**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 **File7RADPFV5.15b** 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 File7 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 File7 (Excel logo **File7RADPFV5.15b** 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 File7 (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 ‘**File7RADPFV5.13.9b.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 File7.

**(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 **File7RADPF05.13.9b.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.