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Re: LC derivation II

Original poster: Terry Fritz <teslalist@xxxxxxxxxxxxxxxxxxxxxxx>


At 12:48 AM 3/23/2005, you wrote:


L = u Nsqrd Area / l

A far better and more accurate formula is the Wheeler equation:


Accurate to a percent...

 r/2l  x  Surface area solenoid = Area

Assuming spherical capacitor for the top end:

Capacitance = surface area sphere x e/R

There are theoretically "Perfect" programs out there now that do toroids and such!!


Placing both of these into:

Freq. = 1 / 2 pi sqrt(LC)

No... The secondary coil itself stands in space and has an associated capacitance as well generally known as the Medhurst capacitance.



If you add the Medhurst capacitance to the top capacitance and use wheeler's for the inductance, you start getting really close!!

But given a coil and top load shape, it is really a problem for finite element analysis (E-Tesla6):

See note far below:


Removing u and e  from the radical as C

Placing an r in the numerator of the radical and pulling an r out of
the radicals denominator. Gives us:

Frequency = C/ wire length x sqrt (  [2 (l sqrd) R r] / [ (Sa sphere x
Sa solenoid)] )

Again the units of 1/s come from the velocity of light divided by the
wire length. The rest of the expression has no units and can be
thought of as a scaling factor.

We have been told numerous times  that we have a pathological fixation
on wire length. Hmmmm,  is that so?

:-) It was popular to find the wire length as the quarter wave frequency

Lwire = C / (4 x Fo)

But the coil is magnetically coupled and the velocity factor in it is far different than a straight wire, so the equation does not have merit. But one "can" use it if you want. It will not work any better or worse... The wire length can be a wide range of lengths usually only limited by what is practical to physically wind on a given coil form. there are impedance issues too, but those are not related to the velocity of light. a nice paper on secondary voltages and current is at:


E-Tesla note:
Terry Fritz <twf@xxxxxxxxxxx>
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:


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 :-))




 Jared Dwarshuis, Larry Morris
 March 05