Numerical Accuracy of Circuit-Based Equations for Antenna Quality Factor

Electrically small antennas (ESAs) are known to have narrow impedance bandwidths. A long-standing means to quantify this trade-off between electrical size and bandwidth has been through the quality factor
$Q$
of the antenna. Chu (L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys., vol. 19, no. 12, pp. 1163–1175, Dec. 1948) quantified this by expanding the wave impedances of spherical waves into lumped RLC circuits, whose
$Q$
can be computed directly by carrying out a circuit analysis for each spherical wave order. Chu also provided approximate expressions for general mode order, though these only converge onto the exact solution in the
$ka\rightarrow 0$
limit, where
$k$
is the wavenumber and
$a$
is the radius of the smallest sphere that circumscribes the antenna. Collin and Rothschild (R. E. Collin and S. Rothschild, “Evaluation of antenna
$\mathrm{Q}$
,” IEEE Trans. Antennas Propag., vol. AP-12, no. 1, pp. 23–27, Jan. 1964) integrated directly over the same spherical waves (after subtracting the radiated fields) to obtain the stored energy of each spherical wave, and hence
$Q$
. Recently, Murray and Iyer (A. B. Murray and A. K. Iyer, “Analytical expressions for spherical wire antenna quality factor demonstrating exact agreement between circuit- based and field integration techniques,” IEEE Trans. Antennas Propag., vol. 72, no. 2, pp. 1636–1647, Feb. 2024) have shown that these two methods produce identical results, for all spherical wave orders
$n$
and electrical sizes
$ka$
.