PA0JOZ webpage on Small Antennas (lots of useful references)
Basic "Chu" limit is Q>[(1/kr)^3 + 1/kr]
The Chu limit addresses Q as the ratio between the energy stored in the antenna (and it's near field) and that energy radiated away into the far field. More precisely: Q = 2*omega*mean electric energy stored beyond the input terminals/ power dissipated as radiation. It's NOT necessarily the Q as if you hooked an impedance meter up to the antenna terminals. That Q could be a lot lower, due, for instance, to resistive losses in the structure. The other thing is that the Chu limit doesn't necessarily assume that the feedpoint impedance is constant. In practice, you'd need some sort of matching network, which may have losses (there's a 1948 paper by R.M. Fano that addresses this aspect).
An important aspect is that to achieve this limit requires creating an arbitrary distribution of current within the sphere, which may not be practical. Another important aspect to Chu is that there's an assumption of a single port. Several authors have come up with schemes with multiple ports which analytically beat the Chu limit. Whether such schemes are practical is another question.
Chu gives 3 assumptions for the most favorable conditions:
Poynting's theorem says that if these conditions are met, the antenna will present a pure resistance at the terminals at the operating frequency (but not necessarily elsewhere, nor is the resistance constant with frequency).
Defining Q is a bit tricky. Are we talking gain bandwidth (driven from a matched source?) or bandwidth driven from a constant impedance source (so mismatch reduces the gain). Walden (link) describes it in terms of the feedpoint impedance (R+jX)
Q = w/2R * [ (dX/dw) + |X/w|] ( I think that w in Walden's paper is really omega)
Gnorm = (beta *R)^2 + 2*(beta*R)
as the maximum gain obtainable using wave functions of order n<=N=beta*R. (going higher in gain is "superdirective"). R is the radius of the smallest sphere that can contain the antenna.
k=2*pi/lambda (also called the wave number), also referred to as beta.
Let's run some typical cases..
half wave dipole r = 0.25*lambda
1/(2*pi/lambda * 0.25*lambda)^3 = (1/2pi)^3 = ----
Looks like a run of the mill dipole meets the Chu limit (substantially!)
how about a 1m diameter compact loop on 20m (r = 0.5m)
1/(2* pi /20 * 0.5) = 258 (54 kHz)
at 40m Q>2064 (3kHz)
Assume the antenna fits in a 50x100x30 foot box (a suburban high density lot with a house). 150,000 cubic feet. (4250 m3, a sphere 14.7 m in radius)
At 1.8 MHz (166.6 m), 3.7 MHz (81.03 m) the Chu Q is 7.3 and 1.4 respectively. The Harrington Gnorm (non superdirective gain) is 1.6 and 5.6 dB respectively.
These are for an antenna in free space. Conceivably, since you're over a (sort-of) reflective ground plane, the volume could be considered as twice that (but, that would only increase the volume by a factor of 2, and the radius by cube root(2)).
The Chu limit tells you want the "best possible" bandwidth will be, for an isotropic radiator. Harrington tells you what the gain will be, while keeping that bandwidth (Gnorm). You can get more directivity by storing more energy in the antenna, which will increase the Q (reducing the bandwidth). The other tradeoff is that increasing the stored energy will also increase the dissipative losses: from the medium the antenna is immersed in, and the resistive losses of the antenna itself.
1/Qoverall = 1/Qradiation + 1/Qdissipation
Q radiation = = Chu Q
(k is wavenumber (2pi/lambda), a is radius)
(from http://www.edn.com/article/CA179573.html Leslie J Reading, Galtronics -- EDN, 11/8/2001 "Designing Dual Band Internal Antennas")
There's a fair amount of literature on physically small antennas in connection with VLF radiators. For instance, Wheeler (1975) describes how the U.S. Navy built two huge antennas (one in Maine, and the other in Northwest Cape, Australia) for 15 kHz. They are about 2.7 km in diameter (about 1/7th of the wavelength). Of some interest is the fact that they were powered with 2MW, radiated about half that into a radiation and loss resistance of about 0.3 ohms, and had 435 MVA of reactive power circulating.
You should be able find any of these in a decent library, or in IEEE Xplore, or, perhaps, by Googling.
Chu, L.J., "Physical Limitations of Omni-Directional Antennas", Journal of Applied Physics ,v19, Dec 1948, pp1163-1175
Harrington, R.F., "Effect of Antenna Size on Gain, Bandwidth, and Efficiency", Journal of Research of the National Bureau of Standards - D. Radio Propagation, Vol 64D, #1, Jan Feb 1960
McLean, J.S., "A Re-Examination of the Fundamental Limits on the Radiation Q of Electrically Small Antennas", IEEE Trans. Ant. Prop., vol 44, #5, May 1996, pp 672-675
Hansen, R.C., "Fundamental limitations in antennas", Proc IEEE, v 69, pp 170-182, Feb 1981
http://www.ee.psu.edu/grimes/publications/PIERS2002.pdf "Low Q, Electrically Small Antennas: Mother Nature Can, Why Can't Electrical Engineers?" Grimes, D.M. and Grimes, C.A. An interesting discussion of the limitations on Chu's work (that there's a fundamental assumption that all the modes inside the sphere are driven in phase, so therefore, you might be able to come up with lower Q)
http://www.es.lth.se/teorel/Publications/TEAT-7000-series/TEAT-7114.pdf Anders Karlsson, "Physical limitations of antennas in a lossy medium"
antenna/phased/small.htm - 6 March 2005 - Jim
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