RE: Mega-Sized Secondaries

[paul-at-xxxxxxxxxxxxxxxxxxx (by way of Terry Fritz <twftesla-at-xxxxxxxxxx>)]

John, All,

Sorry for the delayed reply - work has been hectic this week!

John Couture wrote:

> Checked out the website. Four days to compute one Tesla coil
> parameter! Do you have something much faster that could be used
> in a program that coordinates and calculates about 46 TC parameters?

Gosh, no. It might come down to about 1 day after some software
improvements. Bear in mind this is a 'research' model intended to
explore the secondary physics at high precision. We can hope that
eventually it may be used to validate empirical formulae more suited
to practical design calculations.

> Regarding the charge conservation what is the approximate overall
> percent difference between the current at the bottom of the secondary
> compared to the current at the top of the secondary coil? Could you
> break this up into corona losses and charge conservation?

Our modeling efforts are presently confined to linear steady state AC
small signal conditions. The complicating effects of the voltage and
frequency dependent corona/arc loading can only be accommodated by
estimating in advance the extra effective impedance presented to the
top of the resonator.

> The charge conservation is new to me.

I'm sure it isn't - its just Kirchhoff's first law, itself a corollary
of Ampere's law. As charge flows along the coil, some of it must
'stick' in order to place potential across the distributed coil
capacitance. Apply Kirchhoff's first law to each and every point on the
secondary. At each point you have a current coming in from one end, and
a current going out towards the other end, and a third current,
Maxwell's 'displacement' current, going out 'sideways' into the
E-field.

Terry Fritz wrote:

> As far as the currents go.  "I" would say it is roughly proportional
> to the ratio of self capacitance and top load capacitance compared
> to total capacitance. 

The rough proportionality becomes exact in the lumped model,
 
> My lumped models predict for my big coil:
> 
> Cself 15.7pF
> Ctop 28.5pF
> Ibase 320mA
> Iself = 103mA
> Itop 190mA
> Istreamer 27mA
> 
> Cself + Ctop = 44.2pF
> 
> 15.7 / 44.2 x 320 = 114 mA  (close to Iself)
> 28.5 / 44.2 x 320 = 206 mA  (close to Itop)

If you apply Kirchhoff's first law to the top of the coil, you are
entitled to deduct the Istreamer from the Ibase before sharing the
remaining current between the two capacitances, so the top current,
net of streamer current, is 320 - 27 = 293 mA. 
Then,

 15.7 / 44.2 x 293 = 104 mA  (equal to Iself)
 28.5 / 44.2 x 293 = 189 mA  (equal to Itop)
  
The Iself in this lumped equivalent model is that part of the base
current which would have made it to the top were it not for the need to
displace charge across the distributed coil self capacitance.

The L and C effective in the lumped models will not be the same values
as the DC bulk inductance and capacitance of the resonator, since the
current is not uniform and the voltage profile is not a linear rise.
Typically the effective C is much less than the bulk DC capacitance to
earth of the coil, and interestingly the effective L is sometimes
higher than the DC inductance, due to current circulating around the
internal capacitance of the coil which doesn't appear at the ends but
does add to the total EMF. If the effective lumped equivalent L and C
are calculated accurately enough, eg by a finite element model, then
the exact proportionality mentioned above is obtained. At this stage I
cannot suggest an alternative effective empirical way to estimate these
equivalents, although I'm certain that progress will be ultimately made
along these lines.

John Couture wrote:
 
> Are they the fourth order differential equations that the Corums
> showed in their TCTUTOR?

It must be an 'entertaining' piece of mathematics which obtains a 4th
order equation for a system governed by the 2nd order equations of
Maxwell.

> Are these the equations your Linux computer is using?

No, just real world 2nd order equations, reduced to linear finite
differences in steady state phasor quantities, then solved by gaussian
elimination. No new science.

> I would be interested in the Linux results...

To get a feel for the kind of current and voltage profiles predicted
for a resonating tesla secondary, see the examples in

 http://www.abelian.demon.co.uk/tssp/pn1710/

At a resonant frequency, the differential equations for the
instantaneous I and V of the coil can be reduced to simpler 
differential equations in the just the magnitudes. These are stated
(but not derived) in a postscript document at

 http://www.abelian.demon.co.uk/tssp/pn1310.ps

The bulk of the computer time is spent calculating discrete matrix
forms of the C and Cint functions. If an empirical replacement was
available for this step, then the solution of the differential
equations would I think be within the reach of practical design
programs.

Regards All,
--
Paul Nicholson,
Manchester, UK.
--