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Subject: Re: [TowerTalk] GROUND SYSTEMS
From: jimlux <>
Date: Wed, 21 Jul 2010 10:09:08 -0700
List-post: <">>
Pete Smith wrote:
> Even with the increased price of copper these days, 3/4" thinwall copper 
> tubing is still pretty cheap.  I recall that someone a few years ago 
> had, on a web page, a RF grounding installation made entirely of 3/4" or 
> 1" copper tubing.
> I'm a little surprised, though (not knowing any better) that the 
> inductance of a tube at HF is only as good as the tube flattened 
> (roughly equivalent to 1 1/4" strap - I would have thought it would be 
> closer to strap the size of the circumference of the tube, or twice that.

I've been trying to come up with a good simple conceptual explanation 
for why this is, and I'm going to draw some pictures and post it 
eventually, but here goes, in words alone..

Imagine that the tube is a bunch of parallel wires arranged in a circle. 
  Say there's 8 of them, for now..

Start with one single wire.. it has some inductance L.

Now, our tube is represented by 8 parallel wires.. If all those wires 
were a long distance apart, but in parallel, the overall inductance 
would be L/8  (resistance too..)

But they're not a long distance apart.  The magnetic field from one wire 
interacts with all the other wires, so there's some mutual inductance 
between them.  That is, the inductance of two parallel wires (each of 
inductance L) is not L/2, but somewhat more. (Ltotal = (L+M)/2) The 
closer the wires, the bigger the mutual inductance, M, and the closer 
the inductance gets to just L (if the two wires are on top of each 
other.. not physically possible, but as an example)

OK, back to the tube...

Picking a given wire, say the one at the top (12 o'clock position).. It 
has the least coupling to the one that's at the bottom (6 o'clock), and 
gradually more to the ones closer on each side.  Calculate all the 
parallel inductances and you can work out what it is.

Now squash the tube (from the top), so 12 and 6 o'clock are closer to 
each other.

The Mutual L for 12 and 6 is now greater, BUT, the mutual L for 3 and 9 
o'clock is less.  It doesn't exactly cancel, but the overall net effect 
is "small"

(I would imagine that an exact analytical solution is straightforward, 
assuming that elliptic integrals are your friend, and you KNOW, because 
it has circular/elliptical symmetry, that the solution will be something 
using a Bessel function)

As I recall, Terman or some contemporary of his has a description of 
this too, where he talks about the geometric mean of the distances 
between the wires.  It also comes up when calculating the inductance of 
"bundle" conductors for HV transmission lines, too (not only does using 
a bundle reduce corona losses, it also reduces inductance, which is a 
very good thing for power transmission lines)

(Grover's exact formula (page 31) for inductance between two filaments 
separated by d, of length l, (in cm)
M = 0.002 *l *[ln(l/d+sqrt(1+l^2/d^2))-sqrt(1+d^2/l^2)+d/l]

or for d<<l (the usual case), Grover recommends a series expansion:
M = 0.002 * l *[ln(2*l/d) -1 + d/l - 1/4 *d^2/l^2 + ....)

but remember that Grover was working back in the days of slide rules and 
log tables, when square roots were a pain, etc.)

Here's some quick excel data for a circle of 8 wires and a squashed tube 
of 8 wires.  You can see the total M doesn't change much
                Length>>        100     cm
x       y       dist from 1st   M (uH?) 
1       0                                       3 o'clock
0.707   0.707   0.765309088     0.914686261     
0       1       1.414213562     0.793167183     12 o'clock
-0.707  0.707   1.84761955      0.740562029     
-1      0       2               0.725014038     9 o'clock
-0.707  -0.707  1.84761955      0.740562029     
0       -1      1.414213562     0.793167183     6 o'clock
0.707   -0.707  0.765309088     0.914686261     
                        sum>>   5.621844983
1.571   0                       
0.785   0.1     0.23675715      1.148280776     
0.000   0.1     1.004987562     0.860673365     
-0.785  0.1     1.788196466     0.746982368     
-1.571  0       2.570796327     0.675928883     
-0.785  -0.1    1.788196466     0.746982368     
0.000   -0.1    1.004987562     0.860673365     
0.785   -0.1    0.23675715      1.148280776     
                        sum>>   6.187801903

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