Very interesting discussion... 1. Regarding your earlier question about good sources of information:As Jim mentioned, "Spark Discharge" is very good. Another excellent book is Raizer's "Gas Discharge Physics (GDP)". While Spark Discharge focused primarily on breakdown of long air gaps, GDP covers free electron behavior in oscillating E-fields and RF breakdown and conduction, and the role of "displacement currents". Gas Discharge Physics is now available as a paperback reprint and for free via inter-library loan.
Unfortunately, most papers dealing with long sparks tend to focus on power line frequencies, impulse breakdown, spark gaps/switching, or lightning. There are relatively few that deal with RF discharges at less than 1 MHz. One notable exception is the massive (483 page) 2003 study, "VLF/LF High-Voltage Design and Testing", by P. M. Hansen, A. D. Watt. This study also pulls together much of the previous work reported in the literature regarding RF discharges. Fortunately, it is a free download:
http://www.dtic.mil/dtic/tr/fulltext/u2/a526052.pdf 2. Regarding leader models, the following may help:Although a lossless transmission line model of a leader would have a wave velocity of c, real-world (i.e., lossy) leader channels actually have a measured velocity of about 0.1*c, implying that Sqrt(LC) for long leaders is approximately equal to 10 instead of 1. See Raizer (GDP, section 12.10).
Measured leader linear capacitance ranges from about 0.25 to 0.5 pF/cm. This is about 5 times that of the linear capacitance of individual streamers. This does seem to be in the ballpark of your model.
It also looks like Raizer agrees with you regarding the effect of the space charges surrounding the leader. Per Raizer in GDP, section 12.9.4, the higher capacitance is due to the, "thick "sheath" surrounding the high-conductivity thin leader channel; the sheath is formed of weakly conducting plasma and the space charge injected into it by streamers". In the case of an RF leader, charge is being repetitively injected and removed by the oscillating driving voltage - the source of the "displacement current" being sourced by the HV terminal.
The velocity of a _propagating_ leader is a function of the electrical field at its tip. Since growth involves forming a propagating front of countless avalanches, streamers, and multiple streamer-to-leader transitions as the leader grows, leaders propagate considerably slower than the wave velocity within an existing leader. The higher the tip E-field, the higher the leader velocity. Laboratory leaders typically propagate at between 10E6 - 10E7 cm/s. This can reach 10E7 cm/s for lightning.
Hope this is of some help... Bert -- Bert Hickman Stoneridge Engineering http://www.capturedlightning.com *********************************************************************** World's source for "Captured Lightning" Lichtenberg Figure sculptures, magnetically "shrunken" coins, and scarce/out of print technical books *********************************************************************** Udo Lenz wrote:
Kurt Schraner wrote:This thread is really interesting. Would be nice, to have a formula for spark capacitance, relevant for TC's. Antonio's appoach seems very convincing, because it only relies on basic physics law's and the model of a transmission line. May be, the transmission-line model can be questioned for this case. On the other hand, the formulas given in Wikipediahttp://en.wikipedia.org/wiki/Capacitance seem well established. I see kindof a dilemma.I think Antonio's transmission line approach is correct. Along a straight wire, capacitance and inductance conspire in such a way to make the velocity equal to c. I do have a problem with his assumption of the inductance of 1uH/m being mostly independent of the radii we consider here. The spark model I've described needs a capacitance of about 15pF/m. This could be due to branching sparks. The model doesn't take these into account. I've been wondering about the non branching sparks, which can be generated by DRSSTCs with a slow voltage ramp up. If the conducting spark channel is very thin, I'd expect sideways breakout from it. My assumption is, that the arc pushes out charge carriers during the rise of voltage of each RF cycle. This charge cloud would reduce field strength near the arc, preventing breakout. The charge cloud would also increase arc capacitance. Another problem with the spark model is, that the ratio of capacitive and resistive load seen by the coil is too large compared to experiment. If the model is modified by adding a resistor in series with each capacitor, that ratio can be brought into agreement with observation. I believe a charge cloud would have a similar effect, adding a dissipative component to the capacitors. I'm currently trying to model this, based on drift velocities of charge carriers but don't have any results yet. Udo _______________________________________________ Tesla mailing list Tesla@xxxxxxxxxx http://www.pupman.com/mailman/listinfo/tesla
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