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.