Theoretical maximum gain and bandwidth of an antenna.

Picture of an antenna in a sphere of radius a

g_max = (k a)^2 + 2 k a
Q = (1 / (k a)) + (1 / ((k a)^3))                 BW_max = f / Q <==> Q >> 1
k = 2 pi / lambda                 lambda = c_0 / f
Frequency:f = Hz
Minimum radius of a sphere containing the antenna: a = m
Maximum antenna gain:G < dBi
Maximum quality factor:Q < 1/1
Minimum band width:BW > Hz

Enter the frequency f and one value (either a, G, Q or BW) and click the "Calculate" button next to it to compute the three other values.

These values are only valid for perfect antennas (no losses). For real antennas (with losses) the gain lower and the bandwidth is wider.
Bandwidth can only calculated when the condition Q>>1 is true and it's only valid for linear polarized antennas.

Bibliography:

Gain:   L. J. Chu, Physical limitations of omni-directional antennas, Journal of Applied Physics, vol. 19, pp. 1163-1175, December 1948.
Bandwidth:   R. F. Harrington, Time-Harmonic Electromagnetic Fields, Electrical and Electronical Engineering, McGrow-Hill, New York, 1961.
Newton's method:   J. Rappaz, Analyse numérique, Notes de cours: Leçons 1 à 10, École Polytechnique Fédérale de Lausanne, 1994, pp. 64-66.

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