Tesla Coil Mathematical Modeling

This is some of the discussion between Jim Oliver and Richard Quick
(with my comments added) concerning the possibility of mathematically modeling
and predicting the behavior of tesla coils before construction.

Although some detailed modeling has not been done in the past, I contend 
that it is quite possible to do so. Some may say why bother, since tesla 
coils can and have been sucessfully built wihtout accurate models. But if the 
theory is improved and understood, then much less time will be spent on 
fruitless construction of badly coupled and tuned systems.
RQ> Quoting Jim Oliver:

 JO> It's been said before that coiling as we do it is a "sort of
 JO> art", but it's important in any serious study to separate    
 JO> the art from the science. The construction of Tesla coils    
 JO> should not be seen totally as an "art" in itself, this is    
 JO> obviously untrue as evidenced by the huge number of          
 JO> scientific studies of the "Tesla transformer" and associated 
 JO> hardware. There is ample proof that the operation is easily  
 JO> and accurately predicted by theory given accurate components 
 JO> and construction. 

 JO> If it is regarded as an art, then this shows some reluctance 
 JO> on the part of the constructor to accept the part played by  
 JO> the science and theory of the coil. By understanding this    
 JO> well we can begin to build with real confidence. Of course   
 JO> constructors of long experience probably apply more science  
 JO> than they are aware of, or are able to explain to less       
 JO> experienced people.

RQ>  I don't think that any person would dispute that Tesla was, and
RQ> remains, the supreme master of resonate coil systems. Yet Tesla
RQ> left no math legacy. Tesla was not a mathematician. In fact
RQ> Tesla's math was not a whole lot better than mine, and I am a
RQ> very poor mathematician. Tesla never derived or used any math
RQ> more difficult than algebraic equations and functions, and
RQ> Tesla's math never explained what was going on or how to
RQ> reproduce it. 
RQ> mathematics of Tesla coil components, but nobody has produced a
RQ> set of definitive equations that accurately predict the
RQ> performance of Tesla systems. 

As far as coulpled resonant systems are concerned, I couldn't disagree more.
The mathematics for such systems is fairly straightforward and has been known
since before the turn of the century. The problem is that ANALYTICAL solutions
to such problems given L,C,R (or Q) parameters in terms of finite sums of 
ordinary functions like sin cos and exp are not possible. However,
numerical solutions are rather easy to get using a modest computer program.

Now you can argue, rightly in my opinion, that this is cheating since we want
to determine L,C,Q etc.. from the physical parameters of the system such as
dimensions and material properties of the coils. Here, there has not been much
work done by coilers so far. But that does not mean that it is NOT Possible.
All the resonant properties of tesla coils can be described by properly
applying Maxwell's equations or appropriate approximations for the given 
system of dimensions and materials constants 
(dielectric constants, dissipation factors). There is no magic involved, only
some basic physics. 

RQ> Dr. James and Kenneth Corum are probably the two of the best
RQ> Tesla mathematicians on the planet today. In their book, "VACUUM
RQ> TUBE TESLA COILS" they published third order calculus equations

I have the Corum's book, and I agree that it is more mathematical than
most every other book on the subject of Tesla Coils. But, they actually did
nothing more than try to apply existing equations to try to estimate the
behavior of Tesla's (and other's) reults. I see nowhere in the book where they
actully try to improve and derive equations specifically related to tesla
coils. So, if you are right in assserting that the Corum's are at the forfront
of tesla coil theory, then it is no wonder that our predictive power is so

I have a good math background, but I don't
know what you're refering to  as "3rd order Calculus". Dynamical systems are
often modeled using differential equations. Only the most simple diff eq's 
can be handled by classical calculus methods.

RQ> to explain many functions of Tesla's Colorado Springs Machine.
RQ> But they already knew the answers. Tesla gave the experimental
RQ> results in the CSN in text form, not in third order calculus
RQ> equations. In 1993 The Corums collaborated with William Wysock
RQ> (an experienced Hollywood coiler) of Tesla Technologies out of
RQ> Monrovia California. In July 1994 William Wysock assembled a
RQ> large Tesla Magnifier in Colorado Springs as part of the ITS
RQ> Tesla Symposium. This coil system was mathematically calculated
RQ> and designed by the Corum's and was assembled on location at the
RQ> rodeo arena by William Wysock at the cost of many, many,
RQ> thousands of dollars. It was to be a crowning achievement for
RQ> both the Corum's and Wysock.

Say that they calcuated the system parameters? What equations did they use to
determine the parameters? For example, if they used equation (100,101) of their
book to determine the resonant frequency of thier magnifier, then I would
suggest that they were doomed to failure. This equation was originally derived
for use in the theory of travelling wave tube systems. And there it is only an
approximation. It would take a great leap of faith to  belive that it would 
work for tesla coils with capacitive loading driven with a high harmonic
content spark oscillator.

RQ> The coil was a disastrous failure. It refused to tune. It refused
RQ> to quench. When fired, the entire system with extra coil operated
RQ> as the much simpler 1/4 wave resonator; there was no 1/8 wave
RQ> node on the transmission line; in fact the transmission line
RQ> acted as an uncoupled turn in a larger 1/4 wave system. Repeated
RQ> efforts to make adjustments, bring the coil into tune, etc.,
RQ> resulted in the self-destruction of the large rotary gap
RQ> specially built for the project. The coil was never even
RQ> demonstrated before the public. A hail storm passing over the
RQ> rodeo grounds was blamed for the destruction of the equipment (an
RQ> outright lie) and the public demonstrations of the project were
RQ> canceled.  

RQ> So much for using coil math to directly design coil systems. IMHO

Again, I disagree. This disaster just points to the fact that they were 
overly confident in applying inadquate mathematics (or theories as you like) 
to the problem.

RQ> I could also point out that there is at this moment a HUGE debate
RQ> going on as to the theoretical treatment of Tesla coil systems.
RQ> One side of the debate states emphatically that Tesla coils are
RQ> best described mathematically by transmission line theory. The
RQ> other side states that Tesla coils are best described as lumped
RQ> tuned circuits.

The fact that there is a debate on this issue at all speaks volumes of why the
coiling communtity is behind in the theoretical treatments of tesla
coils. Here's why: 

1) A Tesla Secondary or Extra Coil is obviosly not a lumped
circuit element because a lumped inductor, capacitor and resistor can have 
only one resonant frequency. A helical coil has many (idealy infinite) resonant
modes. Therefore, the lumped model is already inadequate. But the lumped model
is not useless either; it can describe, approximately, the flow of energy from
the primary to the seconday. The lumped model is much more accurate ( but not
perfect) in describing the behavior of the primary circuit. 

2) Is the Seconday and Extra Coil a Transmission line? Yes and no. Is a Tesla
coil accurately described by standard transmission line theory developed for
 uniform transmission lines (eg coax or twisted pair)? The answer is NO. 
Tesla coils have very non-uniform electric and magnetic fields which cannot
 be handled in the uniform theory. 

Also one must model the behavior of the capacitive loading due to the torriod
 and plasma discharge as well as ground. The "simple" uniform line appoach
 just places a capacitor at the end of the line. This approximation does not
 affect the properties of the line (such as propegation velocity) and,
 therefore it cannot be valid in the case of a real Tesla coil. 
 Placing a torriod on top of a real tesla coil affects the
 electric fields surrounding the coil, which in turn have wide ranging effects
on the opertaion of the coil. The current distribution and magnetic fields are
 affected, and therefore the proximity effect and dissipation are altered.
Complex beahvior, YES. Intractable? I don't think so.

RQ> Both sides have valid points. Neither view completely explains
RQ> what is going on. Even basic Tesla coil systems have yet to be
RQ> accurately mathematically described. You simply cannot build an
RQ> entire series of good Tesla coils using math as your primary
RQ> tool. You might mathematically describe one design, but scaling
RQ> the size, or power levels, up and down will throw everything off.

What then is a coiler to do if he wants to scale his coil to a size he has 
no previous experience with? Without a model, he is left with only 
expensive trial and error.

RQ> Despite what anyone says, coil designs that are derived primarily
RQ> from mathematical calculations will not function well.

I think this is true only because no coiler has ever done a proper analysis of
the theoretical problem. I don't understand your insistance that math 
modeling is doomed to failure... what is it based on other than anecdotal 
evidence? BTW, I have noticed that Richard Hull shares your disdain for 
what he calls "Pie in the Sky" theory!

 JO> "Tweaking" in the end will produce results, but this         
 JO> tweaking is always the result of experience _and_ science.   
 JO> One can get there much quicker by accepting and using a      
 JO> certain number of basic formulae.

RQ> Nobody can tweak or tune a coil faster mathematically than I can
RQ> with a glance. This may sound conceited, but I challenge anybody
RQ> to come even close. In my mind, Tesla coils are an art form. Once
RQ> they are up and tuned perfectly, then I can get out the
RQ> calculator to explain what I did.

RQ> I used to run computer simulations of coil designs to better
RQ> understand the system parameters before construction. Waste of
RQ> time.

What computaions and mathematical models were your compuer simultaions
based on? I am certain that it was only a 'waste of time' because your
simulations were not doing a good job of modeling. I have seen simulations
offered on internet sites, and they all have rather ttrivial computations
which could not possibly reflect the interesting and complex behavior
of a tesla coil system.

RQ> I found the best way to design was to experiment. I would
RQ> set up my workstation and fabricate primary coils from lengths of
RQ> old wire. By changing the coil type, spacing, angle of
RQ> inclination, etc., I was able to determine EXACTLY the type,
RQ> size, shape, number of turns, etc., worked best with any given
RQ> secondary design. I was able to do this at various power levels,
RQ> with various secondary types (space wound, close wound, insulated
RQ> wire wound). I charted my data, then compared my charts to the
RQ> computer simulations. Finally I abandoned the computer and relied
RQ> on the charts. I guess it was not long before I memorized and
RQ> interpolated the chart data, next thing I knew I was designing
RQ> freehand with no references or aids at all...

Seeing your data would be extremely helpful in producing good models. Do 
you have any ASCII files?

 JO> (Out of interest some of these <mathematically designed      
 JO> coils?> use up to 100kV into 20 microfarad for the primary   
 JO> circuits. One uses a pressure vessel filled with
 JO> sulphur hexafluoride to 60 psi as an insulator ! Peak        
 JO> voltages are up to 2.5 MV. Capacitors are often of the water 
 JO> type)
 JO> In all cases the theory closely predicts the performance.

RQ> Yet Tesla was exceeding these voltages, (and RMS amps too I would
RQ> guess) in 1899 with wooden coil forms and gutta-percha insulated
RQ> wire, using no more than .12 microfarads tank capacitance shock
RQ> excited at 60kV. To my knowledge Tesla's 9.5 MV peak -at- 1100 RMS
RQ> amps in the helix still stands as a record 96 years later. 

 The fact that Tesla
 achived higher peak voltages and currents says nothing of the efficency. 
The system which Jim is describing achieved efficiencies (power out/power in)
of well over 50% in tightly coupled two coil system. I belive that some coilers
have claimed that this is not possible in a two coil system!

In fact, the efficiency greatly depends on the degree of coupling. With a
coupling of say .1 you can never achive the efficiency of a system which is
coupled at .38  if both systems are optimally tuned. Tesla simply could not
handle the extreme stresses caused by a tightly coupled two coil system. His
resolution was to add the "extra" coil so that primary and secondary could be
highly coupled without too much stress. I have yet to see any reason to belive
that the Magnifier configuration is more efficient than a two coil system given
the same primary secondary coupling.

RQ> In Tesla's case the art form still transcends theoretical science
RQ> despite nearly 100 years of advancement in the later.
I think, instead, that the art of the coiler transcends 
most coiler's theoretical backgrounds.

Things coilers have not typically investigated deeply in their approximations:
(not complete)

The effects of Q, primary frequency, secondary frequency, coupling, and quench
time on the power transfer from primary to secondary.

Effect of skin and proximity losses on Q of primary and secondary (complicated)

Effect dissipation in insulating forms and wire coatings on Q

Self inductance and skin effect on capacitors

Spark gap conductivity in time ( initiation and quenching)

Effect of length, diameter, wire size, winding pitch, insulation, loading, and 
grounding on secondary Q and frequency

-Ed Harris