Re: quarter wave

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

Original poster: "Bob (R.A.) Jones" <a1accounting-at-bellsouth-dot-net> 

Hi,

Comments within the original text

----- Original Message -----
From: "Tesla list" <tesla-at-pupman-dot-com>
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Sent: Sunday, July 25, 2004 10:50 AM
Subject: Re: quarter wave


 > Original poster: "Gerry Reynolds" <gerryreynolds-at-earthlink-dot-net>
 >
 > Hi Paul,
 >
 > More comments interspersed.
 >
 > Gerry R.
 >
 >  > Original poster: "Paul Nicholson" <paul-at-abelian.demon.co.uk>
 >  >
 >  >
 >  > One can certainly begin to picture the energy flow as spiraling
 >  > around the coil following the path of the current -  to a first
 >  > approximation.  Two main components of the field are a B field
 >  > parallel to the axis (due to the circular current motion)
 >
 > from the circular current motion in both the primary and secondary
 >
 >  > and a
 >  > radial E field.  Picture the cross product of these two as a
 >  > field of arrows which would join to make circles around the
 >  > solenoid.

The effect of the spiral motion 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.
Because the pitch is so much smaller than the above the in the average TC
coil the effect would be very small.
 >
snip

 >  > >  > Now there are a few speculative matters worth listing:-
 >  >
 >  > a) As the in-coil wavelength becomes shorter, the total mutual
 >  > coupling affecting a given point on the coil becomes an average
 >  > over more and more wavelengths of the signal and therefore might
 >  > be expected to tend to zero. If so, the propagating wave is
 >  > not able to 'leapfrog' so much, and the velocity comes down to
 >  > that of the wire itself.

The velocity is a primarily a function mutual inductance per m and the self
capacitance per m (ignoring resistance, other losses and longitudinal
capacitance)
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.
The inductive coupling function can be derived from the differential wrt to
length of the standard equation for inductance of a solenoid (approximately
say 2%)
 >From memory 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.
 >From inspection I would expect as the wave length decreases wrt to the mean
inductive coupling function length, the correlation of the current decreases
and hence the retarding induced voltage decreases which  reduces the series
impedance which increases velocity.

As you suggest the retarding voltage induced in adjacent turn are not all in
phase and hence the sum is less.

For the first few mode frequencies the effect is small (becasue the wave
length is large compared to the mutual indictance function)and hence only a
small increase in velocity would be expected.
For signals with wave lengths similar to the width of the inductive function
the may be a beating  effect, which could be observed as ripple in the
velocity or cyclic variation in the position of the modes. This may be a
sensitive method to determine some of the real parameters.

For completeness at the fundamental mode with an l/d of  5 the mutual
inductance/m is approximately the total inductance divided by the length of
the coil.

Incidentally as one contributor suggested the ends of the coil are inductive
coupled. That is true but for the average coil it is less than 1% and in any
case it is cancelled with other coupling terms resonance.
Hence despite of the apparent proximity of the ends at the 1/4 wave mode
frequency no signal reaches the end from the other in less than a 1/4 of a
cycle.

Although I hesitate to disagree with Paul or perhaps I misunderstood the
point he was making (in a previous posting or possibly different thread) I
will add the following:

As most practical components are distributed to some extent its is true that
even the apparently lumped components of the primary are only approaching
the case of theoretical perfect lumped components. However 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. 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).

In addition in the wave solution the response at any frequency is the some
of all the responses of all the modes.

This is not to suggest that in practice that any given system is in a lumped
or a wave mode. Most systems are in what I will call a Maxwell Mode and that
any given system may more or less be modeled as a harmonic oscillator or
wave. Yes it is common practice to fiddle with the L and or C terms so that
even in the more distributed  case the standard harmonic equation with the
2*pi can be used.  I good example of the this is Medhurst C which is not the
intrinsic self capacitance of the coil but is a value of C that must be
added to a much larger Parallel C to get the correct resonant frequency
using the harmonic equation.  From the above it should be obviously that
near resonance (small top load) you can not simple add the top load C to
Medhurst C even if the top load did not effect the self C and that as the
response of a coil is the sum of all the modes then the harmonic equation is
an approximation that can be either made good near one resonance or a single
harmonic response to approximate the low frequency response of all the modes
(the Medhusrt approximation). (Cancellation between poles and zero may make
either approximation better this is an attempt to account for why it has not
been noticed particularly in magnifiers. Though I suspect it has and been
attribute to some other cause)

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. However the equivalent L and C are apparently
closer to the actual total L and C of the coil.   The R is a bit more
problematic unfortunately.

I think its important to understand the difference between the two
solutions.  Not only must you be careful about which L and C you use you
should be careful about what equation you use them in and over what
frequency range. In particular when trying to relate voltages or currents to
stored energy or extrapolate Medhurst frequency (large top load case) to no
top load or small top loads..

An other point with noting.  As you (Garry) suggested there are may well be
frequency dependent ends effects too and non uniform distribution effects
but probably the major effect as is typically with longitude waves is simply
the frequency dependent mutual inductive term in the longitudinal coupling.

It has just occurred to me that the case of a torus coil is particularly
amenable to the above assuming its has rotationally unformity in C and L.

Bob Jones

 >  >
 >  > b) It may turn out that for an infinite solenoid, the velocity
 >  > factor is unity (or some other constant near unity) for all
 >  > frequencies, perhaps for reason (a).
 >  >
 >  > c) If (b) is so, then we might legitimately interpret the trend
 >  > towards effectively higher velocity factors for low frequencies as
 >  > simply an end effect, ie brought on by the finite length of the
 >  > coil interrupting long-range cancellation of mutual coupling when
 >  > below some frequency.
 >
 > This sorta reminds me of the end effect on a propagating light wave
causing
 > defraction.
 >
 >  >
 >  > Another way to present the 'end-effects are the cause' view is to
 >  > picture the waves travelling at 'c' following the wire spiral, but
 >  > allowing that they don't have to complete a full traverse of the
 >  > coil.  The impedance changes as you approach the ends and so a
 >  > travelling wave would see a gradual rather than a sudden sharp
 >  > discontinuity, especially at low frequencies.
 >
 > Could this be the reason that some in the group recommend that the top
turns
 > of the secondary be space wound to provide an impedance transformation?
 >
 > Every time I reread this, I pick up more nuances.  Again, thankyou for all
 > the energy you put into this.
 >
 > Gerry R.
 >
 >
 >