Re: Maximum voltage gain in a Tesla coil

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

Original poster: "Gerry Reynolds" <gerryreynolds-at-earthlink-dot-net> 

Hi Antonio,

I've had this belief that detuning say the primary to a lower frequency was
to compensate for added topload due to streamer formation.  Is this true? I
take it that what you are describing here is another consideration?  How the
"streamer affect" fit in to the optimization.

Gerry R


 > Original poster: "Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br>
 >
 > Hi:
 >
 > Is Marco Denicolai still in the list? He would find this interesting.
 >
 > After a discussion with Jim Lux, I wrote an "optimizer" program for
 > the lumped model of a Tesla coil during the energy transfer transient:
 >
 >      +--R1--+        +--R2--+
 >   +  |      |        |      |   +
 > Vc1 C1     L1 <-k-> L2     C2 Vc2
 >   -  |      |        |      |   -
 >      +--<---+        +-->---+
 >        Il1             Il2
 >
 > The program calculates exact solutions for the transient that starts
 > with an initial Vc1, and tries then to optimize the circuit following
 > several criteria.
 > One of the possibilities is to generate the maximum possible
 > voltage gain. Departing from a coil with the usual tuning relation:
 > T=(L2*C2)/(L1*C1) = 1
 > that has a voltage gain equal to:
 > A=sqrt(L2/L1)=sqrt(C1/C2)
 > the program changes one of the capacitors and k, trying to generate
 > the largest possible Vc2 after an unespecified time.
 > It's well known that it's possible to detune the coil, increasing
 > C1 or decreasing C2, sacrificing complete energy transfer and
 > efficiency, and obtain a small increase in gain before the
 > detuning forces a fall. Adjusting k too allows a slightly
 > larger gain. The objective is to obtain the maximum gain with
 > the same coils. (The new maximum gain sqrt(C1/C2) is not reached.)
 > See the paper by Marco in the Review of Scientific Instruments,
 > 73,8, Sept. 2002, and its references, where this possibility is
 > demonstrated.
 > The largest voltage gain reported is with T=0.541136 and k=0.545659,
 > what generates Vc2/Vc1=sqrt(L2/L1)*1.1802.
 >
 > I was imagining that my program would find this solution. But to my
 > surprise it found another, better: T=0.479222 and k=0.593349, that
 > results in Vc2/Vc1=sqrt(L2/L1)*1.19950, at the first negative peak.
 >
 > This same solution produces a slightly larger positive peak 4
 > cycles later (*1.19957), and further optimization increases this
 > peak to *1.2307 with T=0.4603 and k=0.6054.
 >
 > Similar things happen with smaller ks, but then the gain over the
 > tuned design is smaller.
 >
 > Useful? Probably not. But curious.
 >
 > The program can also optimize the circuit in the presence of losses,
 > represented by R1 and R2, finding solutions with maximum voltage
 > gain, maximum efficiency, and complete energy transfer.
 > It was interesting to see that solutions with complete energy
 > transfer (at some point all the remaining energy is in C2) are
 > possible in the lossy case. The differences between the optimized
 > circuit and the lossless circuit are negligible, however.
 > Example:
 > C1=4.5 nF
 > L1=1 mH
 > L2=30 mH
 > C2=15 pF
 > k=0.18033
 > The voltage gain is 5.477.
 > Adding 20 Ohms as R1: A=4.5766.
 > Optimizing for maximum voltage gain, changing k and L1:
 > k=0.18018, L1=1.0165 mH. The result is A=4.5813.
 > The solution with complete energy transfer is inferior:
 > k=0.1671, L1=0.988 mH, A=4.5367.
 >
 > Antonio Carlos M. de Queiroz
 >
 >