Reply To: Global Warming

[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.