Chu-Harrington limit

The Chu-Harrington limit or Chu limit sets a lower limit on the Q factor for a small radio antenna. The definition of a small antenna is one that can fit inside a sphere $\frac{\lambda }{2 \pi}$. 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 received from small antennas such as are used in mobile phones.

Chu established the limit on Q as: $Q = \frac{1}{(k^3 a^3)} + \frac{1}{(k a)}$ for a linear polarized antenna. $k = \frac{2 \pi}{\lambda}$ A circular polarized antenna can be half the size.^[1] Harrington extended the theory of Chu to the circularly polarized case.^[2] As antennas are made smaller, the bandwidth shrinks, the radiation resistance is smaller, and efficiency reduces. For users this decreases the bitrate, limits range, and shortens battery life.

Proof

Chu expressed the electromagnetic field in terms of evanescent modes with a real component and no propagating modes. The fields were expressed as a spherical harmonic series with the components being Legendre functions and spherical Bessel functions. The impedance could be expressed as a series of a ratio of a derivative of a Hankel function to other Hankel functions An equivalent circuit is a ladder line with the rungs being inductors and the capacitors running in series. The number of elements used in the mathematical series matches the number of capacitor - inductor pairs in the equivalent circuit.^[3]

Comparison of antennas

• Quarterwave whip

Designs close to the limit

• The Goubau antenna from 1976 has a size ratio of 1 and bandwidth of 80%. Q is 1.5 times the limit.^[4]
• The Foltz drawing pin like antenna from 1998 size .62 and 22% bandwidth.
• The Rogers cone from 2001 is .65 size and right on the limit.
• Lina and Choo Planar spirals in size ration range from 0.2 to 0.5
• The fractal Koch curve antenna approaches the limit.
• A meander line antenna optimises the size for narrower bandwidths of the order 10%.^[5]

References

1. ^ Carles Puente Baliarda (November 2000). "The Koch Monopole: A Small Fractal Antenna". pp. 1733–1781. 
2. ^ Jahoda, Joseph R. (August 2006). "JTRS/SINCGARS ultrabroadband airborne blade antenna for subsonic aircraft and helicopters". RFDesign. pp. 20–22. http://rfdesign.com/mag/608RFDEf3.pd. Retrieved 28 August 2011. 
3. ^ R. C. Hansen (February 1981). "Fundamental Limitations in Antennas". Proceedings of the IEEE 69 (2): 170–182. http://www.cs.berkeley.edu/~culler/AIIT/papers/radio/Hansen%201981%20Fundamental%20Limitations%20in%20Antennas.pdf. 
4. ^ "Chu Limit". http://www.skycross.com/technology/chuLimit.html. 
5. ^ Caimi, Frank (August 2002). "Meander Line Antennas". http://www.skycross.com/technology/Whitepapers/Meander_Line_Antennas.pdf. Retrieved 28 August 2011. 

Further reading

• Yazdandoost, K. Yekeh (200x). "Design and Analysis of an Antenna for Ultra-Wideband System". http://www.eurasip.org/Proceedings/Ext/IST05/papers/105.pd. Retrieved 28 August 2011.