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Published by Fusion Energy Division, Oak Ridge National Laboratory
Building 5700 P.O. Box 2008 Oak Ridge, TN 37831-6169, USA
Editor: James A. Rome Issue 116 August 2008
E-Mail: jar@ornl.gov Phone (865) 482-5643
On the Web at http://www.ornl.gov/sci/fed/stelnews
Validation of HF coil location
by field mapping on CTH
Field mapping is typically performed on stellarators to
confirm the existence of closed nested flux surfaces. Qualitative
agreement is often obtained between the simulation
and measured vacuum flux surfaces before the device is
used in further plasma experiments. In this study, field
mapping results on the Compact Toroidal Hybrid (CTH)
are quantitatively compared to simulation results and
through a fitting procedure the simulation coil model is
modified to achieve a better agreement between the experimental
and calculated results.
Description of the CTH facility
CTH [1,2] at Auburn University is a 5 field period, lowaspect-
ratio torsatron with a major radius of R0 = 0.75 m
and a minor radius avessel = 0.29 m. The CTH device is
designed to investigate the MHD stability of currentcarrying
compact stellarator plasmas.
The CTH facility is shown in Fig. 1. The 96-turn helical
field (HF) coil consists of a single = 2, m = 5 coil. The HF
coil was designed to have stellarator symmetry. The helical
path followed by the center of the HF coil is expressed
by radial and toroidal winding laws which are both functions
of the poloidal angle, θ:
(1)
. (2)
Here, rc is the minor radius of the coil center and ϕ is its
toroidal location. The values of the coefficients (ax, bx, af,
bf) of the designed HF coil winding law are listed in
Table 1.
The HF coil was constructed by winding the flexible conductor
into a helical trough surrounding the vacuum vessel
(see Fig. 1), thus defining the toroidal portion of the winding
law. The HF trough dimensions were machined to tolerances
of ±0.5 mm. Additional uncertainties in the
toroidal winding law of Eq. (2) are estimated up to ±1 mm
Fig. 1. The CTH device.
OVF
(red)
TVF
(brown)
HF
(red)
rc(θ) = ax0 + ax1 cos(θ) + ax2 cos(2θ) +…
+ bx1 sin(θ) + bx2 sin(2θ) + …
ϕ θ ( ) 25
= --θ + af0 + af1 cos(θ) + ax2 cos(2θ) +…
+ bf1 sin(θ) + bf2 sin(2θ) + …
In this issue . . .
Validation of HF coil location by field mapping
on CTH
Field mapping results on the Compact Toroidal Hybrid
(CTH) are quantitatively compared to simulation
results. The simulation coil model is modified using a
fitting procedure to achieve better agreement between
the experimental and calculated results. ................. 1
Motojima to Receive FPA Distinguished Career
Award
Professor Osamu Motojima, Director of the National
Institute for Fusion Science (NIFS) in Japan, has been
selected by Fusion Power Associates (FPA) Board of
Directors to receive its 2008 Distinguished Career
Award. .................................................................... 4
Stellarator News -2- August 2008
due to compression of the flexible conductor inside the
trough.
The radial position of the coil was expected to exhibit
greater deviations from the design because the outer radius
of the HF coil is not constrained by the winding frame.
Measurements of the radial build of the HF coil were made
during the construction process to determine deviations
from the radial winding law of Eq. (1). The deviations
generally show similar behavior from one field period to
the next. These measurements were averaged and used to
develop a new field period–symmetric HF winding law
with the coefficients listed in the second column of
Table 1. The measured coil winding law shows a small
deviation from up-down symmetry of the design coil.
Table 1. HF coil winding law coefficients for the designed,
physically measured (before optimization), and optimized
(after optimization) coil model.
Also shown in Fig. 1 are the outer vertical field (OVF) and
trim vertical field (TVF) coils needed to produce the vertical
field necessary for confinement equilibrium. The HF
and OVF coils are electrically connected in series. The
current in the TVF coil is independently controlled for
radial positioning of the plasma. For clarity, the 10 toroidal
field coils, shaping vertical field coil set, and ohmic heating
coil system are not shown in Fig. 1 so that the HF coil
can be more clearly viewed.
Vacuum field mapping
Vacuum field mapping experiments were performed on
CTH to confirm the existence of closed nested flux surfaces
and to compare the shape and rotational transform of
the observed surfaces with calculation. The field mapping
results presented in this note focus only on the HF, OVF,
and TVF coils.
Field mapping measurements were performed with techniques
similar to those used on previous devices [3,4]. A
movable electron beam source is used in conjunction with
a screen or movable wand coated in zinc-oxide viewed by
a digital camera capable of long exposure photographs.
Points of interest in the resulting Poincaré puncture plot
photograph undergo a calibrated transformation to coordinates
in real space using LabView Vision Software® [5].
The measured surfaces are then compared to those predicted
by a field line following code [6], with the goal of
quantifying the discrepancies between the actual coils of
CTH and the simulation coil model used in the calculation,
or other sources of error such as ambient magnetic fields.
While images can be made of the full cross section of flux
surfaces, the analysis in this article makes use of the location
and rotational transform of the magnetic axis because
it is a fixed point, and it can be determined both experimentally
and through simulation.
Multiple field mapping experiments were performed with
DC power supplies at IHF/OVF = 300 A (B0 = 450 G). This
is approximately 7% of the normal operating field used in
plasma experiments. In these studies, the TVF current
ITVF was varied from 9% to 17% of the HF current (B0 =
20 G–38 G). The magnetic axis position and rotational
transform were measured at two toroidal locations, ϕ =
36° and ϕ = 252°. The experimental magnetic axis positions
from one such TVF current scan are shown in Fig. 2.
As expected, the increased vertical field produced by
increasing ITVF shifts the magnetic axis radially inward by
up to 0.15 m. The vertical position of the axis is found to
shift upward, above the midplane by 0.022 m as the ITVF is
increased, indicating that in this regime the magnetic field
structure has an up-down asymmetry.
HF coil
coefficients Designed Measured Optimized
ax1 (m) 0.385 0.3836 0.3826
ax2 (m) 0 0.0000 0.0012
ax3 (m) 0 0.0005 -0.0011
ax4 (m) 0 0.0001 0.0009
bx2 (m) 0 -0.0007 -0.0009
bx3 (m) 0 0.0002 0.0002
bx4 (m) 0 -0.0001 0.0004
af1 (radians) 0 0.0000 0.0002
af2 (radians) 0 0.0000 0.0005
af3 (radians) 0 0.0000 0.0002
af4 (radians) 0 0.0000 0.0013
bf2 (radians) -0.252 -0.2520 -0.2521
bf3 (radians) 0.052 0.0520 0.0530
bf4 (radians) -0.024 -0.0240 -0.0243
Stellarator News -3- August 2008
Simulation Results
The location of the magnetic axis is calculated using a
field line following code and compared with the experimental
field mapping results. With the original HF coil
winding law, the calculated magnetic axis remains near the
midplane regardless of changes in the value of ITVF (Simulation-
1).
When the adjusted HF coil model based on the physical
measurements made during the coil construction is used in
the simulation, the calculated magnetic axis position is
observed to shift above the midplane as ITVF is increased
(Simulation-2). The results of this calculation are similar
to the behavior of the axis seen experimentally, although
the calculated data has a radial offset of ΔR = 0.02–0.03 m
and a vertical offset of ΔZ = 0.005–0.015 m relative to the
experimental data. These discrepancies are well outside
the experimental error bars, suggesting that the coil model
could be further improved to more accurately describe the
results of field mapping.
The discrepancies between the experimental results and
those of Simulation-2 are caused not only by inaccurate
knowledge of the actual “as-built” HF coil winding law
but also by limited knowledge of the background fields
present within the vacuum vessel at the time of the experiment.
Owing to the presence of ferromagnetic material in
the vicinity of CTH (pipes, structural rods in the floor and
ceiling, etc.), the background field (the field not due to the
currents in the coils) was measured to be significantly different
from the expected Earth’s field. Measurements
made with a hand-held Hall probe in a separate experiment
have found that the remnant vertical background field can
be as great as 2.5 G. In addition, the background field has
been shown to be affected by the currents in the CTH coils
and thus can vary depending on the past operational history
of CTH. The initial aim of these studies was to use the
results of field mapping to accurately model the HF coil
winding law along with the other coil positions. With the
discovery of a variable background field, extracting information
about the coils becomes more difficult and now
must include an additional calculation of the background
field itself.
A least squares minimization fitting routine [7] has been
used to modify the coil parameters in the coil model to
minimize the differences between the experimental and
computed magnetic axis positions and rotational transforms.
Here the term “coil parameters” refers not only to
the coefficients in the HF coil winding law [Eqs. (1) and
(2)], OVF coil positions, etc., but also to the background
field which now must be included in the optimization. In
the calculation, the HF coil winding law is assumed to be
field period symmetric. The background field is assumed
to be uniform throughout the volume of the vacuum vessel
with the horizontal and vertical field components to be
determined by the fitting routine.
Starting from the physically measured coil positions, the
coil optimization procedure was applied to deduce a better
coil model. The positions of the magnetic axis computed
from the optimized coil model are shown in Fig. 2 (Simulation-
3). Comparing the calculated axis positions from
before and after optimization, we find the results from the
new coil model agree much better with the experimental
axis positions than those from the mechanically measured
model. The differences in radial position of the axis are
reduced to less than 0.001 m, while the differences in vertical
position are less than 0.002 m.
The coefficients of the slightly modified HF coil model are
given in Table 1. We find that the modifications to the
measured HF coil winding law are less than the uncertainties
of the coil position measurements. Accurate field
mapping measurements thus allow fine adjustments to be
made to the HF coil winding law, below the tolerance level
achieved using physical measurement techniques. The
OVF/TVF coil parameters were left nearly unchanged by
the coil optimization. The coil optimization was also used
to determine the background field in the laboratory during
the experiment. The three components of the earth’s field
(east, north, up) for Auburn, Alabama, are B = (0.0, 0.2,
−0.4) G [8], whereas the background field values com-
Fig. 2. Magnetic axis locations in (R, Z) for a TVF current
scan. The results of Simulation-1 are computed with a
model of the coils based on the original design and the
expected ambient magnetic field of the laboratory. The
results of Simulation-2 are computed with a physically
measured model of the coils (before optimization). The
results of Simulation-3 are computed with an optimized
model of the coils based on the fitting results.
-0.005
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.70 0.75 0.80 0.85 0.90
R (m)
Z (m)
Experiment
Simulation-1
Simulation-2
Simulation-3
ITVF/IHF = .09
ITVF/IHF = .17
ITVF/IHF = .13
Stellarator News -4- August 2008
puted by the coil optimization are significantly larger at B
= (0.1, 0.7, −2.1) G. Overall the new coil model required
no unreasonable modifications to successfully simulate the
experimental results.
In conclusion, the optimization process was able to
account for the observed radial and vertical shift of the
experimental magnetic axis by making small modifications
to the HF coil winding law and background fields.
The optimized HF coil winding law was found to have
only slight deviations from both the designed and measured
winding laws (although these deviations in fact can
break the up-down symmetry of CTH). In addition, the fitting
routine identified a characteristic background field
vector needed to adequately model the experimental
results. The presence of the background field, particularly
a time-varying one, limits the ability to make further
improvements in the geometrical coil parameter values. To
gain further information about the HF/OVF/TVF coils,
further field mapping experiments should be performed
under conditions where the background field is constant or
at least known more accurately.
Acknowledgment
Supported by U.S. DOE Grant DE-FG02-00ER54610
J. T. Peterson, G. J. Hartwell, S. F. Knowlton, J. Hanson
Auburn University, Auburn, AL
E-mail: peterjt@auburn.edu
References
[1] S.F. Knowlton, Stellarator News 69 (2000) p. 2–4.
[2] J.T. Peterson, J. Fusion Energy 26 (1–2) (2007), 145–
148.
[3] G.J. Hartwell, R. Gandy, M. Henderson et al. Rev. Sci.
Instrum. 59 (1988) 460.
[4] Takahashi et al., in Proceedings of the International
Stellarator/Heliotron workshop, IAEA Tech. Comm.
Meeting, Vol. 2 (1986) 220.
[5] LabView 8.0 for Machine Vision © 2008 National Instruments
Corporation.
[6] J.R. Cary and J.D. Hanson, Phys. Fluids 29, 2464
(1986)
[7] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W. T.
Vetterling. Numerical Recipes in Fortran 77: The Art
Of Scientific Computing, 2nd ed. (New York, Cambridge
University Press, 1996), 51–63, 651–700.
[8] http://www.ngdc.noaa.gov/seg/geomag/magfield.shtml.
Motojima to Receive FPA Distinguished
Career Award
Professor Osamu Motojima, Director of the National Institute
for Fusion Science (NIFS) in Japan, has been selected
by Fusion Power Associates (FPA) Board of Directors to
receive its 2008 Distinguished Career Award. The Award
will be presented to Professor Motojima at Fusion Power
Associates Annual Meeting and Symposium in Livermore,
California, December 3–4, 2008.
In selecting Professor Motojima, the FPA Board recognizes
his key roles in the design and construction of a
series of large stellarator facilities and subsequent experimentation
on them, in fostering international cooperation
in fusion research, and his leadership of the NIFS.
Fusion Power Associates Distinguished Career Awards
have been given annually since 1987 to individuals who
have made distinguished lifelong career contributions to
fusion development. A list of previous recipients is posted
at http://fusionpower.org and click on Awards.
Professor Motojima may be contacted at
motojima@lhd.nifs.ac.jp
Stephen O. Dean
Fusion Power Associates
http://fusionpower.org