New model of calculating the energy transfer efficiency for the spherical theta-pinch device

0.38 MB·English

by  G. Xu

New model of calculating the energy transfer efficiency
for the spherical theta-pinch device
G. Xu,1,2,3, C. Hock,3 G. Loisch,3 G. Xiao,1 J. Jacoby,3 K. Weyrich,4 Y. Li,5 and Y. Zhao1
∗
1Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, China
2University of Chinese Academy of Sciences, Beijing, 100049, China
3Plasma physics Group, Institute of Applied Physics,
Goethe University, 60438, Frankfurt am Main, Germany
4Plasmaphysik, GSI-Darmstadt, 64291, Darmstadt, Germany
5 5Institut fu¨r Theoretische Physik, Goethe-Universita¨t Frankfurt am Main, 60438 Frankfurt am Main, Germany
1
(Dated: January 16, 2015)
0
2 Ion-beam-plasma-interaction playsan importantroleinthefieldofWarmDenseMatter(WDM)
n andInertialConfinementFusion(ICF).Asphericalthetapinchisproposedtoactasaplasmatarget
a in variousapplications includinga plasma strippercell. Onekeyparameter for suchapplications is
J the free electron density. A linear dependency of this density to the amount of energy transferred
5 into the plasma from an energy storage was found by C. Teske. Since the amount of stored energy
1 is known, the energy transfer efficiency is a reliable parameter for the design of a spherical theta
pinch device. The traditional two models of energy transfer efficiency are based on assumptions

] whichcomprisetheriskofsystematical errors. Toobtainpreciseresults, thispaperproposes anew
h model without the necessity of any assumption to calculate the energy transfer efficiency for an
p
inductively coupled plasma device. Further, a comparison of these three different models is given
-
m at a fixed operation voltage for the full range of working gas pressures. Due to the inappropriate
assumptions included in the traditional models, one owns a tendency to overestimate the energy
s transfer efficiency whereas the other leads to an underestimation. Applying our new model to a
a
wide spread set of operation voltages and gas pressures, an overall picture of the energy transfer
l
p efficiency results.
.
s
c PACSnumbers: 52.55.Ez,52.80.Tn,52.25.Fi,52.50.Dg
i
s
y
I. INTRODUCTION in 1964 [10].
h
p The key parameter for the plasma stripper is the free
[ In former works Z-pinches have been used as plasma electrondensity. Toachievethe higherfreeelectronden-

targets in scientific fields like the research on Inertial sity, the method is to upgrade the device. However, the
1
v Confinement Fusion, High Energy Density Physics and remaining difficulty is how to forecast the electron den-
5 accelerator development [1–6]. Plenty of achievements sity after upgrading. Fortunately, a linear scaling law
2 have been obtained in these scientific fields. As the Z- between the free electron density and the deposited en-
6 pinch is a capacitive discharge between two electrodes, ergy in plasma was shown in [9]. To derive the electron
3 erosion plays an important role in the lifetime limita- density, the knowledge of the precise energy transfer ef-
0
tion of such devices. Therefore the Z-pinch is not the ficiency values is essential. Reviewing the former energy
.
1 optimal choice for high repetition rate applications e.g. transferefficiency models [8,10–13],allofthem contains
0 as a plasma stripper. However, an inductively coupled aninaccurateassumptionthatthereflectedplasmaresis-
5
plasma device has the advantage to use external coils tant is constant. Such treatment is to solve the differen-
1
withoutcontactwiththeplasmatoinitiateandmaintain tial equation which will be well described in this article.
:
v the gas ionization. The so called theta-pinch combines To obtain the precise results, a new model based on the
Xi the high lifetime with the free electrondensity compara- experimental measurement is proposed.
ble to those of Z-pinches [7–9].
r
a Conventionalsmallscalethetapinchdeviceshavebeen
broadlyinvestigatedinvariousapplications[10,11]. The
II. EXPERIMENTAL SETUP
maximum energy transfer efficiency was determined to
be 59% [12]. However, in 2008, a spherical theta pinch
withthe significantlyadvancedenergytransferefficiency The basic working principle and experimental setup
up to 80% was realized by C. Teske and J. Jacoby [7]. have been described already in preceding papers [7–
Later on, the efficiency was raised to 86% [9]. All these 9, 14, 15]. In this article, to apply as a plasma stripper,
results for the energy transfer efficiency were calculated the spherical theta pinch is reconstructed and upgraded
outbyapplyingthemodelfirstproposedbyS.Aisenberg as shown in fig. 1. The device axis is positioned hori-
zontally to match the ion beam direction. To achieve a
higherelectrondensity, sixcapacitorswitheacha capac-
itance of 25 µF are used, being connected parallelly in
∗ [email protected] three groups of two capacitors connected in series. This
2

15000
U=6kV I(t)
P=30Pa Fit of I(t)
10000
nt I(t)/A 5000 2p=2.35E-4s
e 0
urr
C
-5000
-10000
-15000 0.0 1.0x10-3
Time/s
FIG. 3. Typical current signal shape and corresponding fit
FIG. 1. Spherical theta-pinch setup for the plasma stripper
application

15000
Ll R0 L0 U=6kV Current 200
10000 P=30Pa Intensity
U0,C0 Rp′ A) 5000 100Inte
Switch urrent ( 0 tp 0 nsity (m
C V
FIG.2. LRCequivalentcircuitfortheplasmastripperset-up )
-5000
-100
-10000
-200
setting results in a total capacitance of 37.5 µ F with a
-15000
maximum charging voltage of 18 kV. To satisfy the de- -400 -200 0 200 400 600 800 100012001400
mandofalonglife-timeevenathighrepetitionrates,the Time ( s)
maximum operation voltage is limited to 14 kV. Never-
theless, this means a maximum of 3.7 kJ stored energy FIG. 4. Light signal from the plasma with assumed plasma
inthe capacitorswhichisalmosttwiceasmuchasinthe start time
previous setup [9] . Correspondingly, the thyristor-stack
is alsoimprovedfor adapting the operationvoltageof14
kV.Besides,thebiggerglassvesselofthe6000cm 3 dis- circuit. In this circuit, Ll andR0 areundesiredparasitic
−
inductancesandresistancesrespectively. L istheinduc-
charge volume is closely encircled by a seven-turns coil. 0
TheCapacitors,theswitch,thetransmissionlineandthe tanceofthecoilwhileRp′ representsthereflectedplasma
resistance which is the transformed value from the real
coil form a typical LRC (inductance, resistance and ca-
plasma resistance into the LRC circuit. C is the total
pacitance)circuitwitharesonancefrequencyofabout10 0
capacitance of the capacitor bank with the initial charg-
kHz.
ingvoltageU . Undertheframeofthisequivalentcircuit,
AmixtureofmainlyArgonand2.2%Hydrogenisused 0
thetraditionalandthenewmodelsarederivedasfollows.
as a working gas and the pressure is measured by a full
range pressure gauge. As a diagnostic tool, a fast photo
diode is installed to monitor the plasma. In addition, a
A. Traditional models
Rogowski coil measures the circuit current in the trans-
mission line. These two diagnostic signals are recorded
by an oscilloscope setting at two sample rates of 1 and 1. S. Aisenberg’s model
2.5 MHz.
As Fig. 2 fulfils Kirchhoff’s voltage law, the circuit
equation is expressed as
III. MODELS OF ENERGY TRANSFER
EFFICIENCY t
( L +L )dI(t) +(R +R′)I(t)+ 0 I(t′)dt′ =0.(1)
0 l dt 0 p C
R 0
To calculate the energy transfer efficiency of theta
pinchdevices,S.Aisenbergetal. firstproposedanequiv- To solve this differential equation, R′ is assumed to be
p
alent circuit as Fig. 2 [10, 12]. This is a typical LRC constant during the whole discharge. Then Eq. (1) is
3
Analogously, for a free oscillation, the energy dissi-
pated in the parasitic resistance equals the whole stored
0.018
energy E:
0.017 R0
0.016 Linear Fit of R0 E = +∞I2(t)R dt 1I2R τ , (10)
ce/0.015 Z0 0 ≈ 2 0 0 0
n
a0.014
sist Equation y = a + b*x where 2τ0 is the decay constant for the free oscillation.
Re0.013 WReesigidhutal Sum of Instrum1e.8n5ta9l87 Finally, the energy transfer efficiency η from the ca-
0.012 SAqduj. aRre-Ssquare 0 pacitors to the plasma is
Value Standard Error
0.011 Resistence ISnltoeprceept 0.014580 6.13558E--5- Wp τp
η = =1 . (11)
0.0105 6 7 8 9 10 11 12 13 14 15 E − τ0
Voltage/kV
FIG. 5. Experimental value for the parasitic resistance 2. C.Teske’s modification
In this modification, it is assumed that the plasma
transformed to
start time t indicated by the light emission from the
p
1 plasmaisnotthesameasthecurrent. Hence,theenergy
( L +L )I¨(t)+(R +R′)I˙(t)+ I(t)=0, (2)
0 l 0 p C transferred to the plasma is
0
Generally, there are three different solutions depending +
on the value of ∆=(R +R′)2 4L0+Ll: Wp = ∞I2(t)Rp′dt
0 p − C0 Ztp
1 t 2ωτ
I(t)=I0e−2(RL00++RL′pl)tsin(cid:18)2(L√0−+∆Ll)t(cid:19), ∆<0, (3) = 2I02Rp′τpexp(cid:18)−τpp(cid:19)(cid:20)1+ 4ω2τp2p+1sin(2ωtp)
1
I(t)=I1te−2R(L00++RL′pl)t, ∆=0, (4) −4ω2τ2+1cos(2ωtp) . (12)
p (cid:21)
I(t)=I2e−2R(L00++RL′pl)t e2(L√0+∆Ll)t e2(L−0√+∆Ll)t , ∆>0.(5) Usually,theplasmaignitesafterthefirsthalfwavewhich
−
(cid:20) (cid:21) suggests t π/ω as Fig. 4 shows. Besides, ωτ is as-
p p
≈
Considering the current curve shape in Fig. 3, which is sumedtobemuchgreaterthan1. Asaresult,theenergy
a damped oscillation, Eq. (3) is the only proper solution deposited in the plasma is approximated as
which is simplified to
1 t
W I2R τ exp p (13)
I(t)=I0e−2τtp sin(ωt) (6) p ≈ 2 0 p′ p (cid:18)−τp(cid:19)
with τp = RL00++RL′pl, ω = 2(L√0−+∆Ll), I0 = ω(LU0+0Ll), where Cfoormrrueslapoisndaitntgaliyn,edthaesmodified energy transfer efficiency
2τ is the decay constant while ω is the oscillation fre-
p
quencyofthecircuit. Inthecaseof(R0+Rp′)2 ≪4L0C+0Ll, η = 1 τp exp tp . (14)
ω is approximated as − τ −τ
(cid:18) 0(cid:19) (cid:18) p(cid:19)
1
ω = . (7)
sC0(L0+Ll) B. New model
Further, the energy consumed in the plasma is
In the preceding models, the expressions for the en-
+ ergytransferefficiencyarebothbasedontheassumption
Wp = ∞I2(t)Rp′dt that the reflected plasma resistance is constant. Since
Z0 the light signal oscillates with time shown in Fig. 4, nei-
= 21I02Rp′τp 1− 4ω2τ12+1 . (8) tphlaesrmthaerepsliasstmanacpeacraanmbeetecrosnnstoarntth.eHveanlucee,otfhtehterardefliteioctneadl
(cid:18) p (cid:19)
models containthe risk of deviating from the true value.
In the case of ωτ >5, the approximation for W within
p p To avoid this risk, a new model without any assump-
an error of 1% is
tion is established. The key point for this new model is
1 that the parasitic resistance is constant as Fig. 5 shows,
Wp ≈ 2I02Rp′τp. (9) for the experimentally measured values. Moreover,since
4

15000 1.0
U=6kV I(t) S.Aisenberg’s model
P<1E-3Pa Fit of I(t) C.Teske’s modification U=6kV
10000 0.8 New Model
urrent I(t) (A) 50000 2 =1.02E-3s Efficiency 00..46
C -5000
0.2
-10000
0.0
-15000 0.0 2.0x10-3 4.0x10-3 0 10 20 30 40 50 60
Time t (s) Pressure (Pa)
FIG.6. Decayconstantforthecaseoffreeoscillation at6kV FIG.7. Efficiencies obtainedwiththethreemethodsat6kV
the capacitance and inductance do not consume the en-

ergy,thetotalenergyiscompletelydissipatedinthepar-
1.0
asitic resistance and the reflected plasma resistance. As
C0=37.5 F
a result, the energy deposited in the plasma is 0.8
Wp =E−R0∆t ∞ I2(ti), (15) ency 0.6 6kV
i=1 ci 7kV
where E is the total energy storXed in the capacitors; ∆t Effi 0.4 89kkVV
10kV
is a constant time interval between ti and ti+1; I(ti) is 0.2 11kV
the measured current value at t . Even though ∆t is 12kV
i 13kV
consideredto be infinitesimally small,the sum is usedto 0.0 14kV
takeintoaccountthe finite time resolutionofthe oscillo-
0 20 40 60 80 100 120 140 160
scope. For our experiment, the substitution error is less
Pressure (Pa)
than 0.01%.
For the free oscillation,the total energy which is com- FIG. 8. Energy transfer efficiencies of a complete pressure
pletely dissipated in the parasitic resistance is and voltage set
∞
E =R ∆t I2(t ). (16)
0 0 0 i
i=1 Directly applying Eq. (11) from the S. Aisenberg’s
X
model on these decay constants, the corresponding en-
Here, I (t ) is the measured current value at t . ∆t is
0 i i 0
ergytransferefficienciesforthedifferentgaspressuresare
again a constant time interval between t and t .
i i+1
calculated. For the C. Teske’s modified model, another
Thus, the energy transfer efficiency is written as
parameterbesidesthedecayconstantsistheplasmastart
η =1 ∆t iI2(ti) . (17) timetp. Thisisobtainedfromthedifferencebetweenthe
− ∆t I2(t ) beginning of the light emission in time and the current
0Pi 0 i
starttime. Then,adoptingEq.(14),themodifiedenergy
P transfer efficiencies are also computed. Both of the pre-
IV. RESULTS AND DISCUSSION ceding models need the plotting and fitting procedures
which cause propagating errors. However, for our new
In this article, an operation voltage is applied from 6 model, there is no need to plot and fit the data because
to 14 kV. For each operation voltage, the gas pressure Eq. (17) is straightforwardly applied on the measured
varies in the rangeof 0.6-160Paand anadditional”zero current values from the data sheet. Consequently, the
shot” in the order of 10 3 Pa. This ”zero shot” repre- energy transfer efficiencies are derived from this.
−
sents the free oscillation of the current. The black line A comparison is made among the results from these
in Fig. 6 shows this kind of free oscillation at 6 kV. The models at 6 kV shown in Fig. 7. Their tendencies are
red line displaying a good overlap with the black one is extremely similar. At low pressures, the curves rise very
thefittingcurve. Gainedfromthefittingparameters,the fast. Then they slow down at medium pressures. Fi-
decay constant of this free oscillation is 1.02 ms. Analo- nally, all of them suddenly drop down to the point of
gously,thedecayconstantsforthedifferentpressuresare 60 Pa which is approximated the breakdown threshold.
obtained. The biggest difference among the models can be found
5
at medium pressures over a wide range. The S. Aisen- energytransferefficienciesvary withthe pressuresofthe
berg model shows the highest efficiencies whereas the C. different operation voltages. In our measured pressure
Teske’sgivesthelowestinthispressurerange. Themax- range, the breakdown threshold is only observed at
imum energy transfer efficiency for the S. Aisenberg’s the lower voltage of 6 and 7 kV. For each voltage,
model is 78% while 52% for C. Teske’s modified model. the pressure corresponding to the peak value of the
In our new model, the value for the maximum energy efficiency is defined as the optimal pressure which shows
transfer efficiency is 71%. Due to no assumption made the maximum transfer efficiency. Comparing all of the
in the new model, it is considered reliable and precise. plots in Fig. 8 , it is found that the optimal gas pressure
Hence, the S. Aisenberg’s model overestimates the en- shifts to higher values when the voltage is increased.
ergy transfer efficiency. On the contrary, the C. Teske’s The maximum energy transfer efficiency found here is
modification model underestimates the values. 73% which occurs at around 80 Pa under the maximum
voltage of 14 kV.
Thesedeviationsforthe traditionalmodelsaremainly
caused by the assumption that the reflected plasma re- V. CONCLUSION
sistance is constant, which is untrue. The truth is the
reflected plasma resistance varies with time as the light
Anewmodelisproposedtocalculatetheenergytrans-
signal in Fig. 4 shows. However, this assumed constant
fer efficiency of theta pinch in general and especially for
reflectedplasmaresistanceleadstoanapproximatelyav-
thesphericalthetapinch. Asthisnewmodelcontainsno
erage value for the true time-dependent resistance. Re-
assumptions, the results obtained with it are considered
garding the energy consumed in the plasma, it does not
exact.
only depend on the resistance but also the square of the
ThedeviationsofthetraditionalmodelsofS.Aisenberg
current. Namely, the squareof the currentis the weight-
and C.Teske from the real values are mainly caused by
ingfactoroftheresistanceforcalculatingthetransferred
the improper assumption that the reflected plasma re-
energy. Due to the oscillation decay behavior of the cur-
sistance is constant. The first leads to overestimation
rent, the weighting factorfor the firsthalf waveis bigger
whereas the latter to underestimation. The correspond-
than for the other half waves. Unfortunately, the resis-
ing explanations are given respectively.
tance for the first half wave is almost zero indicated by
The latest setup of the spherical theta pinch that is
thelightsignalshowninFig.4. Consequently,thecalcu-
usedforthe presentedinvestigationshasamaximumen-
lated energy transfer by applying the average resistance
ergytransferefficiencyof73%foranArgon-Hydrogengas
is overestimated. This accounts for the S. Aisenberg’s
mixture whichis promising especially for high repetition
model’s overestimation. For C.Teske’s modified model,
rate applications e.g. flash lamps or a plasma stripping
this first half wave of the current is omitted for calcu-
device.
lating the transferred energy according to the Eq. (12).
However, the calculation of the average resistance still
counts in this first half wave where the reflected plasma
ACKNOWLEDGMENTS
resistance is almost 0. This results in the underesti-
mation of the average resistance. Correspondingly, the
This work is supported by the BMBF (German Min-
transferred energy calculation is underestimated. Be-
istryforEducationandScience)underthecontractnum-
sides, the neglection of the sine term in Eq. (12) also
ber 05P12RFRB8. The authors Ge Xu and Christion
contributes to this energy underestimation. For instance
Hock are supported by scholarships from HGS-HIRe
at 30 Pa and 6 kV, the maximum deviation for this sine
(Helmholtz Graduate School for Hadron and Ion Re-
term is about 3%.
search) for FAIR (Facility for Antiproton and Ion Re-
Due to the advantageof our new model and the faults search). We especially acknowledge Dr. Christian Teske
of the traditional models, our new model is adapted and his colleagues for designing and manufacturing the
to calculate the energy transfer efficiency from the sphericalthetapinchdevice. Wesincerelythankourcol-
measured current values. Fig. 8 shows all the measured leagues who gave us support in this work.
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