Re: quarter wave

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

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

Bob Jones wrote:

 > The effect of the spiral motion [of the energy flow in the field
 > around the coil] is to generate circular polarized waves.  If the
 > pitch is a something like a 1/4 or 1/2 wavelengths it becomes a
 > helical antenna with a peak end on.

Yes, as you say, once the frequency becomes high enough.

To form circularly polarised waves beaming parallel to the axis of
the coil we need some transverse E field, which only begins to occur
noticeably when the circumference of a turn exceeds a few percent of
the free space wavelength.  Below that frequency, the E field is
largely radial and longitudinal rather than transverse - in other
words there isn't much potential difference across the diameter of
any turn.

 > I hesitate to disagree with Paul ...
 > ...in the mathematics we can identify two classes of problem the
 > lumped case and the distributed case.
 > The first case is frequently referred at simple harmonic motion
 > and the second case is solutions to wave equations with particular
 > boundary conditions.

I don't think we disagree on that point - they are certainly quite
different models - although related to the extent that the
distributed model can be phrased in terms of an infinite array of
coupled simple harmonic oscillators. The SHO is like a building
block for systems capable of supporting waves.  Solutions to the SHO
are just simple functions of time, whereas the solutions for the
wave equation are functions of position and time.   We can represent
the latter with the former by placing an SHO at every point x of a
distributed resonator, and allowing for whatever coupling is present
between each pair of SHOs we get out the wave equation for the
system.   Another way to do this is to use an SHO to represent each
normal mode of the resonator. Again an infinite array of oscillators,
but this time we are working in the discrete space of the normal
mode spectrum.  It is with this latter representation that we arrive
at the 'lumped LC model', being the SHO which represents the
fundamental mode.

 > The key and not even subtle different is that for simple harmonic
 > motion the frequency is the reciprocal of 2*Pi*square(L*CO)
 > and in the wave case for the fundamental its the reciprocal of
 > 4*l*squareroot(LL*CC),where l is the length of coil, LL and CC are
 > the mutual inductance per unit length and capacitance per unit
 > length.
 > This is approximately in our case the reciprocal 4*squareroot(L*C).

Yes, they do seem very different.  But they are closely related when
you use an SHO model to represent some resonant mode of a distributed
resonator.

 > The mutual inductance per unit length is the sum of all induced
 > voltage in an arbitrary small section of coil  from the current
 > in all other turns. .  i.e. the convolution of the inductive
 > coupling function wrt to distance and the current profile wrt
 > distance.

You put it very nicely in terms of the convolution - which is easy
to visualise if the mutual inductance function is invariant of
translation, ie M = f(|x-y|) instead of the more general M = f(x,y).
The former would apply when the radius is constant.

 > the inductive coupling function is bell shaped and drops to less
 > than a few percent with in few diameter lengths.  So for an
 > average 1/4 wave coil of l/d ratio of 5 the current profile is
 > strongly correlated (adds up) over the coupling function.

A typical mutual inductance distribution function is shown in
figure 3.1 of
  http://www.abelian.demon.co.uk/tssp/pn2511.html

 > It is also true that the capacitance and mutual inductance is not
 > evenly distributed along the coil and that there is also
 > longitudinal C so even if the wave equation is used it will still
 > only be a uniformly distributed equivalent to the real case.

Yes, the non-uniformity itself isn't much of a problem.  You can
still apply the telegraphist's equation, but with L, C, beta,
velocity factor, Z, etc all functions of position.  The longitudinal
coupling is more of a problem though.  This is discussed in section
4 of pn2511 above.

Thanks for your comments Bob, I agree with them all.  You obviously
follow this stuff pretty closely so I'm pleased you don't find
anything too serious to disagree with.  You finish with

 > probably the major effect as is typically with longitude waves is
 > simply the frequency dependent mutual inductive term in the
 > longitudinal coupling.

That's my feeling too.  If true then an infinite solenoid would
still show dispersion (not unreasonable).  So would a toroidal coil.
We await measurements, I guess.
--
Paul Nicholson
--