**IPFS**

*knowledge should be freely
accessible to all*

Institute for Plasma Focus Studies

Internet Workshop on Numerical Plasma Focus Experiments

Module 1: (Follow the
instructions in the following notes. You may also wish to refer to the
supplementary notes **part1supplementary.htm**

**Summary:**

**(a) Introduction to
the Worksheet**

**(b) Configuring the
Numerical Plasma Focus Laboratory (UPFL)**

**(c) Firing a shot in
NX2**

**(d) Studying the
results**

**(e) An exercise**

**The material:**

You need the latest code **File7RADPFV5.15b**
(Excel format referred to hereafter as File7 or 5.14; any reference to version
5.13.9b should be changed to version 5.14)

You should have File7 on your Desktop for the next step.

**(a) Introduction to the Worksheet**

**(i) 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 File7 (Excel logo **File7RADPFV5.15b**
your Desktop).

Security pop-up screen appears.

**Click** on **enable macros**

Excel spreadsheet appears and should look like Fig.1 below (this page has since been improved, so you may notice slight differences which do not matter):

Fig 1

**(ii) 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: A4-F9 refers to the block of
cells within the rectangle bordered by row A4-F4, column A4-A9, row A9-F9 and
column F4-F9.]

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.**

**(iii) Preliminary orientation** for computed results:

Cells A10-G13: computed
characteristic quantities of the configured plasma focus.

Cells K6-M7: computed neutron yield, component &
total; if operated in deuterium

Cell N6-N7 computed SXR line radiation

Cells H8-N9: computed quantities such as I_{peak},
I_{pinch}, Speed Factor S, peak induced tube voltage, minimum radius
ratio (minimum radius reached /anode radius)

Cells H15-N16: computed durations
of axial phase, radial phase and pinch phase.

Columns A20 to AO20: computed point by point results for:

Time in ms,
total current, tube voltage, axial position, axial speed, time radial phase in
ns, 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, radiation powers,
Joule and radiation energies, plasma self-absorption correction factor,
surface-mode radiation, specific heat ratio and effective charge number etc.
Each computed quantity as a function of time is displayed in a column. The quantity
corresponding to each column is described in Rows A18 & A19. After a run
each of these columns is typically filled to several thousand cells.

Computed results are also
summarized in 8 figures:

Fig 1: (Top left) total discharge current and tube voltage

Fig 2: (Top right) axial
trajectory and speed

Fig 3: radial trajectories

Fig 4: total discharge current and
tube voltage during radial phase

Fig 5: radial speeds

Fig 6: plasma temperature

Fig 7: Joule heat and radiation
energies

Fig 8: Joule power and radiation
powers

An additional Fig on the right
displays the specific heat ratio and effective charge number.

**(b) Configuring the Worksheet for a
specific machine:**

As a first exercise we configure the UPFL
(Universal Plasma Focus Lab) so it operates as the NX2,
the Hi-rep neon focus developed for SXR lithography in

The parameters are:

Bank: Lo=20
nH, Co=28 mF, ro=2.3mOhm

Tube: b=4.1cm,
a=1.9 cm, zo=5cm

Operation: Vo=11kV,

Model: massf=0.0635, currf=0.7,
massfr=0.16, currfr=0.7; these are the mass and current factors for the axial and
radial phases. [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. Exercises in fitting
model parameters will be included in Week 2.]

**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 guide in A4-F4, to key in A5-F5 for the relevant parameters; i.e.
A4 states Lo nH; so fill in below it in A5 20 ; and so on…

For identification purposes key in at B3 ‘NX2’

**(c) Firing a shot:**

Place the
cursor in any blank non-active 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:

Fig 2
Fig 3

In Fig 3 is superimposed a current
waveform (in blue; *you do not have this
waveform*) of the plasma focus short-circuited across its input end
insulator with the current waveform (pink) you have just computed [see your
worksheet Fig.2.] (In a later session you will learn how to do the
short-circuit computation and superimposition).

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 L-C-R discharge without the plasma focus load
(Fig 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 ‘distortion’ contain the information of the plasma
electrodynamics.

*(skip notes 2 -5 and save for later reading)*

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/sec axial speed; or
0.8mW per unit speed of cm/ms. At the peak axial speed of
6.6cm/ms (see Fig2 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 LHS are
positive]

In the section (d) below we will
get from the output figures of the worksheet the following values at around the
time of peak piston speed:

r_{p}~2.4mm, z_{f}~15mm,
dr_{p}/dt~13.5 cm/ms [1.35*10^5 m/s]; dz_{f}/dt~1.7*10^5m/s;

Substituting into expression above,
we get at the time of peak piston speed

dL/dt~280 mW ;
giving us (after considering current factor of 0.7) still around 100 mW of
dynamic resistive loading due to the current sheath motion. This dynamic
resistance (compared to r_{o }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~0.28 W;
multiplying this by 0.7x200kA of current (which is the approx value of current
at this time) gives us just under 40kV. So we note that the dynamics at this
time (just as the radial shock is going on axis) contributes a back voltage of
~40kV through this term.[the other term L*(dI/dt) terms is negative; so the
maximum induced voltage is considerably less than 40kV, as you can see from Fig
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 total power
flow:(d/dt)(0.5*L*I^2) and the inductive power flow: VI=I*(d/dt)(L*I). What do
you notice?

**(d)
Studying the Results:**

(The results are obtained from your Excel Sheet; not from
the above images)

Remember we are operating a Neon
plasma focus.

Computed I_{peak}: cell L17 322kA

I_{pinch}: M17 159kA (pinch current at start of
pinch phase)

Peak
tube voltage:V17 26.1kV

k_{min}: S17 0.078
(min radius=0.078x anode radius=1.48mm)

[you
may check this against Fig 3.]

Durations:
H11-N11 Axial phase ends at 1.172us

Radial phase ends at 1.408us (add 1.172 to 0.236us)

of which the last 26.3ns is the pinch phase

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 Fig 1 and
clicking). Then point cursor arrow at trace near peak and move until point
1.17us appears; that is the end of axial phase which is also the start of the
radial phase.

Note that this point occurs not at
the start of current rollover (start of dip), but a little before that. There
is no distinct indication on the trace that precisely marks this point. The
start of the rollover occurs a little after the end of the axial phase.

Next locate point 1.408 which is
the end of the radial phase. Also locate the point 1.38 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 6.6cm/us

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 210ns and
ends at 236 ns at a further compressed radius of 1.48mm.

Fig 4: Computed waveform of tube
voltage during radial phase. The current waveform is suppressed from this
figure as otherwise the tube voltage will be too small to be seen on the same
scale.

Fig 5: Select Fig 3. Note from the
dark blue curve that peak radial shock speed is 20.6 cm/us 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/us reached just before
the radial shock reaches its peak speed. Yellow curve shows column elongation
speed.

Other Figs: 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

**(e) Exercise 1:**

Fill in the following blanks:
(copy and paste and e-mail to me by 19 April 2008)

Comment: If you know you can do parts
or all of the exercise you can save your time as well as mine by making a
statement here such as: “*I am familiar
with the use of the code and wish to skip this exercise; or partially skip this
exercise. Please ignore all blanks (or blanks to the following: e.g. Q1, Q6…..)*.
“

Q1. The peak temperature reached
is ? K.

Q2: At that temperature the
effective charge number (from small fig) is
? ; and specific heat ratio is ? .

Q3:The temperature approximately
doubles 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

Q6: Line Radiation reached a maximum
value of ?
J

Q7: Peak radiation power reaches a
value of ? W

**Conclusion:**

We had an introduction to the
Worksheet of RADPFV5.13.8

We configured the UPFL as the NX2
at 11kV 2.6Torr Neon

We used properly fitted model
parameters. [fitting model parameters will be covered in a future session].

We noted that the current waveform
is distorted from the sinusoid-like waveform when circuit is short-circuited.

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 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 Part 1