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Re: [TowerTalk] Ground wire

To: Gary Schafer <>
Subject: Re: [TowerTalk] Ground wire
From: Jim Lux <>
Date: Mon, 17 Sep 2007 20:30:00 -0700
List-post: <>
>> Think of it this way.  Say you had 2 wires side by side.. it has some
>> mutual L.  Separate the wires by some distance.. the mutual L is less,
>> so the overall L is less.
> -------------------------------------------------------
> Hi Jim,
> I guess you are describing 2 conductors of the same original size so that
> the overall cross sectional area is doubled. I agree that the inductance
> would be less in that case, however not half. 

No, take one conductor.. it has some inductance and some skin effect 
(i.e. the current distribution favors the outside)

Now, split that conductor into two halves.Leave them in contact with an 
infinitely thin layer of insulation.  Still the same L and Rac, right, 
because the current distribution is identical...

Now move the conductors apart.. the mutual L must decrease (because less 
of the field of one conductor is intercepted by the other),  therefore, 
the overall L is less.  And Rac is less, too.. because the field of the 
"other" conductor isn't as strong, more current will flow in the flat half.

You can continue the exercise with round conductors, etc.

> I thought Jim Miller had asked about using 2 or more wires of smaller size
> to EQUAL the cross section area of a single large wire. 
> That is why I was saying that the overall inductance would be greater than a
> single wire, because of the mutual inductance between the wires when in a
> bundle. Yes, separate them and the mutual inductance is reduced and the
> overall inductance is reduced. 

That's what I was talking about too.. same cross sectional area. 
Another example.. take a round conductor that's solid.  Now, stretch it 
and put a hole in the middle, making a tube.  That's got both lower 
inductance and lower Rac, for the same cross section.

Another example.. take the round wire and smash it flat into a wide flat 
strap.. lower Rac and lower L.

Consider any small subsection of the wire. There's a self L for that 
section, plus the effects of the magnetic fields from all the other 
sections in the wire.  Move the other sections away, and the field is 
reduced (as 1/r), so the contributions of those sections are reduced.

> Two wires of the same size does not reduce the total inductance to 1/2 of
> what a single wire is because of the mutual inductance between them. If they
> are sufficiently separated then it will be 1/2.

It's a bit more complex, especially with round wires, but.. in order of L
one wire has L
two wires, same cross section, adjacent, slightly less than L
two wires, separated by an infinite distance, slightly more than L/2
(because the inductance of a round wire is slightly influenced by 
diameter, getting higher as diameter decreases)

A decent approximation for the inductance of a straight solid wire is
L=0.0002 b * ( (2.303*log10(4b/d)-0.8)
where b is length in mm
and d is diameter in mm
and L is inductance in uH

So, you can see that the inductance goes as log(1/d), so a factor of 10 
in diameter (20 wire gauges!) changes the inductance by about a factor of 2.

Terman has a similar formula, with different units
L=0.2 * b *(ln(4*b/d)-0.75) (b,d in meters, L in uH)
>> The solid conductor is sort of the limiting case where the wires are
>> co-located.
>> You also pick up some benefit from the fact that the "surface area" is
>> larger in the multi strand case, so the impact of skin effect is less.
>> Terman has a description of this, and the equations..
> As I understand it multiple insulated wires are more effective at some low
> frequencies but at most of the lightning spectrum it doesn't really help.

That would be true.. I think Litz wire is used only up to about a MHz. 
Somewhere I have a IEEE paper that talks about this for switching power 
supplies, and they analyze all the various factors.  As the frequency 
goes up, you need smaller conductors to make it worthwhile, which 
reduces the packing density (copper cross section as a fraction of 
overall wire), which means that the length of the wire becomes longer 
(because the physical dimensions of the winding are larger), so it's a 
dimininishing returns thing.

But the inductance reducing aspect (and the reduction in E field) is a 
very big deal for HV transmission lines, since series L is a big, big 
problem in keeping the overall system stable.  The last thing you want 
is that power line being resonant at 60 Hz or a harmonic. That's one of 
the advantages of HVDC transmission.. you care less about L.


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