Wire length, coil geometry, and velocity factor

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

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

Recently Ed Phillips noticed that wire length divided by the free
space wavelength of the quarter wave resonance of an unloaded coil
was a smooth function of the h/d ratio and largely independent of
the turn count.

It is commonly understood (we hope) by coilers that the resonant
frequency of a coil is not given by the quarter wave resonance of
the straight wire from which the coil was wound.   The actual
frequency usually exceeds the straight wire prediction by 50%
or more in typical TC secondaries.

This implies that signals (EM waves) at TC frequencies traverse
the coil faster than they would do if strictly confined to the
helical path of the winding.  This is not unreasonable because
each turn of the coil has some degree of interaction with all
the other turns via their E and H fields (inductive and capacitive
mutual coupling).

We can indicate numerically the extent to which signals 'leapfrog'
the turns of the winding by expressing the apparent or effective
speed of propagation along the wire as a ratio with the speed
of light.   This is the velocity factor of the wire when wound [+].

For example, if a particular coil has a velocity factor of 1.72,
this in effect means that:

*) The 1/4 wave frequency will be 72% higher than that predicted for
its straight wire.

*) A signal entering the coil will appear at the other end as if
it had travelled the wire at 1.72 times the speed of light.

*) If we imagine the EM signal to be spiralling along the coil,
the pitch of this spiral would be 72% greater than that of the
winding.

Ed's observation recognises that the velocity factor for a coil
is a function largely of the overall geometry of the coil and does
not depend very much on how many turns are put in.   This means
that it is worth while defining a geometry factor with which to
relate the velocity factor directly to the h/d ratio of the coil.

The graph

   http://www.abelian.demon.co.uk/tmp/ph1.gif

shows (green crosses) measured velocity factors for a bunch of
coils of varying sizes and turns.   The blue line is a logarithmic
function chosen for a reasonable fit.  The function used is

    Ph1(h/d) = ln(h/d) * 0.39 + 1.19

where ln() is the natural logarithm.  The trial function has been
tested against a larger set of modelled coils, and the results
are plotted in the red dots.

The red dots actually represent 2732 assorted coils ranging from as
few as 50 turns up to 3000 turns.  Resonant frequencies of some of
these coils go up into the Mhz range.  We still see a reasonably
good fit to Ph1(h/d) even when faced with such a wide range of
hypothetical coils.

The residual (the difference between Ph1(h/d) and the measured or
modelled velocity factor) is about +/-5%.  Closer inspection of
the comparison database reveals that about half of this residual
is due to variation of coil base height above the modelled ground
plane.

The function Ph1() can be used to directly estimate the quarter
wave Fres from the h/d ratio and the wire length:=

    Fres = (0.39 * ln(h/d) + 1.19) * 75e6/wire_length  (Hertz)

where wire_length is the total length of wire in metres.  It seems
we should expect this prediction to be within about +/-5% of the
actual frequency for most coils.   This is obviously a very much
more direct approach to Fres than the conventional route via
calculations of inductance and capacitance.

All of the above refers the the fundamental (1/4 wave) free
resonance of the unloaded coil.  Velocity factors for other
frequencies will be different.

The next two overtones, 3/4 wave and 5/4 wave also follow a compact
smooth curve when their velocity factor is plotted against
h/d ratio.  The independence from turns continues.  However, these
overtones don't seem to follow quite the same sort of logarithmic
function of h/d.  (A database of modelled coils is available in
csv format if anyone wants to try for a fit).

It is clear that the function Ph1 (and the coils) tend towards
some velocity factor greater than 2 for large h/d. No measurements
are currently available for h/d > 10 so we can only speculate.

One common factor with all the coils modelled and measured in
this comparison is that they are all fairly close wound.
The smallest modelled spacing ratio is 0.55, and our models
continue to refer to close wound behaviour as we let h/d tend
to infinity.   For this reason we should not be surprised that
the velocity tends to some number rather larger than, say, the
0.95 we would expect from a straight wire.

To reach this straight wire velocity factor, our models and
calculations would have to allow the pitch to increase to
infinity as well as the h/d ratio.  This they do not do.

We might anticipate that the compact distribution of velocity
factors would be lost if we allowed the pitch to increase
substantially.   Unfortunately our software models are not
qualified for large pitch coils since they rely on some
approximations which are only accurate when each turn is
almost a circle.  Helical antenna modelling software might be
tried instead, but those packages tend to bog down when dealing
with large numbers of turns.   For that reason it might have to
be a task for the experimenters to tell us what happens at pitch
angles larger than say 5 or 10 degrees.

[+]
The velocity factor of the wire when straight would be about
0.95, depending on thickness.  The velocity factor of the
winding is given by

    4 * wire_length * Fres / c

where c is 300e6 metres/sec, wire_length in metres, Fres in Hz.
--
Paul Nicholson
--