[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.

[asymmetric_implosion]

I think I better understand your goal, but if you are looking at truly external power should you be looking at the power supply charging the capacitors. The rest of the circuit is difficult to treat as two terminals. I’ve tried similar approaches to the one you describe but there are some problems that are difficult to model. I have attached a poster I gave at ICOPS back in 2012 on the dynamics problem. The snow plow model was used to calculate the important dynamic circuit parameters using both the current and the voltage at the load. This was done for a 230kA plasma focus for a couple hundred shots in D2

The energy in the bank is a surprise to me. When I’ve measured the capacitor voltage on my old machines, I never saw more than 10% of the initial charge voltage on the bank at peak current for optimal conditions. I could always force the device to implode early or late which impacted the energy stored in the bank but that never gave the desired radiation yields.

Attached files

Snow_plow_model_poster_shrunk.pptx (231 B) 

[asymmetric_implosion]

A few things you should know:

1) a pinch device is a dynamic system with variable inductance and resistance. There are baseline inductance, capacitance and resistance in the system which I believe your calculations try to capture.

2) because inductance is changing your basic circuit equation is no longer simple. You have an extra term dL/dt*I (the time derivative of the inductance times current). This term drives some of the more interesting behavior of the system near the pinch time. R is also time changing which leads to some interesting consequences near pinch time.

d/dt(LI) +IR+ int(I/C)=0; => dL/dt*I+L*dI/dt+I*R+int(I/C)=0;

3) The motion of the plasma from the strike along the insulator to the completion of the axial phase is commonly described by a combination of the circuit equation above and an equation of motion typically the momentum equation for the plasma. This combination is commonly referred to as the snow plow model. Lee has a free excel sheet you can download and use to describe the motion of the plasma focus in all stages with various assumptions. Look up Lee plasma focus model to get it. The documentation is pretty good on how to use the sheet. The model is described in the documentation as well.

In your spreadsheet, you calculate that the capacitor voltage is still very high. In reality, it should be very low. If you look at the energy in the system, the initial energy is stored in the capacitor so 0.5*C*V^2, where C is the capacitance and V is the voltage. As the bank discharges, you can show that even for a system with smoothly increasing inductance and resistance like a PF, that the maximum current occurs when the capacitor bank is at or near full discharge. This is because the magnetic energy in the system is 0.5*L*I^2 where L is the inductance and I is the current. If you have very little resistive loss, you can equate the two energies to estimate the voltage to achieve a certain current or current you can expect for a certain bank charge. It is not perfect but the estimation is pretty good. In a properly designed system i.e. gas pressure, electrode geometry and pulse power are matched, one can show the best pinch occurs just after peak current; typically ~100 ns after depending upon your anode diameter. This mean that in a system with a 1-2 us pulse, you little time to extract more energy from the capacitor bank or recharge it in 100 ns. It is possible to design a system with a poor match between the capacitor bank, electrodes and gas pressure but that is probably not that interesting for application and it has very negative consequences on efficiency and heating.

[asymmetric_implosion]

meemoe_uk wrote: Well here’s an amateur’s rough design. If its no good, you can tell me why and I can learn something!

Are we allowed to move the anode away from the cathode?
– The barriers have orifices to allow fluid circulation.
– The cathode edge ( and ideally its whole cylinder ) is positioned out of direct line from the fusion zone
– If need be, the anode can be simply shielded from direct line of the fusion zone too.

You have designed a Z-pinch. You have lost the axial flow of plasma that stabilizes the plasma focus and allows for reproducible operation. The cathode is a lesser problem than the anode in a plasma focus. The anode takes the brunt of the energy from the expanding pinch plasma. This can lead to metal vaporization, melting, etc, of the electrode. The cathode will vaporize but a much smaller extent as the current is distributed over a larger area.

The proximity of the electrodes is a key problem for any pulse power based approach that directly flows current into the plasma. Tokameks and laser systems remove this complication by taking away electrodes. The move to the tokameks was driven by the electrode problems in long, slow Z-pinches. The materials problem of the PF is non-trivial and needs a serious research effort. A few teams have only scratched the surface and as is usually the case, the answer is not the same for every application. For example, the LPP need for high x-ray transmission of the anode is driving the use of Beryllium. Folks working on neutron generators and lithographic x-ray sources are less concerned about x-ray absorption in the anode and favor long lifetime operation so they choose materials like SS-304, molybdenum and tungsten alloys.

The message at the end of the day is the electrodes are a consumable. You can only extend their life so far given the limits of available materials. The alternative is to choose inexpensive materials that can be replaced rapidly. I favor frequent replacement as complex engineering system may not buy much increase in operational lifetime for a significant cost sink. Another trade study to be completed.

[asymmetric_implosion]

@Lerner: I’m still a bit confused. What is 3500 K? If the gas is 3500K I don’t see a problem holding the electrodes as 1100 K or less with some creative thermal engineering. If the metal is 3500 K, I don’t understand the logic. Your calculation of the heat transfer from what I presume is the gas is likely inaccurate. The results appears to assume that you can remove heat at 1.5 kW/cm^2. All heat flow is driven by temperature differences in the system and the convection of the gas if the gas is the hottest element. Radiation is important for a time but convection could also play a role as the gas is extremely turbulent and should be directed at the chamber and the inner anode diameter. The decay of the electron population is also important. Electrons will carry away and radiate away heat.

My piece of advice is design a larger vacuum spool so you can address the shock problem with CF gasket leading to vacuum issues if you still believe it a problem. This will allow for a large gas buffer and throw away volume. If you are interest in dealing with gas based nucleation, I would suggest looked at our semiconductor friends. They have these problems and deal with them. Probably a good start.

[asymmetric_implosion]

A couple of questions/comments:

1) If the operating temperature is 3500 K and the melting point of Be is <1600K and vapor point is 2800K are you planning on vapor electrodes? Not impossible but probably pretty challenging at a high repetition rate. Our Z-pinch friends use gas puff technology all the time.

2) I assume that the generator is still running at 200 Hz. If so, you have a 5 ms window for particle deposition. Most sonic processes occur on the ~1 ms scale so particles may not be too bad. I’d be more worried about chemistry between Boron and Be. Laminar jets are probably going to be terribly upset by the returning shock from the gas returning to the vacuum created by the pinch process.

3) What is the pump lifetime you require? Pumping particles through any turbo is very bad news. Rotors spinning a speed of sound hitting particles is like bullets hitting a wall.

4) Be is typically alloyed with aluminum to increase aluminum’s strength.

[asymmetric_implosion]

vansig wrote: such current thresholds and field strength thresholds are just constraints to be considered inputs to the design.

OK. The constraint is being able to have a pinch at >100 T next to a superconductor with a peak magnetic field of ~10 T on a really good day. Typically that is accomplished with magnetic shield. Now you have material between the pinch the superconductor which carries the current.

Now the real problem, do you really want a superconductor in your system? Sure, losses transfer is cool and all for DC power transmission but it is a good thing for pulse power. At first, sure it seems like a good idea. Who doesn’t want a more efficient system? Well, the capacitors in the system are resistive so you have some loss there. Probably a fairly large part of the controllable loss. OK, the creative soul is not stopped by this detail. Inductive energy storage is the solution and a superconducting coil is the solution. OK. One problem down. Now, you need to transfer that current in either a switch or cause the plasma focus to breakdown. Either gas conductor is resistive. So you still get some loss, but hey it’s better than before….or is it? A plasma focus is an resistive-inductive-capacitive circuit. You need finite resistance to keep the circuit from going crazy and it is more resistance than the plasma alone can supply. On the forward pulse this seems like a great idea at you get more current but on the reverse pulse you have a great deal of voltage and current to deal with that some components aren’t prepared to deal with. Now the pinch should take a lot of energy which helps but you are relying on the pinch. What if you have a back shot and the pinch fizzles? This will happen and if the pulse power can’t handle the full current in reverse it will be damaged.

Now think about the cost. Copper, tungsten and other metals are pretty good conductors that don’t require much to just work and while the costs vary, you can get the materials relatively easily is the shapes of interest. If you think that tungsten is a problem material, you ain’t seen nothin’. Superconductors typically have to be in a nice crystal structure to work at their best. The plasma erodes the surface on each shot so bye-bye nice crystal structure. Then you need a cooling system with an efficiency typically less than 1% (Thank you, Mr Carot). This means that any heat to take out of the superconductor requires 100X the total system power. Hmmmm. X-rays and UV get absorbed into the anode along with particles. Now you have to remove 100X that heat. The compact, efficient plasma focus just became a tokamek. 🙂

[asymmetric_implosion]

There is some truth in people loving their own concept but these days it seems the MCF approaches are banded together as have the ICF approaches in the broader sense e.g. Sandia supporting the NIF “breakeven”.

I’m not saying things are not possible or wrong. I think the progress is significant for the funding, but I believe the true test is receiving the funding. I’ve had many folks claim support for proposals when it was someone else paying. “Great idea and it would be nice…if they pay for it.” Results drive larger budget which drives results… should be the approach to all science. Sadly politics gets in the middle. My point is that the independent review could be taken in a negative light very easily.

I like independent review and the results that come from it when people use it as intended. The Devil’s advocate is truly intended as that. I believe any technologist must be his/her harshest critic while being being the biggest advocate. It’s a bit sick in a way but I believe that drives success in the long run.

[asymmetric_implosion]

OK. Devil’s advocate time.

1. Leading fusion researchers reviewed the LPP concept. From a purely technical standpoint this seems like a good idea, but fusion is far from purely technical. All reviewers are proponents of fusion so would they turn down any fusion concept in the experimental stage? This seems unlikely given their budgets are not impacted. Shouldn’t a technically knowledgeable fusion hater be on the panel. If you think they don’t exist, they do.

2. The technical evaluation largely says that none of the key metrics are met and data doesn’t support them. We support this idea because of….our warm fuzzy feeling about fusion?!?

—

OK. To be fair:

1. The concept of superior compression with higher Z elements is a pretty common belief with some data supporting it already. I don’t know if anyone has compared this fairly on the same machine, but the literature strongly hints at it. Small DPF machines with high Z gases in small concentrations perform better than machines without high Z gases. This is not fully understood but there is experimental supporting that high Z makes life better.

2. Ions hotter than electrons in a pinch is largely proven. It may not be at the level required by LPP but it is possible.

[asymmetric_implosion]

Even without frozen fuel, the interface between a superconductor and the regular conductor is challenging. The plasma will also heat up the metal which will quench the superconductivity. Even if the plasma heat doesn’t do it, the high magnetic fields from the pinch will quench the superconductor.

[asymmetric_implosion]

Long story short: Graphite is a mediocre electrical conductor full of pollutants. It tends to explode at high current and hydrogen likes carbon. Worst case option is the boron carbides with the surface and well… pretty bad conductor. Carbon is good for x-ray transmission but bad for high current.

Tungsten is important in the initial testing to reduce impurities and determine the optimal conditions. Tungsten is the material of choice for DPF electrodes for low rep rate devices. It might not be good in the long haul but it is an important step. Beryllium may or may not pan out for a number of reasons.

[asymmetric_implosion]

Z pinches are one of the most common implosion type radiation devices. The Z-machine at Sandia National lab is the largest such device in the world operating at nearly 26 MA with a rise time of less than 100 ns. A large PF device is 3 MA with a rise time time of up to 10 us. Z-pinches are largely used for nuclear weapons effects simulations and fundamentals studies of dense matter (stellar stuff) as they produce copious amounts of x-ray radiation. The preferred mode of operation is to use wires of high Z metals strung in two layered cylindrical arrays of many fine wires. A typical ring of wires can be more than 200 individual wires. Gas puff technology is viable for such devices and on occasion layered shells of gases are used as the radiating material. Published works suggest the Z-pinch devices are under performing compared to PF devices with a pure deuterium fill. A number of hypothesis exist as to the cause.

The plasma focus was an accidental discovery that happened to mimic the Z-pinch. The advantage of the PF is the less expensive pulse power system as it operates with a slower rise time thus less voltage to drive the plasma. The two primary stages of operation, the axial flow phase and the radial implosion phase, seem to produce a more stable pinch in a plasma focus. The Z-pinch is a pure radial implosion. The PF never took off for a few reasons: some political as the chief advocate for PF devices in the US had personal problems that made him appear unreliable, the PF devices tended to shatter insulators at increasing current which remains an issue in the classic PF design, and historically poor performance compared to Z-pinch devices.

The political issue was unfortunate but people are involved in science and some times our issues get into the work. The shattering insulators was a perception issue as much as a technical issue. No one in the Z-pinch world thinks twice about polishing key components and rebuilding switches are every shot. A drop in piece of ceramic doesn’t seem so bad by comparison. There are also effective alternatives to a solid, plasma facing insulator. More needs to be done but the initial results done in the 1990s were promising. The historically poor performance was a result of poor understanding of the PF process. Most people looked for thermal fusion which the PF does not do well. The non-thermal fusion, fusion driven by fast ions generated in the pinch, was frequently put down by those in the fusion community as unreliable and unpredictable. I think recent work has caught the attention of more open minded folks and showed them that reproducible and predictable are at hand. There is also a bit of a sense of that fusion must be hard. If our best minds have not cracked it yet, it must be very hard. I don’t know that I agree but I think there are options of fusion that are yet to be explored using the best knowledge of modern plasma physics. Is there an approach that is vastly to superior to all other approaches? I don’t know but it seems less and less likely that NIF or ITER are going to reach the goal of fusion energy and be economically viable in the current financial climate.

[asymmetric_implosion]

It looks more like a Z-pinch than hypo-cycloidal pinch. You need three parallel plates for a hypo-cycloidal pinch so you can drive an axial implosion. You are driving a largely radial implosion like in a plasma focus but without the axial phase to stabilize the plasma flow.
Pure radial implosions have a large number of instabilities that you need to over come.

[asymmetric_implosion]

The nuclear physics is pretty well understood. The typical tests involve firing the lighter reaction element at a target made of the heavier element. Hydrogen-hydrogen fusion is a pretty low reaction rate. It is well studied thanks to our solar system’s largest fusion reactor. The magnetic fields effectively increase the travel path of ions which increases the chances of collision and in some cases and change the atomic physics by splitting levels in the electron orbits. I would be hesitant in calling the magnetic fields organized but they are strong.

Most of the neutron producing reactions are driven by by-products and the energetic alpha particles. Compared to D-T fusion the neutron problem is pretty small for p+11B.

[asymmetric_implosion]

You can fire ion beams at each other but you will get very little fusion. The total particle content of the beams is too low. Beam-beam fusion concepts always suffer from the low cross section of fusion. People have looked at them for 50+ years and the fundamental problem is the ability to accelerate ions and keep them going when they don’t fuse.

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