Re: Quarter Wavelength Frequency

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

Original poster: Paul Nicholson <paul-at-abelian.demon.co.uk> 

Jared E Dwarshuis wrote:

 > we believe that an envelope exists between L.C.
 > resonance and wire length resonance.

This seems to be where you're going wrong in your
interpretation of your coil's behaviour.  LC resonance
and 'wire length' resonance are two equivalent
descriptions of the coil's resonant modes, and should
not be thought of as two different modes of resonance
capable of being excited simultaneously in order to
produce beating or interference.

A common factor in the two descriptions is the distributed
inductance and distributed capacitance of the wire.  To
proceed in one direction, you integrate these to produce
overall equivalent 'lumped' L and C values for use in the
'LC resonance' model.  Going the other way, you proceed to
derive a propagation velocity for the coil (from 1/sqrt(LC)
where L and C are the per-unit-length values), and so
deduce the 'wire resonance' modes.   They are of course the
same set of physical resonant modes in both descriptions.

The distributed L and C of the wire depends strongly on
how the wire is arranged with respect to itself and
surroundings.   Both alter in a more or less complicated
way when the straight wire is wound into a coil.  Therefore
you cannot draw upon the reactances (and the corresponding
resonances) of the original straight wire when interpreting
behaviour of the wound structure, since the original straight-
wire distributed reactances were completely lost when the coil
was wound.

During winding, the self inductance of a wire element is
greatly increased by the presence of the neighbouring
turns being brought up against it.   At the same time, the
self capacitance of the wire element is greatly reduced,
because it is now partially shielded by the adjacent conductors
being at almost the same potential.   These two changes occur in
approximately the same ratio, give or take a factor of 2 or so,
resulting in the velocity 1/sqrt(LC) usually being within a factor
of two each way of light speed.

Now the claim that 'wire length' resonance and 'LC resonance'
are occuring simultaneously as physically distinct resonant modes
requires the coil to resonate with its wound resonance, while at
the same time somehow 'remembering' the reactive properties that
the wire once had when straight and resonating in accordance with
those too.  If this were the case, it would be possible to observe
the mode spectrum of a coil to be the union of the free-space straight-
line original wire mode spectrum, plus the normal spectrum of wound
'LC' resonances.

This is never seen, instead we always see a single mode spectrum
whose mode frequencies can be related (equally correctly) via an
LC model or via a wire resonance model, back to the distributed
reactances of the wound wire.

A typical straight solenoid has a fundamental resonant frequency
a little higher than that which the straight wire used to have.

Using the example offered to Shawn,

 > Suppose we make a hypothetical secondary with 1000 turns of
 > 22 gauge around an 8 inch diameter pipe, Medhurst predicts about
 > 11.7 Pf.  Wheelers formula gives .523 Henry while the classic
 > inductance formula gives .594 Henry, then the self resonant
 > frequency of this coil would be between 240,000 and 260,000 Hz
 > But the predicted quarter wave wire length frequency is only
 > 118,000 Hz.

Indeed so (*).  Now to be satisfied that only the 200kHz resonance
is present, it is merely necessary to sweep the coil with a
signal generator to see that there is no mode lower than this, and
in particular there will be no change in the coil's dynamics as
you sweep through the frequency that the wire used to resonate at
before it was wound.

(*) For this coil, assuming 28" wound length and mounted 4"
above a ground plane, base grounded, I get 198kHz for the 1/4 wave,
50.8mH for DC inductance,  41.7mH for the lumped equivalent
inductance of the 1/4 wave resonance, 15.5pF for the corresponding
effective lumped capacitance, and irrelevantly, the Medhurst
capacitance would be 12.7pF.

 > The coil operating at 118,000 Hz will have much larger
 > amplitudes and be easier to tune.

We're supposing here that you mean pulling the 200kHz resonance
down to 118kHz by end loading with topload capacitance.  But there
is no evidence that any special behaviour occurs when this
is done.   We know that to do so results in a satisfactory
proportion of stored charge in the topload of the TC, but there is
no reason either experimental or theoretical, to suppose that
the original straight line wire length resonance is the optimal
target to aim for.  If you were to study the dynamics of this
hypothetical coil in the region between DC and the resonant
frequency of 198kHz, you would not be able to find any measurement
which does not vary smoothly and indifferently as you pass
through the frequency corresponding to the resonance of the
original straight wire.  Likewise if you top-load the resonance
down to lower and lower frequencies - again you will not see
any measurement reach any sort of a peak or optimum as you
load down through the frequency of the original wire resonance.

 > When we run our full wave devices we can only get them to work
 > at the wire length frequency (or multiples).

Perhaps so.  It is quite feasible that winding into a toroidal
coil just happens to leave a unity velocity factor.

Or do you mean that the coils wont perform unless you apply
additional reactance in order to pull the natural resonant
frequency of the toroidal coil down to or up to the frequency
that its wire used to have before it was wound?

 > Changes in top end capacitance do not destroy the resonance;

You mention top-end capacitance, so you have, it seems, added loading.

 > ...it appears to be fixed by the primary L.C. and the wire length
 > of the secondary.

Ok, thats fine.  It might suggest the resonant modes you are exciting
are not strongly coupled to the top capacitance, ie the top-C is
perhaps near a voltage node?  Without data we can only do futile
speculation.

 > When we ran up the Levi configuration for the first time we got
 > a slow beat frequency between the two coils (a slow cycling of
 > spark length).

You must look for a more realistic explanation for this beating, one
which doesn't require radically new physics.  I took a look at
the web page

  http://people.emich.edu/jdwarshui/groundless.html

which gives a few hints as to what you're doing, but it doesn't give
anything like enough info to go on.   Referring to the arrangement
which produced the beats, a circuit diagram would be helpful, and
some indication of how you are driving the coil.  It is difficult
to draw any conclusions from the info given so far.

 > Yes the velocity appears to be very close to, if not exactly,
 > the speed of light. How close? couldn?t say. We have to base our
 > conclusions mainly on observations and calculations.

If so, then you will have measured the wire length, and measured the
resonance frequencies, and then simply calculated

   velocity (along the wire) = wire_length * resonance_freq

for the full wave resonance, etc.

The coil configurations that you're working with look to be quite
interesting and complicated and will be difficult to study. The
fact that you're using three coils, at least two of which appear
to be floating, and two of which may be capacitively as
well as inductively coupled,  makes things trickier still.
The whole system should be measured and studied carefully before
coming to any conclusions about which resonant modes are being
excited to produce the observed spark behaviour.  The explanations
given to us at present seem to be rather vague and partly
based on a familiar myth.  Plus they are not supported by any
measurements, circuit diagrams and dimensions, and so on, which
leaves us, temporarily I hope, unable to offer more reasonable
alternatives.

I think we would like to see first some basic studies of the
toroidal coil resonances themselves, ie for each coil in isolation
we would want to see what its mode spectrum was: the resonant
frequencies, and for each resonance the locations of voltage and
current nodes.  This in itself would be quite a challenge, because
the spectrum and the node locations will be sensitive to symmetry
and balance of the toroid with respect to ground, and so on.
You might be able to observe mode splitting due to asymmetry,
etc.  And you might even be able to obtain a slowly rotating
pattern of nodes by careful excitation of one of these coils
at two frequencies.

Let me thoroughly recommend studying to death just one of these
toroidal coils before even considering exploring its coupling to
other coils.  If this is not done, then when you observe
interesting behaviour of the coupled system, you will have
no firm basis upon which to offer more than speculative
explanations.  I'm sure many list members would, like me, be
interested in a close look at this type of coil.
--
Paul Nicholson
--