Chu–Harrington limit

In electrical engineering and telecommunications the Chu–Harrington limit or Chu limit sets a lower limit on the Q factor for a small radio antenna.[1] The theorem was developed in several papers between 1948 and 1960 by Lan Jen Chu,[2] Harold Wheeler,[3] and later by Roger F. Harrington.[4] The definition of a small antenna is one that can fit inside a sphere whose diameter is ${\displaystyle {\tfrac {1}{\pi }}\lambda }$ (radius ${\displaystyle {\tfrac {\lambda }{2\pi }}}$) – a little smaller than 1⁄3 wavelength in its widest dimension. For a small antenna the Q is proportional to the reciprocal of the volume of a sphere that encloses it. In practice this means that there is a limit to the bandwidth of data that can be sent to and received from small antennas such as are used in mobile phones.

More specifically, Chu established the limit on Q for a lossless antenna as ${\displaystyle Q\geq {\frac {1}{k^{3}a^{3}}}+{\frac {1}{ka}}}$ for a linear polarized antenna, where ${\displaystyle a}$ is the radius of the smallest sphere containing the antenna and its current distribution and ${\displaystyle k={\frac {2\pi }{\lambda }}}$ is the wavenumber. A circular polarized antenna can be half the size[5] (an extension of the theory of Chu by Harrington).[6]

As antennas are made smaller, the bandwidth shrinks and radiation resistance becomes smaller compared to loss resistances that may be present, thus reducing the radiation efficiency. For users this decreases the bitrate, limits range, and shortens battery life.