What is the Chu limit?
The Chu limit is the lower bound on the radiation quality factor of any antenna that fits inside a sphere of radius a. In McLean’s exact form for a linearly polarized antenna radiating only the lowest TM (or TE) spherical mode, Qmin = 1/(ka)³ + 1/(ka); an antenna exciting TM01 and TE01 equally can reach about half this value.[1][2] Since bandwidth is inversely proportional to Q, the limit ties size to bandwidth.
Where does the Chu limit come from?
Chu expanded the fields outside the sphere in spherical modes and represented each mode by an equivalent ladder circuit. Outside the sphere, every mode stores reactive energy; even with no energy stored inside, the external stored energy for the lowest mode alone gives the bound. Higher modes store far more energy for small ka, so a small antenna should radiate only the lowest modes.[1]
What does the limit give numerically?
| ka | Qmin (single mode) | Max FBW at VSWR 2 ≈ 0.707/Q |
|---|---|---|
| 1.0 | 2.0 | ≈ 35% |
| 0.5 | 10.0 | ≈ 7% |
| 0.3 | 40.4 | ≈ 1.7% |
| 0.2 | 130 | ≈ 0.54% |
| 0.1 | 1010 | ≈ 0.07% |
Computed from McLean’s formula and the Yaghjian–Best bandwidth relation.[2][3]
How is efficiency included?
With radiation efficiency η, the bound on loaded Q becomes approximately η·QChu: losses lower Q and broaden bandwidth only by wasting power. So a small antenna claiming wide bandwidth must either be larger than it looks (e.g., using the ground plane) or inefficient.[4][3]
Frequently asked questions
What is ka for a 433 MHz antenna in a 5 cm sphere?
λ ≈ 69 cm, k ≈ 9.1 rad/m, a = 0.025 m gives ka ≈ 0.23 and Q_min ≈ 90.
Can metamaterials beat the Chu limit?
No passive, linear, time-invariant antenna can; claims usually involve losses or larger effective size.
Is the Chu limit achievable?
It can be approached closely by 3D designs such as spherical helices filling the sphere.
See also
References
Online sources accessed October 21, 2026; printed works cited by edition. See all sources.
- “Physical Limitations of Omni-Directional Antennas”. L. J. Chu, Journal of Applied Physics, vol. 19, pp. 1163-1175. December 1948.
- “A Re-examination of the Fundamental Limits on the Radiation Q of Electrically Small Antennas”. J. S. McLean, IEEE Transactions on Antennas and Propagation, vol. 44, no. 5, pp. 672-676. May 1996.
- “Impedance, Bandwidth, and Q of Antennas”. A. D. Yaghjian and S. R. Best, IEEE Transactions on Antennas and Propagation, vol. 53, no. 4, pp. 1298-1324. April 2005.
- “Electrically Small, Superdirective, and Superconducting Antennas”. R. C. Hansen, Wiley. 2006.