Re: Etesla6 math questions

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

Original poster: "Terry Fritz" <teslalist-at-qwest-dot-net>

Hi Peter,

At 04:12 PM 1/31/2003 -0800, you wrote:
>Terry,
>       I've tried once again to figure out how Etesla6 works by trying
>to describe my limited understanding of it to someone who does have a
>fairly good knowlege of E&M.  Here is what we think is true, leading up
>to what we think is the gap in our understanding...
>
>0. Capacitance = charge per volt, we need to compute the charge on the
>    TC (toroid plus secondary) for some given arbitrary voltage on the top.

Yes,  We use Gauss's law to find the charge given an impressed (arbitrary) 
voltage on the coil.  Then, C = Q/V  "All" we need to know is the 
capacitance of this charged object.  E-Telsa's only purpose is to find the 
capacitance of this charged object in a given boundary condition space.


>1. The voltage on the toroid can be chosen arbitrarily and that value
>    will be constant across every point on the toroid, and the voltage
>    on the secondary can be approximated to be linear from the top down
>    to 0 at the base (or a better approximation can come from TSSP).

The voltage along the secondary is curve fitted, first to my random guess, 
but now to TSSP profiles supplied from Paul's work.  The program is not 
super sensitive to the profile but accurate curves have help much.  Paul 
has also shown that the secondary's inductance is "different" due to the 
non-uniform current in it.  E-Tesla now corrects the secondary's inductance 
from meter measured values.


>2. The charge inside an enclosing surface can be computed by summing up
>    the strength of the E-field normal to the surface at all points on the
>    surface.  This calculation will be independent of the shape or size
>    of the surface, in the case of Etesla6 it is a sphere that encloses the
>    entire TC.

Yes, gauss's law used to the extreme.  There is a text file that explains 
this in the program's zip files.  I think it was called "original.txt".


>3. The E-field at any point of the enclosing surface can be computed as
>    the (vector) sum of the E-fields from all the points on the surface of
>    the object(s) inside the enclosing surface.

Yes, but the program just finds the 2D field voltages and does the 
volts/distance thing.  Calculus adds it all up and does a cylindrical sweep 
for the grand total.  The program assumes the coil is uniform about a 
cylindrical axis and does the calculus thing around the center line.


>4. The E-field at a point on the object(s) inside the enclosing surface
>    depends on the charge density at that point.

Proportional.  Not sure that matters.


>Minor questions:
>
>? In statement (3) this is independent of whether the line from a point on
>   the object's surface to the measurment surface crosses through the object
>   or not (we're assuming the object(s) is conductive).

E-Tesla computes the voltage profile around an object.  If there is 
anything in the way, that will be compensated for.  See:

http://hot-streamer-dot-com/andrewb/

so a charged point that is between a grounded surface and the measurement 
surface will be blocked.


>? In statement (3) what if the line crosses significant amounts of dialectric.

E-Tesla assumes either air or conductors.  Dielectric effects are 
considered insignificant.  If half the coil were say in a pool of oil, 
things would change...  It would not be terribly difficult to add dielectrics.


>Major question:
>
>??? It seemed to us that even though the voltage on the toroid is constant
>     across all points on its surface, the charge density would not be, ditto
>     for the secondary solenoid. If the charge density is not constant we 
> have
>     a major problem computing it (and I think that is the "trick" in Etesla6,
>     but I cannot remember what it is), otherwise the above gives a fairly
>     straightforward outline of a numerical analysis approach (except that
>     even if the charge density is constant across the object surface it is
>     not clear what that density would be for a given voltage...).

The charge density certainly is non-uniform.  But that is controlled by the 
shape and E-Tesla does the "E-field relaxation thing" to find the E-field 
(voltages) around the coil. Here we see the charges building up around the 
toroid's outer edge:

http://hot-streamer-dot-com/andrewb/models/soutput.jpg

When we do the relaxation matrix, the charge density on the parts works 
out!!  Sharp edges get high fields do to high densities.  Smooth edges get 
low fields do to low densities.  The "relaxation" does this field density 
adjustment for us.



>     I remember the last time I asked you this question I did not really
>     follow the answer (something about "shrinking the sphere down to the
>     object" in a mathematically smooth way), and my E&M friend could not
>     figure it out either.

Here is the file of a coil with the Gaussean measurement sphere added:

http://hot-streamer-dot-com/andrewb/models/diag_contore.jpg

One just finds the voltage difference across the sphere's surface and adds 
it all up in a spherical integration (easier than it sounds ;-))


>For me part of the joy of coiling is getting to use my machine shop tools,
>part is watching the sparks, and part is learning some new math/physics.
>For the later I'ld really like to understand Etesla6 (and someday TSSP too).

I like machine tools too :-))))  Sparks or sort of dull to me, rather watch 
them on the scope :o)))  Math is something I find myself doing to make 
things better.  I am not real good at fields stuff but Tesla coils force me 
to do field theory ;-))  I still struggle greatly with fields...

I see what Paul's programs do and why, but I never seem to have that 
perfect intuitive view of things.  I really have to think about 
it.  However, it appears Paul's program are truly reflecting what really 
goes on and the results are astounding!!  Paul is far far better than I at 
seeing the full theoretical picture and converting that to program code and 
results!!

I pulled up the "original.txt" file below.  This is what the whole mess is 
based on.

============================
tesla-at-pupman-dot-com
Terry Fritz <twf-at-verinet-dot-com>
New Fo, Cself, Ctotal Program
1/3/99  01:17pM



Hi All,

We have often wanted to know the resonant frequency, self capacitance,
and total capacitance of our secondary coils before they are built.  Wheeler's
formula gives us the secondary inductance to a very good accuracy so 
calculating
the inductance of the secondary has never been a real problem.  The Medhurst
equation supplies us with a number for the secondary self capacitance that is
fairly accurate.  However, once you put a terminal on the top of the 
secondary,
things get bad.  The terminal is placed within the self capacitance space and
has the effect of adding to the self capacitance.  There are rules and ideas
about how to guess at this situation but guesses are all there are.  People 
have
done experiments but the experimental set up never seems to match our systems
well and the results may not be very good.  You won't find a good single 
equation
for this situation.

So.... the real problem is finding the total capacitance of our secondary
systems by calculation rather than building it and seeing how close we 
guessed.
If one thinks about all the variables the problem quickly seems impossible.

However, consider this.  The capacitance of an object is simply the charge in
Coulombs on the object divide by the voltage.  If we know the charge and the
voltage we know the capacitance (and Fo).  The voltage is really easy.  It can
be any arbitrary voltage  ( I use 100 volts... for no real reason).  Then the
problem is simply to find the charge, on the coil system, at that 
voltage.  Sounds
hard to figure out and the mental effort behind the solution is in the 
realm of
genius.  Fortunately, around 200 years ago Karl Friedrich Gauss (1777-1855)
figured it out for us.  It doesn't mater how complex or messy the 
dimensions of
the charged object are.  All that matters is what the field around it looks 
like.
Gauss came up with what is known as Guass's Relation.  It is:

"The total flux passing outward through any closed surface equals (1/eo) times
the total electric charge inside the closed surface."

In other words, if you throw any shaped charged object into a bag with lots of
little electric flux sensors sewn into it.  The charge on the object will be
equal to the sum of what all the sensors measure times eo.  Or...

Q = Sum E x eo

So... That still sounds harder than just building the darn thing and seeing
what happens :-)  However, we now know how a secondary coil's voltage is
distributed.  It is a sine shaped distribution along the length of the coil.
The top of the coil and terminal are at the same potential while the base is
grounded.  Thus we can set up a computer simulation to find the electric field
around the coil given it's dimensions.  The finite element analysis technique
to do this is well known by people who worry about such things.  It is really
very simple but takes a very large number of calculations.  So the computer
can crunch out the field distribution.  Our task (the computer's task) is to
simply place a virtual surface around the coil and add up all the flux passing
through it.  The surface can simply be a sphere with the Tesla coil contained
inside it.  This is the simplest surface to use for our needs.  There are no
unknowns here.  Just Gauss's wonderful relation, some simple math and one heck
of a lot of calculation.  We have the relation,  the math is straight forward,
and modern computers can easily do the calculations in some reasonable time 
frame.
So we have all the parts.  So... would someone please write a program to do 
this?...

Too late! :-))   I couldn't wait.  It is still an alpha version but I
think it works well.  It is called TWFreq and is available at my site:

www.peakpeak-dot-com/~terryf/tesla/misc/twfreq.zip

I'll call this the Alpha version.  It is written in DOS's QBASIC (which is
included since modern OSs don't have it anymore).  It will run on any PC.
It will run in a DOS window on NT and the like.  If it works out, someone can
rewrite it in some nice language since it is short, simple, and 
straightforward.
Programming is not one of my strong points...  I hear there are DOS 
emulators for
Macs.  If so, it should work fine on those too.  This is a straight text based
program with no fancy stuff.  It can be converted to any computer's BASIC
programming language (it needs more than 8k of RAM :-)).  Nothing 
fancy.  Expect
it to take at least a few hours to get down to a stable number.  The extra 
cash
you paid for the faster computer will pay off now.  It writes the voltage 
field
data to disk periodically so you can print the field plots out if you have 
Excel97
or some other program that can do surface plotting.  It can be modified to do
field stress too very easily.  It only does one terminal but two terminals 
or other
configurations would be easy to add.  Just a matter of putting the shape in.

Basic instructions are included and any problems found or suggestions 
should be
sent to me for fixing.  The program works fine on my system and the parts I 
can
mix and match together but only a real field test will insure it "really" 
works.
If you know your system well, please report the accuracy to me so I can 
determine
if there are any weak spots and come up with a good number for claimed 
accuracy.
There are no "fudge" factors in it now but that could change :-))

This program has never been field tested before so the guarantees are zero.
However, it should work.  I hope it works out.  It will fill a one of the few
holes we have left in Tesla coil design for the armchair coiler...

Good luck!    We'll blame Karl if it doesn't work :-))

         Terry
         terryf-at-verinet-dot-com
==================================

Although the program has had many revisions, BASIC, Qbasic, complied BASIC, 
C, C++......  It is still the same idea...  The program has been refined, 
but the basics have never changed.  It is just a computer applying gausses 
law....

Let me know if I can help with further questions.  I spent uncounted hours 
on this stuff, and others did too, to make E-Tesla what it is today.  It is 
a pretty refined program at this point.  If I think a little, I can 
probably recount a long story behind every letter of the code ;-))  But 
remember, Paul's work is far far beyond E-Tesla!!!

Cheers,

         Terry




>thanks,
>Peter Lawrence.