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.

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.

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.

gianfranco wrote: Still impossible to upload Excel files…

?

As a test here are 3 varieties of excel uploaded.

And you can always zip a file if the file extension is not allowed.

If it remains a problem I’ll check the allowed file types later…

disable excel macros. That might be why uploads are being blocked.

And everyone else… be careful opening archive files (rar)… they may contain malicious macros.