Re: Resonance _s_ Re: Quarter Wavelength Frequency

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

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

I thought I'd just write a few lines to try to clarify
some of the notions about resonance which are in circulation.

Someone wrote:

 > Wind the wire up and the stray capacitance and others effects
 > will change the fundamental, the harmincs and add a few nearby
 > ones.

No new resonances are added in the winding up process.

 > Add a capacitance top hat and another set is _added_.

The existing resonances are shifted a bit, but no new ones are
added.   Putting on top capacitance lowers the frequency of each
mode by a factor in some inverse proportion to the number of
quarter waves in the resonance.  This means the lowest mode
suffers the most reduction, and if the added top capacitance
is large compared to the coil's own capacitance, then this
fundamental mode is pulled down much lower than the first or
higher overtones, to the extent that it can be treated as the
only resonance for many practical purposes.

In this regime of the heavily end-loaded quarter wave, the coil
current is almost the same at both terminals, and most of the
capacitive reactance is gathered into one lump. The circuit then
behaves most closely to the idealised abstractions of the
'lumped LC model'.  But the idealised model only has one degree
of freedom, ie it can only represent one resonant mode (per LC
pair).  The real resonator betrays its physical nature by
exhibiting a spectrum of overtones.

It is perhaps because under these conditions the fundamental
mode is pulled down so much lower than the lowest of the overtones,
that many treat it, unjustifiably, as a physically different type
of resonance (sometimes even to the extent, it seems, of
suggesting it can exist simultaneously with the original
unloaded resonance).

A familiar example is the TC primary, which fits into this
'heavily end loaded' class of resonator.  Without the primary
capacitor (ie with the gap open) a typical spiral primary inductor
will have its fundamental (1/4 wave) mode at perhaps a Mhz or so,
along with the usual spectrum of overtones (at distinctly non-
integer multiples of the fundamental).  This spectrum is set by the
distribution of the primary coil's inductance and capacitance,
(or equivalently, by the velocity factor along the wire!). When we
close the gap, the primary capacitance now takes part in the
resonance, and pulls all the coil mode frequencies down some.
(In particular we hope it pulls the primary quarter wave down to
the operating frequency we want.)  The primary capacitor is perhaps
around 100 times the size of the primary coil's self-capacitance,
and so the primary resonator fundamental frequency with the gap
closed will be a tenth that of the unloaded primary, ie it has
been pulled down by 90%, say.  The first overtone of the primary
(the 3/4 wave) might end up only about 30% lower than it was when
unloaded.  The next higher (5/4 wave) may be 10% lower, and so on.

When the primary resonator is fired, the overtones are excited,
not just the fundamental.  An example of this can be seen in

  http://www.abelian.demon.co.uk/tssp/md110701/

which show secondary base currents from Marco's Thor system.
These show evidence of the primary overtones appearing in the
secondary base current. (The extent to which primary overtone
energy might contribute to onset of racing arcs is unknown,
and would be a good research topic.)

But this illustrates that although we tend to treat primary
resonators as 'LC resonators', what we really mean is that we
can easily model them as an 'ideal LC resonator' with good enough
accuracy at the operating frequency.  It doesn't imply that the
primary is resonating in any essentially different way when end
loaded, than it was when unloaded.

If any coilers remain unconvinced by this snippet of EE theory,
then take any resonator and sweep it with a signal generator up
to a few Mhz and make a note of first 3 or 4 or so resonances.
Then add a little end capacitance and take another sweep.  Keep
on doing this until you have enough end capacitance for you
to be happy to call it a lumped resonator.  When you plot the
resonant frequencies, you'll see how they're all pulled down
in the manner described.  You'll be convinced by this procedure
that 'Lumped LC' is just a model set up to represent the
lowest mode of the resonator's spectrum.

Someone wrote:

 > A lumped L-C circuit will have only one resonance.

and a reply was:
 > An ideal one, perhaps...
 > A real one will have another where the coil resonates with
 > self C.

The real one will have lots of resonances (lets try never to
call them harmonics). All the resonances (ie fundamental and
all the overtones) involve both the self C and any end-loading
C that is present.

The 'lumped LC model' models only a single resonance (per LC
section). The 'lumped LC circuit' doesn't exist in nature, only
on paper.  But many resonant circuits look sufficiently lumped
(by the means described above, for example) that they can be
easily represented with sufficient accuracy by an LC *model*.

Those resonators which are not so physically lumpy can still
be represented by an LC model - we just have to be careful to
calculate the correct equivalent L and C values.

I've tried to draw a distinction between the 'lumped LC model'
which is a mathematical thing, and what you might call a
'lumped-looking resonator' which is a real circuit with a spectrum
of modes. Failure to appreciate this used to lead to long debates
where individuals argued that a certain resonator is a lumped or
a distributed resonator.  One enthusiast would try to settle the
matter with phase measurements.  Another would suggest that you
modify your coil to operate in 'distributed mode' rather than
'lumped' mode in order to achieve some remarkable but non-existent
effect.   But this was all futile and a bit silly, because the
choice between 'lumped' and 'distributed' is a free choice of which
*model* you care to apply to the resonator, rather than a switch
between two different physical modes of vibration.  Real resonators
always have many resonances, even the most lumped looking ones.

I'll post a little more on this topic later.
--
Paul Nicholson
--