Lower Bounds on the Q of Electrically Small Dipole Antennas

General expressions are obtained for the lower bounds on the quality factor
$(Q)$
of electrically small electric- and magnetic-dipole antennas confined to an arbitrarily shaped volume
$V$
and excited by general sources or by global electric-current sources alone. The lower-bound expressions depend only on the direction of the dipole moment with respect to
$V$
, the electrical size of
$V$
, and the static electric and magnetic polarizabilities per unit volume of hypothetical perfectly electrically conducting and perfectly magnetically conducting volumes
$V$
. The lower bounds are obtained directly from the electromagnetic field expressions for
$Q$
with the help of current equivalence principles and the uncoupling of Maxwell's equations for electrically small volumes into quasi-electrostatic and quasi-magnetostatic fields.