Re: Mathematics of Tesla coils (was Re: Design tool: TC-plan V1.0)

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

Original poster: "Jim Lux by way of Terry Fritz <twftesla-at-uswest-dot-net>" <jimlux-at-earthlink-dot-net>



Tesla list wrote:
> 
> Original poster: "Loudner, Godfrey by way of Terry Fritz
<twftesla-at-uswest-dot-net>" <gloudner-at-SINTE.EDU>
> 
> Hi Marco
> 
> A great deal is known, but I would like to see someone set up the Maxwell
> field equations for a Tesla coil and solve the resulting boundary value
> problems. Then from the solutions, derive formulas for all the important
> parameters. When I made an attempt, I ran into seemingly impossible problems
> in the theory of elliptic function which I could not solve. Because I am a
> life long student of elliptic modular functions, I was embarrassed. Maybe
> this has already been done, and I have not seen it in the literature. In
> fact I have no access to any research literature.
> 
> Godfrey Loudner
> 

I suspect that you are in a situation like deriving an analytical
expression for the capacitance of a cylinder above a ground plane or the
field between two spheres.  Both rapidly turn into a long series of
elliptic integrals, and are best handled with approximations, either from
empirical data (i.e. Medhurst or spark gap tables), or by numerical
integration.

For the TC case, construction tolerances are big enough that the empirical
approximations (good to a percent or so) are good enough.

Also, once you start throwing air breakdown into the mix, deriving an
analytical expression for a system that is fundamentally statistical is
going to be very tough.  Your best analytical bet is going to be developing
a suitable numerical model, which you can then run a number of times (Monte
Carlo fashion) and from that, develop a "better" set of empirical formulae,
or, alternately, confirm that the existing formulae are accurate enough,
given the inevitable random variations in the system.