase of each element. The expression is derived from the definition of Directivity. As this work does not assume any special lattice of array and aperture distribution, performance of this expression is not limited by these parameters. Simplicity and compactness of this expression saves the runtime, memory requirement and any extra software costs. More, as each element pattern is periodic with Θ and if the Fourier series coefficients of the element pattern are known, the same expression is useful to calculate the radiated power even for real elements. Examples of directivity show exact match between directivity results obtained by the derived analytical expression and numerical method. Simpson's 1 / 3 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rd</sup> rule is the numerical method utilized for numerical estimation of directivity.