IPFS
knowledge should be freely
accessible to all
Institute for Plasma Focus Studies
Internet Workshop on Numerical Plasma Focus Experiments
Module2; (Follow the instructions in the following notes. You may also wish to refer to the supplementary notes part2supplementary.htm
Summary:
For this module we fit model parameters so that computed current waveform matches measured current waveform.
First we configure the UPFL 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; ie some
comparison with experimental results.
A total current waveform of the PF1000 has been published; we have it in digitized form.
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 a Sheet 2 to our numerical plasma laboratory; place the digitized data of the measured trace in Sheet 2 and plot this trace in a chart. 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 PF400.
We then tabulate important results of both machines, and do a sidebyside comparison of a big vs a small plasma focus to obtain important insights into scaling laws/rules of the plasma focus family.
Steps: (a) To configure the code for the PF1000 using trial model parameters
(b) To place a published PF1000 current waveform on Sheet 2.
(c) To place the computed current waveform on Sheet 2 in the same figure
(d) To vary the model parameters until the two waveforms have the best match.
(e) Exercise 2: Tabulate results for PF1000 obtained in our numerical experiment.
(f) Exercise 3; participants to fit the PF400 and tabulate the results for PF400 side by side with the results for PF1000, for a comparative study.
The material:
You need File7RADPFV5.15b (called File7) 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. Use the cursor to drag each file onto your desktop.
(a) Configure the code for PF1000
Double click on File7 (Excel logo File7RADPFV5.15 on your Desktop).
Security popup screen appears.
Click on enable macros
The worksheet opens.
[Type in cell B3: PF1000; for identification purposes.]
[The PF1000, at
40kV, 1.2 MJ full capacity, is one of the biggest plasma focus in the world.
Its 288 capacitors have a weight exceeding 30 tonnes occupying a huge hall. It
is the flagship machine of the ICDMP, International Centre for Dense Magnetised
Plasmas.]
We use the following bank, tube
and operating parameters:
Bank: L_{o}=33.5
nH, C_{o}=1332 mF, r_{o}=6.3mOhm
Tube: b=16
cm, a=11.55 cm, zo=60 cm
Operation: V_{o}=27kV,
For this exercise we do not know the model parameters. We will use the trial model parameters recommended in the
code (See cells P9V9)
Model: massf=0.073, currf=0.7,
massfr=0.16, currfr=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
0.063 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’. (equivalent to 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: File7
contains computed data for PF1000 with the trial model parameters of: massf=0.073, currf=0.7, massfr=0.16,
currfr=0.7
(b)
To place a published PF1000 current waveform on Sheet 2
(i)
The PF1000 current waveform is in the file PF1000data.xls. You now
want to place this data into File7 as Sheet2. With File7 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
‘File7RADPF13.9b.xls. ‘Before Sheet:’ select ‘(move to end)’. Click ‘OK’. You
have copied PF1000data.xls into File7 as Sheet1(2); you might like to rename
Sheet1(2) as Sheet2.
(ii) Chart
the measured current waveform: The
measured current waveform is already charted in Sheet2. You may adjust the size
and position of the chart for your preferred viewing.
(c) To place computed
current waveform on Sheet 2 in the same chart
To place
the computed current waveform on the same chart: Position the cursor on the
chart containing the measured current waveform. Now right click. Popup appears. Select
‘Computed Current in kA’ in the name box.
In the
next steps we will place the computed current data from Sheet 1 into this same
chart in Sheet 2, by the following procedures. Place the cursor in the box
against ‘X values’ by clicking. Then 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’.
The pink
trace (see figure below) is the computed current trace transferred from Sheet 1
(where the time data in us is in column A, from A20Aseveralthousand; and
corresponding computed current data in kA is in column B, from B20 to B severalthousand).
We are selecting the first 5980 points (if that many points have been
calculated) of the computed data; which should be adequate and suitable.
Comparison
of traces: Note that there is very poor matching of the traces; using the first
try model parameters.
(d) Vary model parameters to obtain
matching of computed vs measured current traces. (bank,
tube & operating parameters all given
correctly)
(i) First fit the axial phase:
[suggestion:
read part2supplementary.doc pg 2 bottom para ‘First step is fitting axial
phase’.]
From the
comparison chart on sheet 2,
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*dL/dt=0.5x
L’*dz/dt] is a dynamic resistance loading the circuit
during
the axial phase; here the inductance per unit length L’=(m/2p)*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 Sheet 1 by clicking on ‘Sheet 1’ (just below the worksheet).
Click on
cell A7, and type in 0.1.
Fire the
focus by pressing Ctrl+A.
Program
runs until complete, and results are presented.
Note
TRadialStart (H16) has increased some 0.7 us.
Toggle
to Sheet 2 (ie click on Sheet 2 just below work sheet).
Note
that the computed current dip is now closer to the measured in time (still
short by some 10%; reason being that increasing 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 Sheet 1, type 0.12 in A7. Fire. Back to Sheet 2. 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 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.
(ii) Next, fit Radial phase:
Note
that the computed current dip is not steep enough, 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.
Continue
making increments to massfr (cell C7). When you have reached the massfr 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 considering 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 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 (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, as that will
also move the computed trace in the correct direction.
Try a
slight decrease in massf, say 0.13.
Note
that this change aligned 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_{o}
( or changes to L_{o} or C_{o}; fitting those could be tricky,
and we try to avoid unless there are strong reasons to suspect these values).
Easier to try lowering r_{o} first. Try
changing r_{o} 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, 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 sheet 1. Note from sheet 1 that the radial phase ends at 9.12 ms. Back to sheet 2.
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 fig
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 0.02MA out of a dip of 0.66MA
(or 3%).
The
fitting has already achieved good agreement in all the features (slopes &
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.
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 8.6x10^10 is
in agreement with the reported PF1000 experiments.]
(e) Exercise 2:
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 Sheet 2 here.
Q3
Fill up the following table. Use the file compareblank.xls for this purpose. 













Parameter 

PF1000 









( at 27kV 3.5 Torr D2) 




Stored Energy Eo in kJ 







Pressure in Torr, 








Anode radius a in cm 








c=b/a 









anode length z_{o} in cm 







final pinch radius r_{min }in cm 







pinch length z_{max} in cm 







pinch duration in ns 








r_{min}/a 









z_{max}/a 









I_{peak} in kA 








I_{peak}/a in kA/cm 








S=(I_{peak}/a)/(P_{o}^{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/us 







peak radial shock speed cm/us 







peak radial piston speed cm/us 

















peak temperature in 10^6K 







neutron yield in 10^6 


















[After filling, save
this Excel sheet You will use the same Excel sheet to fill in the results for
PF400 which is the subject of the next exercise.]
(f) Exercise 3: Participant to fit
computed current to measured current waveform of PF400 (bank, tube and operating parameters all correctly given)
In Modulek 1, we worked with the
Singaporean NX2; a 3kJ neon plasma focus designed for SXR lithography. This
week we worked with the Polish PF1000, one of the largest plasma focus (MJ) in
the world. You are now given data for the PF400, a small subkJ plasma focus
operated in
Given: the current
waveform data of the PF400, digitized from a published waveform. The data is in
the file PF400data.xls.
Your job: is to fit
model parameters until the computed current waveform matches the measured waveform.
Some guidance is given below.
Suggested steps to fit PF400:
Copy a clean copy
of File7RADPF05.14.xls
(called File7) from your Reference folder to your Desktop. Open File 7.
Copy PF400data.xls
into Sheet2 of File7. The measured waveform is already precharted
Transfer computed current data
from Sheet 1 onto Sheet 2; using strings: “=sheet1!$a$20:$a$6000”
[without the quotation marks] &: “=sheet1!$b$20:$b$6000”
[without the quotation marks]. No trace of computed current appears yet, since we
have not yet ‘fired’ PF400.
Write
down the bank, tube and operating parameters (from the table in the lower part
of the page, NOT from the top line, which contains some nominal values). Toggle
to Sheet 1.
Configure the Universal Plasma
Focus:
Key in the
following bank and tube parameters and the operating parameters.
Lo(nH) 
Co(uF) 
b(cm) 
a(cm) 
zo(cm) 
ro(mohm) 
40 
0.95 
1.55 
0.6 
1.7 
10 
MASSF 
CURRF 
MASSFR 
CURRFR 
Model
Parameters 







Vo(kV) 

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].
Fire PF400; and see the comparative results
by toggling to Sheet 2.
Fitting the computed current waveform to
the measured waveform:
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 Sheet 1 and locate that position on the Chart in Sheet 2,
using the cursor.
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 PF400 is not very dramatic. Enlarge the
trace so the rollover and the dip can be more clearly compared.
(f) Exercise 3:
Fill in
the following, copy and paste and email to me by 26 April 2008.
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 Sheet 2 here.
Q3:
Complete the Excel Sheet which you started in the last Exercise; to compare a
BIG (~500kJ) 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 PF400 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 next week.
Make up the following table comparing a BIG plasma focus
with a small one. 















Parameter 

PF1000 

Ratio 
PF400 







( at 27kV 3.5 Torr D2) 
PF1000/PF400 
(at 28kV 6.6 Torr D2) 



Stored Energy Eo in kJ 
486 

1313 
0.37 




Pressure in Torr, 

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_{o}^{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/us 


1.24 





peak radial shock speed cm/us 


0.48 





peak radial piston speed cm/us 


0.48 
















peak temperature in 10^6K 


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 








Note: ratios in
orange: values are of the order of 1; ratios
in blue: values are of the order of (ratio of
anode radii) 



or (ratio of Ipeak); ratio of temperature (orange*) is a special case, because of the
difference in values of c. 



[These points are worth thinking
about; with reference to the file on the Theory of the Lee model, 




available from http://www.plasmafocus.net/ 







Look especially at the sections
on the scaling parameters of the axial and radial phases] 















This table summarises the results of our numerical
experiments for Week 2 of the course. 



It could be the start of a compilation
covering all focus machines for which measured current traces are available. 

We could then use the tabulation for several uses including
the following: 




Think of scaling rules, laws: 








1. How does r_{min},
z_{max}, and pinch duration, scale primarily with anode radius 'a'?
Should there be a relationship? 

2. How does the (pinch volume*pinch duration) scale with
'a'? 

Should there be a
relationship? 


3. What is the significance of the Speed Factor S? 







[hint: speed factor S is a measure of the
axial speed; it is also a measure of the energy per unit mass during the
axial phase; 

also a measure of the
energy per unit mass of the radial phase, however the radial phase speeds*** relative
to the axial phase 

speed are modified by a
factor [(c^21)/lnc]^0.5; so for 2 devices if the axial speeds are the same
and c is the same, 

one would expect the radial
speeds to be essentially the same. In that situation the temperatures would also be essentially 

the same. Following this line of argument,
can you see why there should be a big difference between the temperatures 

of PF1000 and PF400?
Which one's temperature should be higher?] 
















4. How should the
neutron yield scale? With storage energy E, with I_{peak}, with I_{pinch}? 




Papers\PP2 with Erratum JoFE
NeutronScalingLawsFromNumericalExperiments.pdf 


***The ratio radial speed/axial speed is:
is _{} http://www.plasmafocus.net/
download the Theory of the model
Conclusion:
In these
two sessions 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 Universal Plasma Focus Laboratory to the PF400.
We
tabulated important results of the two
machines side by side.
We noted
important physics:
that
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 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^21)/lnc]^0.5. For these 2 machines
the factors differ by 1.5; hence explaining the higher radial speeds in PF400; and also the higher temperatures in the smaller PF400.
The pinch dimensions scale with ‘a’ the anode radius. The pinch duration also scales with ‘a’,
modified by the higher T of the PF400, 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
huge machine** filling a huge hall, the other a desk top device. Tabulation of
the results reveal 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 Part 2