[Prev][Next][Index][Thread]

Re: Why Medhurst derived fr is approxiamlty correct and the transmission line equation for the resoance frequency of a 1/4 wave helical resonator or Tesla coil.

Hi Bob,

At 02:23 PM 05/17/2000 -0400, you wrote:
>Hi all,
>
>The following is an updated copy of a previous post that was lost.

Sorry about all that...  Between that virus, some software upgrades, and
mail bounces from defunct accounts...  The servers choked up.  In
desperation, the ISP had to dump the mail early last week and start over... 

>
>I have completed most of my analysis and I have reached my goal of trying
>to understand what initially appeared to me as violations of circuit laws. I
>have now even solved the boundary problem for the current and voltage
>profiles in a Tesla coil.

Wow!  That would be great!  The number crunching programs (especially mine
;-)) could really use such information!

>
>I started off with the view that the lumped tuned circuit equation using
>Medhurst C could not possibly produce the correct frequency. This was due to
>my belief and apparently that of others that Medhurst C was the true self C
>of the coil. It is actually the value you must subtract from a much
>larger(>100x)  tuning capacitor to calculate the measured resonance
>frequency of a coil with one end grounded and the coil isolated. i.e. a
>correction factor for true self C and the internal turn to turn capacitance.
>Where as true self C is the capacitance you would measure with a low
>frequency (<< fr) LC bridge with the coil isolated.  It is the sum of the
>distributed capacitance of each turn to ground and is an intrinsic property
>of the coil.  It is only a function of its geometry and surrounding.  It is
>independent on the mode of oscillation of the coil. For a close wound coil
>its value is equal to the capacitance of a hollow cylinder with the same
>dimensions as the windings.
>
>If you differentiate a formula for Med C it confirms that the true self C is
>approximately constant along the coil with a small increase at the ends.
>Incidentally it also confirms its for an isolated coil.  Then with simple
>circuit analysis referring a constant distributed C to one end shows that
>true self C of the coil is 3 times Med C for very short coils and greater
>than 10x the 1/4 wave resonant frequency,  and  2 times Med. C for  long
>coils and at least 10 times the 1/4 wave resonate frequency.  So a quick bit
>of lumped approximation shows that the lumped equation using Medhurst C
>could produce approximately the correct answer.  Which has been confirmed by
>measurements on a variety coils.

If your new methods can explain why the older methods give the results they
do, that would be a truly remarkable advancement!  It is one thing to find
a new equation or something, but to also be able to explain all the other
equations and their various results is a milestone!

>
>A more accurate analysis using my (not a claim of originality) transmission
>line equation (given below) gives  Med. C as being  2.46 or (PI**2)/4 at the
>1/4 resonate frequency for long coils. I did not realize the significance of
>this when I first  posted the above comments.  It simply means that Med. C
>can only be proved to be valid an theoretical grounds for frequencies at
>least 10x the 1/4 wave resonance of the coil or putting it an other way when
>the propagation time of the coil is insignificant.  Which is precisely what
>Medhurst was using it for.
>
>Please note that I do not necessarily suggest that Med. C when used in a
>practical calculation produces the wrong measured 1/4 wave resonance
>frequency. As you can see from the above for an average coil  the ratio
>between Med. C and true self C must be between 2 and 3 compared to the 2.46
>factor which has no end effects, then add the error reduction effect of
>the squareroot and you have a reasonable  fr from Med. C. There are also
>other effects which could conspire to correct it. This readily accounts
>measurements on coils that are compatible with the lumped tuned circuit
>equation using Med C.

Perhaps your new analysis could be used to go back and regenerate a new
more accurate table for the Medhurst values.  Being able to do that without
having to take any "real data" but just doing it buy had (or computer)
would really be neat!

>
>
>When I first checked the transmission line equation I erroneously used Med C
>which produces the wrong answer not surprising considering the above.
>
>Now if you use the true self C per unit length  and L/length in a
>transmission line equation for fr you do get a reasonable answer.   But I
>now know that the standard transmission line equations are not supposed to
>apply because of the mutual coupling between turns or you should be using
>the inductance per unit length which is zero.  So why the correct answer
>with L/length???

I have always suspected that if one tries regular transmission line
equations, that the L is the measured L.  However, I never knew about this
Ctrue stuff...  I suspect that a much better model that the simple lumped
parameter model will be needed to explain streamer behavior.

>
>I now know from my analysis that the correct parameter that must be used in
>the transmission line equation is the mutual inductance per unit length
>which is a function of frequency.  But for coils greater than 3 diameters
>long this has the same numerical value as the total inductance dived by the
>length of the coil (for wavelengths greater the length of the coil).  Hence
>the equation works.  Which means that if you measure the true L and C of
>your coil the 1/4 wave resonance is the reciprocal of the four times
>squareroot of the product of the L and C measured in H and F respectively.
>Strictly this only true for long  isolated coils (no end effects) close
>wound and at least ten turns per diameter length . In fact there is
>justification for believing that the end effects are small. It also does not
>include the effects of internal C or top load.
>
>The same reason that makes the velocity equation valid also makes the wave
>impedance equation valid again accurate only for long coils (no end
>effects).  I don't know what the end effects will have on the impedance.
>
>Revelation is a wonderful feeling but I feel a little sorry that there are
>no new circuit laws for Tesla coils or at least I know of none.

Personally, I would worry if "new" circuit laws were needed ;-))  The
hundred plus year old laws work for everything else, so Tesla coils should
not be an exception.  The only problem has been figuring out how to apply
them which you seem to have finally done!!  YEA!!

>
>I have also completed my analysis of the simple addition law for top C and
>Med. C and true self C which I will post in due time.

If you could also account for top loads, that would really be great!

>
>This is not attempt  to resurrect the wave/lumped debate.

The wave / lumped thing was just born from our struggle to find a way to
explain what we saw when we really did not have a good way to explain it.
Your analysis should solve that mess by being something we can all (99.9%
of use - there's always somebody... ;-)) finally agree on.  Apparently, it
will explain the observations well and justify or dispute the older ideas
and show why those ideas predicted what they did.

We can then turn our attention to trying to fit various models and
equations to output arc behavior...  It never really ends...  ;-))

>
>It is just the conclusions of my application of circuit laws to Tesla Coils
>on which subject I am happy to receive or invite comment.

The Fo = 1 / (4 x SQRT (L x Creal)) formula is a powerful new tool for us.
I guess we need to find an accurate closed from equation for the
capacitance of a cylinder in free space like we have for a toroid.  Then
one will not need to hang a coil from the ceiling to test it. ;-))  I
assume the length and diameter are the only factors in such an equation as
long as the coil is not extremely space wound.  That equation is probably
sitting in a fields book somewhere on my shelf...  E-Tesla5 could find
Creal but going through trillions of calculations for this "should be
simple" task is not really justified. 

>
>Regards Bob (Robert Alwyn Jones  a circuitlawophile)
>

Fantastic work you are doing here!  Looking forward to hearing more details...

Cheers,

	Terry