Plasma Focus Numerical Experiments
knowledge should be freely accessible to
all
Sor Heoh Saw and Sing Lee in collaboration with Erol Kurt
Reference to the Lee model code should be
given as follows:
Lee S. Radiative Dense Plasma Focus Computation
Package (2011): RADPF www.plasmafocus.net http://www.intimal.edu.my/school/fas/UFLF/
Module 1: Introduction  The
Plasma Focus and the Lee model
Summary
1.1
Description of the plasma focus; how it
works, dimensions and lifetimes of the hot dense plasma
1.2
Scaling properties of the plasma focus
1.3
The radiative Lee model: the 5 phases
1.4
Using the Lee model as reference for
diagnostics
1.5
Insights on plasma focus from numerical
experiments using Lee model code
Description of the plasma focus.
How it works, dimensions and lifetimes of the hot dense plasma
1.1.1
Introduction
The Plasma Focus
is a compact powerful pulsed source of multiradiation [1]. Even a small
table top sized 3 kJ plasma focus
produces an intense burst of radiation with extremely high powers. For example when
operated in neon, the xray emission power peaks at 10^{9} W over a period of nanoseconds. When
operated in deuterium the fusion neutron burst produces rates of neutron
typically 10^{15} neutrons per second over burst durations of tens of
nanosecond. The emission comes from a point source making these devices among
the most powerful laboratory pulsed radiation sources in the world. These
sources are plasmabased.
When matter is
heated to a high enough temperature, it ionizes and becomes plasma. It emits
electromagnetic radiation. The spectrum depends on the temperature and the material.
The higher the temperature and the denser the matter, the more intense is the
radiation. Beams of electrons and ions may also be emitted. If the material is
deuterium, nuclear fusion may take place if the density and temperature are
high enough. In that case neutrons are also emitted. Typically the temperatures
are above several million K and
compressed densities above atmospheric density starting with a gas a hundredth
of an atmospheric density.
One way of
achieving such highly heated material is by means of an electrical discharge
through gases. As the gas is heated, it expands, lowering the density and
making it difficult to heat further. Thus it is necessary to compress the gas
whilst heating it, in order to achieve sufficiently intense conditions. An
electrical discharge between two electrodes produces an azimuthal magnetic
field which interacts with the column of current, giving rise to a self
compression force which tends to
constrict (or pinch) the column. In order to ‘pinch’, or hold together, a
column of gas at about atmospheric density at a temperature of 1 million K, a rather large pressure has to be exerted
by the pinching magnetic field. Thus an electric current of at least hundreds
of kA are required even for a column
of small radius of say 1 mm. Moreover
the dynamic process requires that the current rises very rapidly, typically in
under 0.1 ms in order to have a sufficiently hot and dense pinch. Such
a pinch is known as a superfast superdense pinch; and requires special MA fastrise (ns) pulsedlines. These lines may be powered by capacitor banks,
and suffer the disadvantage of conversion losses and high cost due to the cost
of the high technology pulseshaping line, in addition to the capacitor banks.
A superior
method of producing the superdense and superhot pinch is to use the plasma
focus. Not only does this device produce superior densities and temperatures,
moreover its method of operation does away with the extra layer of technology
required by the expensive and inefficient pulseshaping line. A simple
capacitor discharge is sufficient to power the plasma focus.
1.1.2 The plasma focus
The plasma focus
is divided into two sections. The first is a prepinch (axial) section. The
function of this section is primarily to delay the pinch until the capacitor
discharge (rising in a damped sinusoidal fashion) approaches its maximum
current. This is done by driving a current sheet down an axial (acceleration)
section until the capacitor current approaches its peak. Then the current sheet
is allowed to undergo transition into a radial compression phase. Thus the
pinch starts and occurs at the top of the current pulse. This is equivalent to
driving the pinch with a superfast rising current; without necessitating the
fast line technology. Moreover the intensity which is achieved is superior to
the line driven pinch.
Figure 1. Schematic of the axial and radial phases. The
left section depicts the axial phase, the right section the radial phase. In
the left section, z is the effective
position of the current sheathshock front structure. In the right section r_{s} is the position of the
inward moving shock front driven by the piston at position r_{p}. Between r_{s}
and r_{p} is the radially
imploding slug, elongating with a length z_{f}.
The capacitor, static inductance and switch powering the plasma focus is shown
for the axial phase schematic only.
The twophase
mechanism of the plasma focus [1] is shown in figure 1. The inner electrode (anode) is separated from
the outer concentric cathode by an insulating backwall. The electrodes are
enclosed in a chamber, evacuated and typically filled with gas at about 1/100
of atmospheric pressure. When the capacitor voltage is switched onto the focus
tube, breakdown occurs axisymmetrically between the anode and cathode across
the backwall. The ‘sheet’ of current lifts off the backwall as the magnetic
field (B_{q}) and it’s inducing current (J_{r}) rises to a sufficient
value.
Axial phase: The J_{r} x B_{q} force then pushes the current sheet, accelerating it
supersonically down the tube. This is very similar to the mechanism of a linear
motor. The speed of the current sheet, the length of the tube and the rise time
of the capacitor discharge are matched so that the current sheet reaches the
end of the axial section just as the discharge reaches its quarter cycle. This
phase typically lasts 13 ms for a plasma focus of several kJ.
Radial Phase: The part of the current sheet in
sliding contact with the anode then ‘slips’ off the end ‘face’ of the anode
forming a cylinder of current, which is then pinched inwards. The wall of the
imploding plasma cylinder has two boundaries (see figure 1 radial phase). The
inner face of the wall, of radius r_{s}
is an imploding shock front. The outer side of the wall, of radius r_{p}_{ }is the
imploding current sheet, or magnetic piston. Between the shock front and the
magnetic piston is the annular layer of plasma. Imploding inwards at higher and
higher speeds, the shock front coalesces onaxis and a superdense, superhot
plasma column is pinched onto the axis (see figure 2 [2]). This column stays
superhot and superdense for typically ten ns
for a small focus. The column then breaks up and explodes. For a small plasma
focus of several kJ, the most intense
emission phase lasts for the order of several ns. The radiation source is spotlike (1mm diameter) when viewed endon.
Figure 2. Dense plasma focus device. Image from Glenn Millam.
Source: Focus
Fusion Society For an
animation of the plasma focus click here.
1.1.3 Radial dynamics of the
plasma focus
Figure 3 shows a drawing of
a typical plasma focus, powered by a single capacitor, switched by a simple
parallelplate sparkgap. The anode may be a hollow copper tube so that during
the radial pinching phase the plasma not only elongates away from the anode
face but also extends and elongates into the hollow anode (see figure 2). In figure
3 is shown the section where the current sheet is accelerated axially and also
the radial section. Also shown in the same figure are shadowgraphs [3] taken of
the actual radially imploding current sheetshock front structure. The
shadowgraphs are taken in a sequence, at different times. The times indicated
on the shadowgraphs are relative to the moment judged to be the moment of
maximum compression.
That moment is taken as t=0. The quality of the plasma compression can be
seen to be very good, with excellent axisymmetry, and a very well compressed
dense phase. In the lower left of figure 3 are shown the current and voltage
signatures of the radial implosion [4], occurring at peak current. The
implosion speeds are measured and has a peak value approaching 30 cm/ms.
This agrees with modelling, and by considering shock
wave theory together with modelling [5] of subsequent reflected shock wave and
compressive effects, a temperature of 6 million K (0.5 keV) is estimated
for the column at peak compression, with a density of 2x10^{19} ions
per cm^{3}. The values quoted
here are for the UNU/ICTP PFF 3 kJ
device.
Figure 3.
UNU/ICTP PFF 
Design, Signatures and Dynamics
In
figure 4a is shown the UNU ICTP PFF 3 kJ
device [46] mounted on a 1 m by 1 m by 0.5 m trolley, which was wheeled around the International Centre for Theoretical
Physics (ICTP) for the 1991 and 1993 Plasma Physics Colleges during the
experimental sessions. The single capacitor is seen in the picture mounted on
the trolley. In contrast, figure 4b
shows the PF1000, the 1 MJ device [7]
at the International Centre for Dense Magnetised Plasmas (ICDMP) in
Figure 4a. 3kJ UNU ICTP PFF Figure 4b. 1 MJ PF1000 plasma focus
We show here the characteristics of several plasma
focus devices [7].
Table 1 Characteristics of three plasma focus devices
Plasma Focus Devices 
E_{0} (kJ) 
a (cm) 
Z_{0} (cm) 
V_{0} (kV) 
P_{0} (Torr) 
I_{peak} (kA) 
v_{a} (cm/us) 
ID (kA/cm) 
SF [(kAcm^{1}) Torr^{0.5}] 
Y_{n} (10 ^{8}) 
PF1000 
486 
11.6 
60 
27 
4 
1850 
11 
160 
85 
1100 
UNU ICTP PFF 
2.7 
1.0 
15.5 
14 
3 
164 
9 
173 
100 
0.20 
PF400J 
0.4 
0.6 
1.7 
28 
7 
126 
9 
210 
82 
0.01 
In table 1 we look at the PF1000 and study its
properties at typical operation with device storage at 500 kJ level. We compare this big focus with two small devices at the kJ level.
From table 1 we note:
Voltage and
pressure do not have any particular relationship to E_{0}.
Peak current I_{peak} increases with E_{0}.
Anode radius
‘a’ increases with E_{0}.
ID (current per cm
of anode radius) I_{peak}/a is in a narrow range from 160
to 210 kA/cm
SF (speed or drive factor) (I_{peak}/a)/P_{0}^{0.5} is 82 to 100
kAcm^{1}/Torr^{0.5} deuterium gas [8].
Peak axial
speed v_{a} is in the narrow
range 9 to 11 cm/us.
Fusion neutron yield Y_{n} ranges from 10^{6}
for the smallest device to 10^{11} for the PF1000.
We stress that whereas the ID and SF are practically
constant at around 180 kA/cm and (90 kA/cm)/Torr^{0.5 }deuterium gas throughout the range of small to
big devices, Y_{n} changes
over 5 orders of magnitude.
The data of table 1 is generated from numerical
experiments [5,9] and most of the data has been confirmed by actual experimental
measurements and observations.
Table 2 Properties of three plasma focus devices
Plasma focus Devices 
c= b/a

a (cm) 
T_{pinch} (10^{6}/K) 
v_{p} (cm/ms) 
r_{min} (cm) 
z_{max} (cm) 
Pinch Duration (ns) 
r_{min}/a 
z_{max}/a 
Pinch duration/a (ns/cm) 
PF1000

1.4 
11.6 
2 
13 
2.2 
19 
165 
0.17 
1.6 
14 
UNU
ICTP PFF 
3.4 
1.0 
8 
26 
0.13 
1.4 
7.3 
0.14 
1.4 
8 
PF400J 
2.6 
0.6 
6 
23 
0.09 
0.8 
5.2 
0.14 
1.4 
9 
Table 2 compares the properties of a range of plasma
focus devices. The properties being compared in this table are mainly related
to the radial phase.
From table 2 we note:
i.
The pinch temperature T_{pinch} is strongly correlated to
the square of the radial pinch speed v_{p}.
ii. The radial pinch speed v_{p} itself is closely correlated
to the value of v_{a} and c=b/a; so that for a constant v_{a}, v_{p} is almost proportional to
the value of c.
iii. The dimensions and lifetime
of the focus pinch scale as the anode radius ‘a’.
r_{min}/a (almost constant at
0.140.17)
z_{max}/a (almost constant at 1.5)
iv. Pinch duration has a
relatively narrow range of 814 ns
per cm of anode radius.
v. The pinch duration per unit
anode radius is correlated to the inverse of T_{pinch}.
T_{pinch} itself is a measure of the
energy per unit mass. It is quite
remarkable that this energy density at the focus pinch varies so little (factor
of 5) over a range of device energy of more than 3 orders of magnitude.
This practically constant pinch energy density (per
unit mass) is related to the constancy of the axial speed moderated by the
effect of the values of c on the
radial speed.
The constancy of r_{min}/a suggests that the devices also produce
the same compression of ambient density to maximum pinch density; with the
ratio (maximum pinch density)/ (ambient density) being proportional to (a/r_{min})^{2}.
So for two devices of different sizes starting with the same ambient fill
density, the maximum pinch density would be the same.
From the above discussion, we may put down as
ruleofthumb the following scaling relationships, subject to minor variations
caused primarily by the variation in c.
i.
Axial phase energy density (per unit mass) constant
ii. Radial phase energy density
(per unit mass) constant
iii. Pinch radius ratio constant
iv. Pinch length ratio constant
v. Pinch duration per unit
anode radius constant
1.2.2 Summarising
i.
The dense hot plasma pinch of a small E_{0}_{ }plasma focus and that of a big E_{0}_{ }plasma focus
have essentially the same energy density, and the same mass density.
ii. The big E_{0}_{ }plasma focus has a bigger physical size and
a bigger discharge current. The size of the plasma pinch scales proportionately
to the current and to the anode radius, as does the duration of the plasma
pinch.
iii. The bigger E_{0}, the bigger ‘a’, the bigger I_{peak}, the larger the plasma pinch and the longer the duration of the plasma
pinch. The larger size and longer duration of the big E_{0}_{ }plasma pinch are essentially the
properties leading to the bigger neutron yield compared to the yield of the
small E_{0}_{ }plasma
focus.
The above description of the plasma focus combines
data from numerical experiments, consistent with laboratory observations.
The next section describes the Lee model code.
1.3 The radiative Lee model:
the 5 phases
The Lee model couples the electrical circuit
with plasma focus dynamics, thermodynamics, and radiation, enabling a realistic
simulation of all gross focus properties. The basic model, described in 1984
[1], was successfully used to assist several projects [46]. Radiationcoupled
dynamics was included in the fivephase code, leading to numerical experiments
on radiation cooling [5]. The vital role of a finite small disturbance speed
discussed by Potter in a Zpinch situation [10]
was incorporated together with real gas thermodynamics and radiationyield
terms. This version of the code assisted other
research projects [4,8,11,12] and was web published in 2000 [13] and 2005 [14].
Plasma selfabsorption was included in 2007 [13], improving the SXR yield
simulation. The code has been used extensively in several machines including
UNU/ICTP PFF [3,8,11,12], NX2 [12,1517], and NX1 [15,18] and has been adapted
for the Filippovtype plasma focus DENA [19]. A recent development is the
inclusion of the neutron yield Y_{n} using
a beam–target mechanism [2024], incorporated in recent versions [5] of the
code (versions later than RADPFV5.13),
resulting in realistic Y_{n} scaling
with I_{pinch} [20,21].
The versatility and utility of the model are demonstrated in its clear
distinction of I_{pinch} from
I_{peak} [25]
and the recent uncovering of a plasma focus pinch current limitation effect [22,23],
as static inductance is reduced towards zero. Extensive numerical experiments
had been carried out systematically resulting in the uncovering of neutron [20,21]
and SXR [2633] scaling laws over a wider range of energies and currents than
attempted before. The numerical experiments also gave insight into the nature
and cause of ‘neutron saturation [9,30,34]. The description, theory, code, and
a broad range of results of this “Universal Plasma Focus Laboratory Facility”
are available for download from [5].
A
brief description of the 5phase Lee model is given in the following.
1.3.1 The 5 phases
The five phases (ae) are summarised [5,13,14,27, 31,33,35]
as follows:
a. Axial
Phase (see figure 1 left part)
Described
by a snowplow model with an equation of motion which is coupled to a circuit
equation. The equation of motion incorporates the axial phase model parameters:
mass and current factors f_{m}_{
}and f_{c}. The mass
sweptup factor f_{m} accounts
for not only the porosity of the current sheet but also for the inclination
of the moving current sheetshock front structure, boundary layer effects, and
all other unspecified effects which have effects equivalent to increasing or
reducing the amount of mass in the moving structure, during the axial phase.
The current factor f_{c} accounts
for the fraction of current effectively flowing in the moving structure (due to
all effects such as current shedding at or near the backwall, and current
sheet inclination). This defines the fraction of current effectively driving
the structure, during the axial phase.
Figure 5. Schematic of radius vs time trajectories to
illustrate the radial inward shock phase when r_{s}
moves radially inwards, the reflected shock (RS) phase when the reflected shock
moves radially outwards, until it hits the incoming piston r_{p} leading to the start of the pinch phase (t_{f}) and finally the expanded
column phase.
b. Radial
Inward Shock Phase (see figure 1 right part, also figure 2)
Described by four coupled
equations using an elongating slug model. The first equation computes the radial inward shock speed from the driving
magnetic pressure. The second equation computes the axial elongation speed of
the column. The third equation computes the speed of the current sheath,
(magnetic piston), allowing the current sheath to separate from the shock front
by applying an adiabatic approximation [5,7].^{ }The fourth is the
circuit equation. Thermodynamic effects due to ionization and excitation are
incorporated into these equations, these effects being particularly important
for gases other than hydrogen and deuterium. Temperature and number densities
are computed during this phase using shockjump equations. A communication
delay between shock front and current sheath due to the finite small
disturbance speed [10,35] is crucially implemented in this phase. The model
parameters, radial phase mass sweptup and current factors f_{mr} and f_{cr}
are incorporated in all three radial phases. The mass sweptup factor f_{mr} accounts for all
mechanisms which have effects equivalent to increasing or reducing the amount
of mass in the moving slug, during the radial phase. The current factor f_{cr} accounts for the fraction
of current effectively flowing in the moving piston forming the back of the
slug (due to all effects). This defines the fraction of current effectively
driving the radial slug.
c. Radial
Reflected Shock (RS) Phase (See figure 5)
When the shock front hits the axis, because the focus
plasma is collisional, a reflected shock develops which moves radially
outwards, whilst the radial current sheath piston continues to move inwards.
Four coupled equations are also used to describe this phase, these being for
the reflected shock moving radially outwards, the piston moving radially
inwards, the elongation of the annular column and the circuit. The same model
parameters f_{mr} and f_{cr} are used as in the
previous radial phase. The plasma temperature behind the reflected shock
undergoes a jump by a factor close to 2. Number densities are also computed
using the reflected shock jump equations.
d. Slow
Compression (Quiescent) or Pinch Phase (See figure 5)
When the outgoing reflected shock hits the inward
moving piston, the compression enters a radiative phase in which for gases such
as neon, radiation emission may actually enhance the compression where we have
included energy loss/gain terms from Joule heating and radiation losses into
the piston equation of motion. Three coupled equations describe this phase;
these being the piston radial motion equation, the pinch column elongation
equation and the circuit equation, incorporating the same model parameters as
in the previous two phases. The duration of this slow compression phase is set
as the time of transit of small disturbances across the pinched plasma column.
The computation of this phase is terminated at the end of this duration.
e. Expanded
Column Phase
To
simulate the current trace beyond this point we allow the column to suddenly
attain the radius of the anode, and use the expanded column inductance for
further integration. In this final phase the snow plow model is used, and two
coupled equations are used similar to the axial phase above. This phase is not considered important as it
occurs after the focus pinch.
We
note [31] that in radial phases b, c
and d, axial acceleration and
ejection of mass caused by necking curvatures of the pinching current sheath
result in timedependent strongly centerpeaked density distributions. Moreover
the transition from phase d to phase e is observed in laboratory measurements
to occur in an extremely short time with plasma/current disruptions resulting
in localized regions of high densities and temperatures. These centrepeaking
density effects and localized regions are not modeled in the code, which
consequently computes only an average uniform density, and an average uniform
temperature which are considerably lower than measured peak density and
temperature. However, because the four
model parameters are obtained by fitting the computed total current waveform to
the measured total current waveform, the model incorporates the energy and mass
balances equivalent, at least in the gross sense, to all the processes which
are not even specifically modeled. Hence the computed gross features such as speeds and trajectories and integrated
soft xray yields have been extensively tested in numerical experiments for
several machines and are found to be comparable with measured values.
1.4 Using the Lee model as reference for diagnostics
1.4.1 From measured current waveform to modelling for
diagnostics
The Lee model code [5,13,14] is configured [9] to work as
any plasma focus by inputting:
Bank parameters, L_{0}, C_{0} and stray circuit resistance r_{0};
Tube parameters b, a and z_{0};
Operational parameters V_{0} and P_{0}
and the fill gas.
The computed total
current waveform is fitted to the measured waveform by varying model parameters
f_{m}, f_{c}, f_{mr} and f_{cr}
one by one, until the computed waveform agrees with the measured waveform.
First, the axial model factors f_{m}, f_{c}
are adjusted (fitted) until the features in figure 6: ‘1’ computed rising slope of the total current trace; ‘2’ the rounding off of the peak
current as well as ‘3’ the peak
current itself are in reasonable (typically very good) fit with the measured
total current trace (see figure 6, measured trace fitted with computed trace).
Then we proceed to adjust (fit) the radial phase model
factors f_{mr} and f_{cr} until features ‘4’ the computed slope and ‘5’ the depth of the dip agree with the
measured values. Note that the fitting of the computed trace with the measured
current trace is done up to the end of the radial phase which is typically at
the bottom of the current dip. Fitting of the computed and measured current
traces beyond this point is not done. If there is significant divergence of the
computed with the measured trace beyond the end of the radial phase, this
divergence is not considered important.
In this case, after fitting the five features ‘1’ to ‘5’ above, the following
fitted model parameters are obtained: f_{m}=0.1, f_{c}=0.7, f_{mr}=0.12, f_{cr}=0.68.
From experience it is known that the current trace of the
focus is one of the best indicators of gross performance. The axial and radial
phase dynamics and the crucial energy transfer into the focus pinch are among
the important information that is quickly apparent from the current trace
[2729].
Figure 6. The 5point fitting of computed current
trace to measured (reference) current trace. Point 1 is the current rise slope. Point
2 is the topping profile. Point 3
is the peak value of the current. Point
4 is the slope of the current dip. Point
5 is the bottom of the current dip. Fitting is done up to point 5 only.
Further agreement or divergence of the computed trace with/from the measured
trace is only incidental and not considered to be important.
The exact time profile of the total current trace is
governed by the bank parameters, by the focus tube geometry and the operational
parameters. It also depends on the fraction of mass sweptup and the fraction
of sheath current and the variation of these fractions through the axial and
radial phases. These parameters determine the axial and radial dynamics,
specifically the axial and radial speeds which in turn affect the profile and magnitudes
of the discharge current.
There are many underlying mechanisms in the axial phase
such as shock front and current sheet structure, porosity and inclination,
boundary layer effects and current shunting and fragmenting which are not
simply modeled; likewise in the radial phase mechanisms such as current sheet
curvatures and necking leading to axial acceleration and ejection of mass, and
plasma/current disruptions. These effects may give rise to localized regions of
high density and temperatures. The detailed profile of the discharge current is
influenced by these effects and during the pinch phase also reflects the Joule
heating and radiative yields. At the end of the pinch phase the total current
profile also reflects the sudden transition of the current flow from a
constricted pinch to a large column flow. Thus the discharge current powers all
dynamic, electrodynamic, thermodynamic and radiation processes in the various
phases of the plasma focus. Conversely all the dynamic, electrodynamic, thermodynamic
and radiation processes in the various phases of the plasma focus affect the
discharge current. It is then no exaggeration to say that the discharge current waveform contains
information on all the dynamic, electrodynamic, thermodynamic and radiation
processes that occur in the various
phases of the plasma focus. This explains the importance attached to matching
the computed total current trace to the measured total current trace in the
procedure adopted by the Lee model code. Once matched, the fitted model
parameters assure that the computation proceeds with all physical mechanisms
accounted for, at least in the gross energy and mass balance sense.
1.4.2 Diagnosticstime histories
of dynamics, energies and plasma properties computed from the measured total
current waveform by the code
During every adjustment of each of the model parameters the
code goes through the whole cycle of computation. In the last adjustment, when
the computed total current trace is judged to be reasonably well fitted in all
5 waveform features, computed time histories are presented, in figure 7a7o as an
example, as follows: for the NX2 operated at 11 kV, 2.6 Torr neon [9,33].
Figure 7a. Fitted computed I_{total } Figure 7b. Computed I_{total} & I_{plasma}
Figure 7c. Tube voltage Figure 7d. Axial trajectory and speed
Figure 7e. Radial trajectories Figure 7f. Length of elongating structure
Figure 7g. Speeds in radial phases Figure 7h. Tube inductanceaxial & radial phases
Figure 7i. Total inductive energy Figure 7j. Piston work and DR energy;
both traces overlap
Figure 7k. DR axial and radial
phases Figure 7l. Peak & averaged uniform n_{i}
Figure 7m. Peak & averaged uniform n_{e} Figure 7n.Peak and averaged uniform T
Figure 7o. Neon Soft xray power
1.4.3 Comments on computed quantities by Lee model code
i.
The computed total current trace typically agrees very well with the
measured because of the fitting. The end of the radial phase is indicated in
Figure 7a. Plasma currents are rarely measured. We had done a comparison of the
computed plasma current with measured plasma current for the Stuttgart PF78
which shows good agreement of our computed to the measured plasma current [28].
The computed plasma current in this case of the NX2 is shown in figure 7b.
ii.
The computed tube voltage is difficult to compare with measured tube
voltages in terms of peak values, typically because of poor response time of
voltage dividers. However the computed waveform shape in figure 7c is general
as expected.
iii.
The computed axial trajectory and speed, agree with experimental
obtained time histories. Moreover, the behaviour with pressure, running the
code at different pressures, agrees well with experimental results. The radial
trajectories and speeds are difficult to measure. The computed trajectories figure
7e agrees with the scant experimental data available. The length of the radial
structure is shown in figure 7f. Computed speeds radial shock front and piston
speeds and speed of the elongation of the structure are shown in figure 7g.
iv.
The computed inductance (figure 7h) shows a steady increase of
inductance in the axial phase, followed by a sharp increase (rising by more
than a factor of 2 in a radial phase time interval about 1/10 the duration of
the axial phase for the NX2).
v.
The inductive energy (0.5LI^{2})
peaks at 70% of initial stored energy, and then drops to 30% during the radial
phase, as the sharp drop of current more than offsets the effect of sharply
increased inductance (figure 7i).
vi.
In figure 7j is shown the work done by the magnetic piston, computed
using force integrated over distance method. Also shown is the work dissipated
by the dynamic resistance, computed using dynamic resistance power integrated
over time. We see that the two quantities and profiles agree exactly. This
validates the concept of half Ldot as
a dynamic resistance.
vii.
Dynamic resistance, DR (DR will be discussed in Module 2, Section 2.3;
Note 2). The piston work deposited in the
plasma increases steadily to some 12% at the end of the axial phase and then
rises sharply to just below 30% in the radial phase. Dynamic resistance is
shown in figure 7k. The values of the DR in the axial phase, together with the
bank surge impedance, are the quantities that determine I_{peak}.
viii.
The ion number density has a maximum value derived from shockjump
considerations, and an averaged uniform value derived from overall energy and
mass balance considerations. The time profiles of these are shown in the figure
7l. The electron number density (figure 7m) has similar profiles to the ion
density profile, but is modified by the effective charge numbers due to ionization
stages reached by the ions.
ix.
Plasma temperature too has a maximum value and an averaged uniform
value derived in the same manner; are shown in figure 7n. Computed neon soft
xray power profile is shown in figure 7o. The area of the curve is the soft
xray yield in J. Pinch dimensions
and lifetime may be estimated from figures 7e and 7f.
x.
The model also computes the neutron yield, for operation in deuterium,
using a phenomenological beamtarget mechanism [2527]. The model does not
compute a time history of the neutron emission, only a yield number Y_{n}.
Thus
as is demonstrated above, the model code when properly fitted is able to
realistically model any plasma focus and act as a guide to diagnostics of
plasma dynamics, trajectories, energy distribution and gross plasma properties.
1.4.4 Scaling parameters of the plasma focus pinch
The gross dynamics of the plasma focus is discussed
in terms of phases. The dynamics of the axial and radial phases is computed
using respectively a snowplow and an elongating slug model. A reflected shock
phase follows, giving the maximum compression configuration of the plasma focus
pinch. An expanded column phase is used to complete the postfocus electric
current computation. Parameters of the gross focus pinch obtained from the
computation, supplemented by experiments may also be summarised as follows:
Table 3
Plasma Focus Pinch Parameters 
Deuterium 
Neon (for SXR) 
minimum radius r_{min} 
0.15a 
0.05a 
max length (hollow anode) z 
1.5a 
1.6a 
radial shock transit t_{comp}_{ } 
5x10^{6}a_{} 
4x10^{6}a 
pinch lifetime t_{p} 
10^{6}a_{} 
10^{6}a 
where,
for the times in s, the value of
anode radius, a, is in m. For the neon calculations radiative
terms are included; and the stronger compression (smaller radius) is due to thermodynamic
effects.
1.5 Insights on plasma focus from numerical experiments using Lee model
code
Using
the Lee model code, series of experiments have been systematically carried out
to look for behaviour patterns of the plasma focus.
Insights
uncovered by the series of experiments include:
i.
pinch current limitation effect as static
inductance is reduced;
ii.
neutron and SXR
scaling laws;
iii.
a global scaling
law for neutrons versus storage energy combining experimental and numerical
experimental data; and
iv.
the nature and a
fundamental cause of neutron saturation.
These
significant achievements are accomplished within a period of twenty months of
intensive numerical experimentation in 2008 to 2009. The numerical experimental
research continues in 2010 with widening international collaboration.
[1] Lee, S. Plasma focus model yielding trajectory and structure.
In Radiations in Plasmas, McNamara, B., Ed.; World Scientific,
[2] Glenn Millam Focus Fusion Society
[3] Muhammad Shahid Rafique. PhD thesis (in
preparation) NTU,
[4] S.Lee, S.P.Moo, C.S.Wong, A.C.Chew.
Twelve Years of UNU/ICTP PFF A Review. IC/98/231, ICTP Preprint, International
Centre for Theoretical Physics,
[5] Lee
S Radiative Dense Plasma Focus Computation Package: RADPF http://www.intimal.edu.my/school/fas/UFLF/File1RADPF.htm
http://www.plasmafocus.net/IPFS/modelpackage/File1RADPF.htm
[6] S.Lee, T.Y.Tou, S.P.Moo, M.A.Eissa,
A.V.Gholap, K.H.Kwek, S.Mulyodrono, A.J. Smith, Suryadi, W.Usala, M.
Zakaullah. A simple facility for the
teaching of plasma dynamics and plasma nuclear fusion. Amer J. Phys., USA, 1988, 56: 6268.
[7] http://www.intimal.edu.my/school/fas/UFLF/
[8] S Lee and A Serban A, “Dimensions and lifetime of the plasma
focus pinch,” IEEE Trans. Plasma Sci., 24, no.3, 1996, 11011105.
[9] Lee S. Diagnostics and Insights from Current waveform and
Modelling of Plasma Focus. Keynote address: IWPDA,
[10] Potter,
D. E. The formation of highdensity zpinches. Nucl. Fusion. 1978, 18, 813–823.
[11] M.H.Liu.
Soft XRays from Compact Plasma Focus.
PhD thesis. NTU,
[12] S.Lee, P.Lee, G.Zhang, X. Feng, V.Gribkov,
M.Liu, A.Serban & T.K.S. Wong. High Repetition High Performance Plasma
Focus as a Powerful Radiation Source IEEE Trans Plasma Science 26(4) 11191126
(1998).
[13] S. Lee, 2000–2007.
[Online]. Available: http://ckplee.myplace.nie.edu.sg/ plasmaphysics/
[14] S. Lee, ICTP Open Access Archive, 2005. [Online].
Available:
http://eprints.ictp.it/85/
[15] G.X.Zhang. Plasma Soft
xray source for Microelectronics lithography.
PhD thesis, NTU,
[16] S.Lee, V.Kudryashov, P.Lee, G.Zhang,
A..Serban, X..Feng, M.Liu, and T.K.S.Wong.
Lithography Using a Powerful Plasma Focus Soft Xray Source.
International Congress on Plasma Physics,
[17] D Wong,
P Lee,
T Zhang,
A Patran,
T L Tan,
R S Rawat
and S Lee,
“An improved radiative plasma focus model calibrated for neonfilled NX2 using
a tapered anode,” Plasma Sources Sci. Technol. 16, 2007, pp. 116123.
[18] E P
Bogolyubov, V D Bochkov,
V A
Veretennikov, L T
Vekhoreva, V A Gribkov,
A V Dubrovskii,
Yu P Ivanov,
A I Isakov,
O N Krokhin,
P Lee,
S Lee,
V Ya
Nikulin, A Serban,
P V Silin,
X Feng
and G X Zhang,
“A powerful soft xray source for xray lithography based on plasma focusing” 1998 Phys. Scripta., vol. 57, 1998,
pp. 488494.
[19] V Siahpoush, M A Tafreshi, S Sobhanian and
[20] S. Lee and S.H. Saw “Neutron scaling laws from numerical
experiments,” J. Fusion Energy, 2008,
27, no. 4, pp. 292–295.
[21] S. Lee S. “Current and neutron scaling for megajoule plasma
focus machines,” Plasma Phys. Control. Fusion, 2008, 50, no. 10, p. 105
005 (14pp).
[22] S. Lee and S.H. Saw “Pinch current limitation effect in plasma
focus,” Appl. Phys. Lett., 2008,
92, no. 2, p. 021 503.
[23] S. Lee S, P. Lee, S H. Saw and R.S. Rawat, “Numerical
experiments on plasma focus pinch current limitation,” Plasma Phys. Control.
Fusion, 2008, 50, no. 6, 065 012 (8pp).
[24] V. A. Gribkov, A. Banaszak, S. Bienkowska, A.V. Dubrovsky, I.
IvanovaStanik, L. Jakubowski, L.
Karpinski, R.A. Miklaszewski, M. Paduch, M.J. Sadowski, M. Scholz, A.
Szydlowski and K. Tomaszewski “Plasma
dynamics in the PF1000 device under fullscale energy storage: II. Fast
electron and ion characteristics versus neutron emission parameters and gun
optimization perspectives,” J. Phys. D,Appl. Phys., 2007, 40, no. 12,
3592–3607
[25] S Lee, S H Saw, P C K Lee, R S Rawat and H Schmidt, “Computing
plasma focus pinch current from total current measurement,” Appl. Phys. Lett.
92 , 2008, 111501
[26] Akel M., AlHawat Sh., Lee S. “Pinch
Current and Soft xray yield limitation by numerical experiments on Nitrogen
Plasma Focus”. J Fusion Energy DOI
10.1007/s1089400992386. First online 21 August 2009
[27] Saw S. H., Lee P. C. K., Rawat R. S. & Lee S. 2009
‘Optimizing UNU/ICTP PFF Plasma Focus for Neon Soft Xray Operation’ IEEE Trans on Plasma Sc, 2009, 37, 12761282.
[28] Saw S. H. and Lee S. ^{ “ }Scaling laws for plasma focus machines from numerical
experiments”. Invited
paper: IWPDA,
[29] Saw S. H. and Lee S. “Scaling the plasma focus for fusion energy considerations”.
Tubav Conferences: Nuclear &
Renewable Energy Sources,
[30] Lee S.”Nuclear fusion and the Plasma Focus”, Invited paper Tubav Conferences: Nuclear & Renewable
Energy Sources Ankara, Turkey, 28 & 29 September 2009.
[31] Lee S., Saw S. H., Lee P. & Rawat R. S., “Numerical Experiments on Neon plasma focus soft xrays
scaling”, Plasma Physics and
Controlled Fusion, 2009, 51,
105013 (8pp).
[32] Akel M., AlHawat Sh., Lee S. “Numerical Experiments on Soft
Xray Emission Optimization of Nitrogen Plasma in 3 kJ Plasma Focus SY1
Using Modified Lee Model”, J Fusion Energy DOI 10.1007/s1089400992034 First
online, May 19, 2009.
[33] Lee S., Rawat R. S., Lee
P., S H Saw S. H., Soft xray yield from NX2 plasma focus, JOURNAL OF APPLIED PHYSICS, 2009, 106, 023309.
[34] Lee S. “Neutron Yield Saturation in Plasma FocusA fundamental
cause”
APPLIED PHYSICS LETTERS, 2009, 95, 151503 published online 15 October 2009
[35] Lee S., Saw S. H., Soto L., Moo S. P., Springham S. V., Numerical experiments on plasma focus neutron yield versus
pressure compared with laboratory experiments, Plasma Phys.
Control. Fusion, 2009, 51 075006
[36] Lee,
S (2004) Characterising the Plasma Focus
Pinch and Speed Enhancing the Neutron Yield. In: First
Part 1 Basic Course
Module 2: Universal
Plasma Focus LaboratoryThe Lee model code
Follow
the instructions (adapted to EXCEL 2003) in the following notes. You may also
wish to refer to the supplementary notes SP1.doc. Instructions are given in some details
in order to accommodate participants who may not be familiar with EXCEL. Those
who find the instructions unnecessarily detailed may wish to skip the
unnecessary lines. The code seems to run unreasonably sometimes agonizingly slow
when used with EXCEL 2007. So EXCEL 2003 is strongly preferred.
Summary
2.1 Introduction to the Worksheet
2.2 Configuring the Universal Plasma Focus Laboratory (UPFL)
2.3 Firing a shot in NX2
2.4 Studying the results
2.5 Exercise 1: Interpreting and recording data from the Worksheet
2.6 Conclusion
The material
You
should have RADPFV5.15de.xls (contained
in the efolder “Code and Data” accompanying this file) on your Desktop for
the next step. Please also ensure you have kept an identical original copy in a
RESERVE folder. You are going to work with the desktop copy; and may be
altering it. Each time you need an unaltered copy; you may copy from the
reserve folder and paste it onto the desktop.
2.1 Introduction to the Worksheet
2.1.1 Opening the worksheet
(Note: Click means the ordinary click on the
left button of the mouse; as distinct from the term Right Click, which means
the special click on the right button of the mouse.)
Double click on RADPFV5.15de.xls
Work sheet appears and should look like
Figure 1 [shown only as an example]; for the following please refer not to
Figure 1 but to your worksheet.
Security popup screen appears.
Click on enable macros
[2007 Security
Warning “Macros have been disabled” appears at top left hand corner of
Worksheet with side box “options”.
Click on “options” select the button “Enable this content” click
OK
After this
procedure, the worksheet is macroenabled and is ready for firing. ]
Figure 1.
Appearance of worksheetEXCEL 2003 version; EXCEL 2007 version should not look
too different.
2.1.2 Preliminary
orientation for setting controls
(For the following
instructions, use your Excel Sheet; not the above image)
Device configuration:
(Note: Each Cell of the Excel Worksheet is
defined by a Column alphabet A, B, or C….. and a Row number 1, 2 or 3 etc. The
Column alphabets are shown along the top border of the worksheet. The Row
numbers are shown along the left border of the Worksheet. For example, Cell A4
is located at column A row 4. Another example: A4F9 refers to the block of
cells within the rectangle bordered by row A4F4, column A4A9, row A9F9 and
column F4F9; the larger orangered bordered rectangle, containing 6x6 cells,
near the top left of figure1.)
Locate
Cells A4 to F9. These cells are for setting bank parameters, tube parameters,
operating parameters and model parameters.
Taper: Control Cells for anode taper are normally inactivated by typing 0
(number zero) in Cell H7. Ensure that H7 is filled with 0 (number zero); unless anode taper
feature is needed.
One Click Device: This control cell R4 allows choice of a
specific plasma focus using numbers; currently 3 machines are available chosen
with numbers 1, 2 or 3. Ensure that R4
is filled with the number 0. (Otherwise the code will keep defaulting to
the selected machine ‘1’ or ‘2’ or ‘3’.)
2.1.3 Preliminary
orientation for computed results
Cells A10G13: computed characteristic quantities of the configured plasma focus.
Cells K6M7: computed
neutron yield, component & total; if operated in deuterium
Cell N6N7: computed SXR line radiation
Cells H10N11: computed durations of axial phase, radial phase and pinch phase
and end time of radial phase.
Cells
A15AI17: dataline: contains data on row
17 with corresponding labels (and units) in rows 15 and 16. Data: E_{0}, RESF, c=b/a, L_{0},
C_{0}, r_{0},
b, a, z_{0}, V_{0}, P_{0}, I_{peak},
I_{pinchstart}, T_{pinchmin}, T_{pinchmax}, peak v_{a}, peak v_{s}, peak v_{p},
a_{min} (which is r_{min}), z_{max}, pinch duration, V_{max}, n_{ipinchmax},
Y_{n}, Qsxr, Qsxr%, f_{m}, f_{c}, f_{mr},
f_{cr}, EINP, taxialend, SF, ID
and Q_{line}; others may be
added from time to time.
This is a recently introduced very useful feature; enables
computed data for each shot to be copied and pasted onto another sheet; so
different shots may be placed in sequence, and comparative charts may be made.
Columns A20 to AP20:
computed point by point results (data are correspondingly labeled in row
A18 with units in row 19) for the following quantities respectively:
Time in ms, total current, tube voltage, axial position, axial speed, time
of radial phase in ms measured from the start of axial phase, time of radial phase in ns from the start of the radial phase,
corresponding quantities of current, voltage, radial shock position, radial
piston position, radial pinch length, radial shock, piston and pinch elongation
speeds, reflected shock position, plasma temperature, Joule power,
Bremsstrahlung, recombination, line emission powers, total radiation power,
total power, Joule, Bremsstrahlung, recombintion, line emission energies, total
radiation energy, total energy, plasma selfabsorption correction factor,
blackbody power, specific heat ratio
and effective charge number, number thermonuclear neutrons, number beam target
neutrons, number total neutrons, ion density, volume radiation power, surface
radiation power, plasma selfabsorption
correction factor , radial phase piston work in % of E_{0}, neon SXR energy emission.
Each computed quantity as a function of time (displayed in the
relevant column) is displayed in a column.
After a run each of these columns is typically filled to several
thousand cells.
Computed results
are also summarized in 8 figures:
Figure 1: (Top
left) total discharge current and tube
voltage
Figure 2: (Top
right) axial trajectory and speed
Figure 3: radial
trajectories
Figure 4: total
tube voltage during radial phase
Figure 5: radial
speeds
Figure 6: plasma
temperature
Figure 7: Joule
heat and radiation energies
Figure 8: Joule
power and radiation powers
An additional figure 8a on the right displays the specific heat
ratio and effective charge number during the radial phase.
2.2 Configuring
the Universal Plasma Focus Laboratory (UPFL)
2.2.1 Configuring
the worksheet for a specific machine
As a first exercise we configure the UPFL so it operates as the NX2, the
Highrepetition rate neon focus developed for SXR lithography in
The parameters
are:
Bank: L_{0}=20 nH, C_{0}=28 mF, r_{0}=2.3 mW
Tube: b=4.1 cm, a=1.9 cm, z_{0}=5 cm
Operation: V_{0}=11 kV, P_{0}=2.63 Torr, MW=20, A=10, AtMol=1 (these last 3 defines neon for
the code i.e. molecular (atomic) weight, atomic number and whether atomic or
molecular)
Model:
massf (f_{m})=0.0635, currf (f_{c})=0.7,
massfr (f_{mr}) =0.16, currfr
(f_{cr})=0.7; these are the
mass and current factors for the axial and radial phases.
(Note: 1. These model parameters had been fitted
earlier by us so that the computed total current best fits a measured total
current trace from the NX2.
Note:2.
We will carry out exercises in fitting model parameters in Module 3)
Configuring: Key
in the following: (e.g. in Cell A5 key in 20 [for 20nH], Cell B5 key in 28 [for
28mF] etc.
A5
B5 C5 D5 E5 F5
20
28 4.1 1.9 5 2.3
Then A9 B9 C9 D9 E9
11 2.63 20 10 1
Then A7
B7 C7 D7
0.0635 0.7 0.16 0.7
You may of course find it easier to follow the labels in A4F4, to
key in A5F5 for the relevant parameters; i.e. A4 states L_{0} nH; so fill in below it in A5 20 ; and so on.
For identification purposes key in at B3
‘NX2’
2.3 Firing a shot in NX2
Place the cursor in any blank nonactive space, e.g. G8. (point
the cursor at G8 and click the mouse). Press
‘Ctrl’ and ‘A’ (equivalent to firing a shot).
The programme runs and in less than a minute [provided you are
running on EXCEL 2003, For some reason for EXCEL 2007, the process may take
longer, sometimes much much longer] the run has completed and your worksheet
will look something like figure 2 below:
Figure 2. Appearance of Worksheet after a shot.
Figure 3. Plasma focus current is distorted from
unloaded current waveform.
In figure 3 is superimposed a current waveform (in blue; you do not have this waveform) of the
plasma focus shortcircuited across its input end insulator; with the current
waveform (pink) you have just computed [see your worksheet figure 2] (In a later session you will learn how to do
the shortcircuit computation and superimposition).
Notes:
Note 1
The first important point to stress (and one that should never be
forgotten) is that the plasma focus current waveform is very much distorted
from the damped sinusoid of the LCR
discharge without the plasma focus load (figure 3). The ‘distortions’ are
due to the electrodynamical effects of the plasma motion, including the axial
and radial dynamics and the emission of SXR from the Neon plasma. The way we use the code is based on the premise that the
features of these ‘distortions’ contain the information of the plasma
electrodynamics.
The
plasma focus loads the electrical circuit in the same manner as an electric
motor loads its driving circuit. The loading may be expressed as a resistance.
More specifically we may compute the loading or ‘dynamic’ resistance as follows
in Note 2; which shows that the dynamic resistance due to the motion in the
axial phase is more than the stray resistance of the capacitor bank in the case
of the NX2. Note 3 shows further that the dynamic resistance due to the plasma
motion in the radial phase is so large as to completely
dominate the situation. This causes the large current dip as shown in figure 3.
Note 2 As an example we may estimate the effect of one of
the electrodynamical effects. The quantity (1/2)(dL/dt) is a dynamic resistance. In the axial phase L=(m/2p)*ln(b/a)*z where m is permeability and z is the position of the current sheath. Differentiating, 0.5*dL/dt= 10^{7} *ln(4.1/1.9)*axial
speed~0.8 mW
per 10^{4} m/s axial
speed; or 0.8 mW per
unit speed of cm/ms.
At the peak axial speed of 6.6 cm/ms
(see figure 2 of worksheet), that gives us a circuit loading of ~ 5 mW; which is
reduced to 3.5 mW
when we consider the
effect of the current factor. This
is more than the loading of the stray resistance r_{o} of 2.3 mW.
So the axial motion of the current sheath is an important loading to the
circuit. Note 3 Continuing along this vein we may estimate the
dynamic resistive loading of the current sheath motion in the radial phase
when L = (m/2p)*ln(b/r_{p})*z_{f}, where r_{p}= radial piston
position and z_{f} =
length of the elongating column; both r_{p}
& z_{f} changing with
time. Thus
dL/dt = (m/2p)*ln(b/r_{p})*dz_{f}/dt +(m/2p)*z_{f}*(dr_{p}/dt)/r_{p} = 2*10^{7}*(ln(b/r_{p})*dz_{f}/dt
+z_{f}*(dr_{p}/dt)/r_{p})_{ } [both terms RHS are positive] In the section (2.4) below we will get from the
output figures of the worksheet the following values at around the time of
peak piston speed: r_{p}~2.4 mm, z_{f}~15 mm, dr_{p}/dt~13.5 cm/ms
[1.35*10^{5 }m/s]; dz_{f}/dt~1.7*10^{5 }m/s; Substituting
into expression above, we get at the time of peak piston speed dL/dt~190 mW
; giving us (after considering current factor of 0.7) still around 130 mW of dynamic
resistive loading due to the current sheath motion. This dynamic resistance
(compared to r_{0}_{ }of
just 2.3 mW)
dominates the current profile at this stage. Note 4 d[LI]/dt generates an induced voltage;
with one important component in this situation being I*(dL/dt). Since we
have already estimated that dL/dt~0.19
W; multiplying this by 0.7x200 kA of current (which is the approx value of current at this
time) gives us just under 30 kV.
So we note that the dynamics at this time (just as the radial shock is
going on axis) contributes a back voltage of ~30 kV through this term. The other term L*(dI/dt) terms is negative; so the
maximum induced voltage is considerably less than 30 kV, as you can see from figure 2. Note 5 As a separate exercise which you may like to do one
day: What is the basis for saying that (1/2)*(dL/dt) is a dynamic resistance? Can you show this by examining
the power term in the situation when an inductance is changing? Compare the
inductive power flow: (d/dt)(0.5*L*I^{2}) and the total power flow: VI=I*(d/dt)(L*I). What do you notice?
2.4 Studying the results
(The results are obtained from your Excel Sheet; not from the above images in figure 2)
Remember we are
operating a neon plasma focus.
Here are some
important quantities obtained from the data line in row 17.
Computed I_{peak}: L17 322 kA
I_{pinch}: M17 162 kA (pinch current at start of
pinch phase)
Peak tube voltage: V17 26.1
kV
k_{min}: S17 0.075
(r_{min} or a_{min}/a)
[you may also check this against figure 3 of the worksheet.]
Durations: H11N11
Axial phase ends at 1.172 ms
Radial phase ends at 1.407 ms (add 1.172 to 0.235 ms) of which the last 26.2 ns
is the pinch phase.
Now we study the various figures displayed on the worksheet,
Sheet1 (also shown in figure 2).
Fig 1
Computed current trace; One point of interest is to locate the
ends of axial and radial phases on this trace; as well as the start and end of
the pinch phase. To do this, select Fig 1 (by pointing cursor on figure 1 and
clicking). Then point cursor arrow at trace near peak and move until point 1.17
ms appears; that is the end of axial phase
which is also the start of the radial phase.
Note: This point occurs not
at the apparent start of current dip, but a little before that. There is no
distinct indication on the trace that precisely marks this point. The term
rollover may be a better term suggesting a smooth merging of the axial and
radial phase. The apparent current dip occurs a little after the end of the
axial phase.
Next locate point 1.41 ms which is the end of the radial phase. Also locate the point 1.38 ms which is the start of the pinch phase.
There is no clear indication on the trace to mark this point either.
Fig 2
Select Fig 2 (with cursor) and read off the pink curve that the
peak axial speed reached is 6.6 cm/ms. Confirm this on the data line; Cell P17.
How many km per hour is this? And
what is the Mach Number? 1 cm/ms=36,000 km/hr; so 6.6 cm/ms=237,600 km/hr
Expressing this speed in km/hr is to give an idea of how fast the
speeds are in the plasma focus; it should give the idea also of temperature,
since for strong shock waves (high Mach number motion) there is efficient
conversion of energy from directed to thermal, i.e. from high kinetic energy to
high temperature.
Mach
number=speed/sound speed; finding this number is one of the questions for
Exercise 1 (see below).
Fig 3
Select Fig 3. Read from dark blue curve that piston hits axis
(radius=0) at 178ns after start of
radial phase; and outgoing reflected shock (light blue) hits incoming piston
(pink curve) at 210ns at radius of 2.1mm. The pinch phase starts at this 209ns and ends at 235 ns at a further compressed radius of 1.42mm.
Note the square of the ratio
of pinching a/r_{min} is a measure of the how much the ambient density
has been increased by the pinching effect.
Fig 4
Computed waveform of tube voltage during radial phase. Note the peak value of the tube voltage
induced by the rapid plasma motion.
Fig 5
Select Fig 5. Note from the dark blue curve that peak radial shock
speed is 20.4 cm/ms just before the radial shock hits the
axis at 178 ns after start of radial
phase. Also read from the pink curve that peak piston speed is 14.2 cm/ms reached just before the radial shock reaches its peak speed.
Yellow curve shows column elongation speed. Note that these peak speeds are
also recorded in the data line.
Other figures:
Select Fig 6: and read
the peak temperature reached.
Select Fig 7: and read
the various energies.
Select Fig 8: and read
the various powers
Note that more charts are plotted on Sheet2 of RADPFV5.15de.xls These charts form a
more complete picture of the plasma focus pinch, and may be used as starting
guides for laboratory measurements of the various plasma properties.
2.5
Exercise 1: Interpreting and recording data from the worksheet.
Fill in the
following blanks:
Q0: Given
the speed of sound in neon at room temperature is 450 m/s (1600 km/hr), the
Mach number of the peak axial phase speed, and of the peak radial phase speed
(radially inwards shock speed) are _______ and _______. (Note: The peak axial speed can be found from Fig.2, and the peak radial speed can be
found from Fig.5, for this
particular plasma focus operation, also recorded in the dataline)
Q1: The peak temperature
reached is _______K.
Q2: At
that temperature the effective charge number (from small figure) is _______ and
specific heat ratio has a range as follows _______.
Q3: There
is a moment in time when the temperature jumps by a factor of approximately
2. This is at _______ ns from start of radial phase (Note: this happens at reflected shock according to
the model)
Q4: Joule heating reached
a maximum value of_______ J.
Q5: Total
radiation reached a maximum value of_______ J
(Note: the – sign for the radiation
energy indicates energy taken out of the plasma by emission; ignore the – sign
for this measurement)
Q6: Line Radiation reached
a maximum value of _______ J.
Q7: Peak radiation power
reaches a value of _______ W.
2.6 Conclusion
We had an
introduction to the Worksheet of RADPFV5.15de.xls
We configured the
UPFL as the NX2 at 11 kV 2.6 Torr neon.
We used properly fitted model parameters. (Note: Fitting model parameters will be covered in a future session).
We noted that the current waveform is distorted from damped
sinusoidlike waveform (damped sinusoidlike waveform is the current waveform
when the plasma focus is shortcircuited).
We studied the computed results, including total current, tube
voltage, pinch current, radial and axial trajectories, radial and axial speeds,
plasma temperature and plasma Joule heating and radiation energies.
We also located various points on the current trace including: end
of axial phase/start of radial phase; end of radial phase; start and end of
pinch phase.
Note: This
particular numerical ‘shot’ used properly fitted model parameters. The results
of dynamics, electrodynamics and radiation as seen above are, in our
experience, comparable with the actual experiments conducted at NTU/NIE.
End of Module 2.
Reference to the Lee model code should be
given as follows:
Lee S. Radiative Dense Plasma Focus Computation
Package (2011): RADPF www.plasmafocus.net http://www.intimal.edu.my/school/fas/UFLF/
Part 1 Basic Course
Module 3: I.
Configuring and fitting computed current to measured current
II. Comparing a large PF with a small
PFneutron yield etc
(Follow the instructions in the following notes. You may
also wish to refer to the supplementary notes SP2.doc)
Summary
For
this module we fit model parameters so that computed current waveform matches
measured current waveform.
First
we configure the UPFL (RADPFV5.15de.xls)
for PF1000 operating in deuterium; using trial model parameters. We fire a
shot. We do not know how good our
results are without a reference point; i.e. some comparison with experimental
results.
A
total current waveform of the PF1000 has been published; we have it in
digitized form in a file PF1000data.xls.
We also have the chart of this waveform displayed in this file.
To ensure that our computed results are comparable to experimental
results, the key step is to fit
model parameters, by adjusting the model parameters until the computed total
current trace matches the measured total current trace.
To
do this, we add PF1000data.xls to
our numerical focus laboratory RADPFV5.15de.xls. Next we plot the computed current waveform in
the same chart. The model parameters are varied; at each variation the focus is
fired, and the computed current waveform is compared with the measured
waveform. The process is continued until the waveforms are best matched. A good
match gives confidence that the computed results (trajectories, speeds,
temperature, neutron and radiation yields etc) are comparable with actual
experimental results.
After
the guided fitting of the PF1000, we have a self exercise to fit the Chilean
PF400J.
We
then tabulate important results of both machines, and do a sidebyside
comparison of big versus a small plasma focus to obtain important insights into
scaling laws/rules of the plasma focus family.
Configuring
and fitting computed current to measured current
3.1 Configure the code for the PF1000 using trial model parameters
3.2 Place a measured (published) PF1000 current waveform on Sheet3
3.3 Place the computed current waveform onto the same chart as the measured current waveform in Sheet3
3.4 Vary the model parameters to obtain matching of computed versus measured current traces
II. Comparing a large PF with a small PFneutron yield etc
3.5 Exercise 2: Tabulate results for PF1000 obtained in numerical experiments
3.6 Exercise 3: Fitting the PF400J and tabulate the results for PF400J side by side with the results for PF1000, for a comparative study
3.7 Conclusion
The material
You
need RADPFV5.15de.xls
for the following work. Copy and paste a copy on your Desktop. You also need
the files PF1000data.xls,
PF400data.xls
and compareblank.xls
for this session. Copy and paste a copy of each file onto your desktop.
I. Configuring
and fitting computed current to measured current (guided–4 hrs)
3.1 Configure
the code for the PF1000 using trail model parameters
Double click on RADPFV5.15de.xls
on your Desktop.
Security popup screen appears.
Click on enable macros
The worksheet opens
[EXCEL 2007: Security
Warning “Macros have been disabled” appear at top left hand corner of Worksheet
with side box “options”.
Click on “options” select the button “Enable this content”
click OK]
After this procedure, the
worksheet is macroenabled and is ready for firing.
Type in cell B3: PF1000; for
identification purposes.
The PF1000, at 40 kV,
1.2 MJ full capacity, is one of the
biggest plasma focus in the world. Its 288 capacitors have a weight exceeding
30 tonne occupying a huge hall. It is the flagship machine of the ICDMP,
International Centre for Dense Magnetised Plasmas.
We are now
working on this Plasma Focus.
We use the
following bank, tube and operating parameters for the PF1000:
Bank: L_{0}=33.5 nH, C_{0}=1332 mF, r_{0}=6.3 mW
Tube: b=16 cm,
a=11.55 cm, z_{0}=60 cm
Operation: V_{0}=27
kV, P_{0}=3.5 Torr, MW=4, A=1, AtMol=2
For this exercise we do not
know the model parameters. We
will use the trial model parameters recommended in the code (See cells T9V9)
Model: massf (f_{m})
= 0.073, currf (f_{c })= 0.7,
massfr (f_{mr}) = 0.16,
currfr (f_{cr}) = 0.7; our
first try.
Configuring: Key
in the following:
A5 B5 C5 D5 E5 F5
33.5
1332 16 11.55 60 6.3
Then A9 B9 C9 D9 E9
27 3.5 4 1 2
Then A7 B7 C7 D7
0.073 0.7 0.16 0.7
for first try
Or follow the guide
in A4F4, to key in A5F5 for the relevant parameters.
Fire a shot:
Place the cursor in any blank nonactive space, e.g. G8. (point the
cursor at G8 and click the mouse). Press
‘Ctrl’ and ‘A’. (firing a shot)
The program runs
and results are displayed in columns and also in figures.
Is our simulation any good? Not if there is no reference point!!
To assess how good our simulation is, we need to compare our
computed current trace with the measured current trace, which has been
published.
Note that at this point: the configured RADPFV5.15de.xls contains computed data for PF1000 with the trial
model parameters of: massf (f_{m})
= 0.073, currf (f_{c}) = 0.7,
massfr (f_{mr}) = 0.16,
currfr (f_{cr}) = 0.7
3.2
Place a measured
(published) PF1000 current waveform on Sheet3
3.2.1
The PF1000 current
waveform is in the file PF1000data.xls.
You now want to place this data file as an additional sheet in our RADPFV5.15de.xls workbook.
RADPFV5.15de.xls is already open, click on ‘File’ tab; drop down
appears,
click on
‘Open’.
Look in:
Desktop; select PF1000data.xls; double click to open this file.
Click on
‘Edit’ tab; select ‘Move or Copy Sheet’.
A window
pops out; ‘Move selected sheets To book’; select RADPFV5.15de.xls. ‘Before Sheet:’ select ‘(move to
end)’.
Click
‘OK’.
You have
copied PF1000data.xls into RADPFV5.15de.xls as
Sheet1(2); you might like to rename Sheet1(2) as Sheet3.
[EXCEL 2007: ADPFV5.15de.xls is already open, minimize it by
clicking on top right hand corner tab with the  sign.
Open PF1000data.xls.
Locate tab
Sheet3 on lower left corner of worksheet.
Right click on
tab Sheet3.
Select move or
copy to book RADPFV5.15de.xls
Click on (move
to end)
Tick Create a
copy
Click OK]
3.2.2
With this
procedure you have copied PF1000data.xls
as Sheet3 in RADPFV5.15de.xls. The
chart has already been prepared. The measured current waveform appears in the
chart.
3.3
Place
the computed current waveform onto the same chart as the measured current
waveform in Sheet3
In the next steps we will place the computed current data from Sheet1
into this same chart in Sheet 3, by the following procedure.
Place the cursor
on the chart; click, then right click;
drop down appears click on source data;
click on series. In the series box click on computed current in kA; then in the
box against ‘X Values’ type in the following string: “=sheet1!$a$20:$a$6000” [without the quotation marks]. Next click in
the box against ‘Y Values’ and type in the following: “=sheet1!$b$20:$b$6000” [without the quotation marks]. Click button
‘OK’.
[EXCEL 2007: Position
the cursor on the chart in Sheet3 containing the measured current waveform. Now
right click. Popup appears. Click
on Select data. Select the series “Computed current in kA”. Click on Edit. On the new popup for series the name
“=”Computed current in kA” is already
there.
For Series X
values key in “=Sheet1!$a$20:$a$6000”
For Series Y
values key in “=Sheet1!$b$20:$b$6000”
Click OK; and
click OK.]
The computed
current waveform from Sheet1 is charted in the figure in Sheet3 with the same
time scale and the same current scale.
You can now
compare the computed current trace with the measured current trace.
You should see a pink trace which has just appeared on the chart.
The pink trace (see figure 1 below) is the computed current trace transferred
from Sheet1 (where the time data in ms is in column A, from A20A several
thousand; and corresponding computed current data in kA is in column B, from B20 to B several thousands). We are
selecting the first 5980 points (if that many points have been calculated) of
the computed data; which should be adequate and suitable.
Figure 1. Comparison of traces: Note that there is
very poor matching of the traces; using the first try model parameters.
3.4 Vary the model parameters to obtain
matching of computed versus measured current traces
Note that bank, tube and operating
parameters have all been given correctly.
3.4.1
First fit the axial phase
[suggestion: read SP2.doc pg 3 ‘First step is
fitting the axial phase’.]
From the
comparison chart on Sheet2,
We note: that the computed
current dip comes much too early;
that the computed
current rise slope is only very slightly low;
that the computed current maximum is too low.
All these 3 observations are consistent with a possibility that
the axial speed is too fast; which would cause the radial phase to start too
early. Too high an axial speed would also cause too much loading on the
electrical circuit (similar to the well known motor effect) as the quantity
[0.5 x dL/dt=0.5x L’x dz/dt]
is a dynamic resistance loading the circuit during the axial phase; here the
inductance per unit length L’=(m/2p) x ln(b/a).
This too high speed would also lower the peak current.
To reduce the axial speed, we could increase the axial mass factor.
We note that the axial phase ends too early by some 20%; indicating the axial
speed is too fast by 20%.
In the plasma focus (as in pinches, shocks tubes and other
electromagnetically driven plasma devices) speed~density^{0.5}. So the
correction we need is to increase the axial mass factor by 40%. So try an axial
mass factor of 0.073x1.4~ 0.1.
We toggle to
Sheet1 by clicking on ‘Sheet1’ (just below the worksheet).
Click on cell A7,
and type in 0.1.
Fire the focus by
pressing Ctrl+A.
Program runs
until completed, and results are presented.
Note TRadialStart
(H11) has increased some 0.6 ms.
Toggle to Sheet3
(i.e. click on Sheet3 just below work sheet).
Note that the computed current dip is now closer to the measured
current dip in time (still short by some 10%; reason being that increasing the
axial mass factor reduces the speed which in turn causes a reduced loading.
This increases the current which tends to increase the axial speed so that our
mass compensation of 40% becomes insufficient). The value of the computed peak
is also closer to the measured. So we are moving in the right direction!
But still need to move more in the same direction. Next try axial
mass factor of 0.12. Toggle to Sheet1, type 0.12 in A7. Fire. Back to Sheet3. Note improvement in all 3
features.
In similar fashion, gradually increase the axial mass factor. When
you reach 0.14 you will notice that the computed current rise slope, the
topping profile, the peak current and the top profile are all in good agreement
with the measured. The computed trace agrees with the measured trace up to the
start of the dip. Note that the axial model parameters at this stage of
agreement are: 0.14 and 0.7. You may wish to try to improve further by making
small adjustments to these parameters. Or else go on to fit the radial model
parameters.
3.4.2 Next,
fit Radial phase
Note that the computed current dip is too steep, and dips to too
low a value. This suggests the computed radial phase has too high a speed. Try
increasing the radial mass factor (cell C7), say to 0.2. Observe the
improvement (dip slope becomes less steep) as the computed current dip moves
towards the measured dip.
Continue making increments to mass(f_{r}) (cell C7). When you have reached the mass(f_{r}) value of 0.4; it is
becoming obvious that further increase will not improve the matching; the
computed dip slope has already gone from too steep to too shallow, whilst the
depth of the dip is still excessive.
How to raise the bottom of the dip? Here
we suppose the following scenario:
Imagine if very little of
the current flows through the pinch, then most of the total current will flow
unaffected by the pinch. And even if the pinch were a very severe one, the
total current (which is what we are comparing here) would show hardly a dip. So
reducing the radial current fraction, ie currfr (or f_{cr}) should
reduce the size of the dip.
Let us try 0.68 in cell D7. Notice a reduction in the dip. By the
time we go in this direction until currfr(f_{cr})
is 0.65, it becomes obvious that the dip slope is getting too shallow; and the
computed dip comes too late.
One possibility is to decrease massfr(f_{mr}) (which we note from earlier will steepen the dip
slope); which however will cause the dip to go lower; and it is already too
low. Another possibility is to decrease the axial phase massf(f_{m}), as that will also move
the computed trace in the correct direction.
Try a slight
decrease in axial massf (f_{m}),
say 0.13.
Note that this change aligns the dip better but the top portion of
the waveform is now slightly low, because of the increased loading on the
electrical circuit by the increase in axial speed. This suggests a slight
decrease to circuit residual resistance r_{0}
(or changes to L_{0} or C_{0}; fitting those could be
tricky, and we try to avoid unless there are strong reasons to suspect these
values). Easier to try lowering r_{0}
first. Try changing r_{0} to
6.1 mW.
The fit is quite good now except the current dip could be
steepened slightly and brought slightly earlier in time. Decrease massfr(f_{mr}), say to 0.35. The fit
has improved, and is now quite good, except that the dip still goes too low. At
this stage we check where we are at.
Toggle to Sheet1.
Note from Sheet1 that the radial phase ends at 9.12 ms. Back to Sheet3.
Observe (using cursor) that the point 9.12 is not at the point
where the computed (pink curve) dip reaches its inflection point; but some 0.02
ms before that point (see figure below).
So we note that the computed curve agrees with the measured curve
up to the end of the radial phase with a difference of less than 0.02 MA out of a dip of 0.66 MA (or 3%).
The fitting has already achieved good agreement in all the
features (slopes and magnitudes) of the computed and measured total current
traces up to the end of the radial phase.
Do not be influenced by
agreement, or disagreement of the traces beyond this end point.
Figure
2. The best fit.
So we have confidence that the gross features of the PF1000
including axial and radial trajectories, axial and radial speeds, gross
dimensions, densities and plasma temperatures, and neutron yields up to end of
radial phase may be compared well with measured values.
Moreover the code has been tested for neutron and SXR yields
against a whole range of machines and once the computed total current curve is
fitted to the measured total current curve, we have confidence that the neutron
and SXR yields are also comparable with what would be actually measured.
Having said that, those of
you who have some experience with the plasma focus would note that at the end
of the radial phase, some very interesting effects occur leading to a highly
turbulent situation with occurrence, for example, of high density hot spots.
These effects are not as yet modeled in the code. Despite this drawback, the
postulated beamtarget neutron yield mechanism seems able to give estimates of
neutron yield which broadly agree with the whole range of machines. For
example, the neutron yield computed in this shot of 1.08x10^{11} is in
agreement with the reported PF1000 experiments.
One further note: We
have recently confirmed that the above discussion of fitting applies typically
to machines with low L_{0}, below perhaps 60 nH. For machines above 100
nH another strategy of fitting or even modelling may need to be adopted. This
is related to the comment just above this note.
II. Comparing
a large PF with a small PF  neutron yield etc
3.5
Exercise 2: Tabulate results for PF1000 obtained in numerical experiments
You have been
following the guided steps in the above fitting:
Fill in the
following:
Q1: My best fitted model parameters for PF1000, 27kV, 3.5 Torr deuterium are:
f_{m}=

f_{c}= 
f_{mr}=

f_{cr}= 
Q2: Insert an image of the discharge current comparison chart in
Sheet3 here.
Q3: Fill up the
following table. To help with collecting the data use the ‘dataline’ feature in
Sheet1 which is the line 17 with corresponding descriptors in lines 15 &
16. To tabulate the following, use the
file compareblank.xls already
prepared for this purpose.
Parameter 
PF 1000 (at 27 kV 3.5 Torr D_{2}) 
Stored Energy E_{0} in kJ 

Pressure in Torr, P_{0} 

Anode radius a
in cm 

c=b/a 

anode length z_{0} in cm 

final pinch
radius r_{min}_{ }in
cm 

pinch length z_{max} in cm 

pinch duration
in ns 

r_{min}/a (r_{min} is also called a_{min}) 

z_{max}/a 

I_{peak} in kA 

I_{peak}/a
in kA/cm 

S=(I_{peak}/a)/(P_{0}^{1/2})( kA/cm)/Torr^{1/2} 

I_{pinch} in kA 

I_{pinch}/I_{peak} 

Peak induced
voltage in kV 

peak axial
speed in cm/ms 

peak radial
shock speed cm/ms 

peak radial
piston speed cm/ms 

peak
temperature in 10^{6 }K 

neutron yield
in units of 10^{6} 

[After filling,
save this Excel sheet you will use the same Excel sheet to fill in the results
for PF400J which is the subject of the next exercise.]
3.6
Exercise 3: Fitting the PF400J and tabulate results for
PF400J side by side with the results for PF1000 for a comparative study
Participants are to fit computed current to measured current
waveform of PF400J (bank, tube and operating parameters all correctly given)
In Module 2, we worked with
the Singaporean NX2; a 3kJ neon plasma focus designed for SXR lithography. For
our first fitting exercise we worked with the Polish PF1000, one of the largest
plasma focus (MJ) in the world. You are now given data for the PF400J, a small
subkJ plasma focus operated in
Given: the current waveform data of the PF400J,
digitized from a published waveform. The data is in the file PF400data.xls.
Your task: is to fit model parameters until the
computed current waveform matches the measured waveform. Some guidance is given
below.
Suggested steps to fit PF400J
i.
Make a clean copy of RADPF05.15de.xls from your Reserve
folder to your Desktop. Open this file.
ii.
Copy PF400data.xls as
Sheet3 of RADPF05.15de.xls using
procedure as in section 3.2.1 above. The
measured waveform is already precharted.
iii.
Transfer computed current data from Sheet1 onto the
measured current chart in Sheet3; as in
step in section 3.3 above using
strings: “=sheet1!$a$20:$a$6000”
[without the quotation marks] and “=sheet1!$b$20:$b$6000”
[without the quotation marks]. No trace of computed current appears yet, since
we have not yet ‘fired’ PF400J.
Write down the bank, tube and operating
parameters (from the table in the lower part of Sheet3, NOT from the top line,
which contains some nominal values). Toggle to Sheet1.
iv.
Configure the Universal Plasma Focus with the following
bank, tube and operating parameters for the PF400
L_{0}(nH) 
C_{0}(μF) 
b(cm) 
a(cm) 
z_{0}(cm) 
r_{0}(mW) 
40 
0.95 
1.55 
0.6 
1.7 
10 
massf 
currf 
massfr 
currfr 
model parameters 







V_{0}(kV) 
P_{0}(Torr) 
MW 
At No. 
At1;Mol2 
Operation Parameters 
28 
6.6 
4 
1 
2 

Key in the first try model parameters; [scroll a little to the right and use the suggested
parameters for the UNU ICTP PFF, cells T9V9].
v.
Fire PF400J; and see
the comparative results by toggling to Sheet3.
vi.
Fitting the computed current waveform to the measured
waveform
vii.
Suggested first steps: Fit the axial region by small adjustments
to f_{m} and f_{c}, where necessary. In
fitting the axial phase, the more important region to work on is the later part of the rising slope and the
topping profile towards the end of the axial phase. So each time you should
note the position of the end of the axial phase from Sheet1 and locate that
position on the chart in Sheet3, using the cursor.
viii.
Final steps: When you
have done the best for the axial phase up to the end of the axial phase, then
proceed to fit the radial phase. Tip: The dip for the PF400J is not very
dramatic. Enlarge the trace so the
rollover and the dip can be more clearly compared.
Fill in the following questions, copy and paste and email to me.
Questions
Q1: My best fitted model parameters for PF400J, 28 kV,
6.6 Torr deuterium are:
f_{m}= 
f_{c}= 
f_{mr}=

f_{cr}= 
Q2: Insert an image of the discharge current comparison chart in
Sheet3 here.
Q3:
Complete the Excel Sheet which you started in the last Exercise; to compare a
BIG (~500 kJ) plasma focus with a
small one (~400J). As you fill up,
note particularly each group of ratios (each group is denoted by a different
colour). Note particularly the order of magnitude of the ratios. [use the Excel sheet, rather than this
table].
The ratios below were calculated from the
actual PF1000 and PF400J results; and left here as a check for you. Calculate
your own ratios from your own results. At the end of the exercise save this
Excel Sheet as PFcomparison.xls. It
will be used again if eventually you go to the more advanced exercises of Modules
5.
Make up the following
table comparing a BIG plasma focus with a small plasma focus.
Parameter 
PF1000 ( at 27kV 3.5 Torr D_{2}) 
Ratio PF1000/PF400J 
PF400J (at 28kV 6.6 Torr D_{2}) 
Stored Energy E_{0} in kJ 
486 
1313 
0.37 
Pressure in Torr, P_{0} 
3.5 
0.53 
6.6 
Anode radius a
in cm 
11.55 
19.3 
0.6 
c=b/a 
1.39 
0.54 
2.6 
anode length z_{o}
in cm 
60 
35.2 
1.7 
final pinch
radius r_{min}_{ }in
cm 

26.7 

pinch length z_{max} in cm 

22.2 

pinch duration
in ns 

53 

r_{min}/a 

1.4 

z_{max}/a 

1.16 

I_{peak} in
kA 

14.6 

I_{peak}/a
in kA/cm 

0.76 

S=(I_{peak}/a)/(P_{0}^{1/2})(
kA/cm)/Torr^{1/2} 

1.05 

I_{pinch} in kA 

9.64 

I_{pinch}/I_{peak} 

0.65 

Peak induced
voltage in kV 

2.4 

peak axial
speed in cm/ms 

1.24 

peak radial
shock speed cm/ms 

0.48 

peak radial
piston speed cm/ms 

0.48 

peak temperature
in 10^{6}K 

0.19* 

neutron yield Y_{n} in 10^{6} 

81920 

Measured Y_{n} in 10^{6}: range 
(2  7)E+03 

0.91.2 
Measured Y_{n} in 10^{6}
:highest 
2.0E+04 


We could then use the tabulation for several applications
including the following:
Think of scaling rules, laws:
Q4: What is the significance of the Speed Factors S of PF1000 and PF400J?
Which one’s temperature should be higher?
***The ratio radial speed/axial speed is:
_{}
Note: http://www.plasmafocus.net/; download the Theory
of the model
3.7 Conclusion
In this module we have learned how to fit a computed current trace
with a measured current waveform, given all bank, tube and operational
parameters. For the PF1000 we obtained a good fit of all features from the
start of the axial phase up to the end of the radial phases; giving confidence
that all the computed results including trajectories and speeds, densities,
temperatures and neutron and radiation yields are a fair simulation of the
actual PF1000 experiment.
We also fitted the computed current to the measured current of the
PF400J; thus computing its dynamics and plasma characteristics and neutron yield.
We tabulated
important results of the two machines side by side.
We noted
important physics:
Although the machines differ greatly in storage energy and hence
in physical sizes, the speed factor S is practically the same. This has
given rise to the now wellknown observation that all plasma focus, big and
small, all operate with essentially the same energy per unit mass when
optimized for neutron yield. See e.g.: http://en.wikipedia.org/wiki/Dense_plasma_focus
(The Wikipedia article was written by IPFS and added on by various other groups
and researchers.)
The axial speed is also almost the same; in which case the radial
speeds would have been almost the same, except they (the radial speeds) are
influenced by a geometrical factor [(c^{2}1)/lnc]^{0.5}. For these 2 machines the factors differ by
1.5; hence explaining the higher radial
speeds in PF400J; and also the higher
temperatures in the smaller PF400J.
The pinch dimensions scale
with ‘a’ the anode radius. The pinch duration also scales with ‘a’, modified by the higher T of the PF400J, which causes a
higher small disturbance speed hence a smaller small disturbance transit time.
In this model this transit time is used to limit the pinch duration.
Finally we may note that just by numerical experiments we are able
to obtain extensive properties of two interesting plasma focus machines
apparently so different from each other, one a huge machine filling a huge
hall, the other a desk top device. Tabulation of the results reveals an all
important characteristic of the plasma focus family. They have essentially the
same energy per unit mass (S).
A final question arising from this constant energy/unit mass: Is
this at once a strength as well as a
weakness of the plasma focus?
End of Module 3.
Reference to the Lee model code should be
given as follows:
Lee S. Radiative Dense Plasma Focus Computation
Package (2011): RADPF www.plasmafocus.net http://www.intimal.edu.my/school/fas/UFLF/
Part 1 Basic Course
Module 4: PF1000 neutron
yield versus pressure
(Follow
the instructions in the following notes. You may also wish to refer to the
supplementary notes SP3.doc)
Summary
This
module looks at variation of neutron yield with pressure; running PF1000 from
short circuit (very high pressure), through optimum pressure to low pressure.
The very high pressure of the shortcircuit shot stops all current sheath
motion thus simulating a short circuit. The aim of this shot is just to obtain
shortcircuit current waveform for comparison with the focusing waveforms at
different lower pressures. When the operating (ambient) pressure is low enough
neutrons are emitted. The variation of the yield and other properties with
pressure are compiled together, and presented on one chart in normalized form.
In this way correlation of various quantities may be seen.
4.1
Configure the code for the
PF1000 at 27 kV, 3.5 Torr D_{2 }using model
parameters which we had fitted earlier
4.2 Fire the PF1000 at very high pressure,
effectively a short circuit
4.3 Fire the PF1000 at lower pressures from 19 Torr down to 1 Torr; looking for optimum neutron yield
4.4
Exercise 4: Place the current
waveforms (from section 4.3) at different pressures on the same chart for
comparative study
4.5 Exercise
5: Tabulate results at different pressures for comparative study; including
speeds, pinch dimensions, duration, temperature and neutron yield
4.6 General
notes on fitting, yield scaling and applications of the Lee model code
The material
You
need RADPFV5.15de.xls
for the following work. Copy and Paste on your Desktop. You also need the files
PF1000pressureblank.xls. This file
contains also tabulation blanks for your convenience.
4.1 Configure the code for
PF1000 at 27 kV, 3.5 Torr deuterium using model parameters
which we had fitted earlier
i.
Preparing Sheet3
Open RADPFV5.15de.xls; copy PF1000pressureblank.xls
as Sheet3; using procedure which we have practiced in Module 3.
Examine Sheet3
PF1000pressureblank.xls, copied as Sheet3 has the parameters of PF1000
recorded along the top rows. One set of measured timecurrent data is supplied
at Columns A and B. To save participants some time, timecurrent data for
several traces are already computed and filled in: 3.5 Torr at columns C and D; 19 Torr
at columns G & H; 7.5 Torr at
columns M & N and 1 Torr at
columns S & T. These are already plotted in figure 1. You are required to
fire shots at several other pressures and copy the timecurrent data for these
shots and paste onto the columns reserved for these shots. In this way you fill
up the charts with sufficient traces to cover the optimum pressures for neutron
yield. These pressures are 100,000, 14, 10, 6 and 2 Torr.
Scrolling
to the right you see table 1 with plasma focus properties at various pressures.
The properties corresponding to the shots at 19, 7.5, 3.5 and 1 Torr are already filled in. Data from those other shots to be fired by
the participants are to be filled in to complete the table. Below table 1 is
table 2 with data normalized from table 1 using the shot at 7.5 Torr as the reference shot for
normalizing.
Figure
2 displays the normalized Y_{n},
I_{peak}, I_{pinch} and radial phase
piston work EINP versus pressures
from table 2.
[Note that the curves in figure 2 have places
where they all come to zero. At the start the curves lookvery messy and may
appear strange. That is because the table has not been filled for those shots
yet to be fired. For these shots the data points have been put to zero. The
curves will take on their correct shapes once the data has been correctly
filled in. This is the job for the participant]
ii.
Configure the code for PF1000
Use the data in PF1000
pressureblank.xls to configure.
Bank: L_{0}=33.5
nH, C_{0}=1332 mF, r_{0}=6.1
mW
Tube: b=16
cm, a=11.55 cm, z_{0}=60 cm
Operation: V_{0}=27 kV, P_{0}= ? Torr, MW=4, A=1, AtMol=2
Model: f_{m}=0.13, f_{c}=0.7, f_{mr}=0.35,
f_{cr}=0.65
4.2 Fire
the PF1000 at very high pressure, effectively a short circuit
Key in 100,000 Torr at
B9.
[Note: In the laboratory it
is of course impossible to fire such a shot and a physical shortcircuit may
need to be used at the insulator end of the plasma focus; or fire at the
highest safe pressure in argon. In the lab we have used 50 Torr argon, to
obtain very approximate results.]
[In the numerical experiment
at this high pressure the current sheath only moves a little down the tube,
adding hardly any inductance or dynamic loading to the circuit. So it is
equivalent to shortcircuiting the plasma focus at its input end. In the code
there is a loop during the axial phase, computing step by step the variables
as time is incremented. The loop is broken only when the end of the anode
(nondimensionalised z=1) is reached. In this case we do not reach the end of the
anode. However there is an alternative stop placed in the loop that stops the
run when time reaches nearly 1 full cycle is completed (nondimensionalised
time=6 ie nearly 1 full cycle time, 2p,
of the short circuited discharge).
At the start of the run, the
code computes a quantity ALT= ratio of characteristic
capacitor time to sum of characteristic axial & radial times. Numerical
tests have shown that when this quantity is less than 0.65, the total transit
time is so large (compared to the available current drive time) that the radial
phase may not be efficiently completed. Moreover because of the large deviation
from normal focus behaviour, the numerical scheme and ‘house keeping’ details
incorporated into the code may become subjected to numerical instabilities
leading to error messages. To avoid these problems a timematch guard feature
has been incorporated to stop the code from being run when ALT<0.65. When
this happens one can override the stop; and continue running unless the run is
then terminated by Excel for e.g. ‘overflow’ problems. In that case one has to
abandon the run and reset the code.]
Fire the high pressure shot. The pressure is too high for a normal
run and we are automatically toggled over to the Macro; the Visual Basics Code
appears at Statement 430 Stop; with a warning message that pressure is too
high. In this case we know what we are doing, and override as follows: Click
on ‘Run’ (above the code sheet), and ‘continue’. Another ‘Stop’ appears just
below line 485; with a warning about transit time. Click on ‘Run’ and
‘continue’; another ‘Stop’ appears below Line 488. Click on ‘Run’ and
‘Continue’.
In a little while the run has proceeded and finally the statement “If T > 6 Then Stop”
appears; indicating we have completed nearly one cycle of the capacitor
discharge; and reached the preset time limit.
Now, locate the ‘x’ at the extreme right hand corner of the
screen. Click on this ‘x’; popup appears with the message ‘This command will
stop the debugger’. Click on OK, which toggles us back to the worksheet,
Sheet1.
Copy the data in columns A & B from A20 and B20 to the end of
the computed current data (several thousand cells down); toggle to Sheet3 and
‘paste’ the copied timecurrent data onto Columns E & F (in the labeled
space provided in Sheet3. Locate table 1 by scrolling to the right. Fill in the
value of I_{peak} [read from
figure 1 or from the relevant cell of the dataline] onto the table 1 against
100,000 Torr. Put zero against all
the other quantities (I_{pinch},
peak v_{a}, S, peak v_{s} …. T and Y_{n}….)
4.3 Fire the PF1000 at lower pressures from
19 Torr down to 1 Torr, looking for optimum neutron yield
Fire the next shot at 14 Torr.
As the ALT value is over 0.65, the run proceeds as normal. Copy the
timecurrent data from Columns A & B (from rows 20 down) to Sheet3 columns
I & J. Fill in the table 1 [I_{peak}_{,}
I_{pinch}, peak v_{a}, S, peak v_{s} … T… Y_{n}…n_{i} & EINP taken from the
data line) for the data from shot 14 Torr.
Repeat for pressures 10,
9, 8, 7.5 , 7, 6, 3.5 , 2 and 1 Torr; tabulating the data for all these shots onto table 1; but
copy and paste the timecurrent data for only selected shots of 14, 10, 6 and 2
[in order for figure 1 not to become too crowded]. The list of pressures had been chosen as
above in order to demonstrate the way the neutron yield varies with pressure.
It is clear that Y_{n} increases
rapidly from 14 Torr to 10 Torr. More points are chosen between 10 Torr and 6 Torr and it is obvious that the optimum pressure (for Y_{n}) is between 8 Torr and 7 Torr. The participant will notice this as the shots are fired and
as the Y_{n }data is copied on to the table 1.
4.4 Exercise 4:
Place the current waveforms (from section 4.3) at different pressures on
the same chart for comparative study
Suggested procedure: To save
you time, the comparison chart, figure 1 has already been created for you, and
prefilled with several waveforms namely 19, 7.5, 3.5 and 1 Torr. You only have to fill in the ones
for 100,000 and 14, 10, 6 and 2 Torr
in the correct columns indicated by the column headings already placed on
Sheet3.
You will note that the computed current waveform for 3.5 Torr falls neatly on the measured
current waveform (as you have seen during an earlier exercise precisely with
this PF1000 27 kV, 3.5 Torr current waveform.) You will
recognize that we are using the computed 3.5 Torr shot to fit our UPFL to the PF1000 to obtain the model
parameters for the PF1000.
Thereafter the
assumption is that the model parameters apply for all the other shots.
It would of course be better if for every pressure or every shot
we have a measured current trace to fit the code. However despite this
assumption about the model parameters, the numerical experiment does show some
very interesting features as we proceed below.
4.5 Exercise 5: Tabulate results at different pressures
for comparative study; including speeds, pinch dimensions, duration,
temperature and neutron yield
This tabulation
has already been done as step (4.3) proceeded above.
In order to chart some of the computed data on one comparative
chart, as mentioned already, as you fill in table 1, table 2 is at the same
time filled in with each data column normalized to the data at 7.5 Torr, which was found to be the pressure
with the highest Y_{n}. Thus
the values of all the data in the normalized table is in the region of 1.
Plot normalized Y_{n}, I_{peak}, I_{pinch},
and radial EINP against P_{0}.
[As you fill in table 1, the
normalized quantities are automatically computed, and the chart begins to take
the correct shape. At the start the
chart is in a jumble because many points have not been filled in, and thus
there are erratic zero points all over the place.]
Discussion
Note 1
Look at the change of
current waveforms from very high pressures to low pressures. At very high
pressures the waveform is a damped sinusoid. At 19 Torr the characteristic
flattening of the current waveform due to dynamics is already clearly evident.
The current peak comes earlier and is lower than the unloaded (high pressure)
case, the current then droops until the rollover into the dip (due to the
increased radial phase loading) at around 15 μs.
At lower pressures these characteristics remain the same except that the
current trace is depressed more and more as speed increases. The peaking
(reaching maximum current) also comes earlier and earlier, as does the radial
phase rollover of the current trace. This is characteristic of an LCR circuit
with increasing resistance R, as R increases from light towards critical
damping.
At 2.6 Torr, there is hardly
any droop, the current waveform showing a distinct flat top leading to the
rollover. At 1 Torr the axial speed is now so high that the axial phase is
completed in less than 5 μs
and the current is still rising when it is forced down by the radial phase
dynamics.
Note 2
A very important point to
note in neutron scaling is that there exists some confusion and even misleading
information in published literature because of sloppy practice with regards to
I_{peak} and I_{pinch}. These quantities are sometimes treated
as one and the same or when a distinction is attempted there is then confusion
between the total current at the time of pinch and I_{pinch}. For
example in the case of PF1000, there appears to be some disappointment (in
their publications) that (at 35 kV) with the current at more than 2 MA, Y_{n}
is still at best in the mid 10^11; and not at least an order of magnitude
higher that one might expect for currents around 2 MA. However if you
numerically run PF1000 at 35 kV you will
find that I_{pinch} is only 1 MA; so we are not surprised that the
measured yield is at best an order of magnitude down from what you would expect
thinking that your current is around 2 MA. (scaling at Y_{n}~I^{4}
, a factor of 2 in current gives a factor of 16 in the yield; at Y_{n}~I^{3}
, a factor of 8). So it is important that the thinking of yield should be in
terms of I_{pinch} as the relevant scaling parameter. When using this
model code, the distinction of I_{pinch} and I_{peak} is clear.
Next we look at the detailed tabulations: As P_{0} decreases, I_{peak}
decreases, and continues to decrease, because the increasing axial speed
increases the circuit loading, throughout the whole range of pressures. However
it is noticed that I_{pinch} increases
from high pressures, peaking in a flat manner at 6 Torr and then decreases sharply as pressure is reduced towards 1 Torr. One factor contributing to the
increase is the shift of the pinch time from very late in the discharge (when
discharge current has dropped greatly) to earlier in the discharge (when
current has dropped less). That is the main factor for I_{pinch} increasing despite a decreasing I_{peak}. At low pressures (e.g.
1 Torr), the radial phase now occurs
so early that it is forcing the current down early in the discharge. That
lowers both the I_{peak} as
well as the I_{pinch}. These
points are clear when you look at the comparative chart of current traces at
various pressures.
The radial EINP follows the same pattern as I_{pinch}, and for the same reasons. The radial EINP
computes the cumulative work done by the current sheath (piston) in the radial
phases.
Looking at the other quantities, we note that the speeds (axial,
radial shock and radial piston) and temperature all continue to rise as pressure
lowers; similarly S and maximum
induced voltage V also increase as
pressure is decreased. Pinch length z_{max}
is almost a constant. Minimum pinch radius and pinch duration continue to
decrease; the former due to better compression at higher speeds and the latter
due to the increased T. The number
density progressively drops, due to the decreasing starting numbers, despite
the increasing compression.
From the tabulations of the above numerical experiments, it might
be useful to consider the beamtarget mechanism which we are using to compute
the neutron yield. This is summarized in the following note.
Note 3
(Taken from SP3.doc)
Y_{bt}= C_{n}
n_{i }I_{pinch}^{2}z_{p}^{2}(ln(b/r_{p}))s/V_{max}^{1/2 }
where s is the DD fusion cross
section. In the range we are considering we may take s~V_{max}^{n } where n~23; say we take n=2.5; then we have
Y_{bt }~
n_{i }I_{pinch}^{2}z_{p}^{2}(ln(b/r_{p}))
V_{max}^{2}
The factor z_{p}^{2}(ln(b/r_{p}))
is practically constant.
Thus we
note that it is the behaviour of n_{i
,}I_{pinch}^{ }and
V_{max} as pressure changes that determines the way Y_{n}
increases to a maximum and then drops as pressure is changed.
An additional experiment is suggested, in which you can see how
numerical experiments on Y_{n}
versus operating pressure compare with measured results in the case of
PF400J. This is discussed in Module 7.
4.6 General
notes on fitting, yield scaling and applications of the Lee model code
On fitting: In the numerical experiments we soon learn that one
is not able to get a perfect fit; in the sense that you can defend it as
absolutely the perfect fit. The way to treat it is that one has got a working
fit; something to work with; which gives comparable results with experiments;
rather than perfect agreement. There is no such thing anyway; experiments in
Plasma Focus (i.e. on one PF under consistent conditions) give a range of
results; especially in yields (factor of 25 range is common). So a working fit
should still give results within the range of results of the hardware
experiment.
Even though a fit may only be a 'working' fit (as opposed to the hypothetical perfect fit) when one runs a
series of well planned numerical experiments one can then see a trend e.g. how
properties, including yields, change with pressure or how yields scale with I_{pinch}, or with L_{0} etc. And if carefully
carried out, the numerical experiments can provide, much more easily, results
just like hardware experiments; with the advantage that after proper reference
to existing experiments, then very quickly one can extend to future experiments
and predict probable results.
On scaling: Data used for scaling should be taken
from yieldoptimized (or at least from near optimized) situations. If one takes
from the worst case situations e.g. way out in the high pressure or low
pressure regions, the yield would be zero for a nonzero I_{pinch}. Such data would completely distort the scaling
picture.
Not only should the pressure be changed, but there should be
consideration for e.g. suitable (or even optimized) I_{pinch}/a; as
the value of I_{pinch}/a would affect the pressure at which
optimized S is achieved.
On directions of
work and applications: Efforts
on the model code may be applied in at least two directions. The first
direction is in the further development of the code; e.g. trying to improve the
way the code models the reflected shock region or the pinch region.
The second direction is to apply the model to provide a solution
to a particular problem. An example was when it was applied to look at expected
improvements to the neutron yield of the PF1000 when L_{0} is reduced.
Using the model code it was a relatively easy procedure, firing
shots as L_{0} was reduced in
steps; optimizing the various parameters and then looking for the optimized
neutron yield at the new value of L_{0}.
When this exercise was carried out in late 2007, for PF1000 at 35 kV, unexpectedly it was found that as L_{0} was reduced from 100 nH in steps, in the region around 35 nH, I_{pinch}
achieved a limiting value; in the sense that as L_{0} was reduced further towards 5 nH, whilst I_{peak} continued
to increase to above 4 MA, I_{pinch} dropped slightly from
its maximum value of 1.05 MA to just
below 1 MA. This Pinch Current
Limitation Effect could have considerable impact on the future development of
the plasma focus.
On numerical experiments to
enhance experience and intuition: Moreover the relationship between I_{peak} and I_{pinch} is implicit in the
coupling of the equations of circuit and motion within the code which is then
able to handle all the subtle interplay of static and dynamic inductances and
dynamic resistances and the rapid changes in distributions of various forms of
energies within the system. Whilst the intuitive feel of the experienced focus
exponents are stretched to the limit trying to figure out isolated or
integrated features of these interplays, the simplicity of the underlying
physics is captured by the code which then produces in each shot what the
results should be; and over a series of shots then reveal the correct trends;
provided of course the series is well planned.
So the code may also be useful to provide the numerical
experimenter timecompressed experience in plasma focus behaviour; enhanced
experience at much reduced time. At the same time the numerical experimenter
can in a day fire a number of different machines, without restrictions by time,
geography or expense. The problem then becomes one of too much data; sometimes
overwhelming the experience and intuition of the numerical experimenter.
On versatility:
Your numerical experiments have included examining plasma focus
behaviour comparing BIG, medium size and small plasma focus, looking for common
and scalable parameters. You studied neutron yields as functions of pressure,
comparing computed with experimental data. In 4 modules involving some 12 hours
of handson work you have ranged over a good sampling of plasma focus machines
and plasma focus behaviour.
This was all done with one code the RADPFV5.15de.xls the universal plasma focus laboratory facility. We
should have the confidence that if we explore the open experiments suggested in
the last module of the advanced course below, that could lead us to new areas
and new ideas.
End of Module 4 End of Basic Course in
Plasma Focus Numerical Experiments
[Comments and
interaction on the course work and other matters related to plasma focus are
welcome at anytime]
Reference to this course and the Lee model
code should be given as follows:
Lee S. Radiative Dense Plasma Focus Computation
Package (2011): RADPF www.plasmafocus.net http://www.intimal.edu.my/school/fas/UFLF/
Also see list of papers at end of Module 1 above
S Lee and S H Saw