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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 22, NO. 3, JUNE 2012 4903904 Numerical Simulation of Conventional/ Enhanced Permanent Magnet Method: Inuence of Crack on Accuracy T. Takayama, A. Saitoh, and A. Kamitani Abstract—The conventional and enhanced permanent magnet methods for measuring the critical current density in a high-tem- perature superconducting (HTS) tape have been simulated numerically. Moreover, the inuence of the crack on the accuracy of the conventional method has been investigated. In order to simulate both permanent magnet methods, two types of numerical code have been developed for analysing the time evolution of a shielding current density in an HTS tape. The results of com- putations for the conventional method show that the maximum repulsive force acting on the tape drastically decreases when the symmetric axis of the magnet approaches the crack. On the other hand, the results of computations for the enhanced method show that the electromagnetic force near the center of the tape is roughly proportional to the critical current density. It is also found that the accuracy of the enhanced method compared with the conventional one hardly changes except for near the tape edge. Therefore, the enhanced method is a speedy and efcient method for measuring the spatial distribution of the critical current density. Index Terms—Critical current density, high temperature super- conductors, numerical simulation, surface cracks. I. INTRODUCTION A HIGH-TEMPERATURE superconductors (HTSs) are the application of several devices or systems: (e.g., ywheel, fault current limiter, and SQUID, etc.). To develop the HTS de- vices or systems, it is important to measure a critical current density accurately. The standard four-probe method [1], [2] has been generally used for measuring [3], [4]. In the method, electrodes are deposited with a silver paste to decrease the mea- surement error [3], [4]. In addition, a large current source ows in an HTS sample, and simultaneously can be evaluated ac- curately from nonlinear characteristics. However, HTS Manuscript received September 10, 2011; accepted November 10, 2011. Date of publication November 16, 2011; date of current version May 24, 2012. This work was supported in part by the Japan Society for the Promotion of Science under a Grant-in-Aid for Scientic Research (B) 22360042. A part of this work was also carried out under the Collaboration Research Program (NIFS10KNXN178, NIFS11KNTS011) at the National Institute for Fusion Science (NIFS), Japan. Numerical computations were carried out on Hitachi SR16000 XM1 of the LHD Numerical Analysis Server in NIFS. T. Takayama is with the Department of Informatics, Faculty of Engi- neering, Yamagata University, Yonezawa, Yamagata 992-8510, Japan (e-mail: [email protected]). A. Saitoh is with the Graduate School of Engineering, University of Hyogo, Himeji, Hyogo 671-2280, Japan (e-mail: [email protected]). A. Kamitani is with the Graduate School of Science and Engineering, Yama- gata University, Yonezawa, Yamagata 992-8510, Japan (e-mail: kamitani@yz. yamagata-u.ac.jp). Digital Object Identier 10.1109/TASC.2011.2176300 characteristics may be degraded because of a heat generated be- tween the electrodes and the sample [5]. For above reasons, Claassen et al. have proposed the induc- tive method [6]. By applying an ac current to a small coil placed just above an HTS lm, they monitored a harmonic voltage in- duced in the coil. They found that, only when a coil current ex- ceeds a threshold current, the third-harmonic voltage develops suddenly. They conclude that can be evaluated from the threshold current. On the other hand, Ohshima et al. have pro- posed the permanent magnet method [7], [8]. While moving a permanent magnet above an HTS lm, the electromagnetic force acting on the lm is measured. Consequently, they found that the maximum repulsive force is roughly proportional to . This means that is estimated from the measured value of . The two types of the method [6]–[8] are used for not only the determination of -distributions but also the detection of any cracks in an HTS sample [5], [9]. However, it takes long time to measure any -distributions of a large surface area such as an HTS tape or wire. For resolving the above problem, Ohshima et al. recently en- hanced the permanent magnet method [10]. In the method, after the magnet is located just above an HTS sample at the constant distance between the HTS surface and the magnet bottom, it is moved along the surface. Simultaneously, they measured an electromagnetic force . As a result, they found that a spatial distribution of can be obtained from a measured -distri- bution. In addition, they conclude that the enhanced method is higher speed of the -measurement than the conventional one, and its accuracy hardly changes. In the previous study, a numerical code was developed for analysing the time evolution of a shielding current density in an HTS tape [11]. By using the code, the permanent magnet method for an HTS tape was reproduced for the case without any cracks. The results of computations showed that, even if the magnet is placed near the tape edge, the maximum repulsive force is almost proportional to [11]. From this result, the conven- tional permanent magnet method is applicable to measure near the edge. However, any cracks were not at all taken into consideration. The purpose of the present study is to develop the numerical code for analysing the time evolution of the shielding current density in an HTS tape with a crack and to investigate the ap- plicability of the conventional permanent magnet method to the detection of a crack. Furthermore, we numerically investigate the performance of the enhanced method. 1051-8223/$26.00 © 2011 IEEE

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Page 1: Numerical Simulation of Conventional/Enhanced Permanent Magnet Method: Influence of Crack on Accuracy

IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 22, NO. 3, JUNE 2012 4903904

Numerical Simulation of Conventional/Enhanced Permanent Magnet Method:Influence of Crack on Accuracy

T. Takayama, A. Saitoh, and A. Kamitani

Abstract—The conventional and enhanced permanent magnetmethods for measuring the critical current density in a high-tem-perature superconducting (HTS) tape have been simulatednumerically. Moreover, the influence of the crack on the accuracyof the conventional method has been investigated. In order tosimulate both permanent magnet methods, two types of numericalcode have been developed for analysing the time evolution of ashielding current density in an HTS tape. The results of com-putations for the conventional method show that the maximumrepulsive force acting on the tape drastically decreases when thesymmetric axis of the magnet approaches the crack. On the otherhand, the results of computations for the enhanced method showthat the electromagnetic force near the center of the tape is roughlyproportional to the critical current density. It is also found that theaccuracy of the enhanced method compared with the conventionalone hardly changes except for near the tape edge. Therefore, theenhanced method is a speedy and efficient method for measuringthe spatial distribution of the critical current density.

Index Terms—Critical current density, high temperature super-conductors, numerical simulation, surface cracks.

I. INTRODUCTION

A HIGH-TEMPERATURE superconductors (HTSs) are theapplication of several devices or systems: (e.g., flywheel,

fault current limiter, and SQUID, etc.). To develop the HTS de-vices or systems, it is important to measure a critical currentdensity accurately. The standard four-probe method [1], [2]has been generally used for measuring [3], [4]. In themethod,electrodes are deposited with a silver paste to decrease the mea-surement error [3], [4]. In addition, a large current source flowsin an HTS sample, and simultaneously can be evaluated ac-curately from nonlinear characteristics. However, HTS

Manuscript received September 10, 2011; accepted November 10, 2011.Date of publication November 16, 2011; date of current version May 24, 2012.This work was supported in part by the Japan Society for the Promotion ofScience under a Grant-in-Aid for Scientific Research (B) 22360042. A partof this work was also carried out under the Collaboration Research Program(NIFS10KNXN178, NIFS11KNTS011) at the National Institute for FusionScience (NIFS), Japan. Numerical computations were carried out on HitachiSR16000 XM1 of the LHD Numerical Analysis Server in NIFS.T. Takayama is with the Department of Informatics, Faculty of Engi-

neering, Yamagata University, Yonezawa, Yamagata 992-8510, Japan (e-mail:[email protected]).A. Saitoh is with the Graduate School of Engineering, University of Hyogo,

Himeji, Hyogo 671-2280, Japan (e-mail: [email protected]).A. Kamitani is with the Graduate School of Science and Engineering, Yama-

gata University, Yonezawa, Yamagata 992-8510, Japan (e-mail: [email protected]).Digital Object Identifier 10.1109/TASC.2011.2176300

characteristics may be degraded because of a heat generated be-tween the electrodes and the sample [5].For above reasons, Claassen et al. have proposed the induc-

tive method [6]. By applying an ac current to a small coil placedjust above an HTS film, they monitored a harmonic voltage in-duced in the coil. They found that, only when a coil current ex-ceeds a threshold current, the third-harmonic voltage developssuddenly. They conclude that can be evaluated from thethreshold current. On the other hand, Ohshima et al. have pro-posed the permanent magnet method [7], [8]. While movinga permanent magnet above an HTS film, the electromagneticforce acting on the film is measured. Consequently, they foundthat the maximum repulsive force is roughly proportionalto . This means that is estimated from the measured valueof .The two types of the method [6]–[8] are used for not only the

determination of -distributions but also the detection of anycracks in an HTS sample [5], [9]. However, it takes long timeto measure any -distributions of a large surface area such asan HTS tape or wire.For resolving the above problem, Ohshima et al. recently en-

hanced the permanent magnet method [10]. In the method, afterthe magnet is located just above an HTS sample at the constantdistance between the HTS surface and the magnet bottom, itis moved along the surface. Simultaneously, they measured anelectromagnetic force . As a result, they found that a spatialdistribution of can be obtained from a measured -distri-bution. In addition, they conclude that the enhanced method ishigher speed of the -measurement than the conventional one,and its accuracy hardly changes.In the previous study, a numerical code was developed for

analysing the time evolution of a shielding current density in anHTS tape [11]. By using the code, the permanentmagnetmethodfor an HTS tape was reproduced for the case without any cracks.The results of computations showed that, even if the magnet isplaced near the tape edge, the maximum repulsive force isalmost proportional to [11]. From this result, the conven-tional permanent magnet method is applicable to measurenear the edge. However, any cracks were not at all taken intoconsideration.The purpose of the present study is to develop the numerical

code for analysing the time evolution of the shielding currentdensity in an HTS tape with a crack and to investigate the ap-plicability of the conventional permanent magnet method to thedetection of a crack. Furthermore, we numerically investigatethe performance of the enhanced method.

1051-8223/$26.00 © 2011 IEEE

Page 2: Numerical Simulation of Conventional/Enhanced Permanent Magnet Method: Influence of Crack on Accuracy

4903904 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 22, NO. 3, JUNE 2012

Fig. 1. A schematic view of a permanent magnet method.

Fig. 2. A schematic view of a crack in an HTS sample.

II. GOVERNING EQUATION AND NUMERICAL METHOD

In Fig. 1, we show a schematic view of a permanent magnetmethod. A cylindrical permanent magnet of radius andheight is placed above a rectangle-shaped HTS tape ofwidth and length and thickness . Moreover, the distanceis between the tape surface and the magnet bottom.Throughout the present study, we use the Cartesian coor-

dinate system , where the -axis is parallelto the thickness direction and the origin O is the centroid ofthe tape (see Fig. 1). In terms of the coordinate system, thesymmetry axis of the permanent magnet can be representedas . For the purpose of characterizing thestrength of the magnet, we use a magnetic flux density at

for the case with . Here,is the minimum value of the distance .In Fig. 2, we show a schematic view of a crack in an HTS

sample. Hereafter, the square cross-section of the sample is de-noted by . If a crack is contained in the film, has not only theouter boundary but the inner boundary . Otherwise, theboundary of consists of the outer boundary only. More-over, a normal unit vector and a tangential unit vector on aredenoted by and , respectively.As usual, we assume the thin-layer approximation [12]: the

thickness of the HTS film is sufficiency thin that the shieldingcurrent density can hardly flow in the thickness direction.As is well known, the shielding current density is closelyrelated to the electric field . The relation can be written as

. As a function , we assume the powerlaw , where is the critical electric field.Under the above assumptions, the shielding current density

can be written as , and the time evolution ofthe scalar function is governed by the following integro-differential equation [12]:

(1)

Fig. 3. The movement of magnet for the case with the two types of permanentmagnet method. (a) Conventional method. (b) Enhanced method.

where both and are position vectors in the -plane.and denote an applied magnetic flux density and an electricfield, respectively, and a bracket is an average operator overthe thickness of the HTS tape. In addition, the explicit form of

is given by

The initial and boundary conditions to (1) are assumed asfollows:

(2)

(3)

(4a)

(4b)

Here, is an arclength along . Note that the boundary condi-tions (4a) and (4b) are required to assume any cracks.By solving the initial-boundary-value problem of (1), we can

determine the time evolution of the shielding current densityin an HTS tape. Discretized with the backward Euler method,the initial-boundary-value problem of (1) is reduced to the non-linear boundary-value problem. By using the Newton methodand the finite element method, the problem is transformed bysimultaneous linear equations. Note that the coefficient matrixof the equations becomes non-symmetric if we assume a crackin an HTS tape. Since it takes much too long to solve this equa-tion by Gaussian elimination, we use SuperLU [13]. As a result,the CPU time for solutions of initial-boundary-value problem of(1) can be reduced by a factor of about eight.As mentioned the numerical methods, the two types of the

numerical code are developed for analyzing the time evolutionof in an HTS tape. By using the two codes, we simulate theconventionalmethod and enhanced one. The results of computa-tions are described in the following Sections III and IV (Fig. 3).

Throughout the present study, the geometrical and physicalparameters are fixed as follows: , ,

, , ,, . Also, is assumed to be uniform.

III. SIMULATION OF CONVENTIONAL METHOD

A. Determination of Proportionality Constant

In the conventional method, a permanent magnet is moved upand down by changing a distance which is given by

. Here,

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TAKAYAMA et al.: NUMERICAL SIMULATION OF CONVENTIONAL/ENHANCED PERMANENT MAGNET METHOD 4903904

Fig. 4. Dependence of the critical current density on themaximum repulsiveforce . The inset shows the dependence of the electromagnetic force onthe distance for the case with .

, , and are given by , ,and , respectively.In simulating the conventional method, we should first deter-

mine the value of the proportionality constant . The procedureis as follows:1) By assuming the value of , the code is executed for an-alyzing the time evolution of the shielding current density. An electromagnetic force acting on the film is cal-culated from a -distribution and it is easily calculated by

. As a result, becomes afunction of the distance (see inset of Fig. 4).

2) In order to calculate the maximum repulsive force , anapproximate value of the repulsive force is evaluated at

by extrapolating the electromagnetic force(see inset of Fig. 4). In extrapolating, we use the value ofat time and . If this step is executed forvarious assumed values of , data points are obtained.

3) By means of the least-squares method, a straight line isfitted to data points (see Fig. 4). As a result, the pro-portionality constant is determined numerically, and weget the following formula:

(5)

where is an estimated value of .

Consequently, we getfrom Fig. 4. Note that (5) is applicable only to an HTS tapewithout any cracks. From this result, the critical current den-sity can be evaluated by substituting the maximum repulsiveforce to (5).

B. Applicability to Detection of Crack

Let us investigate the applicability to the detection of a crackby the conventional method. To this end, a crack shape is a linesegment with two end points in the -plane.Here, the crack size is denoted by .In Fig. 5(a) and (b), we show the spatial distribution of the

shielding current density . We see from this figure that, for, the shielding current density flows in a clockwise

direction, and the behavior of hardly affects the crack. In otherwords, the -distribution near the symmetry axis of the magnet

Fig. 5. The spatial distribution of the shielding current density for the case withat time . Here, the thick line indicates the crack.

Fig. 6. Dependence of the maximum repulsive force on the magnet posi-tion for the case with .

is similar without assuming a crack. For , the dis-tortion of -distribution is caused by the influence of the crack.As a result, the behavior of significantly affects the maximumrepulsive force .In Fig. 6, we show the dependence of the maximum repulsive

force on the magnet position . We see from this figurethat decreases drastically when the symmetry axis of themagnet approaches the crack . Partic-ularly, becomes the minimum value at . On theother hand, is constant for . From this result,it is found that, although the size of a crack cannot be measuredquantitatively, its existence can be identified. This result sug-gests that the permanent magnet method can detect a crack.

IV. SIMULATION OF ENHANCED METHOD

In simulating the enhanced permanent magnet method, wechange the movement of the magnet. The magnet is placed justabove an HTS sample at a distance , and simulta-neously the magnet is moved parallel to the -axis:

. Here, the value of is .Let us numerically investigate the legality of the enhanced

permanent magnet method. Firstly, an electromagnetic forceas a function of the magnet position is depicted in Fig. 6. Wesee from this figure that the behavior of remarkably changesnear the tape edge, whereas becomes almost constant nearthe center of tape. Therefore, if is proportional to the criticalcurrent density , it may be possible to measure a -distri-bution from the -distribution except for edge. In the presentstudy, we investigate the relation between and by usingthe value of at . This value is denoted by theelectromagnetic force .

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4903904 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 22, NO. 3, JUNE 2012

Fig. 7. Spatial distribution of the electromagnetic force for the case with. The inset shows the dependence of the critical current den-

sity on the maximum repulsive force .

Fig. 8. Dependence of the relative error on the magnet position for thecase with .

Let us investigate the relation between and . In the insetof Fig. 7, we show the dependence of the critical current densityon the electromagnetic force . We see from this figure that

the force increases in rough proportion to . Therefore, weget the following formula:

(6)

Here, denotes a proportionality constant for the en-hanced method. By applying the least-squares approx-imation to the line, we can obtain

. From these results, wecan determine the accuracy of the conventional and the en-hanced method by using the formulas (5) and (6).Finally, let us investigate the accuracy of the conventional

and the enhanced method. As the measure of the accuracy, wedefine a relative error . In Fig. 8, we show

the dependence of the relative error on the magnet position. This figure indicates that, for two methods, the error in-

creases when the magnet is located near the tape edge. Notethat, in general, the accuracy near the tape edge is negligiblebecause cannot be accurately measured due to the edge ef-fect. In the following, the accuracy of two methods is comparedfor . As a result, it is found that the con-ventional method is higher than the enhanced one. However, theaccuracy of the enhanced method is about 8.1% or less. Fromthis result, the enhancedmethod is a speedy and efficientmethodfor measuring the spatial distribution of the critical current den-sity .

REFERENCES

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[2] R. N. Bhattacharya and M. P. Paranthaman, High Temperature Super-conductors. , Germany: Wiley-VCH, 2010.

[3] A. Maqsood, M. Khaliq, and M. Maqsood, “Role of barium additionon the properties of bismuth-based superconductors,” Jour. Mar. Sci.,vol. 27, no. 19, pp. 5330–5334, 1992.

[4] T. Inada and M. Kuwabara, “Interpretation of a peculiar behaviourof the critical current density with relative sintered density in small-grained ceramics,” Jour. Mar. Sci. Lett., vol. 14, no. 17,pp. 1220–1222, 1995.

[5] S. Ohshima, K. Umezu, K. Hattori, H. Yamada, A. Saito, T. Takayama,A. Kamitani, H. Takano, T. Suzuki, M. Yokoo, and S. Ikuno, “Detec-tion of critical current distribution of YBCO-coated conductors usingpermanent magnet method,” IEEE Trans. Appl. Supercond., vol. 21,no. 3, pp. 3385–3388, June 2011.

[6] J. H. Claassen, M. E. Reeves, and R. J. Soulen, Jr., “A contactlessmethod for measurement of the critical current density and critical tem-perature of superconducting films,” Rev. Sci. Instrum., vol. 62, no. 4,pp. 996–1004, Apr. 1991.

[7] S. Ohshima, K. Takeishi, A. Saito, M. Mukaida, Y. Takano, T. Naka-mura, I. Suzuki, and M. Yokoo, “A simple measurement technique forcritical current density by using a permanent magnet,” IEEE Trans.Appl. Supercond., vol. 15, no. 2, pp. 2911–2914, June 2005.

[8] A. Saito, K. Takeishi, Y. Takano, T. Nakamura, M. Yokoo, M.Mukaida, S. Hirano, and S. Ohshima, “Rapid and simple measurementof critical current density in HTS thin films using a permanent magnetmethod,” Physica C, vol. 426, pp. 1122–1126, Oct. 2005.

[9] S. B. Kim, “The defect detection in HTS films on third-harmonicvoltage method using various inductive coils,” Physica C, vol.463–465, pp. 702–706, Oct. 2007.

[10] K. Hattori, A. Saito, Y. Takano, H. Yamada, T. Takayama, A. Kamitani,and S. Ohshima, “Detection of smaller Jc region and damage in YBCOcoated conductors by using permanent magnet method,” Physica C,vol. 471, no. 21–22, pp. 1033–1035, Nov. 2011.

[11] T. Takayama, A. Kamitani, A. Hattori, K. Saito, and S. Ohshima, “Nu-merical investigation of inductive and permanent-magnet methods tomeasure critical current density of HTS thin films,” IEEE Trans. Appl.Supercond., vol. 21, no. 3, pp. 3360–3363, June 2011.

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[13] J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, and J. W. H. Liu,“A supernodal approach to sparse partial pivoting,” SIAM J. MatrixAnal. Appl., vol. 20, no. 3, pp. 720–755, 1999.