Comparison of measured pinch parameters versus pressure for SABALAN 1 plasma focus facility against computed values using Lee model code

In this paper, we have studied the effect of working gas pressure on plasma pinch properties in a Mather type plasma focus (2 kJ, 20 kV, 10 μF), named SABALAN1. Argon at the various operating pressures ranging from 0.1-1.2 Torr has been used as working gas. The total current waveform has been measured for different pressures. Also, a numerical simulation was done using Lee model code to perform numerical experiments in SABALAN1. The numerical simulation was used to compute argon soft x-ray yield as a function of pressure, to anticipate the maximum soft x-ray efficiency at optimum operating gas pressure. The Lee model was configured for SABALAN1, by fitting a simulated discharge current waveform against a measured discharge current waveform was obtained by numerically integrating the output of a dI/dt calibrated Rogowski coil. The experimental and simulation curves for plasma pinch properties have been plotted. Some of the dynamic parameters of the plasma and the characteristics of the plasma pinch have been displayed at different argon pressures in two tables. In addition, the chart of the normalized parameters of the plasma pinch has been plotted in terms of pressure. The results of experimentally measured and numerical simulation show that better pinches with attention to their plasma pinch characteristics, generally were dependent on pressure and the best pinch is obtained at 0.6-0.7 Torr for argon at 14 kV in SABALAN1. Furthermore, in high pressures (higher than 0.7 Torr in our experiments), the discharge current can behave as short circuit discharge and resembles that of a simple L-C-R discharge which is a damped sinusoidal. Comparison of the experimental and numerical curves shows sensible agreement in some plasma pinch properties such as the time to pinch and depth of focus. The results from charts and tables have been discussed.In this paper, we have studied the effect of working gas pressure on plasma pinch properties in a Mather type plasma focus (2 kJ, 20 kV, 10 μF), named SABALAN1. Argon at the various operating pressures ranging from 0.1-1.2 Torr has been used as working gas. The total current waveform has been measured for different pressures. Also, a numerical simulation was done using Lee model code to perform numerical experiments in SABALAN1. The numerical simulation was used to compute argon soft x-ray yield as a function of pressure, to anticipate the maximum soft x-ray efficiency at optimum operating gas pressure. The Lee model was configured for SABALAN1, by fitting a simulated discharge current waveform against a measured discharge current waveform was obtained by numerically integrating the output of a dI/dt calibrated Rogowski coil. The experimental and simulation curves for plasma pinch properties have been plotted. Some of the dynamic parameters of the plasma and the characteristics of the plasma pinch have be...


I. INTRODUCTION
Over the past sixty years, a particular class of dense magnetized plasma has been produced by a device generally known as Plasma Focus (PF).It was discovered independently in 1965 by Mather in the USA and Filippov in the USSR.Two basic differences between Mather and Filippov machine geometry are the electrode dimensions and the aspect ratio of the inner electrode (diameter/axial length). 1,2This is a two-dimensional Z-pinch, organized on the axis or near that at the end of a coaxial plasma accelerator.In this device, a short duration (few nanoseconds to a few hundred nanoseconds) but high temperature (T e >500 eV) and high density (n e >10 20 cm -3 ) magnetized plasma is created. 3t is known that the plasma focus machine is very compact, cost-effective, versatile and easy to maintain compared with all the dense plasma sources.5][6][7][8][9] In the past decades, the plasma focus device has been instated as an interesting x-ray source for various applications such as lithography, 10 radiography, 11 microscopy 12 and micromachining. 13n the past few years, lots of experimental and theoretical investigations have been performed for the optimization of plasma focus device, to enhance the x-ray emission from this device.In these researches, various experimental parameters such as anode length, 14 anode shape and material, [15][16][17][18] bank energy, 19 insulator sleeve length and material, 16,20 gas composition [21][22][23] and gas pressure [24][25][26] were changed and the results of these changes were examined.Most of these studies have been performed on many plasma focus devices operating over the wide range of bank energies from megajoule and few hundred kilojoule banks of large-sized 27 to kilojoule banks of medium-sized 3,16,20,28 to subkilojoule banks of miniature-sized plasma focus devices. 22,23,29So, these studies have been one of the actively proceed fields of plasma focus researches, due to their immense possible applications.
It should be mentioned that experiment could not be separate from modeling and simulation.The results from the numerical simulation can be used to improve the experiments or scheme new experiments.It is evident that simulation is based on physical models and theories with many assumptions.On the other hand, its validity needs to be verified by experiments, namely, the computed yields must be examined against corresponding measured yields.The experimental investigation to obtain the optimized conditions for maximum x-ray yield is erroneous and time-consuming, hence a reliable focus model and corresponding simulation code are necessary to anticipate the x-ray yield from plasma focus device.
The Lee model code provides a beneficial tool to execute initial studies for a given plasma focus machine.It couples the electrical circuit with plasma focus dynamics, thermodynamics and radiation. 30The basic model (2-phase) with axial phase and radial phase described in 1984. 31,32The model was written as a non-radiative model (3-phase) for an experimental program at 1991. 33The radiation part was included in a 5-phase code in 2000. 30The radiative (slow compression) phase will be added and a reflected shock phase is introduced to give a more reasonable start of the quasiequilibrium.The 5-phase code is not suitable for high static inductance (L 0 >100 nH) plasma focus devices because the computed current waveform in these devices has an extended dip beyond the regular dip so that the simulated current dip appeared less than the measured one.This problem was solved in extended 6-phase code by including the effects of the unusual resistances in a post-pinch phase. 34,35his paper reports the influence of operating gas pressure on plasma pinch properties in a mediumsized Mather type plasma focus (SABALAN1).Argon at various operating pressures was used as working gas.The Lee model code was used to carry out the numerical simulation on SABALAN1 to compute its argon soft x-ray yield as a function of working gas pressure, to predict the maximum soft x-ray yield at optimum filling gas pressure.Some plasma dynamic parameters and plasma pinch characteristics at various argon pressures were tabulated.Also, the experimental and simulation curves have been plotted for plasma pinch characteristics such as focusing amplitude and time to pinch.Finally, the results of this study were analyzed in details.

II. EXPERIMENTAL SETUP
In this research, a medium-sized Mather type plasma focus (SABALAN1) was used.The coaxial electrode structure of this device consists of anode, cathode and an insulator sleeve.The anode is a cylindrical copper rod with 79 mm active length and the outer diameter of 20 mm.The coaxial cathode that symmetrically surrounded the anode, made of 12 copper rods with 120 mm length and the outer diameter of 10 mm which arranged in squirrel cage configuration with a radius of 29 mm.The insulator sleeve made up of Pyrex glass with a breakdown length of 25 mm and 2.6 mm wall thickness, is placed between anode and cathode.The aspect ratio of this device is 2.9.The machine uses a single capacitor bank of 10 µF capacitance with the rated voltage of 20 kV and stored energy of 2 kJ.Both cathode and anode are installed inside a chamber with two glass windows, which is made of stainless steel.The current derivative signal was measured and registered, with a 12-turn calibrated Rogowski coil.The output of the Rogowski coil is connected using a coaxial cable shielded with Aluminum foil, to a 4-channel 100 MHz, 1G-sample per second digital Tektronix oscilloscope (Fig. 1).

III. EXPERIMENTS AND RESULTS
The experiments were accomplished in this device with argon as filling gas.The charging voltage was 14 kV.The pressure was varied within the range of 0.1-1.2Torr.In all the experiments, the current derivative signals were registered by the oscilloscope.Fig. 2 and Fig. 3 denote the discharge current derivative waveforms and their integrations for some of the selected argon operating pressures from 0.1-0.5 Torr.Fig. 2 and Fig. 3 show that: i) the current dip starts earlier at lower argon operating pressure and ii) the peak of total current, I peak , increases with increasing the filling argon pressure.The current dip is correlated with the radial phase, so the shifting of current dip is consistent with higher axial speed.The filling gas density is proportional to pressure for a fixed gas and the axial speed is corresponding to the inverse of gas density.At lower pressure, the gas density is reduced so it is leading to increasing in axial speed and decreasing in axial time.Thus, the radial phase and the current dip start earlier and earlier at lower and lower pressures, with higher and higher axial speeds.Furthermore, increasing I peak is ascribed to decreasing dynamic resistance.The dynamic resistance is obtained by comparing the inductive power flow: and the total power flow: as: The inductance of the plasma tube in axial phase is: so the dynamic resistance in axial phase becomes as: where µ is permeability, a and b are the anode and cathode radius respectively, (dz/dt) is the axial speed and z is the position of the current sheath in the axial phase.From Eq. 5 it is seen that the higher axial speeds lead to higher dynamic resistance and subsequently lower I peak at lower pressure.Also, Fig. 4 shows some of the selected current waveforms at various argon gas pressures in the range of 0.8-1 Torr.
As can be seen in Fig. 4, the discharge current in 0.8-1 Torr has a sinusoidal shape and no current dip is seen.At these pressures, the gas density is more than the optimum gas work density, so that the axial speed becomes very small.Thus the current sheet only shifts a little down the plasma tube and does not reach to the end of the anode at the end of the first half cycle drive, so the radial phase does not start.Hence the discharge current can behave as short circuit discharge and look like that of a simple L-C-R discharge which has a damped sinusoidal waveform.

IV. NUMERICAL EXPERIMENTS
The numerical experiments are used to conclude some features of the pinch dynamics and plasma pinch as a function of operating gas pressure.To start with the numerical simulation, we selected a discharge current signal of SABALAN1, was obtained by numerically integrating the output of dI/dt calibrated Rogowski coil.The selected current derivative signal is of a shot for the argon gas at the pressure of 0.2 Torr and the charging voltage of 14 kV.
Fig. 6 shows the simulated current trace and the measured current waveform, taken by numerically integrating the current derivative signal in Fig. 5, after the fitting process.There are two versions of Lee's computational model for this purpose: version RADPF5.15de(5-phase code) and version RADPF6.1b7][38] The 5-phase code was not suitable for SABALAN1 because of the high static inductance of this device, 34,35 so the 6-phase code was used.To start fitting, the following parameters are used: Anode radius a=1 (cm) Cathode radius b=2.94 (cm) Anode length z 0 =7.9 (cm) Operation Pressure P 0 =0.2 (Torr) Voltage V 0 =14 (kV) Model Axial mass factor f m =0.08 Axial current factor f c =0.7 Radial mass factor f mr =0.16  Radial current factor f cr =0.7 Bank and model parameters: L 0, R 0 , f m , f c , f mr , f cr were changed one by one until the measured current waveform was fitted to the computed current waveform.First, the static inductance and the stray resistance are adjusted, until the current rise time and the current amplitude of the computed current waveform was matched with measured.Then, the axial factors are altered until the rising slope of the computed total current trace, the rounding off the peak current, and the peak current itself were in reasonable agreement with the measured.Finally, we proceed to set the radial factors until the computed slope and depth of the dip concur with the measured.In this case, the following fitted bank and model parameters are obtained: L 0 = 230 (nH), R 0 =11.5 (mΩ), f m =0.069, f c =0.71, f mr =0.09, and f cr =0.4.AIP Advances 8, 075209 (2018) FIG. 5. Selected current derivative signal for the argon gas at the pressure of 0.2 Torr and the charging voltage of 14 kV.FIG. 6.The fitting process of the computed and measured current waveform, numerically integrated from the current derivative signal in Fig. 5, according to the version RADPF6.1b of Lee's computational model (6-phase code).

V. COMPARISON OF NUMERICAL RESULTS WITH EXPERIMENTAL RESULTS
To compare the experimental and the numerical results, Lee's 6-phase code (version RADPF6.1b) is used for argon as filling gas with the following parameters:

Bank
Static inductance L 0 = 230 (nH) Bank capacitance C 0 =10 (µF) Stray resistance R 0 =11.5 (mΩ) Tube Anode radius a=1 (cm) Cathode radius b=2.94 (cm) Anode length z 0 =7.9 (cm) Operation Pressure P 0 =0.1-1.2 (Torr) Voltage V 0 =14 (kV) Model Axial mass factor f m =0.069 Axial current factor f c =0.71 Radial mass factor f mr =0.09 Radial current factor f cr =0.4 The filling argon pressure was altered in the range of 0.1-1.2Torr with an increment of 0.1 Torr while the charging voltage was kept constant.Fig. 7 presents the discharge current waveforms for some of the selected pressures for argon at 14 kV.AIP Advances 8, 075209 (2018) FIG. 7. Plot of discharge current waveforms from numerical experiments performed for argon at some of the selected pressures.Fig. 7 shows, similarly, three features that were seen in Fig. 3 and Fig. 4: i) end of the radial phase (current dip) reduces with lowering the operating argon pressure at 0.2-0.8Torr, ii) the peak of total current increases with increasing the filling argon pressure at 0.2-0.8Torr, iii) the discharge currents have a sinusoidal shape at 1 and 1.2 Torr.
In addition, changes in the time to pinch and the focusing amplitude are plotted in terms of argon gas pressure at 14 kV in Fig. 8 and Fig. 9 respectively.Fig. 8 and Fig. 9 show sensible agreement between the results of numerical simulation and experimentally measured, in focusing amplitude and time to pinch.From Fig. 8 it could understand that the amount of the time to pinch increases with increasing pressure.This increased delay is due to the increase in mass swept by the current sheath.The mass carried by the sheath at position z is proportional to gas density as: The gas density is proportional to operating gas pressure, so increase with increasing pressure.It is leading to increasing the mass entrained by the sheath at position z and subsequently increasing in time to pinch.Furthermore, it is evident from Fig. 9 that the depth of focus increases with increasing pressure peaking at 0.6 Torr in numerical results and 0.7 Torr in experimental results.The magnitude of the dip in the current derivative signal which is corresponding to the focus phase is used as a scale of focusing efficiency of plasma focus machine.It may be noted that an important condition for the formation of the hot and dense plasma (pinch) is the current sheath should arrive the axis in coincidence with or near the first peak of the discharge current.This condition represents the optimization of the energy transmission from the capacitor bank to the plasma pinch.
Some plasma dynamics and plasma pinch parameters in terms of argon gas pressure at 14 kV have been displayed in Table I and Table II.
Parameters used in two tables are: P 0 is the argon gas pressure; E 0 is bank stored energy at charging voltage 14 kV; I peak is the peak value of the total discharge current; T axial end is the end time of axial phase; peak v a, peak v s and peak v p are the peak value of the axial speed, radial shock speed, and radial piston speed, respectively; r min is minimum radius and z max is maximum length of focus pinch at time of maximum comparison; V max is the peak value of the total voltage; I pinch start is the current that flows through the pinched plasma columns at the start of pinch phase; EINP is work done by the dynamic resistance during radial phase expressed as % of E 0; T pinch ,N i pinch , Y sxr , and Z eff are the plasma temperature, ion density, argon soft x-ray yield and effective charge number in the middle of pinch phase, respectively.
Furthermore, the normalized plasma pinch properties plotted as a function of operating gas pressure in Fig. 10.In order to plot all the parameters in one figure, each parameter normalized to its value at optimum (the value obtained for 0.7 Torr).As shown in Fig. 10, Table I and Table II, the I peak , increases with increasing the operating argon pressure, similarly to experimental results in Fig. 3, due to decreasing dynamic resistance as a result of increasing axial speed in higher pressure.The I pinch start , increases with increasing pressure to o.2 Torr.As the pressure increased above 0.2 Torr, the I pinch start starts to decrease with increasing pressure.This is because of that, in lower argon pressure until the pressure nears o.2 Torr, the axial speed is high and the pinch time occurs before the current peak time (as shown in Fig. 3 and Fig. 7).As the pressure is increased above 0.2 Torr, the axial speed is reduced and the pinch time is shifted toward the current peak time.The T pinch are trending downward as pressure is increased.The r min is initially reduced by increasing pressure and has a minimum at 0.6 Torr, then increases with increasing pressure to 0.9 Torr.The N i pinch , increases with increasing pressure peaking around 0.7 Torr then reducing at higher pressure.This shows that as the argon pressure is reduced toward 0.6 Torr, the increasing I pinch start increases the compression sufficiently so that despite the drop in ambient number density, the N i pinch is still able to reach a higher value around 0.7 Torr.As the operating pressure is reduced below o.6 Torr, the increase in I pinch start does not appear to be sufficient to further increase N i pinch or simply even to compress the pinch to a smaller radius than at 0.6 Torr.It seems that the peaking of N i pinch around 0.7 Torr is a significant factor for the peaking of Y sxr at 0.7 Torr.This also confirms the peaking of focusing depth observed in Fig. 9. Accordingly, the optimum operating argon pressure with the maximum soft x-ray efficiency is 0.7 Torr at 14 kV in SABALAN1.

VI. CONCLUSION
The results obtained in this study are: i) end of the radial phase (current dip) reduces with lowering the working argon pressure, ii) the peak of total current, I peak , increases with increasing the filling argon pressure, iii) better pinches regarding their pinch parameters, generally are associated with pressure, iv) comparison the experimental and simulation results show reasonable agreement between the results in time to pinch and focusing depth, v) in the argon pressure higher than 0.7 Torr at 14 kV in SABALAN1, the discharge current can be treated as a simple L-C-R discharge which is a damped sinusoidal shape, vi) the best pinch has been obtained at 0.6-0.7 Torr for argon working gas at 14 kV in SABALAN1, and x) the optimum operating argon pressure with the maximum soft x-ray efficiency is 0.7 Torr at 14 kV in SABALAN1.

FIG. 2 .
FIG.2.Discharge current derivative signals at some selected argon pressures below 0.7 Torr, taken with a calibrated Rogowski coil.

075209- 5 Karimi
FIG. 4. Discharge current waveforms in argon operating pressures over 0.7 Torr, taken by numerically integrating the output of dI/dt calibrated Rogowski coil.

FIG. 8 .
FIG.8.Comparison of experimental and numerical results in time to pinch as a function of operating gas pressure for argon at 14 kV.

TABLE I .
Computed plasma dynamic parameters for different filling gas pressures by the numerical simulation conducted on SABALAN1 plasma focus using Lee model code.

TABLE II .
Computed plasma pinch parameters for different filling gas pressures by the numerical simulation conducted on SABALAN1 plasma focus using Lee model code.
FIG.10.Effect of operating gas pressure on some plasma pinch properties, normalized value at optimum pressure (0.7 Torr).