Date: Thu, 02 Sep 2010 13:22:07
From: Markus Vester
Subject: Re: LF: Earth loop depth

Dear LF,
I recently discovered that I had a misconception regarding the effective area of an earth antenna, which may be interesting to other experimenters as well. It seems that short earth antennas are much more efficient than I had intuitively anticipated.

For small electrode spacing, most of the current returns through the ground in the vicinity of the wire. My understanding was that the effective loop area would then look similar to the a half-circle beneath the baseline, as depicted by the red area in the sketch. This means that for small baselines, effective loop area would scale quadratically with baseline length. This would hold until the baseline is made so long that penetration becomes limited by skin effect in the ground, and one enters a regime of linear scaling of area vs length.

Then I tried to calculate the magnetic moment for the non-skin effect case based on DC current densities in homogeneous halfspace. The curent field is similar to the electrical nearfield of a dipole. Integrating depth-weighted current densities over the halfspace volume should then give the total magnetic moment. But this integral did not converge to an asymptotic limit, but appeared to grow monotonically with integration volume. This implies an infinite effective depth of a DC ground loop!

At first I looked for an error in the integral calculations, but then I noticed that the divergence can be explained by a simple scaling argument along the following lines. At a distance r from the dipole (current Iq times length l), current density J in the ground scales as
J(r) ~ Iq l r^-3.
A large half-shell (green) around the dipole has a perimeter pi r around its equator, so there the total current would be
I(r) ~ Iq l r^-2 dr
The contribution to the magnetic moment of the shell is proportional to its broadside area A ~ r^2, which gives
dM(r) = I A ~ Iq l dr ~ constant.
This means that each additional shell will add the same amount of magnetic moment, and the total moment would indeed grow to infinity if r is not bounded by skin effect. Even though the outer fieldlines (blue) carry only a small part of the current, due to their large cross section they still contribute significantly to the loop area.

This reasoning also falls in line with a much easier analysis for the receive case. Vertically polarized groundwaves have transverse magnetic fields, which must be bounded by radial ground currents (ie in the direction of wave propagation). The finite surface resistance of the ground creates an additional radial electric field, which can simply be tapped by the electrode baseline. The induced voltage (and thus effective loop area) will depend linearly on the baseline length, no matter how short it is. Solving the equations for equivalent depth is straightforward and gives d_eff = (omega mu0 conductivity)^-0.5 = skindepth / sqrt(2).

For a crude experimental test, I took a battery operated notebook to the garden, stuck the two leads of the soundcard input into the soil, and measured the induced voltage from the DHO signal. When going from 1.5 m to 3 m electrode spacing, it went up by 6 dB (and not 12 dB), showing that pickup area scaled linearly and not quadratically with baseline.

Kind regards,
Markus (DF6NM)