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

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 (Q), computed indirectly from the antenna´s input reactance and far field, is compared to twice the inverse of the matched half-power VSWR bandwidth (Q _BW) and to an approximate quality factor (Qz) determined from the frequency derivative of the antenna´s input impedance. The well-established approximate equalities Q ap Q _BW ap Q z 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.