Chu–Harrington limit

Remarks

The idea, due to L. J. Chu, is to expand the field outside the sphere in spherical modes and to compute the stored energy of each mode exactly. Energy stored inside the sphere is ignored, which can only lower $Q$. So the $Q$ of the lowest mode outside the sphere is a lower bound. The closed form above is the exact $Q$ of that mode, as computed by J. S. McLean. Higher-order modes have larger $Q$, so adding them does not help. For an electrically small antenna, $ka \ll 1$, the bound is about $(ka)^{-3}$, and since the fractional bandwidth of a matched antenna is roughly $1/Q$, a small antenna is necessarily narrow-band. Example: for $ka = 0.1$ the bound is $1010$, so the usable bandwidth is about $0.1\%$ of the centre frequency. The bound concerns the field outside the sphere; antennas that excite both a $\mathrm{TM}$ and a $\mathrm{TE}$ mode can reach a lower $Q$ than the single-mode bound.

Going deeper

For a student

The limit expresses a property of Maxwell's equations outside a sphere: the low-order spherical Hankel functions have large reactive parts when $ka$ is small. It sets the target against which small antenna designs in phones and sensors are measured. R. F. Harrington extended the analysis to relate gain, bandwidth and size; later work replaced the sphere by other enclosing shapes and gave bounds through optimization over currents.