# Stored energies and Q-factor of two-dimensionally periodic antenna arrays

## TL;DR

This paper derives a shape-independent, current-based expression for the Q-factor of two-dimensionally periodic antenna arrays, enabling accurate bandwidth prediction and beam steering considerations.

## Contribution

It introduces a novel, shape-independent Q-factor formulation for periodic arrays that directly relates to current density and can be rapidly computed.

## Key findings

- Numerical results show good agreement between the proposed Q-factor and fractional bandwidth.
- The method accurately predicts Q-factor for arrays with Q ≥ 5.
- The approach includes the input-impedance Q-factor by Yaghjian and Best, with consistent results.

## Abstract

The Q-factor for lossless three-dimensional structures with two-dimensional periodicity is here derived in terms of the electric current density. The derivation in itself is shape-independent and based on the periodic free-space Green's function. The expression for Q-factor takes into account the exact shape of a periodic element, and permits beam steering. The stored energies and the radiated power, both required to evaluate Q-factor, are coordinate independent and expressed in a similar manner to the periodic Electric Field Integral equation, and can thus be rapidly calculated. Numerical investigations, performed for several antenna arrays, indicate fine agreement, accurate enough to be predictive, between the proposed Q-factor and the tuned fractional bandwidth, when the arrays are not too wideband (i.e., when $Q\geq 5$). For completeness, the input-impedance Q-factor, proposed by Yaghjian and Best in 2005, is included and agrees well numerically with the derived Q-factor expression. The main advantage of the proposed representation is its explicit connection to the current density, which allows the Q-factor to give bandwidth estimates based on the shape and current of the array element.