Re: Wire length, coil geometry, and velocity factor

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

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

First, some news of more progress...

We know from measurements that signals traverse a solenoid winding
rather faster than they would do if the wire were straight.
Ed Phillips has shown that the apparent speed-up factor relates to
the h/d ratio in the manner plotted in

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

Malcolm Watts has supplied measurements for a coil h/d=17.5, which is
by far the largest h/d coil we've ever looked at.  This accounts for
the lonely green cross way over on the right.  The red dot just
beneath it is from the tssp model of Malcolm's coil.  Clearly, both
model and measurements continue to follow the log(h/d)*0.39 + 1.19
function. Malcolm measured 1.0697MHz, the formula predicts 1039.6Mhz*,
so only -2.8% error from the formula, even at this extreme.

Now a little more expository waffle.  Some of this repeats things
already said, but maybe in a different way...

In a straight wire, the H field lines are circular around the wire,
and the E field lines are radial.  The resulting Poynting vector
(which points out the direction in which the conducted energy is
actually flowing) is parallel to the wire at every point in the
space around the conductor.  This means that the signal energy is
guided by the conductor and is confined to move in exactly the same
direction as the associated current flow.[+]

After the coil is wound, a piece of the wire experiences the E and B
fields from some of the rest of the coil in addition to its own
field.  As a result, the Poynting vectors need no longer be parallel
to the wire - which means that signals will propagate along the coil
at a speed likely to be different to that of the straight wire.

We usually express this field coupling between remote parts of the
coil in terms of mutual inductance and mutual capacitance so that
we can apply circuit theory calculations to the coil instead of
having to deal directly with the fields using Maxwell's equations.
This works well and allows us to confirm by calculation Ed's
relationship for the the change in velocity as the coil geometry
varies.

Without any prior knowledge, we might have guessed that the greatest
influence on signal velocity might have occured at small h/d, in
which mutual coupling of both kinds is at its strongest.
But, interestingly, we see from the graph above that the overall
trend is that the greatest speed-up occurs with larger h/d.

For small h/d, it is apparent that the combined effects of mutual
reactance, despite being stronger, are cancelling each other out to
a large extent, resulting in less deviation of the signal flow from
the path of the wire.

At somewhere around h/d = 0.6 the effects seem to balance to give
unity velocity factor. This value is not too far from the h/d = 0.45
known to maximise the inductance of a given length of wire.

The velocity 'along the wire' is given by 1/sqrt(L*C) where L and
C are the reactances per unit length of wire.  When the wire is
wound, the magnetic field around any piece of it is increased by
the field from nearby turns carrying current in the same direction.
Therefore L is increased quite a lot as a result of winding (no
surprise there) and the velocity is reduced accordingly.

Working against this is the electric field effect from the nearby
turns - their mutual capacitance.  In the straight wire, the E field
lines exit the wire radially and head off to infinity, so determining
the wire's self capacitance.  But in the winding, the proximity of
other conductors at nearly the same voltage produces a shielding
effect - hindering the outflow of E-field lines.   The net result is
less charge is needed for a piece of the wire to reach a given
voltage, ie there is less total capacitance on each bit of the wire
and the velocity thus increases.

Ed's tabulated function shows how these two effects compete with
each other as h/d varies.  At large h/d, increase of L wins and
the velocity is high.  At low h/d, the reduction of C seems to
overcome the increase in L and the velocity is low.

For the 3/4 wave and 5/4 wave overtones, the graphs are

  http://www.abelian.demon.co.uk/tmp/ph3.gif
  http://www.abelian.demon.co.uk/tmp/ph5.gif

in which the functions roughly fitted to the modelled data are

   Ph3(x) = ln(x + 2.7) * 0.6 + 0.21
   Ph5(x) = ln(x + 5) * 0.65 - 0.28

[+] In a real wire, there is usually some resistance and so there
is a longitudinal voltage gradient, ie an E-field component parallel
to the wire.  The effect of this is to make the Poynting vectors
bend inwards slightly towards the wire.  This inflowing energy
component represents the power entering the wire itself to be
dissipated in the I^2R loss.
--
Paul Nicholson
--

 >> * I am sure Paul meant 1.0396Mhz - Terry <<