Thoughts on advanced modeling of sparks and toploads
The simple TC model which has stood the test of time is pretty simple. The coupled LC resonators, with the top load modeled as a single capacitor, and an empirical model of the spark as a 220k resistor with a several pF capacitor. The spark model was developed by Terry Fritz on the TCML in around 2001/2002 by adjusting a SPICE model to match measured data on a test coil, based on some discussions on the TCML back in June 1998.
While this model seems to replicate the primary/secondary behavior as far as currents and voltages pretty well, it doesn't necessarily give much insight into what makes big sparks. That is, there is obviously some fairly complex interaction between the growth of the spark and the flow of charge into the topload from the secondary inductor. If for no other reason, the physical topload is often on the order of a meter across, and it takes finite time for the charge to flow from one side to the other to support the leader growth. For this reason, I propose a somewhat more complex model of the spark and the topload.
First, a model of the spark. Based on the discussion in Bazelyan and Raizer (Spark Discharge, CRC Press, 1998), I think the spark can be modeled as a gradually growing distributed R and C. As described in B&R, a leader grows by charge flowing through the spark channel accumulating at the leader head, which they suggest might be effectively modeled as a 1 cm sphere. Eventually, enough charge accumulates so that the field on that sphere exceeds the local breakdown strength of the air (30kV/cm), and a new extension of the channel occurs. The current flowing in the channel also serves to keep the channel hot and conductive, in the face of radiative and conduction cooling. As long as charge is available to extend the channel, the spark will continue to grow:which is how a multi-meter spark can bridge a distance where the average field is only a few kV/meter, well below the 3MV/meter breakdown for air. In the diagram below, I've indicated spark gaps, but in actuality, I think it would be modeled as a conditional statement in some lines of code, not as trying to actually model a spark gap. Likewise the spark channel is shown as a sort of lossy transmission line RLC, but in reality, some time varying model like that of Goncz is probably more appropriate.
The challenge is, of course, in choosing appropriate values for the resistance, the capacitance, and the "jump distance" or "modeling increment size". The capacitance can probably be selected based on data in B&R.. it will be on the order of a few pF/meter. Likewise, the inductance is on the order of 1 uH/meter. Resistive losses in the channel can also be estimated from the well known properties of sparks with the measured current in the sparks. (e.g. the measurements made by Greg Leyh on Electrum, or data on free burning arcs from Somerville or Cobine). The jump length might be best estimated by using the known propagation velocity of a spark, around 2-3 m/microsecond.
It's also true that on the scale of the spark channel, the top load cannot be considered as a point source capacitor, but rather, it's time domain properties must be considered, perhaps as a network of transmission lines or distributed LC networks. There should be some sort of FDTD Transmission Line model (after Yee) that would be suitable.
tcsparkmodel.htm - Copyright 2011, James
revised 18 May 2011