Lower Bounds on the Q of Electrically Small Dipole Antennas

Abstract

General expressions are obtained for the lower bounds on the quality factor <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(Q)$</tex></formula> of electrically small electric- and magnetic-dipole antennas confined to an arbitrarily shaped volume <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$V$</tex> </formula> 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 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$V$</tex> </formula> , the electrical size of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$V$</tex> </formula> , and the static electric and magnetic polarizabilities per unit volume of hypothetical perfectly electrically conducting and perfectly magnetically conducting volumes <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$V$</tex> </formula> . The lower bounds are obtained directly from the electromagnetic field expressions for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$Q$</tex> </formula> 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.