Hall probe measurements of the poloidal magnetic field in Compact Toroidal Hybrid plasmas B. A. Stevenson, S. F. Knowlton, G. J. Hartwell, J. D. Hanson, and D. A. Maurer Citation: Review of Scientific Instruments 85, 093502 (2014); doi: 10.1063/1.4894209 View online: http://dx.doi.org/10.1063/1.4894209 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Development of internal magnetic probe for current density profile measurement in Versatile Experiment Spherical Torusa) Rev. Sci. Instrum. 85, 11D809 (2014); 10.1063/1.4886429 A magnetically driven reciprocating probe for tokamak scrape-off layer measurements Rev. Sci. Instrum. 82, 123505 (2011); 10.1063/1.3661128 Poloidal mode analysis of magnetic probe data in a spherical tokamak configurationa) Rev. Sci. Instrum. 79, 10F120 (2008); 10.1063/1.2965014 Poloidal beta and internal inductance measurement on HT-7 superconducting tokamak Rev. Sci. Instrum. 78, 093501 (2007); 10.1063/1.2779213 Effect of the vacuum vessel on magnetic measurements in TCABR tokamak Rev. Sci. Instrum. 75, 5038 (2004); 10.1063/1.1807001
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 093502 (2014)
Hall probe measurements of the poloidal magnetic field in Compact Toroidal Hybrid plasmas B. A. Stevenson,1,a) S. F. Knowlton,2 G. J. Hartwell,2,b) J. D. Hanson,2 and D. A. Maurer2 1 2
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Department of Physics, Auburn University, Auburn, Alabama 36849, USA
(Received 30 May 2014; accepted 18 August 2014; published online 5 September 2014) A linear array of 16 Hall effect sensors has been developed to directly measure the poloidal magnetic field inside the boundary of a non-axisymmetric hybrid torsatron/tokamak plasma. The array consists of miniature gallium arsenide Hall sensor elements mounted 8 mm apart on a narrow, rotatable printed circuit board inserted into a re-entrant stainless steel tube sheathed in boron nitride. The sensors are calibrated on the bench and in situ to provide accurate local measurements of the magnetic field to aid in reconstructing the equilibrium plasma current density profiles in fully threedimensional plasmas. Calibrations show that the sensor sensitivities agree with the nominal manufacturers specifications of 1.46 V/T. Poloidal fields measured with the Hall sensor array are found to be within 5% of poloidal fields modeled with a Biot-Savart code. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4894209] I. INTRODUCTION
Reconstruction of the experimental magnetic flux surfaces of confined toroidal plasmas is accomplished using measurements of the local and average magnetic field coupled with numerical equilibrium solvers. Magnetic diagnostics to determine the magnetic field components exterior to the plasma are relatively easy to implement. However, measurements of the field inside the plasma volume should lead to a more accurate reconstruction of the rotational transform profile, or alternatively the profile of the safety factor, which is crucial to understanding of plasma stability in both tokamaks and stellarators. Diagnostics that employ the motional Stark effect or Faraday rotation are often implemented for this purpose in large experiments.1, 2 In colder plasmas with short discharge durations, the spatial variation of the magnetic field can be measured with an array of miniature magnetic loops or Hall probes inserted into the discharge itself. While the presence of the probes may perturb the evolution of the plasma, the information gained is useful in understanding specific processes, e.g., major disruptions in toroidal discharges.3 An array of Hall effect sensors has been constructed for use on the Compact Toroidal Hybrid (CTH) plasma physics experiment.4 The development of internal Hall probe arrays in tokamak plasmas for magnetic equilibrium and fluctuation measurements has previously been reported from experiments on TEXTOR,5 HBT-EP,6 and PEGASUS.7 This paper describes the application of new Hall effect sensors to measure the local poloidal magnetic field in current-carrying plasmas in the CTH torsatron/tokamak to aid in the reconstruction of the experimental plasma flux surfaces. Due to the non-axisymmetric plasma geometry of CTH, reconstructions must be performed with a fully three-dimensional analysis.
a) Research performed while the author was at Auburn University. b) [email protected]
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II. DESCRIPTION OF THE CTH HALL PROBE DIAGNOSTIC A. Overview of design
A Hall effect sensor consists of a semiconductor through which a current density J is passed. A sketch of a conceptual Hall probe sensor in an arbitrarily oriented magnetic field is shown in Fig. 1. When the sensor is placed in a magnetic field, the charge carriers (electrons or holes) experience a Lorentz force perpendicular to both the current and the field to produce a Hall voltage.8 The magnetic field-dependent voltage output of a Hall sensor is expressed as VHall = G(Is , B )B⊥ ,
(1)
where B⊥ and B are the components of the magnetic field perpendicular and parallel to the face of the sensor, and G is the sensitivity defined as the ratio of the output voltage to the magnetic field at the sensor. The magnitude of G is linearly dependent on the source current, Is , passed through the sensor, and must generally be determined by calibration. Ideally, the sensor output is nominally insensitive to the component of the magnetic field parallel to the face of the sensor (B ) illustrated in Fig. 1. However, the possibility of a weak dependence of the sensitivity upon the value of B is allowed for in this expression. Unlike a magnetic pickup coil, the Hall sensor directly measures the time-dependent magnetic field without integrators. Also, Hall sensors can be as small as a few millimeters in size providing highly localized measurements of the magnetic field. The small size facilitates their inclusion into a probe that can be inserted into the plasma. Because the Hall effect sensors are not known to be ultra-high vacuum compatible, the sensors are mounted onto a narrow printed circuit board (PCB) and inserted into a small re-entrant stainless steel tube sealed at one end such that the inserted probe array is at atmospheric pressure. From the point of view of diagnosing the poloidal magnetic field, several non-ideal effects on the output voltage of
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FIG. 1. Conceptual diagram of Hall effect sensor in obliquely oriented magnetic field. Ideally, the output voltage, VHall , is only dependent on the component of B perpendicular to the surface of the sensor.
FIG. 3. The location of the Hall probe array is shown in relation to the CTH vacuum vessel and an example set of magnetic flux surfaces. The probe array can be moved a total of 0.3 m in the radial direction.
Hall sensors must be considered. These include the dependence of the sensitivity on temperature, and the magnitude of the planar Hall effect, which is the dependence of the output Hall voltage on the magnitude of magnetic field components parallel to the face of the sensor. Because the poloidal field component in stellarator and tokamak plasmas is typically an order of magnitude lower than the main toroidal field (TF), the planar Hall effect could clearly be important, and has been documented in previous work.6, 7
sensors are fed into differential amplifiers with a gain of 20, filtered, and then passed to the data acquisition system. The probe array is placed inside a re-entrant stainless steel tube with 8.0 mm inner diameter and 0.9 mm thick walls. The PCB of the array is mounted to a long aluminum strip that allows the array to be pivoted about its axis within the re-entrant tube. The aluminum strip is mounted to a rotary positioner with a scale to align the angular orientation of the array for maximum sensitivity to the poloidal component of the magnetic field. The Hall probe array within the CTH vacuum vessel is shown in Fig. 3. The figure illustrates a cross-section of the toroidal plasma at the location of the Hall probe array. The array is shown in two positions that represent the limits of the radial positioning of the array. Also depicted in the figure are nominal equilibrium magnetic flux surfaces of the CTH plasma. The magnetic axis of the plasma at the center of the flux surfaces is located at a major radius of R = 0.75 m. In practice, the shape and position of the flux surfaces depend on the currents in the external coils and the current carried by the plasma. The poloidal field of the CTH plasma refers to the magnetic field components tangent to the flux surfaces within the plane of this figure. The larger toroidal field component of the CTH equilibrium field is oriented normal to the plane of the figure. The probe array is rotated along its axis to maximize the sensitivity to the local poloidal field, which is nearly vertical at the position of the probe. The toroidal field of the CTH device thus represents a parallel field component to the sensor, in the context of Fig. 1. The outer end of the
B. Implementation
The Hall array built for CTH consists of sixteen GH-701 Gallium Arsenide (GaAs) sensors manufactured by Sypris Test & Measurement.9 The GH-701 sensor has an active area of 1.5 mm × 1.5 mm. The sensors are mounted onto a narrow (PCB) with an 8 mm separation between adjacent sensors, as shown in Fig. 2. When mounted, the sensor surface is nominally parallel to the surface of the PCB. To ensure that the sensitivity of each sensor relative to one another remains fixed, a bias current of 8.18 mA is passed in series through each of the 16 sensors via conductors on the PCB to provide a nominal sensitivity of 1.46 V/T. The bias current is supplied by a TL 783 voltage regulator configured as a constant current source, with a typical regulation of 0.15%. The Hall voltage from each sensor is transmitted by a twisted pair of 36 gauge wires in a 65-conductor Litz wire bundle, chosen to reduce unwanted pick-up and because the bundle was small enough to fit into the probe housing. The output voltages of the Hall
FIG. 2. Array of 16 Hall sensors with twisted pair leads of Litz wire bundle.
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re-entrant tube is connected to a motorized linear slide and welded bellows, allowing the probe assembly to be moved radially a distance of 0.3 m within the vacuum chamber. The linear position of the probe is controlled with a Vexta Model PK266 stepping motor and a Velmex VXM controller with a specified precision of 0.01 mm per step. The position and elevation angle of the linear slide was adjusted with the use of a coordinate-measuring machine (CMM) so as to orient the axis of the probe array to lie along the mid-plane of the toroidal plasma. Once the slide was placed, CMM measurements were made for a series of probe positions to determine the final orientation of the slide. A linear fit was applied to these measurements to determine the precise position of the probe. These positions were later used as inputs for magnetic field calculations that were compared to the measured magnetic fields. A boron nitride (BN) tube surrounds the stainless steel tube housing the probe array to prevent the stainless tube from providing a conducting path across the plasma flux surfaces. The BN tube has an outer diameter of 1.27 cm and is closed on the end inside the plasma. Boron nitride was chosen on the basis of previous results10 that demonstrated excellent thermal and electrical resistance while producing lower plasma density changes than with other probe-shielding materials tested. To prevent metal deposition on the BN shield during glowdischarge cleaning and titanium gettering, the probe assembly can be retracted into a 2.54 cm diameter stainless steel tube. A simple shutter at the end of the tube closes the shielding tube when the probe assembly is fully retracted. C. Calibration
The calibration of the magnetic field response of the Hall sensors is performed in two stages. The sensors are initially calibrated at relatively low magnetic fields to obtain the sensitivity G of each detector, and then the calibrations are checked in situ on the CTH device. While previous work using Indium Antimonide (InSb) Hall sensors indicated that the probes sensitivity to fields perpendicular to the surface varies when a strong planar or parallel field is present,6, 7 this variation of sensitivity due to parallel fields has not been fully specified for Hall sensors fabricated with GaAs. The initial calibration magnetic field is produced by a set of Helmholtz coils. To perform this test, a calibration setup was prepared with two orthogonal sets of Helmholtz coils to provide magnetic fields perpendicular and parallel to the Hall probe surface, as shown in Fig. 4. Both Helmholtz coils in this assembly were calibrated using a Bell Teslameter. The parallel field corresponds to the toroidal field in CTH, while the perpendicular field corresponds to the poloidal field, the component of interest to reconstruction. With this method, the Hall sensors were found to have an average sensitivity of 1.49 V/T. The maximum magnetic field that could be achieved using the Helmholtz coils was 0.035 T. At these field magnitudes, no detectable change in the Hall voltage was recorded when a parallel field component was applied, regardless of the orientation of the parallel field. As this magnetic field strength is less than one tenth the typical toroidal field strength of
Rev. Sci. Instrum. 85, 093502 (2014)
FIG. 4. Schematic diagram of double Helmholtz coil calibration stand.
0.5 T in CTH, further calibrations of the sensors were performed in situ on the CTH device itself. In this case, the steady-state fields of the CTH configuration were calculated from the measured coil currents using the Biot-Savart solving capability of the IFT code.11, 12 The results of an in situ calibration for one detector are shown in Fig. 5. Calibrations are performed for three values of the toroidal field. The measured sensitivity remains in the range 1.45 V/T ± 1%, within the manufacturer’s specifications. The linear sensitivity is found to vary with the magnitude of the field component parallel to the plane of the Hall sensor by 1% over the range of fields in this experiment. Data for other probes show similar trends but with different levels of sensitivity to the magnetic field parallel to the probe. This suggests that the effect is most likely due to misalignments of the probe and not an inherent property of the probe itself. The low sensitivity of the probes to parallel fields allow the probes to be used to measure the poloidal component of the magnetic field despite the presence of a stronger toroidal field. The independent parasitic responses of the sensor to the two possible orthogonal components of the parallel field was not measured in this experiment. Subsequently in this paper, we refer to the poloidal field as the component of the field measured by the sensor array, and the toroidal field as the ambient parallel field.
FIG. 5. Measured Hall sensor output versus applied poloidal (perpendicular to the sensors) magnetic field strength when fields parallel to the Hall sensors are applied with the TF coil set on CTH. Lines represent linear fits to the data, and the legend indicates the fitted linear sensitivity in V/T and the offset voltage.
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The sensitivity of Hall effect sensors is also known to be susceptible to changes in temperature. The sensitivity of InSb Hall sensors were reported to shift by up to 2%/◦ C.6, 7 The GaAs Hall sensors of the CTH array are expected to be less sensitive to temperature changes, with a stated temperature sensitivity of 0.06%/◦ C. Nonetheless, the temperature of the probe array is monitored with two thermistors mounted onto the back side of the PCB. Furthermore, the Hall sensors are also actively cooled by compressed air flowing over the sensors during the operation of CTH. The ability to directly measure the probe temperature makes it possible in principle to adjust the calibrated probe sensitivities to account for temperature changes. Currently, temperature corrections have not been found to be necessary due to the small temperature sensitivity of the GH-701 sensors as well the use of active cooling. III. INITIAL RESULTS
Equilibrium reconstruction on CTH makes use of the V3FIT13 code that minimizes differences between experimentally measured signals and modeled diagnostic signals that depend on the parameterizations of the quantities of interest. The Hall sensor array is designed to measure the poloidal component of the magnetic field along a radial line segment in the mid-plane to provide V3FIT with signals that directly relate to the radial profile of the plasma current density. Unlike in tokamaks, the poloidal field component of the torsatron configuration of CTH, produced by the current in the external magnetic coils alone, typically exceeds the magnitude of the poloidal component resulting from the plasma current. To separate the two effects, the effective mutual inductance, or response functions, of the Hall sensors to the external current sources was determined experimentally. Each independent coil set that produces the magnetic field of CTH was driven with a constant current while the other coil sets were open-circuited to prevent induced current flow in the latter. The energizing current was maintained for time periods longer than required for eddy currents in the vacuum vessel and other nearby conductive supports to decay. The Hall sensor response functions, defined as the ratio of the magnetic field measured by each Hall sensor to the measured coil current, were measured and tabulated for each sensor and independent coil set. By design, the normal direction of the sensor surface should be oriented in the local poloidal direction, as indicated in Fig. 3. Thus aligned, the sensors should exhibit no response to the toroidal magnetic field. In reality, each of the individual sensors may have small alignment errors. The response functions of the sensors to the TF coils of CTH were measured and are shown in Fig. 6. The TF coil set of CTH consists of ten planar coils that produce a magnetic field with only a toroidal component at the location of the probe array. Indeed, the majority of sensors show only a small response in comparison with that to the HF coil set. However, four sensors show responses with magnitudes greater than 5 × 10−7 T/A. Shown in Fig. 7(a) are the response functions of the sensors to the helical field (HF) coil set which includes a helical
Rev. Sci. Instrum. 85, 093502 (2014)
FIG. 6. Hall sensor response functions of the 16 elements for an applied toroidal field, nominally parallel to the Hall sensors.
coil electrically in series with vertical field coil and produces both toroidal and poloidal field components. Positioning the array at four different radial positions and determining the response of all 16 sensors generates the data points in this figure. The measured response functions are fit using a quadratic polynomial shown by the black curve in the figure to compare with the Hall probe response function data. The poloidal magnetic field at the Hall sensor locations can also be modeled using a Biot-Savart type code. The dashed curve in Fig. 7(a) shows the modeled radial distribution of the poloidal magnetic field along the outer mid-plane due to a unit current in
FIG. 7. (a) Hall sensor response to the HF current versus major radial position of the Hall sensors. The major radius of the CTH vacuum vessel is 0.75 m. Data for four separate positions of the array are included. The black curve is a polynomial fit to the data and the dotted-dashed curve is the modeled response. (b) The corrected sensor response using the misalignment angles.
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the HF coil. The modeled curve shows generally good agreement to the Hall probe fit shown with the solid curve. The Hall probe response to the HF current exhibits modest scatter about the fitted polynomial curve. Investigation of this scatter shows that the deviations from the fit depend more on the specific probe than on the position of the array as a whole. The probes with the largest deviations are the same probes with large deviations when measuring the toroidal field as shown in Fig. 6. This leads to the conclusion that the probes are misaligned, picking up some toroidal field, and that the measured signal should be expressed as S = Bpol cos θ + Btor sin θ,
(2)
where θ is the angle between the normal of the Hall probe to the nominal. The angle of each probe can be found from the data used to generate the plot in Fig. 6, taken in the presence of a pure toroidal field. The probe angles can then be used to correct responses to the HF current by subtracting out the contributions due to the toroidal field. The corrected responses, shown in Fig. 7(b), show less deviation from the fit as compared to that shown in Fig. 7(a). The fit and model lines are the same in Figs. 7(a) and 7(b). The Hall probe array has been used to measure the poloidal field in both current-free and current-driven plasma discharges, and also in plasma-free scenarios in which all field coils, including the ohmic heating (OH) transformer coil that produces a time-varying current, were energized. The poloidal field measured by the Hall probe array in a fullfield, no-plasma shot is shown in Fig. 8(a), where the field values have again been corrected for misalignments of the sensor. The calculated magnetic axis of the discharge is located at a major radius R ∼ 0.78 m. The Hall probe array was positioned with the inner-most sensor located at a minor radius of 0.8778 m so as to maximize the number of sensors within the last closed flux surface. The OH transformer was pulsed to generate eddy currents in the vacuum vessel and coil support frames similar to those expected in a plasma discharge even though there was no plasma current induced. The vacuum magnetic equilibrium was calculated with the V3FIT reconstruction code. There is a small offset (≤5%) in the measured poloidal field strengths from the modeled values. By using the modeled and empirically determined set of response functions, the magnetic field contribution from the external coils at each Hall sensor location was subtracted leaving the residual signals shown in Fig. 8(b). These resid-
FIG. 9. Poloidal magnetic as measured by the Hall probe array (diamonds) and modeled with the VMEC (filled squares) and V3FIT code (squares). The location of the plasma edge is estimated from reconstruction to be at R = 0.98 m.
ual signals result from time-dependent eddy currents, and are small in comparison to the full magnitude of the measured poloidal field. The contribution from the external poloidal field can thus be effectively removed from the Hall sensor measurements. After determining the contribution from the external coil currents to the modeled poloidal field, one may then subtract this portion from the total poloidal field to obtain the poloidal field produced by the currents within the plasma during CTH discharges. While the insertion of the probe array well into current-driven discharges may lead to an uncontrolled increase in density during the plasma discharge and sometimes a disruption, measurements with the Hall probe array in such discharges can provide the profile of the poloidal field produced by the plasma current. Fig. 9 shows a comparison between the poloidal field measured with the Hall sensor array and modeled with the VMEC14 and V3FIT codes. In this discharge, the plasma current was Ip = 40 kA, increasing the edge rotational transform from -ι vac (a) = 0.125 to ι-total (a) = 0.45. The VMEC code was used to model the magnetic field at positions inside the last closed flux surface (LCFS) (filled squares), while synthetic V3FIT magnetic diagnostics were used to model the magnetic field outside of this region (unfilled squares). The use of separate codes to model the magnetic fields inside and outside of the LCFS is necessary because VMEC will only give results up to the LCFS. As of this writing, V3FIT can handle synthetic diagnostics inside the LCFS, but singularities are possible if the diagnostic is placed too near VMEC grid positions, whereas outside the LCFS, V3FIT works smoothly. Thus, these data cannot be used as an input to the V3FIT code. We note that the trend between the data and the model is similar and that the measured and modeled fields are within 5% of one another. We do not think that this discrepancy is due to a plasma current measurement error. It is more likely that the radial plasma position is in error because of assumptions made for unmeasured quantities in the reconstruction. IV. SUMMARY
FIG. 8. (a) Poloidal magnetic field measured with the Hall probe array (diamonds) and calculated by equilibrium reconstruction (squares). The measured values have been corrected for misalignments of the sensors. (b) Residual poloidal field after externally generated field is subtracted for Hall probe data (diamonds) and reconstructed equilibrium fields (squares).
A new type of Hall probe array for use inside toroidal plasmas has been constructed with the use of GaAs sensors. These transducers are found to be less sensitive to parallel field components and temperature variation that sensors
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of other material used for the same application. The Hall probe array has been tested on the CTH torsatron to measure both vacuum and plasma-generated magnetic fields. While the array can perturb the evolution of the plasma discharge if inserted too deeply, it provides measurements of the local poloidal magnetic field that may be used in reconstruction of the plasma current profiles and associated magnetic flux surface geometry. ACKNOWLEDGMENTS
Discussions with M. Bongard of the Pegasus group at the University of Wisconsin are gratefully acknowledged. We also thank John Dawson for his technical assistance with this project. This work is supported by (U.S.) Department of Energy (DOE) Grant No. DE-FG02-00ER54610. 1 F.
M. Levinton, L. Zakharov, S. H. Batha, J. Manickam, and M. C. Zarnstorff, Phys. Rev. Lett. 72, 2895 (1994).
Rev. Sci. Instrum. 85, 093502 (2014) 2 D.
L. Brower, Y. Jiang, W. X. Ding, S. D. Terry, N. E. Lanier, J. K. Anderson, C. B. Forest, and D. Holly, Rev. Sci. Instrum. 72, 1077 (2001). 3 I. H. Hutchinson, Phys. Rev. Lett. 37, 338 (1976). 4 J. Peterson, J. Hanson, G. Hartwell, and S. Knowlton, Phys. Plasmas 17, 032505 (2010). 5 I. Duran, ˇ J. Stöckel, G. Mank, K. H. Finken, G. Fuchs, and G. V. Oost, Rev. Sci. Instrum. 73, 3482 (2002). 6 Y. Liu, D. A. Maurer, G. A. Navratil, and N. Rivera, Rev. Sci. Instrum. 76, 093501 (2005). 7 M. W. Bongard, R. J. Fonck, B. T. Lewicki, and A. J. Redd, Rev. Sci. Instrum. 81, 10E105 (2010). 8 I. H. Hutchinson, Principles of Plasma Diagnostics, 2nd ed. (Cambridge University Press, 2002), p. 440. 9 See http://fwbell.com/products/hall-effect-sensors.aspx for information on the GH-701 sensors. 10 P. D. Coleman, B. Blackwell, M. Kristiansen, and M. Hagler, IEEE Trans. Plasma Sci. 9, 123 (1981). 11 J. R. Cary and J. D. Hanson, Phys. Fluids 29, 2464 (1986). 12 J. D. Hanson and J. R. Cary, Phys. Fluids 27, 767 (1984). 13 J. D. Hanson, S. P. Hirshman, S. F. Knowlton, L. L. Lao, E. A. Lazarus, and J. M. Shields, Nucl. Fusion 49, 075031 (2009). 14 S. Hirshman, W. van RIJ, and P. Merkel, Comput. Phys. Commun. 43, 143 (1986).
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 093502 (2014)
Hall probe measurements of the poloidal magnetic field in Compact Toroidal Hybrid plasmas B. A. Stevenson,1,a) S. F. Knowlton,2 G. J. Hartwell,2,b) J. D. Hanson,2 and D. A. Maurer2 1 2
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Department of Physics, Auburn University, Auburn, Alabama 36849, USA
(Received 30 May 2014; accepted 18 August 2014; published online 5 September 2014) A linear array of 16 Hall effect sensors has been developed to directly measure the poloidal magnetic field inside the boundary of a non-axisymmetric hybrid torsatron/tokamak plasma. The array consists of miniature gallium arsenide Hall sensor elements mounted 8 mm apart on a narrow, rotatable printed circuit board inserted into a re-entrant stainless steel tube sheathed in boron nitride. The sensors are calibrated on the bench and in situ to provide accurate local measurements of the magnetic field to aid in reconstructing the equilibrium plasma current density profiles in fully threedimensional plasmas. Calibrations show that the sensor sensitivities agree with the nominal manufacturers specifications of 1.46 V/T. Poloidal fields measured with the Hall sensor array are found to be within 5% of poloidal fields modeled with a Biot-Savart code. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4894209] I. INTRODUCTION
Reconstruction of the experimental magnetic flux surfaces of confined toroidal plasmas is accomplished using measurements of the local and average magnetic field coupled with numerical equilibrium solvers. Magnetic diagnostics to determine the magnetic field components exterior to the plasma are relatively easy to implement. However, measurements of the field inside the plasma volume should lead to a more accurate reconstruction of the rotational transform profile, or alternatively the profile of the safety factor, which is crucial to understanding of plasma stability in both tokamaks and stellarators. Diagnostics that employ the motional Stark effect or Faraday rotation are often implemented for this purpose in large experiments.1, 2 In colder plasmas with short discharge durations, the spatial variation of the magnetic field can be measured with an array of miniature magnetic loops or Hall probes inserted into the discharge itself. While the presence of the probes may perturb the evolution of the plasma, the information gained is useful in understanding specific processes, e.g., major disruptions in toroidal discharges.3 An array of Hall effect sensors has been constructed for use on the Compact Toroidal Hybrid (CTH) plasma physics experiment.4 The development of internal Hall probe arrays in tokamak plasmas for magnetic equilibrium and fluctuation measurements has previously been reported from experiments on TEXTOR,5 HBT-EP,6 and PEGASUS.7 This paper describes the application of new Hall effect sensors to measure the local poloidal magnetic field in current-carrying plasmas in the CTH torsatron/tokamak to aid in the reconstruction of the experimental plasma flux surfaces. Due to the non-axisymmetric plasma geometry of CTH, reconstructions must be performed with a fully three-dimensional analysis.
a) Research performed while the author was at Auburn University. b) [email protected]
0034-6748/2014/85(9)/093502/6/$30.00
II. DESCRIPTION OF THE CTH HALL PROBE DIAGNOSTIC A. Overview of design
A Hall effect sensor consists of a semiconductor through which a current density J is passed. A sketch of a conceptual Hall probe sensor in an arbitrarily oriented magnetic field is shown in Fig. 1. When the sensor is placed in a magnetic field, the charge carriers (electrons or holes) experience a Lorentz force perpendicular to both the current and the field to produce a Hall voltage.8 The magnetic field-dependent voltage output of a Hall sensor is expressed as VHall = G(Is , B )B⊥ ,
(1)
where B⊥ and B are the components of the magnetic field perpendicular and parallel to the face of the sensor, and G is the sensitivity defined as the ratio of the output voltage to the magnetic field at the sensor. The magnitude of G is linearly dependent on the source current, Is , passed through the sensor, and must generally be determined by calibration. Ideally, the sensor output is nominally insensitive to the component of the magnetic field parallel to the face of the sensor (B ) illustrated in Fig. 1. However, the possibility of a weak dependence of the sensitivity upon the value of B is allowed for in this expression. Unlike a magnetic pickup coil, the Hall sensor directly measures the time-dependent magnetic field without integrators. Also, Hall sensors can be as small as a few millimeters in size providing highly localized measurements of the magnetic field. The small size facilitates their inclusion into a probe that can be inserted into the plasma. Because the Hall effect sensors are not known to be ultra-high vacuum compatible, the sensors are mounted onto a narrow printed circuit board (PCB) and inserted into a small re-entrant stainless steel tube sealed at one end such that the inserted probe array is at atmospheric pressure. From the point of view of diagnosing the poloidal magnetic field, several non-ideal effects on the output voltage of
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FIG. 1. Conceptual diagram of Hall effect sensor in obliquely oriented magnetic field. Ideally, the output voltage, VHall , is only dependent on the component of B perpendicular to the surface of the sensor.
FIG. 3. The location of the Hall probe array is shown in relation to the CTH vacuum vessel and an example set of magnetic flux surfaces. The probe array can be moved a total of 0.3 m in the radial direction.
Hall sensors must be considered. These include the dependence of the sensitivity on temperature, and the magnitude of the planar Hall effect, which is the dependence of the output Hall voltage on the magnitude of magnetic field components parallel to the face of the sensor. Because the poloidal field component in stellarator and tokamak plasmas is typically an order of magnitude lower than the main toroidal field (TF), the planar Hall effect could clearly be important, and has been documented in previous work.6, 7
sensors are fed into differential amplifiers with a gain of 20, filtered, and then passed to the data acquisition system. The probe array is placed inside a re-entrant stainless steel tube with 8.0 mm inner diameter and 0.9 mm thick walls. The PCB of the array is mounted to a long aluminum strip that allows the array to be pivoted about its axis within the re-entrant tube. The aluminum strip is mounted to a rotary positioner with a scale to align the angular orientation of the array for maximum sensitivity to the poloidal component of the magnetic field. The Hall probe array within the CTH vacuum vessel is shown in Fig. 3. The figure illustrates a cross-section of the toroidal plasma at the location of the Hall probe array. The array is shown in two positions that represent the limits of the radial positioning of the array. Also depicted in the figure are nominal equilibrium magnetic flux surfaces of the CTH plasma. The magnetic axis of the plasma at the center of the flux surfaces is located at a major radius of R = 0.75 m. In practice, the shape and position of the flux surfaces depend on the currents in the external coils and the current carried by the plasma. The poloidal field of the CTH plasma refers to the magnetic field components tangent to the flux surfaces within the plane of this figure. The larger toroidal field component of the CTH equilibrium field is oriented normal to the plane of the figure. The probe array is rotated along its axis to maximize the sensitivity to the local poloidal field, which is nearly vertical at the position of the probe. The toroidal field of the CTH device thus represents a parallel field component to the sensor, in the context of Fig. 1. The outer end of the
B. Implementation
The Hall array built for CTH consists of sixteen GH-701 Gallium Arsenide (GaAs) sensors manufactured by Sypris Test & Measurement.9 The GH-701 sensor has an active area of 1.5 mm × 1.5 mm. The sensors are mounted onto a narrow (PCB) with an 8 mm separation between adjacent sensors, as shown in Fig. 2. When mounted, the sensor surface is nominally parallel to the surface of the PCB. To ensure that the sensitivity of each sensor relative to one another remains fixed, a bias current of 8.18 mA is passed in series through each of the 16 sensors via conductors on the PCB to provide a nominal sensitivity of 1.46 V/T. The bias current is supplied by a TL 783 voltage regulator configured as a constant current source, with a typical regulation of 0.15%. The Hall voltage from each sensor is transmitted by a twisted pair of 36 gauge wires in a 65-conductor Litz wire bundle, chosen to reduce unwanted pick-up and because the bundle was small enough to fit into the probe housing. The output voltages of the Hall
FIG. 2. Array of 16 Hall sensors with twisted pair leads of Litz wire bundle.
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re-entrant tube is connected to a motorized linear slide and welded bellows, allowing the probe assembly to be moved radially a distance of 0.3 m within the vacuum chamber. The linear position of the probe is controlled with a Vexta Model PK266 stepping motor and a Velmex VXM controller with a specified precision of 0.01 mm per step. The position and elevation angle of the linear slide was adjusted with the use of a coordinate-measuring machine (CMM) so as to orient the axis of the probe array to lie along the mid-plane of the toroidal plasma. Once the slide was placed, CMM measurements were made for a series of probe positions to determine the final orientation of the slide. A linear fit was applied to these measurements to determine the precise position of the probe. These positions were later used as inputs for magnetic field calculations that were compared to the measured magnetic fields. A boron nitride (BN) tube surrounds the stainless steel tube housing the probe array to prevent the stainless tube from providing a conducting path across the plasma flux surfaces. The BN tube has an outer diameter of 1.27 cm and is closed on the end inside the plasma. Boron nitride was chosen on the basis of previous results10 that demonstrated excellent thermal and electrical resistance while producing lower plasma density changes than with other probe-shielding materials tested. To prevent metal deposition on the BN shield during glowdischarge cleaning and titanium gettering, the probe assembly can be retracted into a 2.54 cm diameter stainless steel tube. A simple shutter at the end of the tube closes the shielding tube when the probe assembly is fully retracted. C. Calibration
The calibration of the magnetic field response of the Hall sensors is performed in two stages. The sensors are initially calibrated at relatively low magnetic fields to obtain the sensitivity G of each detector, and then the calibrations are checked in situ on the CTH device. While previous work using Indium Antimonide (InSb) Hall sensors indicated that the probes sensitivity to fields perpendicular to the surface varies when a strong planar or parallel field is present,6, 7 this variation of sensitivity due to parallel fields has not been fully specified for Hall sensors fabricated with GaAs. The initial calibration magnetic field is produced by a set of Helmholtz coils. To perform this test, a calibration setup was prepared with two orthogonal sets of Helmholtz coils to provide magnetic fields perpendicular and parallel to the Hall probe surface, as shown in Fig. 4. Both Helmholtz coils in this assembly were calibrated using a Bell Teslameter. The parallel field corresponds to the toroidal field in CTH, while the perpendicular field corresponds to the poloidal field, the component of interest to reconstruction. With this method, the Hall sensors were found to have an average sensitivity of 1.49 V/T. The maximum magnetic field that could be achieved using the Helmholtz coils was 0.035 T. At these field magnitudes, no detectable change in the Hall voltage was recorded when a parallel field component was applied, regardless of the orientation of the parallel field. As this magnetic field strength is less than one tenth the typical toroidal field strength of
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FIG. 4. Schematic diagram of double Helmholtz coil calibration stand.
0.5 T in CTH, further calibrations of the sensors were performed in situ on the CTH device itself. In this case, the steady-state fields of the CTH configuration were calculated from the measured coil currents using the Biot-Savart solving capability of the IFT code.11, 12 The results of an in situ calibration for one detector are shown in Fig. 5. Calibrations are performed for three values of the toroidal field. The measured sensitivity remains in the range 1.45 V/T ± 1%, within the manufacturer’s specifications. The linear sensitivity is found to vary with the magnitude of the field component parallel to the plane of the Hall sensor by 1% over the range of fields in this experiment. Data for other probes show similar trends but with different levels of sensitivity to the magnetic field parallel to the probe. This suggests that the effect is most likely due to misalignments of the probe and not an inherent property of the probe itself. The low sensitivity of the probes to parallel fields allow the probes to be used to measure the poloidal component of the magnetic field despite the presence of a stronger toroidal field. The independent parasitic responses of the sensor to the two possible orthogonal components of the parallel field was not measured in this experiment. Subsequently in this paper, we refer to the poloidal field as the component of the field measured by the sensor array, and the toroidal field as the ambient parallel field.
FIG. 5. Measured Hall sensor output versus applied poloidal (perpendicular to the sensors) magnetic field strength when fields parallel to the Hall sensors are applied with the TF coil set on CTH. Lines represent linear fits to the data, and the legend indicates the fitted linear sensitivity in V/T and the offset voltage.
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The sensitivity of Hall effect sensors is also known to be susceptible to changes in temperature. The sensitivity of InSb Hall sensors were reported to shift by up to 2%/◦ C.6, 7 The GaAs Hall sensors of the CTH array are expected to be less sensitive to temperature changes, with a stated temperature sensitivity of 0.06%/◦ C. Nonetheless, the temperature of the probe array is monitored with two thermistors mounted onto the back side of the PCB. Furthermore, the Hall sensors are also actively cooled by compressed air flowing over the sensors during the operation of CTH. The ability to directly measure the probe temperature makes it possible in principle to adjust the calibrated probe sensitivities to account for temperature changes. Currently, temperature corrections have not been found to be necessary due to the small temperature sensitivity of the GH-701 sensors as well the use of active cooling. III. INITIAL RESULTS
Equilibrium reconstruction on CTH makes use of the V3FIT13 code that minimizes differences between experimentally measured signals and modeled diagnostic signals that depend on the parameterizations of the quantities of interest. The Hall sensor array is designed to measure the poloidal component of the magnetic field along a radial line segment in the mid-plane to provide V3FIT with signals that directly relate to the radial profile of the plasma current density. Unlike in tokamaks, the poloidal field component of the torsatron configuration of CTH, produced by the current in the external magnetic coils alone, typically exceeds the magnitude of the poloidal component resulting from the plasma current. To separate the two effects, the effective mutual inductance, or response functions, of the Hall sensors to the external current sources was determined experimentally. Each independent coil set that produces the magnetic field of CTH was driven with a constant current while the other coil sets were open-circuited to prevent induced current flow in the latter. The energizing current was maintained for time periods longer than required for eddy currents in the vacuum vessel and other nearby conductive supports to decay. The Hall sensor response functions, defined as the ratio of the magnetic field measured by each Hall sensor to the measured coil current, were measured and tabulated for each sensor and independent coil set. By design, the normal direction of the sensor surface should be oriented in the local poloidal direction, as indicated in Fig. 3. Thus aligned, the sensors should exhibit no response to the toroidal magnetic field. In reality, each of the individual sensors may have small alignment errors. The response functions of the sensors to the TF coils of CTH were measured and are shown in Fig. 6. The TF coil set of CTH consists of ten planar coils that produce a magnetic field with only a toroidal component at the location of the probe array. Indeed, the majority of sensors show only a small response in comparison with that to the HF coil set. However, four sensors show responses with magnitudes greater than 5 × 10−7 T/A. Shown in Fig. 7(a) are the response functions of the sensors to the helical field (HF) coil set which includes a helical
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FIG. 6. Hall sensor response functions of the 16 elements for an applied toroidal field, nominally parallel to the Hall sensors.
coil electrically in series with vertical field coil and produces both toroidal and poloidal field components. Positioning the array at four different radial positions and determining the response of all 16 sensors generates the data points in this figure. The measured response functions are fit using a quadratic polynomial shown by the black curve in the figure to compare with the Hall probe response function data. The poloidal magnetic field at the Hall sensor locations can also be modeled using a Biot-Savart type code. The dashed curve in Fig. 7(a) shows the modeled radial distribution of the poloidal magnetic field along the outer mid-plane due to a unit current in
FIG. 7. (a) Hall sensor response to the HF current versus major radial position of the Hall sensors. The major radius of the CTH vacuum vessel is 0.75 m. Data for four separate positions of the array are included. The black curve is a polynomial fit to the data and the dotted-dashed curve is the modeled response. (b) The corrected sensor response using the misalignment angles.
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the HF coil. The modeled curve shows generally good agreement to the Hall probe fit shown with the solid curve. The Hall probe response to the HF current exhibits modest scatter about the fitted polynomial curve. Investigation of this scatter shows that the deviations from the fit depend more on the specific probe than on the position of the array as a whole. The probes with the largest deviations are the same probes with large deviations when measuring the toroidal field as shown in Fig. 6. This leads to the conclusion that the probes are misaligned, picking up some toroidal field, and that the measured signal should be expressed as S = Bpol cos θ + Btor sin θ,
(2)
where θ is the angle between the normal of the Hall probe to the nominal. The angle of each probe can be found from the data used to generate the plot in Fig. 6, taken in the presence of a pure toroidal field. The probe angles can then be used to correct responses to the HF current by subtracting out the contributions due to the toroidal field. The corrected responses, shown in Fig. 7(b), show less deviation from the fit as compared to that shown in Fig. 7(a). The fit and model lines are the same in Figs. 7(a) and 7(b). The Hall probe array has been used to measure the poloidal field in both current-free and current-driven plasma discharges, and also in plasma-free scenarios in which all field coils, including the ohmic heating (OH) transformer coil that produces a time-varying current, were energized. The poloidal field measured by the Hall probe array in a fullfield, no-plasma shot is shown in Fig. 8(a), where the field values have again been corrected for misalignments of the sensor. The calculated magnetic axis of the discharge is located at a major radius R ∼ 0.78 m. The Hall probe array was positioned with the inner-most sensor located at a minor radius of 0.8778 m so as to maximize the number of sensors within the last closed flux surface. The OH transformer was pulsed to generate eddy currents in the vacuum vessel and coil support frames similar to those expected in a plasma discharge even though there was no plasma current induced. The vacuum magnetic equilibrium was calculated with the V3FIT reconstruction code. There is a small offset (≤5%) in the measured poloidal field strengths from the modeled values. By using the modeled and empirically determined set of response functions, the magnetic field contribution from the external coils at each Hall sensor location was subtracted leaving the residual signals shown in Fig. 8(b). These resid-
FIG. 9. Poloidal magnetic as measured by the Hall probe array (diamonds) and modeled with the VMEC (filled squares) and V3FIT code (squares). The location of the plasma edge is estimated from reconstruction to be at R = 0.98 m.
ual signals result from time-dependent eddy currents, and are small in comparison to the full magnitude of the measured poloidal field. The contribution from the external poloidal field can thus be effectively removed from the Hall sensor measurements. After determining the contribution from the external coil currents to the modeled poloidal field, one may then subtract this portion from the total poloidal field to obtain the poloidal field produced by the currents within the plasma during CTH discharges. While the insertion of the probe array well into current-driven discharges may lead to an uncontrolled increase in density during the plasma discharge and sometimes a disruption, measurements with the Hall probe array in such discharges can provide the profile of the poloidal field produced by the plasma current. Fig. 9 shows a comparison between the poloidal field measured with the Hall sensor array and modeled with the VMEC14 and V3FIT codes. In this discharge, the plasma current was Ip = 40 kA, increasing the edge rotational transform from -ι vac (a) = 0.125 to ι-total (a) = 0.45. The VMEC code was used to model the magnetic field at positions inside the last closed flux surface (LCFS) (filled squares), while synthetic V3FIT magnetic diagnostics were used to model the magnetic field outside of this region (unfilled squares). The use of separate codes to model the magnetic fields inside and outside of the LCFS is necessary because VMEC will only give results up to the LCFS. As of this writing, V3FIT can handle synthetic diagnostics inside the LCFS, but singularities are possible if the diagnostic is placed too near VMEC grid positions, whereas outside the LCFS, V3FIT works smoothly. Thus, these data cannot be used as an input to the V3FIT code. We note that the trend between the data and the model is similar and that the measured and modeled fields are within 5% of one another. We do not think that this discrepancy is due to a plasma current measurement error. It is more likely that the radial plasma position is in error because of assumptions made for unmeasured quantities in the reconstruction. IV. SUMMARY
FIG. 8. (a) Poloidal magnetic field measured with the Hall probe array (diamonds) and calculated by equilibrium reconstruction (squares). The measured values have been corrected for misalignments of the sensors. (b) Residual poloidal field after externally generated field is subtracted for Hall probe data (diamonds) and reconstructed equilibrium fields (squares).
A new type of Hall probe array for use inside toroidal plasmas has been constructed with the use of GaAs sensors. These transducers are found to be less sensitive to parallel field components and temperature variation that sensors
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of other material used for the same application. The Hall probe array has been tested on the CTH torsatron to measure both vacuum and plasma-generated magnetic fields. While the array can perturb the evolution of the plasma discharge if inserted too deeply, it provides measurements of the local poloidal magnetic field that may be used in reconstruction of the plasma current profiles and associated magnetic flux surface geometry. ACKNOWLEDGMENTS
Discussions with M. Bongard of the Pegasus group at the University of Wisconsin are gratefully acknowledged. We also thank John Dawson for his technical assistance with this project. This work is supported by (U.S.) Department of Energy (DOE) Grant No. DE-FG02-00ER54610. 1 F.
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