Re: slow-wave helical resonator

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

Original poster: "Paul Nicholson by way of Terry Fritz <twftesla-at-qwest-dot-net>" <paul-at-abelian.demon.co.uk>

Gary wrote:
> I would appreciate the list's critical comments regarding the
> accuracy of the following definition

> slow-wave helical resonator
> A resonant circuit consisting of a single-layer cylindrical
> inductor wound in the form of a helix, usually including
> capacitive end loading.  The speed of a wave disturbance along
> the axis of the helix is significantly less than the propagation
> of an electromagnetic wave in free space.  A Tesla coil's
> secondary and a magnifying transmitter's base-driven extra coil
> can both behave as a slow-wave helical resonators, the latter
> propagating the wave at around 17% the speed of light in free
> space.

I'd agree pretty much with that definition.  I'm not sure that the
term 'slow-wave' adds any value, but the wave is certainly slow!
Much slower than say the wave velocity along other (looser)
helicals, eg those of a helical antenna which generates circularly
polarised EM waves beamed more or less along it's axis.  The waves
radiating from a TC solenoid tend to radiate radially outwards
from the coil instead.  The essential difference being that the
E-field direction in a single layer solenoid is largely along the
axis of the coil, whereas when the pitch or diameter starts to get
near a free space wavelength, the significant E-field instead lies
across the diameter of the coil, and the E-field direction across
the diameter also rotates as the wave ascends the coil.  To radiate
effectively along the axis of the coil, the not-so-slow, wide-pitch
antenna-type helix must be pitched so that the wave rotates
its E-field through a full circle in the time taken for the wave
to travel an 'in-coil' wavelength.  This difference between 
radio antenna helicals and close-wound solenoids is really a
difference in resonant mode, rather than simply a consequence of
the pitch.  One has a fixed axial E-field, the other a rotating
radial E-field. If you take a TC solenoid and drive it at a very
high frequency (much higher than the quarter wave resonance that
we normally use), it will start to behave more in a transverse
(radial) E-field mode than an axial E-field mode.  Hope that makes
sense!  This difference means that you cannot simply extrapolate
helical antenna design formulas for use with TC solenoids.

So instead of slow-wave helical, I'd say let's call it an
axial-mode helical resonator, which implies that it's slow, (like
all helicals), but it adds a more significant thing, because it
says that you intend to resonate it so that it creates a big axial
E-field, with little or no far field, instead of a radial mode
which generates a long narrow far field beam along the axis.
Without this extra qualification, pretty much any coil could
be described as a slow-wave helical.

The 17% in-coil velocity is a little quick.  A typical TC
secondary would have a so-called velocity factor more like 0.1%,
compared with 95% for an open wire and perhaps some 10%-90% for
a helical antenna.

For magnifier operation, the extra coil combined with the secondary
must have a total electrical length of 1/4 wave.  I'm sure the
question of what proportions of length the secondary and extra
should contribute is the subject of much debate.  Those that I've
seen appear to have most of their electrical length in the
extra coil and topload.  You sometimes see the magnifier mistakenly
described as a 5/8 wave or something along those lines, perhaps
because it is thought that taking a quarter wave resonator (ie
the extra coil and its topload) and adding the extra length of 
the driver (secondary) coil, then it must be longer.   Of course,
what happens is that the combination simply has to be driven at a
lower frequency so that the whole thing together becomes a quarter
wave and the extra coil is accounting for perhaps 1/8th of a wave,
and the driver coil and topload each provide around 1/16th wave. 

Incidentally, for practical construction it is probably better to
calculate the effective L and C of the coils involved and treat
the thing as a lumped network, and design using Antonio's formulas.

Note that in any case, the electrical length of a piece of a wire
when wound into a solenoid, is very different from it's electrical
length when stretched out in free space.  Therefore to calculate
the electrical length of a coil, you cannot just take it to be
that of the coil's wire stretched out straight.  

The reason for this is straightforward.  In any wire, straight
or coiled, the charges moving in one part of the wire are
influenced by the fields from charges moving elsewhere in the wire.

When the wire is straight, the influence is at its least possible
value, so that the EM waves associated with the moving charges
are only a little slowed from their free space values.  

When coiled, a charge in the wire is affected by other charges
much further away down the wire, because now they are much closer
physically.  In electrical terms we say that mutual inductance and
mutual capacitance have increased.  If the two just happened to
increase in the same proportions (at every point along the wire),
then the wave velocity would stay the same and calculations based
on straight-line velocity would work.  In practice, this is not
generally the case and the difference can be a factor of two or
more.  In helical solenoids, the wound velocity is faster than
the straight-line velocity, so Fres is quite a bit higher than
you would expect.  The opposite seems to be true for flat spirals.

Some have suggested that the proper way to load a TC secondary is
to add sufficient top C so that Fres is brought down to the value
it would have been if the coil's wire had been run out straight.
There is nothing to gain from this from a theoretical point of
view, although as a rule of thumb it does seem to put you in the
right ballpark for a reasonable amount of toploading.
--
Paul Nicholson
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