Reply To: Global Warming

[asymmetric_implosion]

An interesting work, but you have made the exact opposite assumption of most pulse power for Z-pinches and PF devices. The most common assumption is fixed resistance and dynamic inductance. The reason is driven by the physical system. The dynamic resistance typically comes from changes in the switches, changes in plasma temperature and the location of the plasma flow along the length of the electrode. The switches, by design, go from insulating to highly conducting quickly on the time scale of a PF device. Therefore, most treatment of the switches is neglected and they are assumed to be a constant, typically negligible resistance. The plasma is typically the most likely candidate for change in resistance. The initial breakdown of the system has a relatively low plasma temperature which increases as the plasma flows down the electrodes. Typical conditions show a change in the plasma resistance from 2 mOhm to 1 mOhm. Probably not too bad considering resistance in a typical PF device is ~5-10 mOhm. The change in resistance due to a change in conducting path length tends to be much less than 1 mOhm.

You have touched on one of the more recent literature discussions about plasma focus power systems; how do you optimize the pulse power design? Electrodes play an important role as they are the source of the dynamic inductance. For practical reasons, you have to limit your axial plasma flow speed to ~100 km/s near the tip of the anode. If you select your electrodes (anode radius as the primary input), the current and gas pressure can be optimized using Lee’s drive parameter (I*a/sqrt(P) where I is the peak current, a is the anode radius and P is the fill pressure) which maximizes at 75 to 90 kA-cm/Torr ^0.5. Now it is not as simple as this. The LPP approach is to minimize the anode radius which fixes the cathode at ~2-3X the radius of the anode in most PF devices. With the anode radius defined you can only adapt pressure and current. You typically design for a current as you wish to achieve a radiation yield that scales with current to some power. For D-D reactions the published literature suggests an exponent of 3.3 to >5. The theoretical calculation is an exponent of 4 and is the most commonly adopted value for the power law. OK, how do you get the I. Well, that depends upon how efficient you want your system to be. The calculated optimum inductance for a PF device was calculated by Trunk back in the 1970’s. Trunk suggests nearly optimum matching between the pinch and driver when the driver inductance is 60% of the peak axial phase inductance. So, you can calculate your axial phase inductance from the electrode geometry. Based upon your work and my guessing at anode length, the inductance is roughly 20 nH so the optimum driver is 14 nH. Most drivers at this size can reach 14 nH with ease. So why don’t we go lower. Well, you have to transfer power to the pinch so you want to maximize this transfer. This condition seems to maximize power transfer but the optimum appears to be broad while the driver inductance is greater than the axial phase inductance. Very few machines have a driver inductance less than the axial phase inductance.

OK, you have constrained your inductance, minimize resistance when possible and design your electrodes which is optimized with gas pressure. The rest of the system is the caps and their voltage. You can do a great many things in this respect. Larger caps at low voltage lead to lower impatience due to sqrt(L/C) but you have less voltage to drive the system. Smaller caps at high voltage can present engineering challenges. The published data does not show a significant advantage of one over the other. Both system are able to produce high quality pinches. One has to be reasonable about the definitions of high voltage and low voltage. You cannot produce a 100 V, 2 MA PF device. The impedance with charging the inductor alone is ~10 mOhm. For a 2 MA system, you require a minimum of 20 kV and you can’t get around it. 45 kV is probably a middle of the road voltage. The highest voltages are ~100 kV for a PF at 2 MA. The design philosophy tends to fit available resources and personal experience with high voltage. I always operated low voltage, large bank systems because of resources and other operational constraints.

Various design approaches are described in published literature for the last fifty years and this is just a summary. I would encourage those most interested to read a review by a colleague, M. Krishnan, in IEEE Transactions on Plasma Science from last year (Sorry, gotta pay for it). It discusses these points and cites references that are relevant to this problem.