Dear Markus (and LF Group)

Thank you very much for your very useful comments. I have a few further comments.

(a) On item 3. below: It is useful to note that reflection at the walls of a waveguide and on the surface of a real earth depend on the polarisation of the wave (and the angle of incidence). Also there is a useful analogy with the voltage and current waves reflected at the end of an open-circuit, short-circuit, or imperfectly matched, transmission line. When we consider these reflection boundary conditions we find that mu defined as B/H is not equal to mu(0) just above a conducting or high permeability material ( just as you say). Above a dielectric material the difference between mu and mu(0) is indeed negligible. However then the difference in epsilon (defined as D/E) is large. We therefore have to consider mu and epsilon separately initially, and then combine them together progressively as we move away from the boundary surface. The ‘local’ impedance is usefully defined as Zl = (mu/epsilon)^0.5 and is not in general 377 ohms until ‘free-space’ is reached, away from the boundary. I wish this was better explained in the books! Actually the EMC engineers appear to have a better understanding of this than the antenna engineers.

(b) Item 5 really is the heart of the matter. This discrepancy and disagreement between the two theoretical approaches has existed mainly unnoticed, and certainly unresolved, for about fifty years - since Chu (and Wheeler) presented the well known ‘Q is inversely proportional to volume’ assertion that you quote. I have gone on record as challenging, and continue to challenge, the Chu assertion. The mathematics for this is impeccable but the ‘physics’ and physical assumptions are not. My objections include: no consideration of (analytic) continuity of the fields at the boundary of the sphere containing the antenna; no proof that the Chu assertion excludes the possibility of additional antenna radiation modes for which the Chu assertion does not apply; no consideration of the effects on the large (magnetic and electric) displacement currents in the space surrounding the sphere containing the physical antenna or the possibility (/probability) that such currents themselves radiate; and the sometimes large discrepancies between predicted and measured (loop) antenna efficiency results. I therefore strongly defend the ‘antenna capture cross-section related to antenna pattern and defining the antenna limiting Q’ view against the Chu-Wheeler (and Kraus electric dipole) formulas for antenna Q and radiation efficiency.

(c) I agree that earth reflections can double the radiation resistance under the right assumptions. However some surfaces and wave polarisation can actually halve the observed radiation resistance. The philosophical question then is whether the ground under the antenna is actually itself radiating or just reflecting.

(d) A further point is that some antennas close to ground, but not necessarily conductively connected to it, can launch strong ground waves and this in general will appear as an additional increase in radiation resistance at the antenna terminals.

In conclusion the story (or is it saga?) of small (loop) antennas is not yet finished. We also have to remember that field detectors are ‘reciprocal’ devices and are themselves small antennas. In my view there is considerably more to be found out on all these, particularly when used in a real environment just above ground. .

73 and regards to all.

Mike – G3LHZ

-----Original Message-----
From: owner-rsgb_lf_group@blacksheep.org [mailto:owner-rsgb_lf_group@blacksheep.org] On Behalf Of MarkusVester@aol.com
Sent: 16 March 2005 23:19
To: rsgb_lf_group@blacksheep.org
Subject: Re: LF: Untuned and tuned loops

Dear Mike and LF Group,

thank you for your inspiring comments. I would like to add in a few remarks; hope they will not be regarded as too theoretical...

>> 1. My strong support for what Jim has said. The immediate environment always affects all field measurements, not just magnetic field measurements.

The good side for loop measurements is that the magnetic field is much less influenced by conducting or dielectric surroundings than the electric field. It is easy for a tree or stone wall to shunt E to ground, whereas distorting H requires a significant current flow, which can only be supported by a large sheet of metal or a nearby wire loop. A nonresonant short pole has no effect at all. And this is "the secret" why TX loops work so well in forested terrain.

>> 2. Marcus’s formula is correct for free-space and it is in all the books. Wire loss is not an issue for a typical open-circuit loop

The formula connecting U to B is just an embodyment of Maxwell's
rot E = -dB/dt ,
it should very generally give the correct answer for B and (excluding ferromagnetic cores) also H. It's true even for an electrically loaded loop (see 4. below). However the often-used relationship between the magnetic and electric field (E/H = Zo = 377ohm) is restricted to the undisturbed far field zone.

>> 3. In an ‘environment’ near real ground, buildings etc., the ‘physics’ insists that such a loop gives an open circuit voltage proportional to B = mu * H. The presence of the environment means that mu is not equal to the free-space value of mu(0).

Yes, but this difference is mostly neglegible. Though our environments may have a lot of dielectric and or conducting objects around, they are hardly magnetic, and µr = 1 is an excellent approximation in almost any situation. At RF, even a steel wall looks like a short circuit (due to the eddy currents) rather than a "magnetic wall", which would have high surface impedance. The exeption of course are nonconducting ferromagnetics, like ferrites.

>> 4. The current in a short-circuit lossless loop is directly proportional to the H field and it does not depend on the area. This is not in ‘the antenna books’, but it is in some physics books dealing with super-conductivity.

The short-circuit current does depend on the effective length of the magnetic flux lines. For the schoolbook case of a long cylindrical coil (n turns, arbitrary core material), we have
U = - n area j omega µ H, L = n^2 µ area / length,
n I = n U / (j omega L), = - H / length ,
easily memorized by the unit "amps per meter" for H. For a "short" ring coil however, the magnetic length is a function of both the wire and loop diameters.

The original induction rule is still correct in the short circuit case, as the current creates a secondary field such that the total flux and the (averaged) H passing through the loop area is indeed nulled.

>> 5. ...What the system Q ends up being does not matter for this proof. Some may have noticed however that it is a matter of considerable (unnecessary) dispute, as reported in the pages of RadCom. However loop (or any antenna) losses do matter, because the cross-section area of an antenna is reduced in proportion to its (in)efficiency. This point is in most of the antenna books. After real conductor losses have been minimised and factored out we find practical outdoor Qs of 100 to 350. In a screened room the loop Q can exceed 600.

This raises a fundamental issue on the efficiency limit of a small antenna. For a magnetic antenna represented by a cylinder coil in free space, we would get
   Rrad = Zo/(6pi) * area^2 * (2pi/lambda)^4
   X = omega L = 2pi / lambda * c * µo * area / length,
and using Zo/c = µo we get a radiation Q-factor
   Qrad = X / Rrad = 3/(2pi)^2 * lambda^3 / volume.

For an electrical antenna represented by a plate capacitor, we have
Rrad = Zo/(6pi) * height^2 * (2pi/lambda)^2
X = 1/omega/C = height / (omega*epsilon*area) ,
and with 1/Zo/c = epsilon we again get the very same value
Qrad = 3/(2pi)^2 * lambda^3 / volume.

If we have a lossy antenna confined in a small volume, with a given Q<<Qrad we get a maximum possible efficiency of
eta = Q/Qrad = 13.2 * Q * volume / lambda^3,
no matter whether it is an electric or magnetic antenna! The energy storage is necessary to support a reactive dipole field (r^-3) between the physical antenna volume and the radiation zone, beginning at a distance lambda/2pi ("CFA"-promoters will disagree here...).

Above earth, the radiation resistances and efficiencies for both types of antenna is doubled, which is plausible if we include the image antenna volume below ground. Somewhat surprisingly, this still holds true for loops above weakly conducting ground, as long as the skin depth (a few 10m at LF) is still small compared to a freespace wavelength.

For a thin wire antenna, the "width" of the effective volume is not given by the wire diameter itself but by the capacitance of the wire. For the rule-of thumb 5 pF/m wire, we would have to use approximately half the height of the wire times its length as the effective cross section. My TX antenna (220pF at 10 m above ground, Q = 200) has an effective volume of 2483m^3 and an efficiency of 0.093%

So much for now, it has become late...
73 and good night

Markus, DF6NM

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