Fundamentals of Antenna Bandwidth and Quality Factor

Abstract

After a brief history of the development of quality factor, useful expressions are derived for the robust input-impedance Qz(w) quality factor that accurately determines the VSWR fractional bandwidth of antennas for isolated resonances and a small enough bandwidth power drop. For closely spaced multiple resonances/antiresonances, a definitive formula is given for the increase in fractional bandwidth enabled by Bode-Fano tuning. Methods are given for determining the conventional and complex-energy quality factors of antennas from RLC circuit models. New field-based quality factors Q(w) are derived for antennas with known fields produced by an input current. These Q(w) are remarkably robust because they equal Qz(w) when the input impedance is available. Like Qz(w), the field-based Q(w) is independent of the choice of origin of the antenna fields and is impervious to extra lengths of transmission lines and surplus reactances. These robust field-based quality factors are used to derive new lower bounds on the quality factors (upper bounds on the bandwidths) of spherical-mode antennas that improve upon the previous Chu/CR (Collin-Rothschild) lower bounds for spherical modes. A criterion for antenna supergain is found by combining the Harrington maximum gain formula with the recently derived formula for the reactive power boundaries of antennas. Maximum gain versus minimum quality factor for spherical antennas are determined using the improved lower bounds on quality factor for different values of electrical size ka. Lastly, reduced antenna quality factors allowed by dispersive tuning overcome the traditional Chu/CR lower bounds for lower radiation efficiencies and small enough bandwidth power drops.

Figures (9)

• Figure 1: Maximum possible bandwidth factor increase $f_{\rm BF}$ (over ordinary single-resonance bandwidth) produced by Bode-Fano multiple resonance/antiresonance tuning vs the bandwidth power-drop parameter $\alpha$.
• Figure 2: Schematic of a general transmitting antenna, its matched single-mode feed waveguide, its shielded power supply (${\mathcal{V}}_p$), and a series tuning reactance $X_s$. The volume ${\cal V}_p$ is contained within the surface of the shielded power supply plus feed waveguide up to the reference plane $S_0$ (waveguide port). The infinite volume outside the volume ${\cal V}_p$ is denoted by ${\cal V}_o$, which contains the material of the antenna ${\cal V}_a$. The volume ${\cal V}_a$ of the antenna includes the element of the series tuning reactance $X_s$.
• Figure 3: Gain $G_{\rm dk}$ vs $ka_0$ (dash line) of the circular disk with radius $a_0$ of uniform electric and magnetic surface currents (high-gain "ordinary" antenna) compared to the supergain lower limit vs $ka_0$ on the right-hand side of (\ref{['sgreal']}) (solid line) and that of Kildal and Best KB (dash-dot line).
• Figure 4: Maximum $N$-degree spherical-mode gain $G$ versus the minimum quality factor for the Fante quality factors $Q_{\rm F}(ka)$ ($---$) and the more accurate $Q_{\rm A}(ka)$ quality factors (------) at the three different values of $ka = 0.2$, $0.5$, and $1.0$. The lines extrapolate between the discrete points (circle markers) for $N = 1, 2$ ($G = 3, 8$). The maximum-gain versus minimum-quality-factor curves obtained in Passalacqua-2023 but with our more accurate quality factors in (\ref{['HGQA']}) are shown by the dash-dots ($-\!$ - $\!-$).
• Figure 5: The input impedance for a dispersively tuned $Q = Q_Z = 10$ antenna with $50$% efficiency and reduced fractional quality factor $\rho = 0.5$ (bandwidth doubled, that is, $Q_0 = Q_{0Z} = 20$ for the nondispersively tuned antenna with the same $50$% efficiency). Note the single isolated resonance, which confirms that, unlike Bode-Fano tuning, the Lorentz dispersive tuning does not produce multiple resonances/antiresonances or increase the group delay.