Re: Inductance of wire

"Benj" <bjacoby@xxxxxxxxxxx> wrote in message news:7696d1ca-be7f-4237-a82c-7972543e4388@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Jun 30, 10:31 pm, " Don Kelly" <d...@xxxxxxx> wrote:
---------
In fact, there is no need to go to a huge matrix because the factors that
you don't know limit the possible accuracy. So , do you consider 100
filaments, 10 filaments or?? knowing that a final value within 2-5% of the
actual value is better than expected?
> -------------

What you say is true as far as practical calculations go because
normal construction variations mean that usually as you say 2-5% is
really good enough as if there is a need for accuracy it's adjusted
manually during final phases of construction etc. But my thoughts
were more along theoretical lines. The open question goes: "Since
inductance depends only on geometry, what would be the formula one
would use that starting with an arbitrary shape of conductor in space
with some terminals on it, that using field theory, would give the
inductance one will measure at those terminals?"

A couple of observations are noted, of which I'm sure you are already
aware. One is that presently the question is for the most part,
answered only for a few rather special shapes. Coils of wire, bars of
certain shapes, solenoids etc..A general formula is not presented (as
say is done by Biot-Savart for the magnetic fields around current
elements or the Neumann formula for mutual inductance between
conductors). One problem is that in the general case, the mathematics
even if written down in general form become too complex for a manual
solution. And a major problem is that references (such as Grover and
other classics) were all done back in the days of log tables, slide
rules and manual mathematics.

So what I, Distinti, and some others are trying to do is to try to
update the ideas by including modern computers. To say that modern
numerical calculations here might have some value and we don't have to
restrict ourselves to "easy" mathematics that only humans can do
anymore. So the idea is that you can simply specify mathematical
shapes and some subdivision parameter and the computer turns the crank
for you. The limiting factor of subdivisions is just a matter of CPU
time which with modern PCs is not going to be significant unless some
extreme accuracy is needed for some complicated geometry. In essence
it is the same idea that circuit network calculations have done with
computer simulators for complex situations.



The math is there for what might be considered a multi-conductor group- if end conditions can be ignored. Once you get into end conditions- then the model will , of necessity involve some assumptions. The computer model won't get rid of all of these assumptions but may allow elimination of some of them. Grover's equation is quite similar to one that Rudenberg developed for resistance of a buried rod. I suspect that much of Grover's work was based on earlier work by Rudenberg.
In your case, it appears that the filament of concern would have to be smaller than the skin effect depth at the frequency of concern- that is- small enough that the variation of current density across the filament due to skin effect or proximity effect is negligable (and as the inductance calculation involves a logarithm - this might not be all that bad ). This might (I suspect that much of Grover's work was .
Note that work done by Grover, etc., came from a mathematical analysis which had original assumptions and then simplified things so that a useful expression could be obtained. Their analysis wasn't based on the calculation tools that were on hand although the final expressions, as simplified, were affected (as well as calculated examples).

The use of the computer vs log tables etc would not have changed the basic assumptions- but will allow fewer simplifications and possibly fewer initial assumptions. The latter appears to be your aim. You won't get a theoretical result- but possibly a better approximation.
--
Don Kelly
dhky@xxxxxxxxxxxx
remove the x to reply






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