Re: Series resonance/Was: Waveguide TC

["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>

Jim wrote:

 > I'll bet a typical TC secondary, running over a ground plane,
 > has a varying L/C ratio (impedance) as you move along it, making
 > the transmission line analogy a bit more strained (it's a "tapered
 > transmission line" with a tapered propagation speed, as well)

Yes. No need for the ground plane - the solenoid itself is not
described accurately by the telegraphists equation.  The
predicted resonant modes are at the wrong frequencies.  A couple of
extra terms are necessary - terms which take account of mutual
coupling (capacitive and inductive) between remote parts of the coil.
When these are included, the predicted spectrum of the solenoid
accurately matches the measured values.  So too do the predicted
voltage and current profiles, for example see Terry's measurements:

  http://www.abelian.demon.co.uk/tssp/pn2510/
  http://www.abelian.demon.co.uk/tssp/tfcp260302/

The long range mutual coupling qualitatively alters the current
distribution, shifting the current max some way above the grounded
end of the coil, and creates a point of inflection in the voltage
profile.  Neither feature is predicted by the unmodified
telegraphist's equation.

In transmission lines familiar to RF engineers, such 'longitudinal'
coupling is negligible and the differential equations for the line
need only relate the V and I at a point to the rate of change of
V and I at that same point.  However when longitudinal mutual
coupling is included, the extra terms involve integrals over the
whole line.

In view of this, IMO, the most elegant mathematical description of
the transmission line is based on integral operators over the
vector space of voltage and current distributions.  You can
start out with a lumped LC circuit model, with its associated
network equations, and systematically replace each L and C with
its corresponding integral operator, and at the same time replace
each nodal voltage and mesh current with a V or I vector.

For my attempts to write about this, see
  http://www.abelian.demon.co.uk/tssp/pn1401.html

but it really needs the attention of a mathematician to present these
concepts clearly.

The solenoid exhibits a significant variation of velocity factor
with frequency, and therefore there is a lot of dispersion. A step
impulse into the coil is reduced to a jangling mess after just a few
transits of the coil.

At much higher frequencies than TCs use, the long range mutual
coupling is of less importance and the short range coupling between
neighboring turns begins to determine the electrical properties.

As a further complication, the characteristic impedance is complex
rather than real, and varies with frequency.

As the solenoid gets longer and thinner, the relative effects of
the longitudinal mutual coupling get weaker and the behaviour
gradually becomes more like that of the vanilla transmission line
familiar to RF engineers.

 > When working with systems with potentially large amounts of
 > standing waves (antennas with large reactive feed point impedances),
 > I tend to use lumped models, more akin to a power engineer working
 >  with reactive and active power.

Demonstrating the interchangeability of the lumped and transmission
line concepts.  The RF engineer will happily speak in terms of
capacitance and inductance when presented with a reactive feed point,
and might use transmission line stubs to manufacture the necessary
conjugate effective lumped reactances to achieve desired matchings.

The (vanilla) telegraphist's equation is easily treated analytically
to produce a nice simple formula to calculate the impedance seen into
the end of a transmission line.  Because of the additional terms, the
solenoid equation doesn't seem to accomodate the same process and
a numerical solution has to be made.
--
Paul Nicholson
Manchester, UK
--