Re: quarter wave

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

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

Gerry wrote (on the 19th):

 > Is the original difference of opinion that Jared is taking
 > a length of wire, calculating the resonance based on its
 > length (1/4 wave antenna theory), and then coiling it up
 > and thinking the resonance will be the same?

Yes.   At least he could have used 0.95 as the velocity factor
for the original straight wire, but he seems fixed on 1.0, a priori.
He might be right, but the point is he doesn't seem to have measured
it.  This is a pity - and I'll try to show why in this post.

 > is he also saying the original straight wire resonance will
 > remain and a new coiled resonance will be introduced

Yes too.  He was observing slow cycling of the spark output at
some point during tuning and attributed this to beating between two
resonances, one of which was taken to be 'wire resonance' and the
other to be 'LC resonance' (each presumably with an independent pair
of storage mechanisms, the lack of which was stalling Jared's
further progress with that hypothesis).

 > Seems like Jared believes the path of energy is conducted
 > thru the wire as oppose to thru the air being carried by
 > EM fields.

Yes, a common enough belief. (After all, didn't 'Tesla Himself'
believe such things?)

 > My cut on this (helical coil) is the entire coil is being
 > bathed by EM energy propagating thru the air at light speed
 > so it doesn't follow the wire path.

Yes, there's no need to suggest the field is confined to propagating
along the wire (the charges are confined to the wire, but not
necessarily the energy flow or signal flow associated with them -
any more than they are in say a transformer. Energy and signals are
exchanged between the windings without a charge transfer.)

One can certainly begin to picture the energy flow as spiraling
around the coil following the path of the current -  to a first
approximation.  Two main components of the field are a B field
parallel to the axis (due to the circular current motion) and a
radial E field.  Picture the cross product of these two as a
field of arrows which would join to make circles around the
solenoid.

At the same time, we can cross product the radial E field with the
circular B field due to the net flow of charge up and down the coil.
The result is energy flow arrows pointing parallel to the axis.

These up and down flow arrows added to the circling arrows give
an overall spiral with similar or perhaps the same pitch as the
turns of wire.

To complete the picture, we must also look at the remaining field
components.  We have a vertical E-field, ie potential differences
along the coil, whose cross product with the circling B field
component points radially inwards and outwards from the coil,
representing the stored energy flowing to and fro into the
near field.

Putting all this together, we might try to visualise the overall
effect as a spiralling energy flow, swelling outwards and collapsing
twice each RF cycle.  We know that for low frequencies at least,
the effective pitch of this spiral must be greater than the winding
pitch because signals from one end of the coil arrive at the other
end well before they would if confined to the same pitch as the
wire.  In terms of inductance and capacitance, we might say that
mutual coupling is allowing the signals to leapfrog the turns to
some extent (equivalently, those axial-pointing Poynting vectors).

It is interesting to look at the velocity factor (with respect to
the wire) for a set of resonances.  With figures from one of
Marc Metlicka's coils (h/d=4.66, 2898 turns) with 2080 metres of
wire, we find

Mode    Freq       Free space length    Velocity factor of wire
1/4     61.9 kHz   4847m * 1/4 = 1212m    2080/1212 = 1.72 [+]
3/4    157.9 kHz   1900m * 3/4 = 1425m    2080/1425 = 1.46
5/4    229.7 kHz   1306m * 5/4 = 1633m    2080/1633 = 1.27
7/4    294.4 kHz   1019m * 7/4 = 1783m    2080/1783 = 1.17
9/4    355.6 kHz    844m * 9/4 = 1899m    2080/1899 = 1.10

For shorter and shorter wavelengths in the coil, we find the wire
velocity factor coming down towards unity, as if the pitch of the
'field spiral' was becoming more closely aligned with that of the
wire spiral.  Terry Fritz also provided some figures for a coil
(h/d=2.92, 1000 turns), this one with 819 metres of wire,

Mode     Freq       Free space length    Velocity factor of wire
1/4     148.4kHz     2022m * 1/4 = 506m      819/506 = 1.59 [+]
3/4     353.4kHz      849m * 3/4 = 637m      819/637 = 1.29
5/4     513.8kHz      584m * 5/4 = 730m      819/730 = 1.12
7/4     666.4kHz      450m * 7/4 = 788m      819/788 = 1.04
9/4     819.8kHz      366m * 9/4 = 824m      819/824 = 0.99
11/4    977.4kHz      307m * 11/4 = 844m     819/844 = 0.97
13/4   1133.1kHz      265m * 13/4 = 861m     819/861 = 0.95

We see with this coil the along-wire velocity is pulled right
down to a value 0.95 which would be a typical factor for a
straight wire.  The coil is behaving at these higher frequencies as
if it were not wound at all.

Now there are a few speculative matters worth listing:-

a) As the in-coil wavelength becomes shorter, the total mutual
coupling affecting a given point on the coil becomes an average
over more and more wavelengths of the signal and therefore might
be expected to tend to zero. If so, the propagating wave is
not able to 'leapfrog' so much, and the velocity comes down to
that of the wire itself.

b) It may turn out that for an infinite solenoid, the velocity
factor is unity (or some other constant near unity) for all
frequencies, perhaps for reason (a).

c) If (b) is so, then we might legitimately interpret the trend
towards effectively higher velocity factors for low frequencies as
simply an end effect, ie brought on by the finite length of the
coil interrupting long-range cancellation of mutual coupling when
below some frequency.

With these kind of considerations in mind, it would be interesting
to see what the velocity factors were for each mode of a toroidal
coil so that we are free of major end effects.  It may turn out that
as Jared was assuming, the velocities measured along the wire are
all close to unity for that type of coil.

On the other hand, it may turn out that the toroidal coil mode
spectrum also exhibits the same trend from high velocity factor down
towards unity with increasing frequency that we always see with
solenoid coils.

So there's an experiment to try.  The result would determine whether
we should view the departures from unity velocity factor as

1) something inherent in the helical winding - a consequence of
frequency dependence of the mutual coupling.

and/or

2) something resulting from the finite lengths of our solenoids, so
that an infinite solenoid would show no dispersion (ie a constant
velocity <= 1.0 at all frequencies).

In other words, when we like to say that L and C are altered in a
complicated way during winding such that their product LC does not
stay constant, we could then say whether this was actually due to
the winding itself, or due to the winding coming to an abrupt end,
or maybe some mixture of both.

Another way to present the 'end-effects are the cause' view is to
picture the waves travelling at 'c' following the wire spiral, but
allowing that they don't have to complete a full traverse of the
coil.  The impedance changes as you approach the ends and so a
travelling wave would see a gradual rather than a sudden sharp
discontinuity, especially at low frequencies.  Thus some of the wave
energy would begin to reflect early and so the effective axial
length of the coil for that particular mode would be reduced below
the overall physical length of the coil.

The 'mutual coupling averages to zero at HF' hypothesis would be an
effect dependent on the mean distance spanned by the mutual coupling
from any given turn.  We might picture the distribution of mutual
coupling (both L and C) as say a roughly triangular shaped function.
As soon as the along-axis wavelength gets down to this length or
shorter, we can imagine that any positive contribution from one turn
to another would likely be matched by a roughly equal negative
contribution from some other turn within the range of the mutual
coupling.  If this case applies, then we should still see a pattern
of dispersion in a coil without end effects.

Well, some experiments there which would help us decide how to
view the coil's behaviour: Does an infinite or toroidal solenoid
show the same pattern of dispersion as we see documented above with
Marc and Terry's coils?

(It's typical that we can predict by calculation all the above
measured overtone frequencies quite accurately, and we can get the
computer to plot the fields and so on for any given coil and
frequency.  But that doesn't necessarily help us to decide how to
interpret or visualise the general behaviour in terms of the
underlying physics.  Here is a clear case where experiment can help
us to decide which physical principles form the best foundation for
our mental visualisation of how the energy propagates along the
coil.)

[+]  We can note that these figures for the quarter wave velocity
factor are in good agreement with the table provided recently by
Ed Phillips in another thread.  I'll write about those later.
--
Paul Nicholson
--