  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.
