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*To*: "Tesla Coil Mailing List" <tesla@xxxxxxxxxx>*Subject*: [TCML] Spark models, revisited*From*: "Udo Lenz" <udo_lenz@xxxxxxxxxxxxxx>*Date*: Wed, 17 Oct 2012 16:45:18 +0200*Delivered-to*: teslaarchive@xxxxxxxxxx*Delivered-to*: tesla@xxxxxxxxxx*List-archive*: <http://www.pupman.com/pipermail/tesla>*List-help*: <mailto:tesla-request@pupman.com?subject=help>*List-id*: Tesla Coil Mailing List <tesla.pupman.com>*List-post*: <mailto:tesla@pupman.com>*List-subscribe*: <http://www.pupman.com/mailman/listinfo/tesla>, <mailto:tesla-request@pupman.com?subject=subscribe>*List-unsubscribe*: <http://www.pupman.com/mailman/listinfo/tesla>, <mailto:tesla-request@pupman.com?subject=unsubscribe>*Reply-to*: Tesla Coil Mailing List <tesla@xxxxxxxxxx>*Sender*: tesla-bounces@xxxxxxxxxx

I've tried to make the spark model proposed by Jim Lux http://www.pupman.com/listarchives/2011/May/msg00081.html a bit more quantitative by considering only QCW i.e. stationary sparks. I left out the spark gaps and the inductances in his model so the spark is described by a series of resistors and capacitors. Going to the limit of a continuous distribution of resistors and capacitors, the voltage V along the spark can be described by a second order differential equation: d/dx ( 1/R * dV/dx ) = C * dV/dt where R is the resistance of the arc (ohms/m) and C the capacitance (F/m). I've assumed C to be constant and R to be dependent on the time averaged power dissipation. R depends then on the position along the arc. A simple guess for R is to make it inversely proportional to some power of the power dissipation, i.e. 1/R ~ P^n For n>0 the differential equation has a simple solution. The voltage amplitude and the power dissipation along the arc are proportional to V ~ (L-x)^(1/n) P ~ (L-x)^(2/n) where L is the length of the arc and x the position along it, beginning from the breakout point. The length of the arc and its total power consumption Ptot are L ~ Vinp^n * f^(n/2-1/2) Ptot ~ L^(1+2/n) * f^(1-1/n) where Vinp is the coils voltage and f its frequency. The best match to experimental observations seems to be the case of n=2: The power consumption is then proportional to L^2, which is in agreement with the Freau formula and e.g. a measurement made by Gary Johnson. http://www.eece.ksu.edu/~gjohnson/TeslaCoilImpedance.pdf The power dissipation along the arc drops linearly with the distance from the breakout point. The power needed for a given arc length rises with the sqrt(f) making low frequency coils more efficient regarding spark length. The arc length grows with the square of Vinp. I believe Steve Ward has seen a rather sudden growth of spark length with voltage and my own observations confirm this. For a given voltage the spark length increases with sqrt(f). So a lower voltage is sufficient to achieve a certain length when the frequency is higher. There are mainly two effects, which generate conductivity in arcs, thermal ionisation and ionisation by electric fields. Thermal ionisation has a very strong dependence on the temperature since the air molecules must have enough energy to ionise on impact. Conductivity begins at around 6000K and rises up steeply. See e.g the diagram on plasma conductivity in http://www.zeuz.ch/doc/itet/sem5/ensy/227-0122-00-autography_systemtechnology.pdf on page 21. I'm no expert on plasma physics and I don't have a clear view of where a balance between conductivity, temperature and power dissipation lies. Maybe some of you can help there. Ionisation by electric fields requires fields of about 30kV/cm at room temperature and pressure, which is much more than the the voltage drop along RF arcs. But, as Paschens law states, the breakdown voltage is proportional to the density of the gas and that depends on the temperature. At 6000K, which is 20 times room temperature, the gas density is 20 times lower, reducing the field requirement to about 1.5kV/cm, which is much closer to the observed values. A rough estimate of this effect is: V ~ density of gas ~ 1/T (Paschens law and gas law) P ~ T^4 (Power needed due to loss by thermal radiation) V^2 / R = P (Power dissipated by the arc) Putting these together results in: 1/R ~ P^(3/2) This amounts to a n of 3/2. I have some doubts, though, whether the T^4 law really holds for gases, which are no black radiators as the Stefan-Boltzmann law demands. Possibly the temperature dependence is slower, maybe T^3 or T^2. That would make n larger. Anyway, at voltage drops in the arc below 1.5kV/cm electrical ionisation cannot be anymore the main source of conductivity. I believe, that there is a mix of both thermal and electrical ionisation in a typical RF arc. There are some problems with the n=2 model: 1) I've measured a power consumption of about 20kW at 70kVpk for a 70cm arc. This is close to Gary Johnsons empirical relation of L = 0.17 inch * sqrt(Ptot/W). Both of us ran at about 200kHz. Wards tesla gun achieved (I believe) about 60 inches, which would result in a power requirement of 120kW, which is not realistic from what I know about his electronics. From my discussion above, the higher frequency he works with should make the power needed even higher. 2) The model predicts a phase shift between the voltage of the coil and the current drawn by the arc. The phase shift is by itself of some interest, since it determines the capacitive load on the coil. The shift is given by tan(phi) = sqrt(n+1) For n=2 that would be 60 degrees. I've measured a value of about 45 degrees. The difference is a lot bigger than I expect my experimental errors to be. I'm interested in your comments. Udo _______________________________________________ Tesla mailing list Tesla@xxxxxxxxxx http://www.pupman.com/mailman/listinfo/tesla

**Follow-Ups**:**Re: [TCML] Spark models, revisited***From:*Jim Lux

**Re: [TCML] Spark models, revisited***From:*Steve Ward

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