Chu–Harrington limit

Method of proof

Chu expressed the electromagnetic field in terms of evanescent modes with a real component and non-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 shunts (rungs) being inductors and the capacitors running in series (railings). The number of elements used in the mathematical series matches the number of capacitor-inductor pairs in the equivalent circuit.

Practical implications

In practice an electrically small antenna is one that is operated at a frequency below its natural resonance. Small antennas are characterised by low radiation resistance and relatively high reactance, so that a tuning component must be added in series with the antenna to cancel its reactance and assist matching to the circuit to which it is connected. The addition of this extra component creates a tuned circuit, with a Q-factor that potentially limits the instantaneous bandwidth available for signals passing through the antenna. This is a fundamental limit that sets a minimum size for any antenna used at a given frequency and with a given required bandwidth.

The Chu limit gives the minimum Q, and by implication the maximum bandwidth, for an antenna of a given size on the assumption that it is lossless. However, any antenna can be made to show a larger bandwidth than suggested by the Chu limit if there is additional resistance present to reduce the Q, and this has led to claims for antennas that have breached the limit, but none has so far been substantiated.

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.
• The Foltz drawing pin like antenna from 1998 size 0.62 and 22% bandwidth.
• The Rogers cone from 2001 is size 0.65 and right on the limit.
• Lina and Choo planar spirals in size ratios range from 0.2 to 0.5
• The fractal Koch curve antenna approaches the limit.
• A meander line antenna optimizes the size for narrower bandwidths of the order 10%.
• Underhill and Harper claim that an electrically small loop antenna can violate the Chu limit

References

References

1. Jahoda, Joseph R.. (August 2006). "JTRS/SINCGARS ultrabroadband airborne blade antenna for subsonic aircraft and helicopters". RFDesign.
2. (February 1981). "Fundamental limitations in antennas". Proceedings of the IEEE.
3. Hansen, R.C.. (2006). "Electrically Small, Superdirective, and Superconductive Antennas". John Wiley & Sons.
4. McLean, James S.. "A re-examination of the fundamental limits on the radiation ''Q'' of electrically small antennas".
5. "Chu Limit".
6. Carles Puente Baliarda. (November 2000). "The Koch monopole: A small fractal antenna". IEEE Transactions on Antennas and Propagation.
7. Caimi, Frank. (August 2002). "Meander Line Antennas".
8. Underhill, M.J. (2003). "Small antenna input impedances that contradict Chu-Wheeler ''Q'' criterion". Electronics Letters.