Q-Bounded Maximum Directivity of Self-Resonant Antennas

The upper limit on the directivity of self-resonant antennas that fit within a minimum sphere is determined for a given quality factor. This formulation is obtained by analytically solving a rigorous convex problem and is expressed as a rapidly converging analytical series. The total quality factor, the inverse of the relative frequency bandwidth, is formulated by considering the quality factors of individual spherical waves. From the exact series, approximate closed-form formulas have been derived, which exhibit high accuracy in complementary ranges of the minimum circumscribed sphere’s radius. These ranges encompass small antennas as well as intermediate to large antennas. Special emphasis is given to small antennas, where the solution is interpreted as a combination of dipolar and quadrupolar Huygens’ source contributions with appropriate closed-form coefficients. The solution in this range provides continuity to the maximum directivity between 3 and 8 maintaining a constant Q.