Limitations in Relating Quality Factor to Bandwidth in a Double Resonance Small Antenna

Abstract

We examine the behavior of a matched electrically small antenna that exhibits two impedance resonances within its defined voltage standing wave ratio (VSWR) bandwidth. The exact quality factor ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> ), computed indirectly from the antenna's input reactance and far field, is compared to twice the inverse of the matched half-power VSWR bandwidth ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BW</sub> ) and to an approximate quality factor ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Qz</i> ) determined from the frequency derivative of the antenna's input impedance. The well-established approximate equalities <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> ap <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BW</sub> ap <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</i> for antennas exhibiting an isolated single impedance resonance within their operating band are shown to become highly inaccurate for an electrically small antenna exhibiting two closely spaced impedance resonances.