**IPFS**

*knowledge
should be freely accessible to all*

**Institute for Plasma
Focus Studies**

**Internet Workshop on
Numerical Plasma Focus Experiments**

Module3; You may also wish to
refer to the supplementary notes **part2supplementary.htm**.

**Summary:**

This module is a consolidation of Modules 1 & 2. Module 3
is divided into two parts..

For the first part we look at
a commonly encountered situation when L_{o} is given only as a nominal
or very approximate value and r_{o} is not even mentioned. Then there
are 6 fitting parameters; and the process becomes more involved. Nevertheless
we have found that, despite that, it is still possible to get a reasonable fit.
In these sessions participants will be taken through that experience which will
enhance our ability and confidence to fit.

In the second part is an
exercise in fitting the DPF78, with bank, tube and operating parameters all
provided; but with L_{o} nominal and r_{o} not given. The
participant will fit the current curves. The properties of the DPF78 will then
be placed on the comparison Excel Sheet PFcomparison.xls which you have saved
from last week’s work.

Steps: Part 1:

(a) To configure the code for the PF1000 using nominal L_{o},
trial r_{o} and 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 L_{o}, r_{o} and
the model parameters until the two waveforms achieve the best match.

Part 2: Exercise 4: Given measured current waveform
data for the DPF78; given bank (with nominal L_{o} and no r_{o})
, tube and operating parameters; participant will fit computed to measured
current waveform. Then tabulate DPF78 computed properties into the **PFcomparison.xls** file which was saved
from Week 2; or use attached ** PFcomparisonpf1000pf400.xls **provided
for your convenience

**The material:**

You need **RADPF5.15dd.xls**
for the following work. You should have a clean copy in your Reference Folder.
Copy and Paste a clean copy on your Desktop. You should have RADPF5.15dd.xls on your
Desktop before the next step.

You need the file **PF1000dataNom.xls** and **DPF78dataNom.xls.** You
also need the file **PFcomparison.xls**,
saved from last week’s work*.[or the one
attached for your convenience PFcomparisonpf1000pf400.xls]*

**(a) Configure the code for PF1000**

Double click on RADPF5.15dd.xls (Excel
logo **RADPF5.15dd.xls**
on your Desktop).

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. It
is the flagship machine of the International Centre for Dense Magnetised
Plasmas. On their website, inductance was quoted as 9nH for short circuit.

We searched
through PF1000 publications and found figures for L_{o} of ‘around
20nH’ mentioned. For this work we assume we are looking at PF1000 for the first
time and all we got for L_{o} is the figure L_{o}=20nH. There
is no mention of r_{o}. So we
use a starting value of r_{o}=0.4mW; this being 0.1 of the bank impedance (L_{o}/C_{o})^0.5
taking L_{o} as 20nH.

We use the
following bank, tube parameters and operating conditions.

Bank: L_{o}=20 nH (nominal), C_{o}=1332
mF, r_{o}=0.4mW (guess value).

Tube: b=16
cm, a=11.55 cm, z_{o}=60 cm

Operation: V_{o}=27kV, _{o}

We **assume that we are starting to look at
PF1000 for the first time**; and
that we do not know the model
parameters. We will use the trial model parameters recommended in the code (See
cells P9-V9)

Model
Parameters: massf=0.073,
currf=0.7, massfr=0.16, currfr=0.7;
first try.

**Configuring:** Key
in the following: (e.g. in cell A5 key in 33.5 [for 33.5nH], in cell B5 key in
1332 [for 1332mF] etc)

A5 B5 C5 D5 E5 F5

**20** 1332 16 11.55 60 0.4

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

**Fire the PF1000** with these parameters.

**(b) Place the published
PF1000 current waveform on Sheet 2**

We repeat the
procedure to place the published PF1000 current waveform on sheet 2.

With RADPF5.15dd.xls (fired as PF1000
with first try parameters) open; open PF1000data.xls; click the Edit Tab;
scroll down and click 'Move or Copy file'. A window pops out. In the 'To book:
choose ‘**RADPF5.15dd.xls’**; then
choose ‘move to end’; click ‘OK’. Rename ‘Sheet1(2)’ as Sheet2.

The measured current waveform is now displayed in the chart
in Sheet2 of RADPF5.15dd.xls.

**(c) Place
the computed current waveform on Sheet 2 in the same figure**

Place the computed current waveform on the same chart
following the same procedure we did in Part 2; using the strings: “**=sheet1!$a$20:$a$6000**” [without the
quotation marks] and “**=sheet1!$b$20:$b$6000**”
[without the quotation marks].

The pink trace is the computed current trace transferred from
Sheet 1.

Comparison of traces: Note that there is very poor matching
of the traces; using nominal L_{o}, guessed r_{o} and the first
try model parameters.

**(d) Varying model
parameters and L _{o} and r_{o} to obtain better matching of
computed current to measured current traces**

To vary model parameters:

(i)
Note: that the computed current dip comes much
too early;

that the computed current rise
slope much too high;

that
the computed current maximum is much too large.

**Suppose** we do not know
that L_{o} is not a correct value.

Try varying axial model parameters, which as we know control
the current trace up to nearly the start of the roll-over region of the current trace. To make the dip
come earlier try increasing f_{m}; which will slow down the axial speed
(but as we know now, that will also reduce the circuit loading, leading to an
even larger current; we got to try something anyway). The deviation is very
large, so take a large step; say put f_{m}=0.8 [note: max

allowed value of f_{m} is 1]. That improves the time
position of the dip, but as we expected the current got even bigger. Next try
increasing f_{c}, which will increase the dynamic loading effect of the
dynamics on the circuit. Put f_{c} to its max allowed value of 1

The time position of the dip is now good and the peak current
has improved, but is still way too large. There is not much else we can do with
f_{m} and f_{c}. (you could try reducing them, but you know by
now that you are not going to see any improvement). Perhaps we could increase r_{o};
which will lower the whole current profile. Again large difference, need large
change. Try r_{o}=2mW. Improvement, but
not enough. Try r_{o}=10mW. Possible
improvement, but looks like we have gone beyond. Next try 7mW.

The topping profile
deviation has now improved, even touching the measured current profile
at one place. But the top is too droopy; and the decreased current has pushed
the dip too late. At the same time the current rise rate is still too high. Try
reducing f_{m} to 0.4.

There are now points of agreement; but the current rise slope
is still too steep and the topping profile is still too droopy.

**It is now clear
that in all the things we have tried, the rising slope of the current profile
is still too steep. How do we reduce the slope?** From capacitor
discharge behaviour, we know that increasing L_{o} would do it. (So would increasing C_{o};
but in this case we are fairly sure that the given value of C_{o} is
more reliable than the nominal value of L_{o}.) So let’s try L_{o}=25nH;
at last we see the slope beginning to match. Next try 30nH; even better. We can
see now that at last we are getting onto a better track. It is therefore better
to go back to more normal values of f_{m} and f_{c} ( rather
than the unusual values we tried in our desperation) Go back to f_{m}=0.15
and f_{c}=0.7. The matching is improving, but there is still that extra
slight droop at the top. Try reducing r_{o} to 6mW.

It looks like we are getting there, but the rising slope
could on average be improved by a larger L_{o}, which would also lower
the top. Try L_{o}=33nH. The slope match is now pretty good on average,
top still too high. Making small changes to L_{o} and r_{o},
one comes to a final best fit for these two bank parameters which will not be
too far away from 33nH and 6mW. The rising
slope profile and the topping profile up to the rollover region of the current
trace are now fairly well fitted.

Next make adjustments to f_{m} and f_{c}
until the final best fit is obtained for the axial phase up to the region of
rollover from the current top to the dip.

However we note that the radial phase is yet to be fitted and
currently has f_{mr}=0.16 and f_{cr}=0.7. [We have already done
this part of the fitting in S3S4 when we fitted the same curve for PF1000,
except that then we were given the correct value of L_{o}, which in
that case made the fitting of the axial phase much more simple. The fitting of
the radial phase as suggested below should sound familiar]

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, 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. 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. To decrease the
depth of the dip try reducing f_{cr} to say 0.68. Notice a reduction in
the dip. By the time we go in this direction until 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. Try 0.35

The fit is quite good now except the current dip could be
steepened slightly and brought slightly earlier in time. Try decreasing massfr,
say to 0.35.

The fit has improved, and is now quite good, except that the
dip still goes too low.

However we can check the position of the end of radial phase
which is at time=9.12 us. Putting the cursor on the pink curve at the point
t=9.12, we note that the agreement of the computed curve with the measured
curve up to this point is fair.

The best fit? Anyway, a good working fit!

So after finding the correct values of Lo and r_{o} and
fitting the model parameters, we should have gained more confidence in the
ability of this method of finding a good fit. We repeat that after this fit 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 the end of the 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.

For example, the neutron yield computed in this shot of
8.6x10^10 is in agreement with the reported PF1000 experimental experiments;
(range of 2-7x10^10 with best shots at 20x10^10).

**(e) Exercise 4:**

We are given the following parameters for the DPF78, operating at 60kV, 7.5 Torr D2.

L_{o}=44.5nH (nominal) C_{o}=17.2uF b=5 cm,
a=2.5 cm z_{o}=13.7cm,

The DPF78 was a high voltage
plasma focus operated at the IPF at **DPF78dataNom.xls**)
was provided recently by H Schmidt.

Use our Universal Plasma
Focus Laboratory code **RADPF5.15dd.xls**
to configure the DPF78. Add the DPF78 data to Sheet 2. Then fit the computed
current waveform to the measured.

*[ Hint
1: you need to assume a try value of r_{o} in the same way we did
for PF1000; ie try ro= 0.1*(L_{o}/C_{o})^0.5; which will print
out in cell F13 RESF=0.1 where RESF=r_{o}/(L_{o}/C_{o})^0.5.*

*Hint **2. the value of RESF very seldom goes below 0.05; so don’t put r _{o}
so small that RESF (F13) goes below 0.05.
Hint 3. The current rise
slope is most controlled by value of L_{o} (also by C_{o}, but
in this case we are given a reliable value of C_{o}). Hint 4. Increasing f_{m} has the effect of reducing axial speed and
increasing I_{peak}; reducing f_{c} produces similar effects. ]*

After you are satisfied with the
fit, add the DPF78 properties to the comparison tabulation that was saved from
last week. **PFcomparison.xls**. **Or**
use the one provided for your convenience: *PFcomparisonpf1000pf400.xls.*

Fill in the following, copy and paste and e-mail to me by 26
April 2008.

Q1: My best fitted values 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.

[Copy the Chart and
paste onto a fresh Excel workbook (with just the chart on one worksheet). Save
this workbook and then paste the workbook
here.

Q3. Add the newly computed properties of DPF78 to the file **PFcomparison.xls** saved from last
week. Or use the provided**
PFcomparisonpf1000pf400.xls.** That file already contains the properties
of PF1000 and PF400. You may also calculate the ratio of PF1000/PF78 for each
of the properties; as we did last week for PF1000/PF400. In other words we are
using PF1000 as the reference; comparing PF400 as well as DPF78 with it.

**Conclusion: **

In these two sessions we experienced a common fitting
situation when the given L_{o} is either nominal or wrong and r_{o}
is not given. Despite having to fit these two additional parameters we found
that a reasonable fit could still be achieved. The participant then proceeded
to fit a similar situation with the DPF78. The properties of the DPF78 obtained
in the numerical experiment are then added to the comparative tabulation
obtained earlier for the PF1000 and PF400. saving the file as **PFcomparisonpf1000pf400dpf78.xls.**

We note that the DPF78 was a high voltage plasma focus,
obviously designed to test higher voltage, higher speed operations, resulting
in an unusually high value of S; which is about a factor of 1.5 higher than the
average value of S (close to 100) for most neutron-optimized plasma focus
machines.

Study the comparative data in the light of the discussions
last week, to strengthen and consolidate the main ideas**.

This table could be kept and added to from time to time with
data from other plasma focus which you may be able to compute. Such comparative
data could be useful for theses and publications.

*[Suggestion: You
are invited to fit your own plasma focus and add the data to PFcomparison.xls.
I would appreciate a copy of all your fitted (and nominal) parameters, current
trace comparison, and your PFcomparison.xls;
to add to our database, which will be made available to all for downloading]*

**Some notes
(edited) kindly summarized by a participant:

As ‘a’ increases,
r_{min}, z_{max}, and pinch duration
increases; approximately linear dependence; seen in these
numerical experiments as well as in agreement with general and theoretical
observations.

As ‘a’ increases, (pinch volume*pinch
duration) increases; approximately to the 4th power of 'a'; ( 1 power from
each dimension). Why is this factor important to think about?

S factor:
additional note in comparing PF1000 to PF400:

The ratio of
radial speed/axial speed depends on a factor of [(c^2-1)/lnc].

This factor [(c^2-1)/lnc]~0.92/0.32~**2.9 for PF1000**; and ~5.8/0.96~**6 for PF400**; PF400 will have 2x radial speed as PF1000 (since axial
speeds nearly the same] ; .and for supersonic plasmas: Temp~speed^2 that is the main reason why PF400
has several times higher temperature than PF1000; although same speed factor.

In other same S
means approx same axial speed; and also approx same radial speed; and also
approx same temperature for cases where 'c' is the same. In this example, 'c'
is not the same and favours higher radial speeds and T in PF400.

Y_{n}
scales with I_{pinch}, because
it is I_{pinch} that basically powers the pinching processes during
which the neutrons are produced.

(You might wish to add other points.)

End of Part 3.