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Analysis of Langmuir probe measurements from the Tandem Mirror ExperimentUpgrade (TMXU)D. Buchenauer and A. W. Molvik Citation: Review of Scientific Instruments 59, 1887 (1988); doi: 10.1063/1.1140043 View online: http://dx.doi.org/10.1063/1.1140043 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/59/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Characterization of titanium films in the Tandem Mirror Experiment Upgrade (TMXU) J. Vac. Sci. Technol. A 2, 1222 (1984); 10.1116/1.572707 Measurement of sloshingion spatial profiles in end cell of tandem mirror experimentupgrade (TMXU) Phys. Fluids 26, 2335 (1983); 10.1063/1.864434 Initial wall conditioning for the TMXU fusion experiment J. Vac. Sci. Technol. A 1, 916 (1983); 10.1116/1.572150 Fast pressure measurements for the TMXU fusion experiment J. Vac. Sci. Technol. A 1, 1293 (1983); 10.1116/1.572092 The LLNL tandem mirror experiment (TMX) upgrade vacuum system J. Vac. Sci. Technol. 20, 1177 (1982); 10.1116/1.571513
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Analysis of Langmuir probe measurements from the Tandem Mirror Experiment Upgrade (TMX .. U)
D. Buchenauer
Sandia National Laboratories, Livermore, California 94550
A.W. Moll,lik
Lawrence Livermore National Laboratory, University a/California, Livermore, California 94550
(Presented on 16 March 1988)
A nonlinear least-squares curve-fitting routine has been used to analyze single-tip Langmuir probe measurements from the Tandem Mirror Experiment Upgrade during high-density ( ;:::; 1 X 1012 cm - 3
) operation. This procedure provided estimates of uncertainties (variances) in the electron temperature and plasma density due to noise in the probe current. The electron temperature and plasma potential inferred from the fit were found to increase with the upper cutoff voltage used above a certain voltage. This effect appears to be due to a departure of the electron current from an exponentially increasing function for probe positions inside the limiter radius only. The fitted values for electron temperature and plasma density are consistent with previous measurements obtained with a double-tipped probe and the fitted values of space potential indicate a linear relationship with electron temperature.
INTRODUCTION
Langmuir probe measurements have become a common diagnostic for edge plasmas in many fusion devices, including the Tandem Mirror Experiment Upgrade (TMX-U), Using a double-tipped Langmuir probe, measurements of the electron temperature and density were used to model the "halo" region which surrounds the core plasma in TMX-U. I
•2 The advantage of this technique over the single
tipped operation is primarily in the analysis of the currentl voltage (l IV) characteristic. For a double-tipped probe, the I I V relationship near zero current can be approximated by a straight line,
dUI Te = 0.5I.at - , 'dI 1=0
(1)
the slope of which can easily be found from plotting the data. Here Te is the electron temperature, npl is the plasma density, I sat , is the ion saturation current, andA; is the ion collection area. For a single-tipped analysis, the exponential increase in electron current with voltage requires a logarithmic plot to allow a similar analysis. Not only does the technique break down when the fluctuations in the current become negative, but also analytical estimates of the errors in the temperature and density are not available. Use of a nonlinear least-squares fit to the data overcomes both of these difficulties and provides an improvement in the automation of the data analysis. The main consideration in determining th~ parameters is then reduced to picking an upper cutoff voltage used in the fit. It will be shown below that use of too high a cutoff voltage can lead to erroneously high electron temperatures even when the cutoff voltage is well below the regime of electron saturation current.
I. LANGMUIR PROBE AND ELECTRONICS
TMX-U is a tandem mirror magnetic configuration consisting of a 0.3-T field strength, nonaxisymmetric central cell with two 0.5-T field strength end cells ( 4: 1 mirror ratio). The upgrade in TMX-U refers to the sloshing ions and electron cyclotron heating applied in the end cells which form a thermal barrier reducing axial ion losses. Recent operation has been concerned with increasing the density at which this improvement in axial confinement, known as plugging, can be obtained. 3,4 The Langmuir probe llsed in these studies was a double probe located in the central cell ofTMX-U, 66 cm from the midplane of the machine. Each probe tip has an exposed section 2 mm in length Up ) and 1 mm in diameter (dp ). The tips were made of tungsten with an electrically isolated tantalum shield that defined the exposure region. The double-tipped probe was operated as a single probe by aligning the two tips (separation 1.5 em) across the magnetic field to reduce the perturbing effect of the unused tip. Measurements obtained during high-density conditions will be presented in Sec. III. .
Four KEPCO bipolar operational amplifiers were used in series to supply a swept voltage of ± 400 V (maximum 1 A) at 200 Hz to the probe tip. The sweep voltage was monitored by a 10:1 voltage divider, and the probe current was measured by the voltage drop across a 50-0, series resistance. The voltage and current signals were fed into high commonmode rejection isolators, were amplified, digitized at 40 kHz, and then recorded by the computer. The entire current monitor circuitry had a typical dynamic range of about 100.
11.11 VCHARACTERISTIC FIT
In the TMX-U halo plasma, the electron gyroradius and the Debye length are small compared with the probe dimen-
1881 Rev. Scl.lnstrum. 59 (8), August 1988 0034-6748/88/081887-£13$01.30 © 1988 American Institute of Physics 1887
••••••••••••• -.-.~.-,-.-.-•••• , ••• _.,.,., •••••• ~ •••• , •••• , •••• :.:.;-:.;, ••• , ••••• ' ••••••••••••• r •••••• , ...... _ •• ,_,_, ' ••••• , ••• ~ r. , • ~ ~
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sions, while the ion gyroradius is comparable to the probe diameter. The collection areas for ions and electrons are, therefore, Ai = rrdplp = 6.16X 10-2 cm2 and Ae = 2dplp = 3.92X 1O.- 2 cm2
• HereA; is the true area of the probe, and Ae is the projected area along the magnetic field. The function to be fitted (l fit) isjust the ion saturation current (with T; ~ Te) added to a Boltzmann distribution of electrons
(2)
(3)
I fit is a fit to the probe current data {Ij} in the least-squares sense obtained by varying the fit parameters {am}, i.e., Te (electron temperature), npl (plasma density), and Vsp
(space potential), and observing N
X2 = 2: {Ij-lfit(Gj)P, X~==x2(min). (4) j= I
The minimum of X2 is found using a coarse-grid search and then a combination gradient-expansion method to step toward X~ .5 The latter technique uses a combination of a gradient search of the chi-squared surface which works wen far from the minimum with a linear expansion of the fitting function which works well close to the minimum.
Finally the variances of Te , npl ' and ~,p' i.e., {uam }, are estimated using two techniques. First, by approximating the chi-squared surface as a parabola in the vicinity of the minimum, the variances can be found from the inverse of the curvature matrix (a mn ) :
(5)
1 aX~ E=a- 1, am" = - (6) 2 Jam Jan
Here v is the number of degrees offreedom (v = N - 3) and a mn is calculated analytically. This technique is equivalent to successively moving along each oftlie parameter axes until the reduced X2 increases by 1. This distance is then equal to the variance.
The second method is to expand the fitting function as a linear function of the fitted parameters near the minimum of X2 and then use the ordinary linear least-squares determination of the variances. Keeping only the linear terms, x2 can be approximated by
(7)
Minimizing X2 with respect to the parameter increments ( {8a m }) gives a set of M simultaneous equations
M
f3n = 2: 8am a mn , n = 1, ... ,1\1, m=1
(8)
and the inverse of the curvature matrix can be found as before. Since this latter technique gave larger variances in the vast majority of cases, these variances are the ones plotted with the data (see Sec. III).
1888 Rev. Sci.lnstrum., Vol. 59, No.8, August 1988
.(if
8' 0.04
$
x~= 2.Em x 10 5
T. = 6.91 ± 021/025 eV H" ~ 49.90 ± Uq/t64 x 1010 em-:l
V .. = 3148 ± 021}"0.39 volts
• -0.02 -f-.-......... ..-,-~~,..,...,~-,-.~...,....,~~-r-~.,....,....~~,........,~-+
-80 -60 ···40 ··20 0 20 40 60 80
PROBE VOI~rAGE (volts)
FIG. 1. Typical J I V characteristic during high-density operation at a probe radius of 25 cm. The limiter radius in TMX-U is 24 cm, which maps to a radius of 23 cm at the z = 66 cm axial position of the Langmuir probe. Shown above the characteristic fit and data are the deviations ofthe current values from the fit. The variances are from the parabolic and linear models, respectively.
ill. DATA REDUCTION
Figure 1 shows a typical single-tipped I I V characteristic measured with the central cell Langmuir probe in TMX-U during neutral beam heating. Shown above the fit to the current are the deviations for each data point. The probe ra,;iius was 25 cm, just 2 cm outside the projected limiter radius, and the fitted values of the electron temperature and density are similar to those measured with the double-tipped method used previously. 1 In addition, the fitting routine provides an estimate of the space potential. It is apparent that the fluctuations in the electron saturation current are much larger than those in the ion current. This effect has also been observed in the PDX tokamak,6 but the ratio of electron saturation current to ion saturation current (1. t II ) in PDX . sa e sat,
was much smaller. The ratio of Isate to I'8ti
in TMX-U was typically about 10 and did not vary significantly with the probe's radial position.
Using fits like that in Fig. 1, it is possible to determine the effect of varying the upper cutoff voltage (Vcut ) on the parameters. Figure 2 shows the variation of electron tem-
b) 1
I I IT
JII:!:" II '~ a'+4-r~~--~~-r-+ +-~~--~~~-4~
Co 1.0
~ 08
::> 0.6 u +
+ + + + + +
+ +
+ W 0.4
~ 02 + + I=:,~ + ~mumruldiJ ~ + + r· ++, ~.
o.o~1-'24_'-"!"'O,.........-r,~<~" ...--,.~,.......-l-5 _+2...,....;1;-, ~O.........,.,~2~' "-'--r-"""+ V -v v -v -""''1<1':-' ---"'i<t--'-'
FIG. 2. Electron temperature and normalized probe eurre~t (In ) at the upper cutoff voltage (Vent) plotted against the nonnalized upper cutoff voltage. Ca) Probe radius of21 em, (b) probe radius of25 cm (the projected limiter radius is 23 cm).
Particle. photon-based diagnostics 1888
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perature from the fit and a normalized probe current, In =1 I Usat, - I,at,), with the normalized upper cutoff voltage. Here VI is the floating potential. The arrows on the plots indicate the upper 'cutoff voltage for which the deviations shown in Fig. 1 remain centered ~bout the fit. For voltages exceeding these values, the deviations tend to become unipolar in different voltage ranges and the model is inadequate. Using too low of an upper cutoff voltage results in larger variances in the electron temperature since the curvature of the current is within the noise in this region of the I I V characteristic. The values of the fitted parameters and the probe current at this "optimum" voltage, i.e., just below the onset of unipolar deviations, will be used in the remainder of this analysis.
It is apparent from Fig. 2 that the electron temperature deduced from the fit can become erroneously high if too large an upper cutoff voltage is used. The fitted values for the space potential behave similarly. It can also be seen that the optimum upper cutoff voltage is not necessarily related to the voltage required to obtain the electron saturation regime [Fig. 2 (a) ] . In Fig. 3 (a), this effect is shown as a function of the probe radius. It is apparent that the probe current at the optimum upper cutoff voltage is dependent on whether the probe position is within the limiter radius while the value of the optimum cutoff voltage varies by only 15%. This radial dependence of the optimum cutoff current is caused by a reduction of the electron current from an exponentially increasing function at large voltages for probe positions within the radius of the limiter.
Using the fits obtained at the optimum upper cutoff voltage, Figs. 3 (b) and 3 (c) show the radial profile of electron
~ 1.0
i e.- 0.8
~ ~ 0.6
0:: [3 0.4
§ 0.2
g: 0.0
>-t: ffl Z W P ...: ::>l ffl j Il.
o
160
120
80
40
o 16
+ + + +
I I
I I
x 10lD cm-3
:I: I :I I
~'O)Ecled limite~ ~rn.<lius . ~-____
18 20 22 24
PROBE RADIUS (em)
+ a) +
b)
I I
c)
:E
~ -
26 28
FIG. 3. Normalized ion current Cal, electron temperature (b), and plasma density (c) plotted as a function of the probe radius during a radial probe scan.
1889 Rev. Sci.lnstrum., Vol. 59, No.8, August 1988
temperature and plasma density obtained during a sequence of high-density plasma shots. The temperature profile is flat within the estimated variances, while the density profile appears to be fiat or slightly increasing within the plasma radius. If the profile is extrapolated to the plasma center, the line integral of the density points is in. good agreement with the line-averaged density measured by several microwave interferometers. A comparison of the electron temperature and the space potential also shows a high degree of correlation ( V.
P 'Z 4kTe , R = 0.91 ), as would be expected for these
high-density plasmas.
IV. CONCLUSION
Although the physical measurement of the I I V characteristic is relatively straightforward, there are many references in the literature which discuss the difficulties associated with interpreting the characteristic information. Recent measurements made on JET7 have also shown that use of too large a cutoff voltage can lead to erroneously high electron temperatures. The effect is explained as a reduction of the electron current due to the strong magnetic field. Although theory suggests that a normalized cutoff voltage above zero may lead to this effect, the TMX-U data would indicate that normalized voltages above ::::: 2 still yield accurate fits of Te . In any case, the optimum values of Te , npl' and Vsp can be determined through an investigation of the randomness of the deviations in the fit. This provides an optimum upper cutoff voltage above which the fit becomes inadequate.
ACKNOWLEDGMENTS
The authors wish to acknowledge useful conversations with Dr. Wen L. Hsu. We are also indebted to Dr. Thomas C. Simonen and the TMX-U group for their help in making this work possible. This work was supported by the U.S. DOE under Contract Nos. DE-AC04-76DP00789 and W-7405-ENG-48.
lW. L. Hsu, W. Bauer, R. A. Kerst, K. L. Wilson, T. C. Simonen, J. H. Foote, and W. L. Pickles, J. Vac. Sci. Techno!. A 3, Il7S (1985) .
2W. L. Hsu, J. Nucl. Mater. 128-129, 500 (1984). 3T. C. Simonen et al., in Proceedings afthe lIth International Conference of Plasma Physics and Controlled Nuclear Fusion Research, Kyoto, Japan, 1986 (IAEA. Vienna, 1987), Vol. II, p. 231.
4T. C. Simonen et al., Report No. UCRL-96484, 1988; IEEE Trans. Plasma Sci. (to be published).
5D. W. Marquardt, J. Soc. Ind. App!. Math. 11,431 (1963). 6D. M. Manos, J. Vac. Sci. Techno!. A 3,1059 (1985). 7J. A. Tagle, P. C. Stangeby, and S. K. Erents, Plasma Phys. Controlled Fusion 29, 297 (1987).
Particle, photon-based diagnostics 1889
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