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.
Highlights
T HE Q-factor of an oscillating system is an indirect measure of the width of the system’s resonance
The goal of this section is to illustrate the practical use of the proposed Q-factor expression
For the case of narrow-band arrays, we illustrate that all Q-factors give similar results and that they predict the fractional bandwidth
Summary
T HE Q-factor of an oscillating system is an indirect measure of the width of the system’s resonance. The proposed expression is a step toward developing fundamental bounds, obtainable by current density optimization, similar to the case of isolated antennas [19]–[25]. An approach to obtain fundamental bounds from the here proposed Q-factor by optimizing current densities in a unit cell is presented in [29, Ch. 4] This optimization is done in conjunction with constraints on, for example, conductive losses which lead to tradeoff relations between the Q-factor and efficiency of an array. The periodic Q-factor expression, which is the main theoretical result of this article, is derived in Sections III and IV: in Section III, we represent the stored energies in terms of electric scalar and magnetic vector potentials.
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