Re: Quarter Wavelength Frequency

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

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

Gerry wrote:
 > The "1/4 wave" resonance (if we choose to use this name) must
 > then refer to what resembles a standing wave in the distribution
 > of currents and voltages along the length of the coil.

It's a good name to use, as is any similar sort of n/4 or n/2
term, etc.  It classifies the resonance in a very general way,
summing up immediately the number of nodes and antinodes
to be expected, and gives a qualitative impression
of the spectrum.   The classification applies to fairly
evenly distributed structures, such as coax and open wires
and so on, and equally well to the more clumpy distributions
that people like to think of as 'lumped circuits'.  We might
perhaps treat these terms as 'topological' descriptions
of each resonant mode, summarising characteristics which are
invariant to network-preserving deformations of the reactance
distributions.

Gerry wrote:
 > If the voltage profile (unloaded) is close to sinusoidal,
 > then the voltage gradient must look like the current profile
 > (the greatest turn to turn stress at the bottom).

Yes, dV/dx is roughly proportional to current at x.

(If we take account of mutual inductive coupling, dV/dx is
proportional to a weighted integral of the currents all along
the coil, the weighting function being concentrated in the
neighborhood of x.  The weighting function is based on the
inductance matrix of the coil L(x,y) which gives the EMF induced
at x by a current at y.  Similarly, the current gradient, dI/dx
is given by a weighted integral of the voltage along the coil,
this time the weighting function comes from the capacitance
matrix C(x,y) which gives the charge induced at x per volt at y.)

 > Could you explain why the turn to turn stresses seem to go
 > up and often results in racing arcs at about the 2/3 point
 > up from the bottom when the coil is run out of tune?

But there's not only the turn-turn field to consider, there's
also the radial field, and the actual breakdown would
depend on the vector sum of these.  For example, these
are animated for an in-tune system of k=0.1 in

  http://www.abelian.demon.co.uk/tssp/pn040502/tfsm1-h0.grad.gif

You can see the vertical (turn-turn) surface gradient has a lot
of HF ripple but the max value is broadly the same all the
way up - consistent with the roughly linear voltage rise with
height of this toploaded coil.  At the same time the radial
component of the surface field is steadily rising as you go
up the coil.  Therefore the region of highest total surface
stress is more likely to be in the upper half than the
lower half.

There's lots of possibilities for racing arcs, I listed a
heap of them in

  http://www.pupman-dot-com/listarchives/2002/July/msg00429.html

but I don't really know which of these, if any, might
apply to any particular case.

FWIW, my gut feeling on racing arcs leans towards the HF
components of various origins, rather than the lower modes.
--
Paul Nicholson
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