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Eddy current experiments with closely placed solid boxes simulating a next step fusion device
Fusion Engineering and Design 60 (2002) 179– 190 www.elsevier.com/locate/fusengdes
Eddy current experiments with closely placed solid boxes simulating a next step fusion device Mitsushi Abe a,*, Takehiro Ooura a, Akira Doi a, Masataka Nakahira b, Satoshi Nishio b a
Power and Industrial Systems R&D Laboratory, Hitachi Ltd., 7 -2 -1, Omika-cho, Hitachi-shi, Ibaraki 319 -1221, Japan b Naka Fusion Research Establishment, Japan Atomic Energy Research Institute, 801 -1, Mukoyama, Naka-machi, Ibaraki 311 -0193, Japan Received in revised form 25 January 2002; accepted 15 February 2002
Abstract Eddy current experiments with closely placed solid boxes driven by a coil alternative current were carried out. The experiment was to designed to identify approaches to the problems about: (1) computing accuracy of the eddy current calculation with thin plate approximation; and (2) modeling baselines for a computational study of eddy currents in a super conducting (SC) fusion device. The following conclusions were drawn from experimental data and their comparison with eddy current analyses: (1) the skin effect is a major cause of computational error; (2) if the error due to the skin effect cannot be ignored, the double shell model calculates accurate eddy currents; (3) the mesh size with a narrow gap should be small enough (for example smaller than five times the gap width) so as not to cause additional error. The Finite Element Method (FEM) model for the eddy current computation is well modeled using these conclusions. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Eddy current; Super conducting fusion device tokamak; Experiment; FEM
1. Introduction Two possible options for the next step in fusion research are realization of a high performance plasma with steady state operation by using a super conducting (SC) fusion device, and realization of a burning plasm. Design activities are already being carried out on SC fusion devices, such as ITER [1] and others [2 – 5]. * Corresponding author. Tel.: +81-294-53-3111x5212; fax: +81-294-52-9287. E-mail address: mitsushi – [email protected] (M. Abe).
Several SC fusion devices are already in operation [6–9]. They feature more complicated electrically conducting structures than existing normal conducting fusion devices. Their conducting structures include a cryostat, heat shields, toroidal field coils (TFCs) or helical coil structures, poloidal field coil (PFC) structures, radiation shields, a double wall vacuum vessel (VV), in-vessel structures and coil conductors. Eddy currents can flow on such conducting structures, causing problems in relation to joule heats on cryogenic structures, an undesired magnetic field in the plasma discharge area, electromagnetic forces, and so on.
0920-3796/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 0 - 3 7 9 6 ( 0 2 ) 0 0 0 5 2 - 2
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Eddy current codes were developed in order to discuss the electromagnetic force on the VV with a thin shell assumption. The eddy current on the VV has been discussed, not only experimentally, but also computationally for normal conducting tokamaks [10–15]. The computations were done with the thin plate approximation [10– 13] and the electromagnetic force on the VV due to eddy current was well taken into account in the VV design of fusion devices. The validity of the analysis was confirmed by a rather simple torus model [13]. For SC tokamaks, some issues related to eddy currents have been reported from TRIAM-1M [16–18], mainly from experimental investigations. These issues were related to the magnetic field on the plasma area and joule heat load on cryogenic temperature (4 K) structures due to eddy currents. These reports suggested that a precise study of eddy currents is necessary for designing SC tokamaks. Design studies for eddy currents in SC fusion devices [1–5,19,20] face problems, since the devices have complicated and thick structures. The next step SC fusion devices also have complicated structures and require consideration of the cryostat, heat shields, TFC structures, radiation shields and double wall VV. In order to understand the eddy current characteristics, it is necessary to make an FEM (Finite Element Method) model which considers them all. Due to this complication, a solid FEM model is not applicable, so that researches have usually turned again to the thin shell model for the design studies of eddy currents [21–23]. Another problem related to the skin depth of the eddy currents. The structural material can be thicker than the skin depth and the simple thin shell approximation may cause computational error. These situations make the number of the FEM elements large and the thin approximation wrong, degrading the computational accuracy. In order to obtain guidelines to keep computational accuracy in eddy current studies, we carried out experiments using thick conducting boxes on which eddy currents are driven by a coil with alternating current. This paper describes the results of the experiments and modeling guidelines
we identified for the SC fusion device eddy current analysis.
2. FEM modeling problems Existing SC fusion devices, such as the tokamaks TRIAM-1M [7], TORE SUPRA [8] and HT-7 [9], and the helical device LHD [6], have more complicated structures than normal conducting fusion devices. Fig. 1 shows a schematic drawing of a next step SC fusion device like ITER [21]. It has a cryostat and heat shields, which are not found in existing normal conducting tokamaks, and solid coil support structures to withstand strong magnetic fields. Moreover, since burning plasma studies are to be done, it has radiation shields. The characteristics of the SC fusion device structures are considered as:
Fig. 1. Poloidal cross section of a typical next step SC fusion device. The structures are more complicated than an existing normal conducting device.
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1. solid structures to withstand strong magnetic fields; 2. closely set structures, for example, a double wall VV, radiation and heat shields; and 3. a large number of structures for which eddy currents cannot be ignored. Thin shell FEM computer codes like ECTAS [21] are usually used for the eddy current study of SC fusion devices. Since the computed results depend on the validity of the thin shell approximation and FEM modeling, we have to discuss the computational errors related to the three stated characteristics of SC fusion device structures, i.e. computational errors due to thickness related to (1) and complexity of the structures related to (2) and (3). Origins of the errors can be considered as follows. With regard to (1), thicknesses of the structures can be up to a few tens of centimeters, for example, the cross section of TFC cases. This thick part creates a problem related to the skin depth. The IP of 10 MA order can decay in 10 ms, making it the fastest phenomenon to be considered in a fusion device. This decay can be considered as a quarter cycle of a 25 Hz wave, but the waveform is not a simple trigonometric function. Then, it is necessary to consider a higher frequency than 25 Hz. We consider 50 Hz, at least, as the frequency to which the eddy currents should be computed accurately. The skin depth for 50 Hz with stainless steel (SS) resistivity of 7× 10 − 7 Vm is 64 mm. Then, this leads to problems of shell element position and also effective thickness for resistance, related to the validity of the thin shell approximation. These problems degrade the thin shell FEM computational accuracy. In relation to (2) and (3), problems about FEM modeling occur. The complicated structures (TFC structures, VV, cryostat and so on) have large sizes of roughly 10 m, but there are also fine structures, such as thin insulation layers or a narrow gap between conducting walls. This presence of both large and fine structures increases the number of meshes in the FEM model and sets a limit for computing ability for a large scale fusion device. A small mesh size model needs a large computer memory size, which makes the computation impossible, while a large mesh size model increases computational error.
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Then, we decided to carry out eddy current experiments using a simple setup, in which the features of thick structures and narrow gapped structures were included.
3. Experimental setup The setup for the eddy current experiments (Fig. 2) included SS boxes with rather thick walls and a solenoid coil. The component sizes were as given in the figure and Table 1. The SS boxes were placed side by side so that eddy currents on them coupled with each other electromagnetically. This placement represents the thick structures and narrow gapped structures in SC fusion devices. The primary box current was inductively activated by the solenoid coil current i.e. induced by the flux in the solenoid coil, while the secondary box current was induced by the return magnetic field flux of the coil and reduced by the flux from the primary box current. The solenoid coil was connected to an IGBT (Insulated Gate Bipolar Transistor) switching power supply, which supplied an alternating voltage of 9 500 V with very short switching time of about 1 ms. Then the applied voltage had a well shaped rectangular waveform. An additional feature was that the alternating frequency could be changed between 50 Hz, 500 Hz and 4.9 kHz frequencies. This made it possible to discuss the problem related to skin depth. The gap of the two boxes was changed from 1.7 to 32.1 mm. This made it possible to discuss the computational error due to close placement of the structures and mesh sizes. The Rogowski coils were placed to measure eddy current distribution as shown in Fig. 2. They measured not only total current of the boxes, but also the eddy current distribution. There were three vertically directed holes and four horizontally directed holes to measure the current distribution in the thickness direction and height direction, respectively. The holes had a 3 mm diameter, while their depths were 30 and 260 mm, respectively. They were placed separately, so we expected little effect on the current density distribution.
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Fig. 2. Setup of eddy current experiment with two SS boxes and one coil.
The signals were filtered (5 ms) in order to reject the high frequency component related to IGBT switching characteristics. The signals for the 50 and 500 Hz frequencies were digitized with 50 and 100 k samples per second, respectively, and then integrated digitally. For the 4.9 kHz frequency, the signals were integrated directly by analogue circuits, in order to avoid possible errors during digital integrations of high frequency components and they were sampled at 500 kHz.
4. Eddy current experiments and FEM modeling guidelines The experimental data were compared with computational results in order to get FEM model-
ing guidelines. However, the objectives of eddy current study in a SC fusion device are to understand: 1. magnetic fields due to eddy currents in plasma discharge area for equilibrium control system design; 2. electromagnetic forces due to eddy currents for structural design; and 3. Joule heat production on the cryogenic temperature structures. Then, precise distributions of the eddy currents in structures are not a main concern. We first compared the eddy total box currents with calculated results by a simple single shell model. Following this, the current density distributions and eddy current in narrow gap case were examined to confirm modeling guidelines regarding the skin effect and complicated structures.
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Fig. 3. Typical experimental results of an eddy current experiment (a) and a computational single shell FEM model (b), with surface gap distance of 1.7 mm. The calculated eddy currents were obtained using ECTAS with the experimental coil current and locating the shell mesh at the center of the SS box wall thickness. The 510 turn coil was modeled filament currents at each windings (5 ×102 windings).
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4.1. Experiment for error e6aluation of the thin shell approximation Fig. 3(a) shows a set of typical experimental waveforms with calculated eddy currents, which were calculated by a model shown in Fig. 3(b) and measured coil current as input data. The FEM model is a single layer shell model. The elements are at the center in the SS box wall thickness direction. The box resistances were calculated with the real thickness of 30 mm. A alternative coil current of 285 A (14.5 kA.T) was applied. The coil is modeled as 510 filament loop current placed at each winding. The mesh size is roughly same size as the box wall thickness. The shell model assumes no skin effect and uniform current distribution in thickness direction. Then, we believe that typical length to describe the distribution is longer than the wall thickness and the mesh size much smaller than the thickness is not adequate. This mesh size produces enough current distribution as far as skin effect is ignorable. The experiment was done with surface gap of 1.7 mm, then 31.7 mm gap in FEM model. The calculated currents were a little smaller than the experimental measurements. The difference was considered as mainly due to the current distributions in the wall thickness direction. The skin depth for SS using the 50 Hz frequency was calculated as 64 mm which was twice as deep as the experimental box 30 mm wall thickness. HowTable 1 Equipment specifications for eddy current experiment Parts
Specifications
Power supply Voltage waveform Voltage Frequency
9450 V 50, 500, 4.9 kHz
Coil Coil winding Current
510 turns, 14 mH, 0.35 V Triangle shape
Boxes Material Shape
Square
30 mm thick stainless steel (SS) 240 mm long, 360 mm square cross section pipe (Fig. 2)
Fig. 4. Dependence of the calculated eddy current magnitude on the FEM shell model position. The vertical axis value of 1.0 means an accurate computation.
ever, the skin effect could not be ignored because the coil current waveform was triangular shaped and the high frequency components were contained in the waveform. The calculated eddy current on the primary box had an amplitude which agreed well with the experimental one, however during the steep time derivative phase of the eddy current, the experimental current was larger than the calculated one. The experimental eddy current on the secondary box was larger than the calculated values. These computational errors could be explained by taking into account the skin effect. As a general characteristic of eddy currents, they tend to flow where the inductances are small and small inductances make currents larger. However, the computational model had eddy currents at the centers of the wall thickness and did not include the skin effect. This FEM model had larger inductances than the real eddy currents when the time derivative of the currents was steep. Then, the computation tended to compute smaller eddy currents. Other than the skin effect, a very narrow gap between the boxes or thick SS walls can be a cause of a computational error, as pointed out in Section 1. We considered the guidelines for mesh size and mesh position in the thickness direction. Fig. 4 plots the ratio of the calculated primary box current amplitude normalized by the experi-
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mental one, against shell element position d in the thickness direction. The position parameter d was measured from the center of the thickness directed outward as shown in the Fig. 4. The ratio 1.0 means that the computed results well describe experiments. The ratio was roughly 1.0 for a model with centered shell position (0 mm) at 50 Hz. At 4.9 kHz, the model with the inside surface shell (d = − 15 mm) well described the experimental result and the center shell model calculated an 18% smaller current. Two lines are plotted as proportional to 1/(2d +L) and 1/(2d +L)2, where L is a typical size of the structure, and in this experiment L=330 mm. As a general characteristic, the current is inversely proportional to inductance for the high frequency case and to resistance for the low frequency one. Inductance is roughly proportional to the area enclosed by the current path, then to (2d +L)2 and the amplitude Ieddy is expected to be, Ieddy 81/(2d + L)2,
(1)
for the high frequency (4.9 kHz) case. Resistance is proportional to the length of the current path, then to (2d +L) and Ieddy is, Ieddy 81/(2d + L),
(2)
for the low frequency (50 Hz) case. The current was expected to flow with a uniform current density (d =0 mm) without the skin effect at 50 Hz and the calculation with d = 0 mm gave a good result. However, the eddy current flowed along the inside surface due to the skin effect at 4.9 kHz, then d = − 15 mm gave a good result and the calculation with d =0 gave 18% smaller eddy current. The discussion of Fig. 4 led us to the idea that the error due to the skin effect is due to the error from the eddy current path assumption in the computational model. Usually, we produce the FEM model with centered shell elements (d = 0 mm) and for the low frequency case, the eddy current calculations are expected to give good results. For the high frequency case, the calculation error occurs through inductance of the eddy current as discussed in Fig. 4 and Eq. (1). The largest error is expected at the surface current, i.e.
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2d is equal to the wall thickness, and the computational error related to the skin depth can be as small as, ErrorB 1− 1/(2d/L+ 1)2 : 2t/L,
(3)
where the wall thickness t is usually much less than L. We considered Eq. (3) as a computational error magnitude and in the case that the error of Eq. (3) cannot be ignored, double shell model can be applicable as discussed in Section 4.2.
4.2. Experiment for skin effect Section 4.1 showed the skin effect is a major cause of the computational error. However, there is a technique to avoid the large computational error. The technique is discussed here. The eddy currents distribute in the direction of both thickness and height due to the skin effect. This skin effect is a common characteristic for the two boxes. Then, we used the primary box to discuss the current distributions, because the primary box had a larger current and better measurement accuracy than the second box. The skin effect made it easy for eddy current to flow at the edge area in the height direction and the inner side (coil facing) surface in the thickness direction. The skin effect makes the current distribution change as a function of the frequency and the single shell model cannot represent the various distributions. One idea was multi shell placement to reconstruct the various current distributions. However, placement of a many layer shell made the number of FEM elements large and caused a computational difficulty. Then, we tried a multi shell model. The eddy current distributed continuously in the depth direction and the multi shell model could not reconstruct the distribution precisely. We expected the double shell model improved the computational accuracy of the total current and the magnetic field, because the change of the current center due to the skin effect could be represented between the two shells. Fig. 5 shows total currents in the primary box at 4.9 kHz. Two computational models were applied. They are shown at the top of the figure. One was a model with single shell and the other was a model with a multi shell. The latter model
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had two layers in the thickness direction, and it had additional shells at the top and bottom. On the additional shells, eddy currents could flow due to the skin effect and the shells had 6.3 mm thickness resistance, roughly equal to skin depth. Other than top and bottom shells, the shells had a resistance of 15 mm thickness. The total current computed by the multi shell model correctly agreed with the experiment in not only the amplitudes as shown in Fig. 4, but also in waveforms, while the single shell model calculated smaller eddy current than the experimental value as discussed in Section 4.1. The current waveform was a triangle form with 200 A amplitude and the averaged current density was 2.6× 104 A/m2. The triangle waveform suggested that the current was inductively driven and there was a strong skin effect. Fig. 6 plots current density in height (Fig. 6(a)) and in thickness (Fig. 6(b)) directions, comparing the experiment with the multi shell calculation. The current density distribution in the height
Fig. 6. Current density distributions in height (a) and thickness (b) directions. The computed densities are plotted every mesh point and the measured data are plotted at every measured point.
Fig. 5. Total current waveforms of 4.9 kHz case. Experimental and computational data with double shell model are plotted.
direction was measured at four intervals (every 32.5 mm) above the mid plane. The measured and calculated data were plotted as density (A/m2). The measured density at the edge was 4.4× 104 A/m2 and other data were around 2×104 A/m2. This situation was consistent with the small skin depth of 6.4 mm at 4.9 kHz. Other than the edge region, the density was around 2× 104 A/m2 for both calculation and experiment. At the edge, using the two data from 0 to 32.5 mm, the calculated currents were averaged as 5.2×104 A/m2, which was a little larger than the experiment. We considered this was due to the placement of FEM meshes at the top surface, as discussed for Fig. 4. However, this error creates
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only 4.5% of the total current and it can be improved easily by replacing the FEM position at a suitable position as discussed in Fig. 4. Even removing the top and bottom shells, the computational error should be small because the remaining shells can reconstructed in the current distribution in the height direction. This point is discussed with Fig. 7. Fig. 6(b) plots current densities in the thickness direction. The densities were averaged in height direction. The calculation was on two points in the thickness direction because of the multi shell model, while experimental data were for every 1/4 part of the thickness. Taking into account these averaging procedures, we thought the computation well reconstructed the experimental results within the error discussed for Fig. 6(a). The discussions on Figs. 5 and 6 can be summarized as eddy current characteristics can be well reconstructed by the multi shell model even when the skin effect can not be ignored. Fig. 7 shows the dependencies of the amplitude ratio on frequencies of 50, 500 and 4.9 kHz. Two FEM models were tested. One was a double shell model, a model like the multi shell model in Fig. 5 except that the top and bottom shells were removed. The double shell model had two shells at inner and outer surfaces in this computation. The other FEM model was a single shell model.
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At 50 Hz, computations with both models could well describe the experimental results. However, at high frequency the double shell model gave more accurate results than the single shell model. At 4.9 kHz and 500 Hz, the skin depths were 6.4 and 20 mm, respectively. They were thinner than the box wall thickness. Then, we concluded that the double shell model well described the skin effect. For structures thicker than the skin depth, the double shell model calculated eddy currents well for a wide range of frequencies. The shell position of the double shell model was at the surface, however the discussion of Fig. 4 showed the surface shell model produced smaller inductance and resistance than the actual values. This was why the double shell model tended to give a little larger eddy current amplitude than the real one (amplitude ratio \ 1.0) and from the experimental results, it was apparent that the shell position should not be on the surface. The shell model computational analysis is mainly for low frequency condition. Then, the shell should be at center of the wall in thickness direction as discussed in Fig. 4 for single shell model. In case that double shell model the shells should be at centers of the each half thickness of the wall that means shells at 1/4 and 3/4 of the thickness direction. The error due to the shell position could be evaluated using Eq. (3), but the thickness should be half of the real thickness. Then, the error was evaluated as roughly half of the single shell model. However, the accuracy was kept in a wide range of the frequencies.
4.3. Experiment for closely placed conducting walls
Fig. 7. Dependence of the calculated eddy current amplitude on the frequency. The amplitude is normalized by the experimental amplitude and the vertical axis value of 1.0 means an accurate computation. Two FEM models were tested.
Fig. 8 compares the calculated and experimental secondary box currents, as a function of the gap between the box walls. As could be recognized from Fig. 3 waveforms, the secondary box current had more high frequency components than the primary box current. This allowed discussion of the accuracy for the eddy currents on walls, which were very close to each other from Fig. 8. The experiment was done with gaps of 1.7, 6.8 and 32.1 mm at 50 and 500 Hz. The single shell model gave smaller amplitudes than experimental data, while the double shell model gave larger
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results. Since the 6.8 mm case did not have the numerical error, we decided to do the eddy current computations with FEM element sizes smaller than five times the shell gap.
4.4. Guidelines for thin shell FEM modeling The single shell model well calculated the total eddy current when the skin depth was larger than the wall thickness. Then we can say: 1. The single shell model is applicable for small skin depth cases. The multi shell model well calculated not only the total box current, but also the current center for a wide range of the frequencies. Then, we can say: 2. Multi shell model is applicable for thick structure cases. The computer code well calculated box current when the gap between two shell was larger than 1/5 of the mesh size. Then, we can say: 3. The mesh size should be smaller than five times the gap. Guideline (3) is for our computer code ECTAS only. This accuracy should be examined again using other codes. These are guidelines for thin shell FEM modeling of SC fusion devices. Using these guidelines, we can calculate eddy currents within the following error: ErrorB 2t%/L, Fig. 8. Dependence of the calculated eddy current magnitude on the gap between two boxes. Two FEM models were tested.
amplitudes. However, the computational error was smaller with the double shell model than with single shell model for 6.8 and 32.1 mm gap cases. An important point was that the error became large at a small gap of 1.7 mm for the double shell model. The mesh size was 30 mm for these computations. The error was considered to be due to a numerical error during the calculation in the code. This meant that the eddy currents should not be calculated for walls with a 1.7 mm shell gap using the 30 mm size mesh. The single shell model which had shell elements at the center of the thickness had a 31.7 mm shell gap in the FEM model for the 1.7 mm gap case and had the same level error as large gap
(4)
where t% is a thickness represented by a shell and L is a typical size of the structure.
5. Summary Eddy current experiments were carried out in order to obtain information about: (1) accuracy of eddy current computation with thin plate approximation; (2) modeling baselines for computational study of eddy currents in a SC fusion device, in which structures are solid, closely set and a large number of structures cannot be ignored. Experimental data and their comparison with eddy current analyses led to the following three results:
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1. Skin effect is a major cause of computational error, the magnitude of which is roughly described by Eq. (3), in the single shell model computation. 2. When the error due to the skin effect cannot be ignored, the double shell model calculates accurate eddy currents. 3. The mesh size with a narrow gap should be smaller than five times the gap width using ECTAS code. Based on these guidelines, FEM models for the computational eddy current studies of a SC fusion device will be well constructed.
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Eddy current experiments with closely placed solid boxes simulating a next step fusion device Mitsushi Abe a,*, Takehiro Ooura a, Akira Doi a, Masataka Nakahira b, Satoshi Nishio b a
Power and Industrial Systems R&D Laboratory, Hitachi Ltd., 7 -2 -1, Omika-cho, Hitachi-shi, Ibaraki 319 -1221, Japan b Naka Fusion Research Establishment, Japan Atomic Energy Research Institute, 801 -1, Mukoyama, Naka-machi, Ibaraki 311 -0193, Japan Received in revised form 25 January 2002; accepted 15 February 2002
Abstract Eddy current experiments with closely placed solid boxes driven by a coil alternative current were carried out. The experiment was to designed to identify approaches to the problems about: (1) computing accuracy of the eddy current calculation with thin plate approximation; and (2) modeling baselines for a computational study of eddy currents in a super conducting (SC) fusion device. The following conclusions were drawn from experimental data and their comparison with eddy current analyses: (1) the skin effect is a major cause of computational error; (2) if the error due to the skin effect cannot be ignored, the double shell model calculates accurate eddy currents; (3) the mesh size with a narrow gap should be small enough (for example smaller than five times the gap width) so as not to cause additional error. The Finite Element Method (FEM) model for the eddy current computation is well modeled using these conclusions. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Eddy current; Super conducting fusion device tokamak; Experiment; FEM
1. Introduction Two possible options for the next step in fusion research are realization of a high performance plasma with steady state operation by using a super conducting (SC) fusion device, and realization of a burning plasm. Design activities are already being carried out on SC fusion devices, such as ITER [1] and others [2 – 5]. * Corresponding author. Tel.: +81-294-53-3111x5212; fax: +81-294-52-9287. E-mail address: mitsushi – [email protected] (M. Abe).
Several SC fusion devices are already in operation [6–9]. They feature more complicated electrically conducting structures than existing normal conducting fusion devices. Their conducting structures include a cryostat, heat shields, toroidal field coils (TFCs) or helical coil structures, poloidal field coil (PFC) structures, radiation shields, a double wall vacuum vessel (VV), in-vessel structures and coil conductors. Eddy currents can flow on such conducting structures, causing problems in relation to joule heats on cryogenic structures, an undesired magnetic field in the plasma discharge area, electromagnetic forces, and so on.
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Eddy current codes were developed in order to discuss the electromagnetic force on the VV with a thin shell assumption. The eddy current on the VV has been discussed, not only experimentally, but also computationally for normal conducting tokamaks [10–15]. The computations were done with the thin plate approximation [10– 13] and the electromagnetic force on the VV due to eddy current was well taken into account in the VV design of fusion devices. The validity of the analysis was confirmed by a rather simple torus model [13]. For SC tokamaks, some issues related to eddy currents have been reported from TRIAM-1M [16–18], mainly from experimental investigations. These issues were related to the magnetic field on the plasma area and joule heat load on cryogenic temperature (4 K) structures due to eddy currents. These reports suggested that a precise study of eddy currents is necessary for designing SC tokamaks. Design studies for eddy currents in SC fusion devices [1–5,19,20] face problems, since the devices have complicated and thick structures. The next step SC fusion devices also have complicated structures and require consideration of the cryostat, heat shields, TFC structures, radiation shields and double wall VV. In order to understand the eddy current characteristics, it is necessary to make an FEM (Finite Element Method) model which considers them all. Due to this complication, a solid FEM model is not applicable, so that researches have usually turned again to the thin shell model for the design studies of eddy currents [21–23]. Another problem related to the skin depth of the eddy currents. The structural material can be thicker than the skin depth and the simple thin shell approximation may cause computational error. These situations make the number of the FEM elements large and the thin approximation wrong, degrading the computational accuracy. In order to obtain guidelines to keep computational accuracy in eddy current studies, we carried out experiments using thick conducting boxes on which eddy currents are driven by a coil with alternating current. This paper describes the results of the experiments and modeling guidelines
we identified for the SC fusion device eddy current analysis.
2. FEM modeling problems Existing SC fusion devices, such as the tokamaks TRIAM-1M [7], TORE SUPRA [8] and HT-7 [9], and the helical device LHD [6], have more complicated structures than normal conducting fusion devices. Fig. 1 shows a schematic drawing of a next step SC fusion device like ITER [21]. It has a cryostat and heat shields, which are not found in existing normal conducting tokamaks, and solid coil support structures to withstand strong magnetic fields. Moreover, since burning plasma studies are to be done, it has radiation shields. The characteristics of the SC fusion device structures are considered as:
Fig. 1. Poloidal cross section of a typical next step SC fusion device. The structures are more complicated than an existing normal conducting device.
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1. solid structures to withstand strong magnetic fields; 2. closely set structures, for example, a double wall VV, radiation and heat shields; and 3. a large number of structures for which eddy currents cannot be ignored. Thin shell FEM computer codes like ECTAS [21] are usually used for the eddy current study of SC fusion devices. Since the computed results depend on the validity of the thin shell approximation and FEM modeling, we have to discuss the computational errors related to the three stated characteristics of SC fusion device structures, i.e. computational errors due to thickness related to (1) and complexity of the structures related to (2) and (3). Origins of the errors can be considered as follows. With regard to (1), thicknesses of the structures can be up to a few tens of centimeters, for example, the cross section of TFC cases. This thick part creates a problem related to the skin depth. The IP of 10 MA order can decay in 10 ms, making it the fastest phenomenon to be considered in a fusion device. This decay can be considered as a quarter cycle of a 25 Hz wave, but the waveform is not a simple trigonometric function. Then, it is necessary to consider a higher frequency than 25 Hz. We consider 50 Hz, at least, as the frequency to which the eddy currents should be computed accurately. The skin depth for 50 Hz with stainless steel (SS) resistivity of 7× 10 − 7 Vm is 64 mm. Then, this leads to problems of shell element position and also effective thickness for resistance, related to the validity of the thin shell approximation. These problems degrade the thin shell FEM computational accuracy. In relation to (2) and (3), problems about FEM modeling occur. The complicated structures (TFC structures, VV, cryostat and so on) have large sizes of roughly 10 m, but there are also fine structures, such as thin insulation layers or a narrow gap between conducting walls. This presence of both large and fine structures increases the number of meshes in the FEM model and sets a limit for computing ability for a large scale fusion device. A small mesh size model needs a large computer memory size, which makes the computation impossible, while a large mesh size model increases computational error.
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Then, we decided to carry out eddy current experiments using a simple setup, in which the features of thick structures and narrow gapped structures were included.
3. Experimental setup The setup for the eddy current experiments (Fig. 2) included SS boxes with rather thick walls and a solenoid coil. The component sizes were as given in the figure and Table 1. The SS boxes were placed side by side so that eddy currents on them coupled with each other electromagnetically. This placement represents the thick structures and narrow gapped structures in SC fusion devices. The primary box current was inductively activated by the solenoid coil current i.e. induced by the flux in the solenoid coil, while the secondary box current was induced by the return magnetic field flux of the coil and reduced by the flux from the primary box current. The solenoid coil was connected to an IGBT (Insulated Gate Bipolar Transistor) switching power supply, which supplied an alternating voltage of 9 500 V with very short switching time of about 1 ms. Then the applied voltage had a well shaped rectangular waveform. An additional feature was that the alternating frequency could be changed between 50 Hz, 500 Hz and 4.9 kHz frequencies. This made it possible to discuss the problem related to skin depth. The gap of the two boxes was changed from 1.7 to 32.1 mm. This made it possible to discuss the computational error due to close placement of the structures and mesh sizes. The Rogowski coils were placed to measure eddy current distribution as shown in Fig. 2. They measured not only total current of the boxes, but also the eddy current distribution. There were three vertically directed holes and four horizontally directed holes to measure the current distribution in the thickness direction and height direction, respectively. The holes had a 3 mm diameter, while their depths were 30 and 260 mm, respectively. They were placed separately, so we expected little effect on the current density distribution.
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Fig. 2. Setup of eddy current experiment with two SS boxes and one coil.
The signals were filtered (5 ms) in order to reject the high frequency component related to IGBT switching characteristics. The signals for the 50 and 500 Hz frequencies were digitized with 50 and 100 k samples per second, respectively, and then integrated digitally. For the 4.9 kHz frequency, the signals were integrated directly by analogue circuits, in order to avoid possible errors during digital integrations of high frequency components and they were sampled at 500 kHz.
4. Eddy current experiments and FEM modeling guidelines The experimental data were compared with computational results in order to get FEM model-
ing guidelines. However, the objectives of eddy current study in a SC fusion device are to understand: 1. magnetic fields due to eddy currents in plasma discharge area for equilibrium control system design; 2. electromagnetic forces due to eddy currents for structural design; and 3. Joule heat production on the cryogenic temperature structures. Then, precise distributions of the eddy currents in structures are not a main concern. We first compared the eddy total box currents with calculated results by a simple single shell model. Following this, the current density distributions and eddy current in narrow gap case were examined to confirm modeling guidelines regarding the skin effect and complicated structures.
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Fig. 3. Typical experimental results of an eddy current experiment (a) and a computational single shell FEM model (b), with surface gap distance of 1.7 mm. The calculated eddy currents were obtained using ECTAS with the experimental coil current and locating the shell mesh at the center of the SS box wall thickness. The 510 turn coil was modeled filament currents at each windings (5 ×102 windings).
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4.1. Experiment for error e6aluation of the thin shell approximation Fig. 3(a) shows a set of typical experimental waveforms with calculated eddy currents, which were calculated by a model shown in Fig. 3(b) and measured coil current as input data. The FEM model is a single layer shell model. The elements are at the center in the SS box wall thickness direction. The box resistances were calculated with the real thickness of 30 mm. A alternative coil current of 285 A (14.5 kA.T) was applied. The coil is modeled as 510 filament loop current placed at each winding. The mesh size is roughly same size as the box wall thickness. The shell model assumes no skin effect and uniform current distribution in thickness direction. Then, we believe that typical length to describe the distribution is longer than the wall thickness and the mesh size much smaller than the thickness is not adequate. This mesh size produces enough current distribution as far as skin effect is ignorable. The experiment was done with surface gap of 1.7 mm, then 31.7 mm gap in FEM model. The calculated currents were a little smaller than the experimental measurements. The difference was considered as mainly due to the current distributions in the wall thickness direction. The skin depth for SS using the 50 Hz frequency was calculated as 64 mm which was twice as deep as the experimental box 30 mm wall thickness. HowTable 1 Equipment specifications for eddy current experiment Parts
Specifications
Power supply Voltage waveform Voltage Frequency
9450 V 50, 500, 4.9 kHz
Coil Coil winding Current
510 turns, 14 mH, 0.35 V Triangle shape
Boxes Material Shape
Square
30 mm thick stainless steel (SS) 240 mm long, 360 mm square cross section pipe (Fig. 2)
Fig. 4. Dependence of the calculated eddy current magnitude on the FEM shell model position. The vertical axis value of 1.0 means an accurate computation.
ever, the skin effect could not be ignored because the coil current waveform was triangular shaped and the high frequency components were contained in the waveform. The calculated eddy current on the primary box had an amplitude which agreed well with the experimental one, however during the steep time derivative phase of the eddy current, the experimental current was larger than the calculated one. The experimental eddy current on the secondary box was larger than the calculated values. These computational errors could be explained by taking into account the skin effect. As a general characteristic of eddy currents, they tend to flow where the inductances are small and small inductances make currents larger. However, the computational model had eddy currents at the centers of the wall thickness and did not include the skin effect. This FEM model had larger inductances than the real eddy currents when the time derivative of the currents was steep. Then, the computation tended to compute smaller eddy currents. Other than the skin effect, a very narrow gap between the boxes or thick SS walls can be a cause of a computational error, as pointed out in Section 1. We considered the guidelines for mesh size and mesh position in the thickness direction. Fig. 4 plots the ratio of the calculated primary box current amplitude normalized by the experi-
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mental one, against shell element position d in the thickness direction. The position parameter d was measured from the center of the thickness directed outward as shown in the Fig. 4. The ratio 1.0 means that the computed results well describe experiments. The ratio was roughly 1.0 for a model with centered shell position (0 mm) at 50 Hz. At 4.9 kHz, the model with the inside surface shell (d = − 15 mm) well described the experimental result and the center shell model calculated an 18% smaller current. Two lines are plotted as proportional to 1/(2d +L) and 1/(2d +L)2, where L is a typical size of the structure, and in this experiment L=330 mm. As a general characteristic, the current is inversely proportional to inductance for the high frequency case and to resistance for the low frequency one. Inductance is roughly proportional to the area enclosed by the current path, then to (2d +L)2 and the amplitude Ieddy is expected to be, Ieddy 81/(2d + L)2,
(1)
for the high frequency (4.9 kHz) case. Resistance is proportional to the length of the current path, then to (2d +L) and Ieddy is, Ieddy 81/(2d + L),
(2)
for the low frequency (50 Hz) case. The current was expected to flow with a uniform current density (d =0 mm) without the skin effect at 50 Hz and the calculation with d = 0 mm gave a good result. However, the eddy current flowed along the inside surface due to the skin effect at 4.9 kHz, then d = − 15 mm gave a good result and the calculation with d =0 gave 18% smaller eddy current. The discussion of Fig. 4 led us to the idea that the error due to the skin effect is due to the error from the eddy current path assumption in the computational model. Usually, we produce the FEM model with centered shell elements (d = 0 mm) and for the low frequency case, the eddy current calculations are expected to give good results. For the high frequency case, the calculation error occurs through inductance of the eddy current as discussed in Fig. 4 and Eq. (1). The largest error is expected at the surface current, i.e.
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2d is equal to the wall thickness, and the computational error related to the skin depth can be as small as, ErrorB 1− 1/(2d/L+ 1)2 : 2t/L,
(3)
where the wall thickness t is usually much less than L. We considered Eq. (3) as a computational error magnitude and in the case that the error of Eq. (3) cannot be ignored, double shell model can be applicable as discussed in Section 4.2.
4.2. Experiment for skin effect Section 4.1 showed the skin effect is a major cause of the computational error. However, there is a technique to avoid the large computational error. The technique is discussed here. The eddy currents distribute in the direction of both thickness and height due to the skin effect. This skin effect is a common characteristic for the two boxes. Then, we used the primary box to discuss the current distributions, because the primary box had a larger current and better measurement accuracy than the second box. The skin effect made it easy for eddy current to flow at the edge area in the height direction and the inner side (coil facing) surface in the thickness direction. The skin effect makes the current distribution change as a function of the frequency and the single shell model cannot represent the various distributions. One idea was multi shell placement to reconstruct the various current distributions. However, placement of a many layer shell made the number of FEM elements large and caused a computational difficulty. Then, we tried a multi shell model. The eddy current distributed continuously in the depth direction and the multi shell model could not reconstruct the distribution precisely. We expected the double shell model improved the computational accuracy of the total current and the magnetic field, because the change of the current center due to the skin effect could be represented between the two shells. Fig. 5 shows total currents in the primary box at 4.9 kHz. Two computational models were applied. They are shown at the top of the figure. One was a model with single shell and the other was a model with a multi shell. The latter model
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had two layers in the thickness direction, and it had additional shells at the top and bottom. On the additional shells, eddy currents could flow due to the skin effect and the shells had 6.3 mm thickness resistance, roughly equal to skin depth. Other than top and bottom shells, the shells had a resistance of 15 mm thickness. The total current computed by the multi shell model correctly agreed with the experiment in not only the amplitudes as shown in Fig. 4, but also in waveforms, while the single shell model calculated smaller eddy current than the experimental value as discussed in Section 4.1. The current waveform was a triangle form with 200 A amplitude and the averaged current density was 2.6× 104 A/m2. The triangle waveform suggested that the current was inductively driven and there was a strong skin effect. Fig. 6 plots current density in height (Fig. 6(a)) and in thickness (Fig. 6(b)) directions, comparing the experiment with the multi shell calculation. The current density distribution in the height
Fig. 6. Current density distributions in height (a) and thickness (b) directions. The computed densities are plotted every mesh point and the measured data are plotted at every measured point.
Fig. 5. Total current waveforms of 4.9 kHz case. Experimental and computational data with double shell model are plotted.
direction was measured at four intervals (every 32.5 mm) above the mid plane. The measured and calculated data were plotted as density (A/m2). The measured density at the edge was 4.4× 104 A/m2 and other data were around 2×104 A/m2. This situation was consistent with the small skin depth of 6.4 mm at 4.9 kHz. Other than the edge region, the density was around 2× 104 A/m2 for both calculation and experiment. At the edge, using the two data from 0 to 32.5 mm, the calculated currents were averaged as 5.2×104 A/m2, which was a little larger than the experiment. We considered this was due to the placement of FEM meshes at the top surface, as discussed for Fig. 4. However, this error creates
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only 4.5% of the total current and it can be improved easily by replacing the FEM position at a suitable position as discussed in Fig. 4. Even removing the top and bottom shells, the computational error should be small because the remaining shells can reconstructed in the current distribution in the height direction. This point is discussed with Fig. 7. Fig. 6(b) plots current densities in the thickness direction. The densities were averaged in height direction. The calculation was on two points in the thickness direction because of the multi shell model, while experimental data were for every 1/4 part of the thickness. Taking into account these averaging procedures, we thought the computation well reconstructed the experimental results within the error discussed for Fig. 6(a). The discussions on Figs. 5 and 6 can be summarized as eddy current characteristics can be well reconstructed by the multi shell model even when the skin effect can not be ignored. Fig. 7 shows the dependencies of the amplitude ratio on frequencies of 50, 500 and 4.9 kHz. Two FEM models were tested. One was a double shell model, a model like the multi shell model in Fig. 5 except that the top and bottom shells were removed. The double shell model had two shells at inner and outer surfaces in this computation. The other FEM model was a single shell model.
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At 50 Hz, computations with both models could well describe the experimental results. However, at high frequency the double shell model gave more accurate results than the single shell model. At 4.9 kHz and 500 Hz, the skin depths were 6.4 and 20 mm, respectively. They were thinner than the box wall thickness. Then, we concluded that the double shell model well described the skin effect. For structures thicker than the skin depth, the double shell model calculated eddy currents well for a wide range of frequencies. The shell position of the double shell model was at the surface, however the discussion of Fig. 4 showed the surface shell model produced smaller inductance and resistance than the actual values. This was why the double shell model tended to give a little larger eddy current amplitude than the real one (amplitude ratio \ 1.0) and from the experimental results, it was apparent that the shell position should not be on the surface. The shell model computational analysis is mainly for low frequency condition. Then, the shell should be at center of the wall in thickness direction as discussed in Fig. 4 for single shell model. In case that double shell model the shells should be at centers of the each half thickness of the wall that means shells at 1/4 and 3/4 of the thickness direction. The error due to the shell position could be evaluated using Eq. (3), but the thickness should be half of the real thickness. Then, the error was evaluated as roughly half of the single shell model. However, the accuracy was kept in a wide range of the frequencies.
4.3. Experiment for closely placed conducting walls
Fig. 7. Dependence of the calculated eddy current amplitude on the frequency. The amplitude is normalized by the experimental amplitude and the vertical axis value of 1.0 means an accurate computation. Two FEM models were tested.
Fig. 8 compares the calculated and experimental secondary box currents, as a function of the gap between the box walls. As could be recognized from Fig. 3 waveforms, the secondary box current had more high frequency components than the primary box current. This allowed discussion of the accuracy for the eddy currents on walls, which were very close to each other from Fig. 8. The experiment was done with gaps of 1.7, 6.8 and 32.1 mm at 50 and 500 Hz. The single shell model gave smaller amplitudes than experimental data, while the double shell model gave larger
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results. Since the 6.8 mm case did not have the numerical error, we decided to do the eddy current computations with FEM element sizes smaller than five times the shell gap.
4.4. Guidelines for thin shell FEM modeling The single shell model well calculated the total eddy current when the skin depth was larger than the wall thickness. Then we can say: 1. The single shell model is applicable for small skin depth cases. The multi shell model well calculated not only the total box current, but also the current center for a wide range of the frequencies. Then, we can say: 2. Multi shell model is applicable for thick structure cases. The computer code well calculated box current when the gap between two shell was larger than 1/5 of the mesh size. Then, we can say: 3. The mesh size should be smaller than five times the gap. Guideline (3) is for our computer code ECTAS only. This accuracy should be examined again using other codes. These are guidelines for thin shell FEM modeling of SC fusion devices. Using these guidelines, we can calculate eddy currents within the following error: ErrorB 2t%/L, Fig. 8. Dependence of the calculated eddy current magnitude on the gap between two boxes. Two FEM models were tested.
amplitudes. However, the computational error was smaller with the double shell model than with single shell model for 6.8 and 32.1 mm gap cases. An important point was that the error became large at a small gap of 1.7 mm for the double shell model. The mesh size was 30 mm for these computations. The error was considered to be due to a numerical error during the calculation in the code. This meant that the eddy currents should not be calculated for walls with a 1.7 mm shell gap using the 30 mm size mesh. The single shell model which had shell elements at the center of the thickness had a 31.7 mm shell gap in the FEM model for the 1.7 mm gap case and had the same level error as large gap
(4)
where t% is a thickness represented by a shell and L is a typical size of the structure.
5. Summary Eddy current experiments were carried out in order to obtain information about: (1) accuracy of eddy current computation with thin plate approximation; (2) modeling baselines for computational study of eddy currents in a SC fusion device, in which structures are solid, closely set and a large number of structures cannot be ignored. Experimental data and their comparison with eddy current analyses led to the following three results:
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1. Skin effect is a major cause of computational error, the magnitude of which is roughly described by Eq. (3), in the single shell model computation. 2. When the error due to the skin effect cannot be ignored, the double shell model calculates accurate eddy currents. 3. The mesh size with a narrow gap should be smaller than five times the gap width using ECTAS code. Based on these guidelines, FEM models for the computational eddy current studies of a SC fusion device will be well constructed.
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