Maximum Gain of Lossy Antennas Without and With Q-bounds

Authors of this paper have recently formulated a maximum bound of super-directivity of self-resonant antennas for a given minimum Q (maximum frequency bandwidth). This paper complements the above work treating the influence of the losses. The problem is faced by assuming small losses in terms of surface resistance over the metalized surface of the minimum sphere circumscribing the antenna. The final closed form formula shows that the maximum gain is obtained by a summation that resembles the well-known Harrington’s sum for maximum directivity, except that the expansion coefficients are weighted by the radiation efficiency of each spherical harmonic. The formulation is next generalized to the case of self-resonant antenna, providing a tighter bound for any losses. For small antennas, we provide a simple interpretation of the field corresponding to the maximum gain in terms of dipolar and quadrupolar source contributions, weighted by the appropriate efficiency, offering a physical insight into the phenomenon. The formulation is then extended to also account for a Q-bound, deriving a final series expression as a function of the loss resistance and of the antenna electrical size. This expression seamlessly merges to the previously derived Q-bounded maximum directivity as losses tends to zero and converges to Q-unbounded maximum Gain for Q that tends to very large values.