5th International Conference on the Frontiers
of Plasma Physics and Technology
18-22 April 2011, Singapore
The Plasma Focus- Numerical Experiments
Leading Technology
Sor Heoh Saw 1,2 and Sing Lee 1,2,3
1INTI
International University, 71800 Nilai, Malaysia
2Institute for Plasma Focus Studies, 32 Oakpark Drive, Chadstone, VIC 3148, Australia
3Nanyang Technological University, National Institute of Education, Singapore 637616
e-mails: sorheoh.saw@newinti.edu.my; leesing@optusnet.com.au
The Plasma Focus- Numerical Experiments
Leading Technology
Outline of talk:
Numerical experiments assist design; provide reference for diagnostics and provide
guidance for implementation of technology for Plasma Focus devices.
Pointed the way to development of the Nanofocus, extending proven scalable range of
DPF devices from 0.1J to MJ over unprecedented 7 orders of magnitude in storage energy
Important guidance uncovered by numerical experiments include:
(1) Plasma current limitation effect, so show that it is futile to lower static inductance
below around 20 nH
(2) Scaling laws of neutron yield and soft x-ray yield as functions of E0 & I
(3) Deterioration of scaling laws due to dynamic impedance; so need to go to yield
enhancement techniques: high-Z seeding, higher voltage operation & circuit manipulation
The Plasma Focus
1/2
 Plasma focus: small fusion device, complements international
efforts to study nuclear fusion
 Multi-radiation device - x-rays, particle beams and fusion
neutrons
 Soft XR applications include microelectronics lithography and
micro-machining
 Large range of device-from J to thousands of kJ
 Experiments-dynamics, radiation, instabilities and non-linear
phenomena
3 kJ machine
Small Plasma Focus
1000 kJ machine
Big Plasma Focus
These two Images are shown to geometrical scale:
Energy scaling : 300 times
Radial Compression (Pinch)
Phase of the Plasma Focus
Our early numerical work [Lee & Serban IEEE Trans
Plasma Sci 24 (1996) 1101] already showed the
following scaling rules-of-thumb
Axial phase energy density (per unit mass)
constant
Radial phase energy density (per unit mass)
constant
Pinch radius ratio
constant
Pinch length ratio
constant
Pinch duration per unit anode radius
constant
The constant energy density scaling pointed the
way to development of so-called nanofocus
Nanofocus: an ultra-miniature
dense pinch plasma focus device
with operating at 0.1 J
pinch
L Soto et al- Plasma Sources Sci
& Tech 18 (2009) 015007
Sub-mm anode
Scaling Rules-of-thumb predicted numerically
are used to design nanofocus- DPF’s operate
over 7 orders of magnitude: 0.1J to 1MJAxial phase energy density (per unit mass)
constant
Radial phase energy density (per unit mass)
constant
Pinch radius ratio
constant
Pinch length ratio
constant
Pinch duration per unit anode radius
constant
Extending DPF’s to operate over an
unprecedented scalable 7 orders of magnitude:
0.1J to 1MJ-
This is an example of:
Numerical Experiments Leading Technology
The Plasma Focus
Axial Phase
Radial Phases
The 5-phases of Lee Model code
Includes electrodynamical- & radiation- coupled equations
to portray the REGULAR mechanisms of the:
axial (phase 1)
radial inward shock (phase 2)
radial RS (phase 3)
slow compression radiation phase (phase 4)
the expanded axial post-pinch phase (phase 5)
Crucial technique of the code: Current Fitting
The Lee Model codeComprehensive Numerical Experiments
The approach of the Lee Model code
To model the plasma dynamics & plasma conditions
Then obtain insights into scaling properties
Then uncover scaling laws
Critical to the approach:
Model is linked to physical reality by the current
waveform
Insights 1/2
The Lee model code has produced groundbreaking insights no other plasma focus codes
has been able to produce
These insights have led to other examples of
Numerical Experiments leading Technology
Ground-breaking Insights published- Numerical
Experiments leading Technology
Limitation to Pinch Current and Yields- Appl Phys Letts. 92
(2008) S Lee & S H Saw:
an unexpected, important result
Neutron Yield Scaling-sub kJ to 1 MJ-J Fusion Energy 27 (2008)
S Lee & S H Saw- multi-MJ- PPCF 50 (2008) S Lee
Neon Soft x-ray Scaling- PPCF 51 (2009) S Lee, S H Saw, P Lee, R S
Rawat
Neutron Yield Saturation- Appl Phys Letts. 95 (2009) S Lee
Simple explanation of major obstruction to progress
From Measured Current Waveform to
Modelling for Diagnostics
1/2
Procedure to operate the code:
Step 1: Configure the specific plasma focus
Input:
Bank parameters, L0, C0 and stray circuit
resistance r0;
Tube parameters b, a and z0 and
Operational parameters V0 and P0 and the
fill gas
Step 2: Fitting the computed current waveform to the
measured waveform-(connecting with reality) 2/2
A measured discharge current Itotal waveform for the specific plasma focus is
required
The code is run successively. At each run the computed Itotal waveform is fitted
to the measured Itotal waveform by varying model parameters fm, fc, fmr and fcr
one by one, one step for each run, until computed waveform agrees with
measured waveform.
The 5-Point Fit:
First, the axial model factors fm, fc are adjusted (fitted) until



(1) computed rising slope of the Itotal trace and
(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 Itotal trace.
Next, adjust (fit) the radial phase model factors fmr and fcr until
-
(4) the computed slope and
(5) the depth of the dip
agree with the measured Itotal waveform.
Example : NX2-Plasma SXR Source
NX2
11.5kV, 2 kJ
16 shots /sec; 400 kA
20J SXR/shot (neon)
109 neutrons/shot (D)
1/4
Example of current fitting: Given any plasma
focus : e.g. NX2 16 shots/sec Hi Rep 2/4
Bank parameters: L0=15nH; C0=28uF; r0=2 mW
Tube parameters: b=4.1 cm, a=1.9 cm, z0=5cm
Operation parameters: V0=11kV, P0=2.6 Torr in Neon
The UPFLF (Lee code) is configured (by keying figures into the
configuration panel on the EXCEL sheet) as the NX2
INPUT:
OUTPUT: NX2 current waveform
NX2 dynamics & electrodynamics
NX2 plasma pinch dimensions & characteristics
NX2 Neon SXR yield
Fitting computed Itotal waveform to measured
Itotal waveform: the 5-point fit
3/4
Once fitted: model is energy-wise & mass-wise
equivalent to the physical situation
4/4
All dynamics, electrodynamics, radiation,
plasma properties and neutron yields are
realistically simulated; so that the code output
of these quantities may be used as reference
points for diagnostics
Numerical Diagnostics- Example of NX2
Time histories of dynamics, energies and plasma properties
computed by the code
1/3
Last adjustment, when the computed Itotal trace is judged to be reasonably well fitted in all 5 features, computed times histories are
presented (NX2 operated at 11 kV, 2.6 Torr neon)
Computed Itotal waveform fitted to measured
Computed Itotal & Iplasma
Computed Total Current & Plasma Current
Input: Measured Total Current; shown with fitted
Current in kA
350
300
Measured current
kA
Computed current
kA
250
200
150
100
end of radial phase
50
0
0
0.5
1
1.5
Time in microsec
2
Current in kA
computed total current
400
400
350
300
Total Current kA
Plasma Current
250
200
150
100
50
0
0.0
2.5
Computed Tube Voltage
1.5
2.0
2.5
Computed Axial Trajectory & Speed
Breech Voltage kV
15
10
5
0
0.0
0.5
1.0
1.5
2.0
2.5
Position in cm,
Speed in cm/usec
12
20
Voltage in kV
1.0
Time in microsec
25
-5
0.5
10
8
6
4
2
Axial position
Axial Speed
0
0.0
Time in microsec
Computed Tube voltage
0.5
1.0
1.5
2.0
Time in microsec
Computed axial trajectory & speed
2.5
Numerical Diagnostics- Example of NX2
Computed total inductive energy as % of stored energy
Computed tube Inductance (axial + radial)
80
70
60
50
40
30
20
10
0
% of stored energy
Inductance in nH
30
25
20
15
Plasma Inductance
10
5
0
Inductive energy
0.0
0.0
0.5
1.0
1.5
2/3
2.0
0.5
1.0
1.5
Time in microsec
2.5
2.0
2.5
Time in microsec
Compare energy dissipated by 'dynamic resistance' with piston work
Dynamic Resistance in mOhm
120.0
30
100.0
25
DR in mOhm
% of stored energy
35
20
15
10
energy dissipated by 0.5Ldot
Piston Work
5
80.0
60.0
40.0
0
0.0
0.0
0.5
1.0
1.5
Time in microsec
2.0
2.5
0.0
0.5
1.0
1.5
2.0
2.5
Time in microsec
Computed Radial trajectory, Shock & Reflected Shock
Computed Length of Radial Structure
20
30
15
25
10
Radial Inward Shock
Radial Piston
5
Radial Reflected Shock
0
Length in mm
Radial position in mm
DR in mOhm
20.0
20
15
10
Length of Radial Structure
5
0.0
0.5
1.0
1.5
-5
Time in microsec
2.0
2.5
0
0.0
0.5
1.0
1.5
Time in microsec
2.0
2.5
Numerical Diagnostics- Example of NX2
Computed averaged and peak plasma temperature
Computed Radial speeds, Shock, Reflected Shock & elongation
30
3.0
Radial Piston
Elongation Speed
10
0
0
-10
1
1
2
2
3
Temp (10 6 K)
Speeds in cm/usec
Radial Inw ard Shock
20
3/3
2.0
1.0
Averaged uniform T
Peak (temporal & spatial) T
-20
0.0
-30
0.0
Time in microsec
Computed averaged & peak ion number density
number density 1023 m-3
30
25
20
15
10
Averaged uniform ni
nimax, full shock jump
5
0
0.0
0.5
1.0
1.5
Time in microsec
2.0
2.5
Computed SXR Power in GW
SXR Power in GW
3.0
2.0
SXR emission power
1.0
0.0
0.0
0.5
1.0
1.5
Time in microsec
2.0
2.5
0.5
1.0
1.5
Time in microsec
2.0
2.5
More on the
The Lee Model Code
1/3
 Realistic simulation of all gross focus properties
 Couples the electrical circuit with plasma focus
dynamics, thermodynamics and radiation (Lee
1983, 1984)
 5-phase model; axial & radial phases
 Includes plasma self-absorption for SXR yield
(Lee 2000)
 Includes neutron yield, Yn, using a beam–target
mechanism (Lee & Saw 2008, J Fusion energy)
Numerical Experiments leading Technology
Pinch current limitation effect in plasma focus
(S. Lee and S. H. Saw, Appl. Phys. Lett. 92, 021503 (2008), DOI:10.1063/1.2827579)
Pinch current limitation effectIpinch does not increase beyond a certain value
however low Lo, the static inductance is
reduced to.
Decreasing the present Lo of the PF1000
machine will neither increase the pinch current
nor the neutron yield, contrary to expectations.
Results from Numerical Experiments with PF1000
- For decreasing Lo - from 100 nH to 5 nH
As Lo was reduced from 100 to 35 nH - As expected


Ipeak increased from 1.66 to 3.5 MA
Ipinch also increased, from 0.96 to 1.05 MA
Further reduction from 35 to 5 nH


Ipeak continue to increase from 3.5 to 4.4 MA
Ipinch decreasing slightly to - Unexpected
 1.03 MA at 20 nH,
 1.0 MA at10 nH, and
 0.97 MA at 5 nH.
11
Yn also had a maximum value of 3.2x10 at 35 nH.
Energy distribution in the system at the end of the
axial phase and at the end of the pinch-(1/2)
The energy equation describing this current drop is written as follows:
2

2
2
2
0.5Ipeak (Lo + Lafc ) = 0.5Ipinch (Lo/fc + La + Lp ) + cap  plasma
Where
La = inductance of the tube at full axial length zo.
= energy imparted to the plasma as the current sheet moves to the pinch position
plasma= integral of 0.5(dL/dt)I2
~ 0.5LpIpinch2 (an underestimate for this case)
=energy flow into or out of the capacitor during this period of current drop.
cap= 0 (capacitor is effectively decoupled-duration of the radial phase is short compared
to the capacitor time constant)
2
2
 Ipinch = Ipeak (Lo + 0.5La)/(2Lo + La + 2Lp)
2
(Note : fc=0.7, fc ~0.5)
Pinch Current Limitation Effect - (1/3)
 Lo decreases higher Ipeak bigger a zp longer
bigger Lp
 Lo decreases shorter rise time shorter zo
smaller La
Lo decreases, Ipinch/Ipeak decreases
Pinch Current Limitation Effect - (2/3)
Lo decreases, L-C interaction time of capacitor decreases
Lo decreases, duration of current drop increases due to
bigger a
Capacitor bank is more and more coupled to the inductive
energy transfer
 cap  0
Pinch Current Limitation Effect - (3/3)
A combination of two complex effects
•
•
Interplay of various inductances
Increasing coupling of Co to the inductive energetic
processes as Lo is reduced
Conclusions – (1/2)
Several sets of Numerical results For PF1000 with
different damping factors indicate
•
•
Optimum inductances are around 30-60 nH with Ipinch
decreasing for Lo below optimum value
Reducing Lo from its present 20-30 nH will increase neither
Ipinch nor Yn
Conclusions – (2/2)
•
•
For a fixed Co powering a plasma focus, there exist an
optimum Lo for maximum Ipinch
Reducing Lo will increase neither Ipinch nor Yn
Because of the Pinch Current Limitation Effect
The numerical experiments and discussions – 4/7
Figure 2. PF1000 current waveforms (computed) at 35 kV, 3.5 Torr D2
for a range of Lo.
The numerical experiments and discussions – 6/7
Figure 3. Effect on currents and current ratio (computed) as Lo is reduced-PF1000, 35
kV,3.5 Torr D2.
 Adapted from Beam-target neutron generating
mechanism
(ref Gribkov et al)
A beam of fast deuteron ions close to the anode
Interacts with the hot dense plasma of the focus pinch
column
Produces the fusion neutrons
 Given by:
Yb-t= Cn niIpinch2zp2(ln(b/rp))σ /U0.5
where
ni = ion density
b = cathode radius,
rp = radius of the plasma pinch column with length zp,
σ = cross-section of the D-D fusion reaction, n- branch,
U= beam energy, and
Cn = calibration constant
Note:
•
•
•
•
The D-D cross-section is sensitive to the beam energy in the
range 15-150 kV; so it is necessary to use the appropriate
range of beam energy to compute σ.
The code computes induced voltages (due to current motion
inductive effects) Vmax of the order of only 15-50 kV. However
it is known, from experiments that the ion energy responsible
for the beam-target neutrons is in the range 50-150keV, and
for smaller lower-voltage machines the relevant energy could
be lower at 30-60keV.
In line with experimental observations the D-D cross section σ
is reasonably obtained by using U= 3Vmax.
The model uses a value of Cn =2.7x107 obtained by
calibrating the yield at an experimental point of 0.5 MA.
Neon SXR energy generated YSXR = Neon line radiation QL
QL calculated from:
where :
Zn = atomic number,
ni = number density ,
Z = effective charge number,
rp = pinch radius,
zf = pinch length and
T = temperature
QL is obtained by integrating over the pinch duration.
NOTE
Note:
The SXR yield is the reduced quantity of generated energy after plasma
self-absorption which depends primarily on density and temperature
The model computes the volumetric plasma self-absorption factor A
derived from the photonic excitation number M which is a function of the
Zn, ni, Z and T.
In our range of operation the numerical experiments show that the self
absorption is not significant.
Liu Mahe (1999) first pointed out that a temperature around 300 eV is
optimum for SXR production. Shan Bing’s (2000) subsequent work and
our experience through numerical experiments suggest that around 2x106
K (below 200 eV) or even a little lower could be better.
Hence for SXR scaling there is an optimum small range of temperatures (T
window) to operate.




Computed Total Current versus Time
For L0 = 30nH; V0 = 20 kV; C0 = 30 uF; RESF = 0.1; c=1.5
Model parameters : fm = 0.06, fc = 0.7, fmr =0.16, fcr = 0.7
Optimised a=2.29cm; b=3.43 cm and z0=5.2 cm.
Ysxr scales as:
•E01.6 at low energies in
the sub-kJ to several kJ
region.
•E00.76
at high energies
towards 1MJ.
• Scaling with currents
• Ysxr~Ipeak3.2 (0.1–2.4 MA)
and
• Ysxr~Ipinch3.6 (0.07-1.3 MA)
• Black data points with fixed
parameters RESF=0.1; c=1.5;
L0=30nH; V0=20 kV and model
parameters fm=0.06, fc=0.7,
fmr=0.16, fcr=0.7.
• White data points are for specific
machines with different values for
the parameters :c, L0, V0 etc.
 The scaling laws obtained (at optimized condition)
for Neutrons:
 Yn~E02.0 at tens of kJ to
 Yn~E00.84 at the highest energies (up to 25MJ)
 Yn =3.2x1011Ipinch4.5 (0.2-2.4 MA)
 Yn=1.8x1010Ipeak3.8 (0.3-5.7MA)
 The scaling laws obtained (at optimized condition)
for neon SXR:
 Ysxr~E01.6 at low energies
 Ysxr~E00.8 towards 1 MJ
 Ysxr~Ipeak3.2 (0.1–2.4 MA) and
 Ysxr~Ipinch3.6 (0.07-1.3 MA)
Global Scaling Law for Neutrons
Combining experimental & numerical
Experimental data from 0.4kJ to 1 MJ & beyond
What causes the deterioration of Yield scaling?
The axial speed loads the discharge circuit with a
dynamic resistance
The same axial speed over the range of devices means
the same dynamic resistance constituting a load
impedance DR0
Small PF’s : have larger generator impedance
Z0=[L0/C0] 0.5 than DR0
As energy is increased by increasing C0, generator
impedance Z0 drops
At E0 of kJ and tens of kJ the discharge circuit is
dominated by Z0
Hence as E0 increases, I~C0-0.5
At the level typically of 100 kJ, Z0 has dropped to the
level of DR0; circuit is now no longer dominated by
Z0; and current scaling deviates from I~C0-0.5,
beginning of current scaling deterioration.
At MJ levels and above, the circuit becomes
dominated by DR0, current saturates
Axial phase dynamic resistance causes current scaling
deterioration as E0 increases
Yn~Ipinch4.5
Yn~Ipeak3.8
Hence deterioration of scaling of Ipeak will
lead to deterioration of scaling of Yn.
Analysis using the Lee model code has thus
shown that the constancy of the dynamic
resistance causes the current scaling
deterioration resulting in the deterioration of
the neutron yield and eventual saturation.
This puts the global scaling law for neutron
yield on a firmer footing
Into the Future
Beyond Saturation Plasma Focus?
Current Stepped pinch: b= 12cm, a= 8cm, z0= 2cm; 2 capacitor banks: L1= 30nH, C1=
8uF, r0=6mW, V1= 300kV; L2= 15nH, C2= 4 uF, r0=6.3 6mW, V2= 600kV; P0= 12 Torr
D
C2 switched after radial start when r=0.8a,Yn= 1..2E12; r=0.6a, Yn= 1.5E12; r=0.5a, Yn=
1.8E12; r=0.4a, Yn= 1.9E12
IPFS-INTI Series 10, 10 October 2010 RADPF15.15d CS
Other methods
For example:
Enhanced compressions by:
Thermodynamics effects
Radiative cooling and radiative collapse
(already covered by S Lee & S H Saw in earlier
paper)
Further steps: will be discussed by P Lee in his
paper tomorrow
Conclusions
Numerical experiments
• Assist design; provide diagnostic reference and guidance for technology of PF’s
• Pointed the way for development of Nanofocus, extending proven scalable range
of devices over unprecedented 7 orders of magnitude in storage energy
• Uncovered Plasma current limitation effect as inductance is lowered,
• Extended Scaling laws of neutron yield & SXR yield as functions of E0 & I
• Found that Deterioration of scaling laws due to dynamic impedance;
• Suggests yield enhancement techniques:, higher voltage operation &
circuit manipulation; thermodynamic enhancement and radiation collapse.
5th International Conference on the Frontiers
of Plasma Physics and Technology
18-22 April 2011, Singapore
The Plasma Focus- Numerical Experiments
Leading Technology
Sor Heoh Saw 1,2 and Sing Lee 1,2,3
1INTI
International University, 71800 Nilai, Malaysia
2Institute for Plasma Focus Studies, 32 Oakpark Drive, Chadstone, VIC 3148, Australia
3Nanyang Technological University, National Institute of Education, Singapore 637616
e-mails: sorheoh.saw@newinti.edu.my; leesing@optusnet.com.au
 S Lee and S H Saw, “Pinch current limitation effect in plasma focus,” Appl. Phys. Lett. 92,
2008, 021503.
 S Lee and S H Saw, “Neutron scaling laws from numerical experiments,” J Fusion Energy 27,
2008, pp. 292-295.
 S Lee, P Lee, S H Saw and R S Rawat, “Numerical experiments on plasma focus pinch
current limitation,” Plasma Phys. Control. Fusion 50, 2008, 065012 (8pp).
 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.
 S Lee, “Current and neutron scaling for megajoule plasma focus machine,” Plasma Phys.
Control. Fusion 50, 2008, 105005, (14pp).
 S Lee and S H Saw, “Response to “Comments on ‘Pinch current limitation effect in plasma
focus’”[Appl. Phys. Lett.94,076101 (2009)],” Appl. Phys. Leet.94, 2009, 076102.
 S Lee, S H Saw, L Soto, S V Springham and S P Moo, “Numerical experiments on plasma
focus neutron yield versus pressure compared with laboratory experiments,” Plasma Phys.
Control. Fusion 51, 2009, 075006 (11 pp).
 S H Saw, P C K Lee, R S Rawat and S Lee, “Optimizing UNU/ICTP PFF Plasma Focus for
Neon Soft X-ray Operation,” accepted for publication in IEEE Trans. on Plasma Science.
 Lee S, Rawat R S, Lee P and Saw S H. “Soft x-ray yield from NX2 plasma focus- correlation
with plasma pinch parameters” (to be published)
 S Lee, S H Saw, P Lee and R S Rawat, “Numerical experiments on plasma focus neon soft xray scaling”, (to be published).
Lee S, Rawat R S, Lee P and Saw S H. “Soft x-ray yield from NX2 plasma focus”
International, Journal of Applied Physics, 106, 30 July 2009.
S Lee & S H Saw, “Neutron scaling laws from numerical experiments,” J Fusion
Energy 27, 2008, pp. 292-295.
S Lee, “Current and neutron scaling for megajoule plasma focus machine,” Plasma
Phys. Control. Fusion 50, 2008, 105005, (14pp).
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.
S Lee and S H Saw, “Pinch current limitation effect in plasma focus,” Appl. Phys.
Lett. 92, 2008, 021503.
S Lee, P Lee, S H Saw and R S Rawat, “Numerical experiments on plasma focus
pinch current limitation,” Plasma Phys. Control. Fusion 50, 2008, 065012 (8pp).
S Lee, “Plasma focus model yielding trajectory and structure” in Radiations in
Plasmas, ed B McNamara (Singapore: World Scientific Publishing Co, ISBN 9971966-37-9) vol. II, 1984, pp. 978–987
S Lee S et al, “A simple facility for the teaching of plasma dynamics and plasma
nuclear fusion,” Am. J. Phys. 56, 1988, pp. 62-68.
T Y Tou, S Lee and K H Kwek, “Non perturbing plasma focus measurements in the
run-down phase,” IEEE Trans. Plasma Sci. 17, 1989, pp. 311-315.