There are some interesting tradeoffs when you start looking at antennas with multiple elements, phased arrays in particular. If you make the individual elements small (compared to a wavelength) then you reduce the element to element coupling, which makes the feed networks simpler.

Of course, you also reduce the efficiency, but that might not be a big problem for HF. On receive, you're generally atmospheric noise limited, so a preamp can make up the difference. On transmit, more power is an option; subject to licensing limits and high power breakdown. Physically small antennas tend to reactive, and tuning out the reactance with a passive device results in a high Q tuned circuit, so voltages and currents may be quite high. (This aggravates the loss problems)

Physically small radiators using reactive networks tend to be narrow band (unless resistively loaded and even more inefficient), however, since amateur applications are narrow band, this shouldn't be a problem as long as you can tune them reasonably quickly. Once again, the difference between receive and transmit means that one could spoil the Q with resistive loading on receive for "broadband receive monitoring" kinds of applications.

D. Jefferies suggests that for small antennas, the dominant efficiency effects will be from the surroundings (ground loss, etc.).

Here, I'm going to assume that the physically small antenna is tuned by some reactive component that is adjusted. I calculate efficiency as the ratio between the radiation resistance and the sum of radiation and loss resistances.

Rr = 20 * pi^2 *[D/lambda*pi]^4

=19227 * (D/lambda)^4

D. Jeffries provides the following little table:

D/lambda | R(ohms) about |

1/10 | 1.9 |

1/15 | 0.38 |

1/20 | 0.12 |

1/30 | .024 |

1/50 | .0031 |

(after D. Jefferies: http://www.ee.surrey.ac.uk/Personal/D.Jefferies/antennexarticles/cloops.htm)

Losses will be primarily in the IR loss of the loop. The loss in the capacitor will likely be fairly low.

http://www.tcibr.com/PDFs/612625webs.pdf Loop arrays for receiving from TCI

(monopole over a ground plane, short compared to wavelength, so uniform current assumed)

Rr = 1580 * (L/lambda)^2

(after T. Rauch at
http://www.w8ji.com/radiation_resistance.htm

see also:
http://www.ee.surrey.ac.uk/Personal/D.Jefferies/antennexarticles/radimp.htm)

Updating our little table:

H/lambda | R(ohms) about |

1/10 | 15.8 |

1/15 | 7.02 |

1/20 | 3.952 |

1/30 | 1.75 |

1/50 | 0.632 |

Here the loss is primarily in the loading coils.

More math and algorithms

The ARRL antenna book includes a program called Mobile.exe, a DOS application which does some sorts of calculations for short whips. I assume (but don't know, because there's no documentation) that it uses the equations in the antenna book.

C = 17 *length/( (ln(24*l/D)-1)*(1-(f*length/234)^2) ) (feet, inches)

(converting to consistent unit of measure)

C = 5.18 * length / ( (ln(2*length/D)-1)*(1-(f*length/(767.52))^2) )

(I'm sure that there's a cleaner equation, in terms of epsilon0. Certainly, the f*length/constant term is reminiscent of the usual length/lambda (= length / (300/f)) kind of thing. Maybe 2*pi*length/lambda?

Here's the next set of design equations

Eq 4: Rrad = h^2/312

h in degrees, valid only for sinusoidal current distribution and no reactive loading.

omitting the derivation in the handbook, Eq 9 reads:

Rrad = .0128 * A^2

where Eq 10 gives:

A = .5 * ( h1*(1-cos(h1) + h2*cos(h1))

where h1 is elec length of base in deg, h2 elec length of top in degrees

Eq 11, giving loading coil reactance

Xl = -j Km cot(h)

Km is mean characteristic impedance (whatever that is?)

Eq 12: Km = 60 ((ln(2H/a)-1)

H and a are height and radius of the antenna, respectively, (in the same units)

Eq 13, for a loading coil in the middle:

Xl = j * Km2*cot(h2) - j Km1 * tan (h1)

where Km2 and Km1 are the "characteristic impedances" of the top and base, respectively.

Eq 17 gives efficiency as:

Rrad*100/(Rrad + Rgnd + Rcoil*(cos(h1)^2) )

radio/antenna/small.htm - 19 Aug 2004 - Jim
Lux

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