Re: max L with given length of wire

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "harvey norris by way of Terry Fritz <twftesla-at-qwest-dot-net>" <harvich-at-yahoo-dot-com>


--- Tesla list <tesla-at-pupman-dot-com> wrote:
> Original poster: "Jim Lux by way of Terry Fritz
> <twftesla-at-qwest-dot-net>" <jimlux-at-earthlink-dot-net>
> 
> This came up recently in some posts...
> 
> Turns out Maxwell (of equation fame) solved this
> problem and found that the 
> optimum coil has a square cross section with mean
> diameter 3.7 times the 
> dimension of the square... 2a = 3.7c.  Brooks and
> others refined the 
> estimate and recommended 2a = 3c as the optmimum
> shape.  L = 1.353* mu0*a * N^2
> 
>   a is the mean radius, b is the thickness of the
> coil (along the center 
> axis), c is the width of the windings.
> 
I wasnt going to cite a reference here, since
evidently you are speaking of a "square coil" and not
the common circular coil, but since the circular coil
in comparison might be relevant, I found this info to
be quite informative;
Maximum Inductance of a Coil for a given Wire Length
and Turn Spacing (By Mike Hollingsworth)
http://groups.yahoo-dot-com/group/teslafy/message/10

To add further to this info, the following fact seems
to become very relevant. Using Wheelers equation as
Mike has supplied it;

The Wheeler equation is normally used in coil
design to calculate inductance. 
L = R2*N2 / (9*R + 10*B) uF
R = radius, inches
N = number of turns
B = coil length, inches

The added dimension of thought that can be added here
is the fact that if the same length of wire is used,
(and also the same spacing between turns, but I'm not
sure making this quick posting if the second
requirement is absolutely necessary, which I am almost
sure it is not!), the top term R2N2 will be a
CONSTANT.

This implies that when we change a coils radius to
make a new geometry, but still using the same length
of wire; we can quickly find the value of B, the coils
length so that the denominator, (9*R + 10*B) former,
will equal (9*R + 10*B) latter, meaning the for any
coil we can quickly find another coil of equal
inductance, but different geometry. Since the top term
remains constant, we only need to find another value
for B to make the bottom term equal, for both coils to
be equal inductance by wheelers equation.

It takes a little more work to show that we can also
add the requirement that the new coil ALSO have equal
spacing between wires, and also equal length of wire,
but the solution in that case can also be found.  The
calculations I did many years ago for my primitive
secondary of height/ diameter ratio of 2, (not a
typical secondary, where coilers use a minimum of h/d
ratio of 3): but my calculations showed that for my
secondary at least, the new coil of equal inductance
would have a diameter about pi times the old diameter.

For my case then the two coils of equal inductance,
equal spacing of winds, and equal lengths of wire came
out looking like a ring shaped coil around a shortened
secondary. Now since magnifier systems can use such a
shortened "extra coil" (where h/d =2)the whole
geometry of these coil relationships starts looking
somewhat like the geometries that Tesla used at
Colorado Springs.

Mike's conclusion at the end of his great posting was
the following, where if you read his post he sent me
in private mailing; that I reposted to teslafy list,
because I felt it to be of great relevance to coil
builders, for the coil of max L per given length of
wire;

R = 10/9 * B

or 

R = 1.1111*B

Note: This result is in close agreement with that
found in “Electrical Engineers’ Handbook , Electric
Communications and Electronics,Edited by Pender and
McIlwain, copyright 1936, John Wiley and Sons Inc. 
which states that the ratio of the diameter to the
coil length should be 2.45 for maximum inductance

Mike also does all the math to show how this is true.

Now another question comes up, especially for the
building of "inductive only" magnifier systems as
Tesla shows in Sept 19th CSN references: here (in this
example) since the coils have equal wire length, AND
also equal inductance, will they also logically have
equal resonant frequencies ?  Here the mystery of
Tesla's cryptic comment of Sept 19th, where he says
"for obvious reasons", (obvious to who?) one coil
should be tuned to 1/4 wavelength, but the other to
3/4 wavelength. The typical argument that some
"experts" make for line connected magnifier systems is
that the TOTAL length of wire for both secondary and
extra coil is the predominant consideration to
initially base the tuning parameter on. This may be
true for the typical "line connected" magnifier
system, but the inductive magnifier system Tesla is
refering to contains no such line connections, unless
perhaps if they had a common ground, which is not
shown by diagram in those Sept 19 notes.

So what is the big mystery here? If the coil systems
are equal length, and also equal inductance, why
wouldnt they have equal resonant frequencies? (note
Tesla did NOT use this idea of mathematically equal
inductances made by Wheelers equation,to my knowledge,
but I am only citing some cooincidences of geometry
here) The reason that the coil systems would not have
equal resonant frequencies should be obvious for the
fact that the H/d ratios are different, but perhaps
more significant is the fact that since the outer coil
has a little over three times the diameter, it will
also have a little over three times the voltage
between winds, per equal impressed emf's! ,(it being
less then three times the amount of winds).  This
starts to explain why a larger h/d ratio for
secondaries of tesla coils works better, for the sole
reason that that up to a certain point of geometry
past that h/d ratio of 3, the voltage between winds is
minimized to partially do away with the hindrance of
internal capacity present as that voltage between
winds!

Thus to conclude here, IF the coils were equal lengths
of wire, and equal inductance, AND equal spacing
between winds,(this is why Tesla's CSN constructions
do not apply as an analogy because the outer coil was
wound with large spacing between winds): then we could
conclude that the outer coil would have at least 3
times the internal capacity as the inner, and this is
the reason those coils would not also have equal
resonant frequencies. In fact I could go on here to
show some scopings of spirals compared to solenoids,
where the spirals containing more internal capacity,
will have their resonant frequencies quite reduced
because they consist of flat braided wire in adjacent
layerings, allowing for more voltage potential to be
placed between winds: but the solenoids come out
exactly as modern predictions make those resonant
frequencies, but all that should be unnecessary, since
it should be generally agreed to that adding internal
capacity will reduce the resonant frequency.

Thus to finally conclude here, even though the coils
might have equal wire lengths and equal inductances;
if an outer ring coil can be shown to have at least
better than three times the internal capacity, we
might also expect a three fold reduction in its
resonant frequency, and tuning wise this would imply
that one coil should be tuned as if it were a third
harmonic of the other, FOR OBVIOUS REASONS! 

Sincerely Harvey D Norris








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