References: <AANLkTimUDj5oaFLwDGHhjjjFLFLsNVAhjorRAi+zs1LT@mail.gmail.com>
To: rsgb_lf_group@blacksheep.org
Date: Thu, 02 Sep 2010 13:12:08 -0400
In-Reply-To: <AANLkTimUDj5oaFLwDGHhjjjFLFLsNVAhjorRAi+zs1LT@mail.gmail.com>
MIME-Version: 1.0
From: Markus Vester <markusvester@aol.com>
Message-Id: <8CD18D91C21534A-1370-2635@Webmail-d118.sysops.aol.com>
Subject: LF: Earth loop depth
Content-Type: multipart/alternative; 
 boundary="--------MB_8CD18D91C2AD8CE_1370_4051_Webmail-d118.sysops.aol.com"
Sender: owner-rsgb_lf_group@blacksheep.org
Precedence: bulk
Reply-To: rsgb_lf_group@blacksheep.org


----------MB_8CD18D91C2AD8CE_1370_4051_Webmail-d118.sysops.aol.com
Content-Transfer-Encoding: quoted-printable
Content-Type: text/plain; charset="us-ascii"


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 groun=
d in the vicinity of the wire. My understanding was that the effective loo=
p 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 base=
lines, effective loop area would scale quadratically with baseline length.=
 This would hold until the baseline is made so long that penetration becom=
es 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-wei=
ghted current densities over the halfspace volume should then give the tot=
al magnetic moment. But this integral did not converge to an asymptotic li=
mit, but appeared to grow monotonically with integration volume. This impl=
ies an infinite effective depth of a DC ground loop!

At first I looked for an error in the integral calculations, but then I no=
ticed that the divergence can be explained by a simple scaling argument al=
ong the following lines. At a distance r from the dipole (current Iq times=
 length l), current density J in the ground scales as=20
 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 it=
s broadside area A ~ r^2, which gives
 dM(r) =3D 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 bo=
unded 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 rece=
ive 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 elec=
trode baseline. The induced voltage (and thus effective loop area) will de=
pend linearly on the baseline length, no matter how short it is. Solving=
 the equations for equivalent depth is straightforward and gives=20
 d_eff =3D (omega mu0 conductivity)^-0.5 =3D 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 mea=
sured 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 pic=
kup area scaled linearly and not quadratically with baseline.

Kind regards,
Markus (DF6NM)



----------MB_8CD18D91C2AD8CE_1370_4051_Webmail-d118.sysops.aol.com
Content-Transfer-Encoding: quoted-printable
Content-Type: text/html; charset="us-ascii"

<font color=3D'black' size=3D'2' face=3D'arial'>
<div>Dear LF,</div>


<div>&nbsp;</div>


<div>I recently discovered that I had a misconception regarding the effect=
ive area of an earth antenna, which may be interesting to other experiment=
ers as well. It seems that short earth antennas are much more efficient th=
an I had intuitively anticipated.</div>


<div>&nbsp;</div>


<div>For small electrode spacing, most of the current returns through the=
 ground in the vicinity of the wire. My understanding was that the effecti=
ve loop area would then look similar to the a half-circle beneath the base=
line, as depicted by the red area in the sketch. This means that for small=
 baselines, effective loop area would scale quadratically with baseline le=
ngth. 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.</div>


<div>&nbsp;</div>


<div>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 dep=
th-weighted current densities over the halfspace volume should then give=
 the total magnetic moment. But this integral did not converge to an asymp=
totic limit, but appeared to grow monotonically with integration volume.=
 This implies an infinite effective depth of a DC ground loop!</div>


<div>&nbsp;</div>


<div>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 argume=
nt along the following lines. At a distance r from the dipole (current Iq=
 times length l), current density J in the ground scales as <br>
&nbsp;J(r) ~ Iq l r^-3.<br>
A large half-shell (green) around the dipole has a perimeter pi r around=
 its equator, so there the total current would be<br>
&nbsp;I(r) ~ Iq l r^-2 dr<br>
The contribution to the magnetic moment of the shell is proportional to it=
s broadside area A ~ r^2, which gives<br>
&nbsp;dM(r) =3D I A ~ Iq l dr ~ constant.<br>
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 bo=
unded 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.</div>


<div>&nbsp;</div>


<div>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 direct=
ion of wave propagation). The finite surface resistance of the ground crea=
tes an additional radial electric field, which can simply be tapped&nbsp;=
 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 giv=
es </div>


<div>&nbsp;d_eff =3D (omega mu0 conductivity)^-0.5 =3D skindepth / sqrt(2)=
 .</div>


<div>&nbsp;</div>


<div>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 tha=
t pickup area scaled linearly and not quadratically with baseline.</div>


<div>&nbsp;</div>


<div>Kind regards,<br>
Markus (DF6NM)<br>
</div>


<div style=3D"FONT-SIZE: 10pt; COLOR: black; FONT-FAMILY: arial,helvetica"=
>

<div id=3DAOLMsgPart_2_50ca1a4a-e30f-4696-be2e-0f899db8bbb4>&nbsp;</div>
<!-- end of AOLMsgPart_2_50ca1a4a-e30f-4696-be2e-0f899db8bbb4 --></div>
</font>

----------MB_8CD18D91C2AD8CE_1370_4051_Webmail-d118.sysops.aol.com--