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Shielding current analysis by current-vector-potential method: Application to HTS film with multiply-connected structure
Physica C 494 (2013) 168–172
Contents lists available at SciVerse ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Shielding current analysis by current-vector-potential method: Application to HTS film with multiply-connected structure A. Kamitani a,⇑, T. Takayama a, S. Ikuno b a b
Yamagata University, 4-3-16, Johnan, Yonezawa, Yamagata 992-8510, Japan Tokyo University of Technology, 1404-1, Katakura, Hachioji, Tokyo 192-0982, Japan
a r t i c l e
i n f o
Article history: Received 24 January 2013 Received in revised form 8 April 2013 Accepted 15 April 2013 Available online 29 April 2013 Keywords: Critical currents Cuprates Superconducting films
a b s t r a c t The performance of the virtual voltage method is compared with that of the conventional method in which integral forms of Faraday’s law along crack surfaces are treated as natural boundary conditions. As a result, it is found that there is a significant difference between numerical solutions by the two methods. In this sense, not the conventional method but the virtual voltage method should be employed to the shielding current analysis in a high-temperature superconducting film with cracks. By means of the virtual voltage method, the influence of a crack on the inductive method is investigated numerically. The results of computations show that, if the threshold current changes remarkably from its ambient value, a part of a crack is contained in the projection of the field-generating coil onto the film surface. Furthermore, the applicability of the inductive method to the crack detection is investigated numerically. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction
2. Shielding current analysis
The inductive method [1,2] has been widely used for measuring the critical current density jC of high-temperature superconducting (HTS) films. In the method, while applying an ac current I(t) = I0 sin 2pft in a coil placed just above an HTS film, the third-harmonic voltage V3 sin (6pft + h3) induced in the coil is monitored. According to the experimental results, V3 abruptly develops when I0 exceeds a threshold current IT [1,2]. By assuming the critical state model, Mawatari et al. [2] performed a theoretical analysis to get the following equation: jC = 2F(rm)IT/b. Here, F(rm) denotes the maximum of the primary coil-factor function and b is the thickness of an HTS film. By substituting the measured value of IT into this equation, jC can be estimated. This is the basic idea of the inductive method. On the other hand, it is not clear whether or not the inductive method is applicable to an HTS film containing cracks. In order to calculate the shielding current density in an HTS film with cracks, the authors proposed the virtual voltage method [3,4]. Although the virtual voltage method was successfully applied to the numerical simulation of the permanent magnet method [5], its performance has not yet been investigated in detail. The purpose of the present study is to assess the performance of the virtual voltage method and to numerically investigate the influence of a crack on the inductive method.
2.1. Initial-boundary-value problem
⇑ Corresponding author. Tel.: +81 238 26 3331; fax: +81 238 26 3789. E-mail address: [email protected] (A. Kamitani). 0921-4534/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physc.2013.04.066
A schematic view of the inductive method is shown in Fig. 1. In the inductive method, an M-turn coil is placed above an HTS film of thickness b so that the symmetry axis of the coil may be vertical to the film surface. By taking the thickness direction as z-axis and choosing the centroid of the film as the origin, let us use the Cartesian coordinate system hO: ex, ey, ezi. In terms of the coordinate system, the symmetry axis of the coil can be written as (x, y) = (xA, yA). In the following, n and t denote a normal unit vector and a tangent unit vector on an arbitrary curve in the xy plane, respectively. Also, x and x0 are position vectors of two points in the xy plane. We first assume that an HTS film has a square cross section X of side length a and that it contains a crack whose cross section is a line segment connecting two points, (xc, yc ± Lc/2), in the xy plane. For this case, the boundary @ X of X is composed of the outer boundary C0 and the inner boundary C1 (see Fig. 1). Apparently, C1 is the crack surface. We further assume that the coil has a rectangular cross section of inner radius Rin, outer radius R and height H. Also, the distance between the coil bottom and the film surface is denoted by L. As is well known, the shielding current density j and the electric field E are closely related to each other in HTS films. As the relation, the following power law [6–10] is assumed:
E ¼ EðjjjÞ½j=jjj;
EðjÞ ¼ EC ½j=jC N :
A. Kamitani et al. / Physica C 494 (2013) 168–172
Fig. 1. A schematic view of the inductive method.
Here, EC and jC denote the critical electric field and the critical current density, respectively, and N is a positive constant. Under the thin-plate approximation, there exists a scalar function S(x, t) such that j = (2/b)[r (Sez)] and its time evolution is governed by the following integrodifferential equation [3,4,6,11]:
c SÞ þ ðr EÞ ez ¼ @ t hB ez i; l0 @ t ð W
ð1Þ
c is an where h i is an average operator over the thickness and W operator defined by
cS W
Z Z
2
Q ðjx x0 jÞSðx0 ; tÞ d x0 þ
X
2Sðx; tÞ : b
Here, the function Q(r) is given by 2
2
Q ðrÞ ¼ ðpb Þ1 ½r 1 ðr 2 þ b Þ1=2 : The initial and boundary conditions to (1) are assumed as follows [3,4]:
S ¼ 0 at t ¼ 0;
c Sðn1Þ hBðnÞ Bðn1Þ i ez and X ¼ X [ @ X. Also, the where u ¼ l0 W function space HðXÞ is defined by HðXÞ fwðxÞ : w ¼ 0 on C 0 ; @w=@s ¼ 0 on C 1 g. After a straightforward calculation, we get the weak form that is equivalent to (2) and (4). Note that (4) is incorporated into the weak form. In other words, (4) is treated as a natural boundary condition. As is well known, a natural boundary condition is not exactly satisfied by a numerical solution of a weak form. Hence, even if the weak form is numerically solved with the essential boundary condition (3), the resulting numerical solution does not accurately satisfy (4). In order to investigate the residual in c[E], the numerically evaluated value N[E] of c[E] is determined by using the numerical integration. The results of computations show that N[E] does not vanish but take a relatively large value [3]. Hereafter, the numerical method for solving the weak form with (3) is called the conventional method. For the purpose of having N[E] = 0 exactly fulfilled, the authors proposed the virtual voltage method [3,4]. The basic idea of this method is explained as follows: a virtual voltage /V is applied along the crack surface C1 so as to have N[E] = 0 satisfied. The detailed explanation of the virtual voltage method is given in Appendix A. Let us compare the performance of the virtual voltage method with that of the conventional method. To this end, we numerically R evaluate the toroidal current IT defined by IT b C j n ds. Here, the integration is carried out along an arbitrary curve C connecting the outer boundary C0 with the inner boundary C1 (see Fig. 1). The time dependence of IT is determined by means of the two methods and the results of computations are depicted in Fig. 2. For the case with 0.1 < mod (ft, 1) < 0.3 or 0.6 < mod (ft, 1) < 0.8, there is a significant difference between the toroidal currents determined by the two methods. This result suggests that an accurate solution cannot be obtained by means of the conventional method. In fact, j-distributions by the two methods show slightly different patterns especially near both ends of the crack (see Figs. 3(a) and 3(b)).
3. Numerical simulation of inductive method
S ¼ 0 on C 0 ; @S ¼ 0 on C 1 ; @s I
c½E
169
E t ds ¼ 0;
C1
where s is an arclength. Note that c[E] = 0 is the integral form of Faraday’s law on the crack surface C1. By solving (1) together with the initial and boundary conditions, we can determine the time evolution of the shielding current density j. Throughout the present study, the parameters are assumed as follows: a = 20 mm, b = 600 nm, jC = 1 MA/cm2, EC = 1 mV/m, N = 20, (xc, yc) = (0 mm, 0 mm), f = 1 kHz, M = 400, Rin = 1.5 mm, R = 2.5 mm, H = 2 mm, and L = 0.5 mm.
On the basis of the virtual voltage method, a numerical code has been developed for analyzing the time evolution of the shielding current density. By executing the code, the threshold current IT can be easily evaluated. In this section, we numerically investigate the following two problems: How is the inductive method affected by a crack? Is the inductive method applicable to the crack detection?
2.2. Virtual voltage method Throughout this section, the superscript (n) denotes a value at time t = nDt, where Dt is a time step size. If the initial-boundaryvalue problem of (1) is discretized with the backward Euler method, S(n) becomes a solution S of the following nonlinear boundaryvalue problem:
c S þ Dt ðr EÞ ez u ¼ 0 in X; GðSÞ l0 W
ð2Þ
S 2 HðXÞ;
ð3Þ
c½E ¼ 0;
ð4Þ
Fig. 2. The time dependence of the toroidal current IT for the case with I0 = 400 mA, Lc = 16 mm and (xA, yA) = (0 mm, 0 mm).
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Fig. 3. Spatial distributions of the shielding current density j at time t = 1.2/f for the case with I0 = 400 mA, Lc = 16 mm and (xA, yA) = (0 mm, 0 mm). Here, the j-distribution is determined by using either (a) the conventional method or (b) the virtual voltage method. In this figure and Fig. 4, cracks are denoted by thick line segments.
Fig. 4. Spatial distributions of the shielding current density j at time t = 1.2/f for the case with I0 = IT and (xA, yA) = (0 mm, 0 mm). Here, (a) Lc = 2 mm and (b) Lc = 5 mm.
3.1. Influence of crack on inductive method First, let us investigate the influence of the crack size Lc on the inductive method for the case with (xA, yA) = (xc, yc). To this end, jdistributions are numerically determined for various values of Lc and are depicted in Figs. 4(a) and 4(b). For the case with Lc = 2 mm, the j-distribution is little affected by the crack and is almost axisymmetric. In contrast, for the case with Lc = 5 mm, the j-distribution is strongly influenced by the crack so that its axisymmetry is lost considerably. These results indicate that, for the case with Lc J 3 mm (= 2Rin), the axisymmetric j-distribution will be deformed. Hence, the accuracy of the inductive method is expected to be degraded for this case. In order to quantitatively investigate the accuracy degradation due to a crack, we determine the dependence of the accuracy on the crack size. As a measure of the accuracy, the following relative error is adopted: jjC jC j=jC . Here, jC denotes an approximate value of jC that is estimated from IT. The relative error is evaluated as a function of the crack size Lc and is depicted in Fig. 5. For the case with Lc [ 2Rin, the relative error takes an almost constant value and, hence, it hardly depends on Lc. In contrast, for the case with Lc J 2Rin, an increase in Lc will raise drastically. Apparently, Lc > 2Rin is the necessary
Fig. 5. Dependence of the relative error (xA, yA) = (0 mm, 0 mm).
on the crack size Lc for the case with
and sufficient condition for a crack overlapping the projection of the coil onto the film surface (the shaded regions in Figs. 4(a) and 4(b)). Therefore, the above results suggest that, if the crack overlaps the projection of the coil, the accuracy of the inductive method is remarkably degraded. Hereafter, the projection of the coil onto the film surface is called a shadow region of the coil and is denoted by Ds(xA, yA). Specifically, Ds(xA, yA) is given by
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part of a crack. This property is applied to the crack detection in Section 3.2. 3.2. Crack detection using inductive method
Fig. 6. Dependence of the relative error Lc = 5 mm and yA = 0 mm.
on the coil position xA for the case with
In general, jC is unknown before the measurement by the inductive method and, hence, cannot be evaluated in the actual exper iment. In order to approximately calculate ; jC is first evaluated as a function of (xA, yA) and, subsequently, its maximum over X is adopted as jC. By means of this method, we can determine the approximate distribution of that is employed in the crack detection. Figs. 5 and 6 also indicate that, if 6 8% is satisfied, any part of a crack is not contained in the shadow region Ds. On the basis of this tendency, we can approximately determine the position and the shape of a crack. The method for determining the region Dc, that contains a crack, is composed of the following three steps: (1) The relative error is evaluated as a function of coil position (xA, yA) and, subsequently, the region S = {(xA, yA) 2 Uj (xA, yA) < 8%} is determined numerically. Here, U is a subset of X. (2) When (xA, yA) moves in S, all points in Ds(xA, yA) constitute the region T that never contains a crack. The region T is determined by using
T¼
[
Ds ðxA ; yA Þ:
ðxA ;yA Þ2S
(3) The region Dc is determined as Dc = U \ Tc. Here, Tc denotes a complement of T. By means of the above three steps, the region Dc is numerically determined in the 1st quadrant and is depicted in Fig. 7. The contours of the relative error are also shown in Fig. 7. Since the crack is assumed to be a line segment, x = 0 mm and jyj 6 2.5 mm, it is completely contained in Dc. In addition, the area of Dc is extremely small as compared with that of X. From these results, we conclude that a crack can be accurately detected by using the inductive method. 4. Conclusion
Fig. 7. The region Dc and contours of for the case with Lc = 5 mm. Here, the region Dc is denoted by a shaded region.
n o Ds ðxA ; yA Þ ¼ ðx; yÞj R2in < ðx xA Þ2 þ ðy yA Þ2 < R2 : Next, we investigate the effect of the crack position on the inductive method. To this end, the dependence of the relative error on the crack position is numerically determined for yA = yc and the results of computations are depicted in Fig. 6. For the case with 2.5 mm 6 xA 6 7 mm, the relative error remains almost constant and it takes 8% or less. In contrast, for the case with 0 mm 6 xA < 2.5 mm, it drastically increases with a decrease in xA. Since the crack position is fixed as xc = 0 mm, the inequality, 0 mm 6 xA < 2.5 mm, is equivalent to jxA xcj < R. On the other hand, jxA xcj < R is the necessary and sufficient condition for a crack overlapping the shadow region Ds. Hence, the above result also implies that, if the crack overlaps Ds, the accuracy of the inductive method is remarkably degraded. From the above results, we can conclude that, if IT changes significantly from its ambient value, a shadow region Ds contains a
We have investigated the performance the virtual voltage method numerically. As a result, it turns out that the virtual voltage method is a powerful tool for accurately analyzing the timedependent shielding current density in an HTS film containing cracks. By using the virtual voltage method, we have investigated the inductive method numerically. Conclusions obtained in the present study are summarized as follows. When a crack overlaps the projection of a coil onto the film surface, the accuracy of the inductive method is remarkably degraded. This accuracy degradation is attributable to the deformation of the axisymmetric j-distribution. The shape and the position of a crack can be accurately determined by means of the inductive method.
Acknowledgments This work was supported in part by Japan Society for the Promotion of Science under a Grant-in-Aid for Scientific Research ((B) No. 22360042, (C) No. 24560321). A part of this work was also carried out with the support and under the auspices of the NIFS
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Collaboration KNXN237).
Research
program
(NIFS11KNTS011,
NIFS12
Appendix A. Virtual voltage method In the virtual voltage method, a virtual voltage /V is applied along the crack surface C1 so that N[E] = 0 can be satisfied. Specifically, the boundary condition (4) is replaced with the following two conditions:
c½E ¼ /V ; N½E ¼ 0;
ðA:1Þ ðA:2Þ
where /V is an unknown constant. The resulting nonlinear boundary-value problem is composed of (2), (3), (A.1) and (A.2). In the present study, the nonlinear problem is solved for S and /V with the Newton method. In this appendix, the method for solving the nonlinear problem is explained in detail. If the Newton method is applied to the above nonlinear problem, the solutions, S and /V, are iteratively determined by using the following two steps. (1) The linear boundary-value problem:
dG ¼ G½S;
ðA:3Þ
dS 2 HðXÞ;
ðA:4Þ
c½dE d/V ¼ ðc½E /V Þ;
ðA:5Þ
N½dE ¼ N½E;
ðA:6Þ
is solved to get the corrections, dS and d/V. (2) The solutions, S and /V, are updated by means of S :¼ S + dS and /V :¼ /V + d/V. Here, dG and dE are Fréchet derivatives of G(S) and E, respectively, and their explicit forms are given by
c dS þ Dt ðr dEÞ ez ; dG ¼ l0 W 2 dE ¼ PðjÞ ðrdS ez Þ: b In addition, P(j) is a 2nd-order tensor defined by
PðjÞ
d EðjjjÞ j j EðjjjÞ I; þ djjj jjj jjj jjj
where I denotes an identity tensor. The above two steps are repeated until both kdSk/kSk and jd/Vj/j/Vj become negligibly small. Incidentally, in solving the linear boundary-value problem, (A.5) and (A.6) are treated as a natural boundary condition and an essential boundary condition, respectively. In other words, the weak form equivalent to (A.3) and (A.5) is first derived and it is subsequently solved with the essential boundary conditions, (A.4) and (A.6). In the numerical code, the finite element method is applied to the discretization of the weak form and the essential boundary conditions. References [1] J.H. Claassen, M.E. Reeves, R.J. Soulen Jr, A contactless method for measurement of the critical current density and critical temperature of superconducting films, Rev. Sci. Instrum. 62 (1991) 996–1004. [2] Y. Mawatari, H. Yamasaki, Y. Nakagawa, Critical current density and thirdharmonic voltage in superconducting films, Appl. Phys. Lett. 81 (2002) 2424– 2426. [3] A. Kamitani, T. Takayama, Numerical simulation of shielding current density in high-temperature superconducting film: influence of film edge on permanent magnet method, IEEE Trans. Magn. 48 (2012) 727–730. [4] A. Kamitani, T. Takayama, A. Saitoh, H. Nakamura, Accurate and stable numerical method for analyzing shielding current density in hightemperature superconducting film containing cracks, Plasma Fusion Res. 7 (2012) 2405024. [5] S. Ohshima, K. Takeishi, A. Saito, M. Mukaida, Y. Takano, T. Nakamura, I. Suzuki, M. Yokoo, A simple measurement technique for critical current density by using a permanent magnet, IEEE Trans. Appl. Supercond. 15 (2005) 2911– 2914. [6] A. Kamitani, T. Takayama, A. Tanaka, S. Ikuno, Numerical simulation of inductive method for determining spatial distribution of critical current density, Physica C 470 (2010) 1189–1192. [7] Y.B. Jung, S.J. Salon, Calculation of ac losses in bulk HTS rods from the combined action of transport current and external field, IEEE Trans. Magn. 43 (2007) 1397–1400. [8] R. Fresa, G. Rubinacci, S. Ventre, F. Villone, W. Zamboni, Fast solution of a 3-D integral model for the analysis of ITER superconducting coils, IEEE Trans. Magn. 45 (2009) 988–991. [9] G.P. Lousberg, J.F. Fagnard, M. Ausloos, P. Vanderbemden, B. Vanderheyden, Numerical study of the shielding properties of macroscopic hybrid ferromagnetic/superconductor hollow cylinders, IEEE Trans. Appl. Supercond. 20 (2010) 33–41. [10] R. Brambilla, F. Grilli, L. Martini, Integral equations for computing AC losses of radially and polygonally arranged HTS thin tapes, IEEE Trans. Appl. Supercond. 22 (2012) 8401006. [11] Y. Yoshida, M. Uesaka, K. Miya, Magnetic-field and force analysis of high-Tc superconductor with flux-flow and creep, IEEE Trans. Magn. 30 (1994) 3503– 3506.
Contents lists available at SciVerse ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Shielding current analysis by current-vector-potential method: Application to HTS film with multiply-connected structure A. Kamitani a,⇑, T. Takayama a, S. Ikuno b a b
Yamagata University, 4-3-16, Johnan, Yonezawa, Yamagata 992-8510, Japan Tokyo University of Technology, 1404-1, Katakura, Hachioji, Tokyo 192-0982, Japan
a r t i c l e
i n f o
Article history: Received 24 January 2013 Received in revised form 8 April 2013 Accepted 15 April 2013 Available online 29 April 2013 Keywords: Critical currents Cuprates Superconducting films
a b s t r a c t The performance of the virtual voltage method is compared with that of the conventional method in which integral forms of Faraday’s law along crack surfaces are treated as natural boundary conditions. As a result, it is found that there is a significant difference between numerical solutions by the two methods. In this sense, not the conventional method but the virtual voltage method should be employed to the shielding current analysis in a high-temperature superconducting film with cracks. By means of the virtual voltage method, the influence of a crack on the inductive method is investigated numerically. The results of computations show that, if the threshold current changes remarkably from its ambient value, a part of a crack is contained in the projection of the field-generating coil onto the film surface. Furthermore, the applicability of the inductive method to the crack detection is investigated numerically. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction
2. Shielding current analysis
The inductive method [1,2] has been widely used for measuring the critical current density jC of high-temperature superconducting (HTS) films. In the method, while applying an ac current I(t) = I0 sin 2pft in a coil placed just above an HTS film, the third-harmonic voltage V3 sin (6pft + h3) induced in the coil is monitored. According to the experimental results, V3 abruptly develops when I0 exceeds a threshold current IT [1,2]. By assuming the critical state model, Mawatari et al. [2] performed a theoretical analysis to get the following equation: jC = 2F(rm)IT/b. Here, F(rm) denotes the maximum of the primary coil-factor function and b is the thickness of an HTS film. By substituting the measured value of IT into this equation, jC can be estimated. This is the basic idea of the inductive method. On the other hand, it is not clear whether or not the inductive method is applicable to an HTS film containing cracks. In order to calculate the shielding current density in an HTS film with cracks, the authors proposed the virtual voltage method [3,4]. Although the virtual voltage method was successfully applied to the numerical simulation of the permanent magnet method [5], its performance has not yet been investigated in detail. The purpose of the present study is to assess the performance of the virtual voltage method and to numerically investigate the influence of a crack on the inductive method.
2.1. Initial-boundary-value problem
⇑ Corresponding author. Tel.: +81 238 26 3331; fax: +81 238 26 3789. E-mail address: [email protected] (A. Kamitani). 0921-4534/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physc.2013.04.066
A schematic view of the inductive method is shown in Fig. 1. In the inductive method, an M-turn coil is placed above an HTS film of thickness b so that the symmetry axis of the coil may be vertical to the film surface. By taking the thickness direction as z-axis and choosing the centroid of the film as the origin, let us use the Cartesian coordinate system hO: ex, ey, ezi. In terms of the coordinate system, the symmetry axis of the coil can be written as (x, y) = (xA, yA). In the following, n and t denote a normal unit vector and a tangent unit vector on an arbitrary curve in the xy plane, respectively. Also, x and x0 are position vectors of two points in the xy plane. We first assume that an HTS film has a square cross section X of side length a and that it contains a crack whose cross section is a line segment connecting two points, (xc, yc ± Lc/2), in the xy plane. For this case, the boundary @ X of X is composed of the outer boundary C0 and the inner boundary C1 (see Fig. 1). Apparently, C1 is the crack surface. We further assume that the coil has a rectangular cross section of inner radius Rin, outer radius R and height H. Also, the distance between the coil bottom and the film surface is denoted by L. As is well known, the shielding current density j and the electric field E are closely related to each other in HTS films. As the relation, the following power law [6–10] is assumed:
E ¼ EðjjjÞ½j=jjj;
EðjÞ ¼ EC ½j=jC N :
A. Kamitani et al. / Physica C 494 (2013) 168–172
Fig. 1. A schematic view of the inductive method.
Here, EC and jC denote the critical electric field and the critical current density, respectively, and N is a positive constant. Under the thin-plate approximation, there exists a scalar function S(x, t) such that j = (2/b)[r (Sez)] and its time evolution is governed by the following integrodifferential equation [3,4,6,11]:
c SÞ þ ðr EÞ ez ¼ @ t hB ez i; l0 @ t ð W
ð1Þ
c is an where h i is an average operator over the thickness and W operator defined by
cS W
Z Z
2
Q ðjx x0 jÞSðx0 ; tÞ d x0 þ
X
2Sðx; tÞ : b
Here, the function Q(r) is given by 2
2
Q ðrÞ ¼ ðpb Þ1 ½r 1 ðr 2 þ b Þ1=2 : The initial and boundary conditions to (1) are assumed as follows [3,4]:
S ¼ 0 at t ¼ 0;
c Sðn1Þ hBðnÞ Bðn1Þ i ez and X ¼ X [ @ X. Also, the where u ¼ l0 W function space HðXÞ is defined by HðXÞ fwðxÞ : w ¼ 0 on C 0 ; @w=@s ¼ 0 on C 1 g. After a straightforward calculation, we get the weak form that is equivalent to (2) and (4). Note that (4) is incorporated into the weak form. In other words, (4) is treated as a natural boundary condition. As is well known, a natural boundary condition is not exactly satisfied by a numerical solution of a weak form. Hence, even if the weak form is numerically solved with the essential boundary condition (3), the resulting numerical solution does not accurately satisfy (4). In order to investigate the residual in c[E], the numerically evaluated value N[E] of c[E] is determined by using the numerical integration. The results of computations show that N[E] does not vanish but take a relatively large value [3]. Hereafter, the numerical method for solving the weak form with (3) is called the conventional method. For the purpose of having N[E] = 0 exactly fulfilled, the authors proposed the virtual voltage method [3,4]. The basic idea of this method is explained as follows: a virtual voltage /V is applied along the crack surface C1 so as to have N[E] = 0 satisfied. The detailed explanation of the virtual voltage method is given in Appendix A. Let us compare the performance of the virtual voltage method with that of the conventional method. To this end, we numerically R evaluate the toroidal current IT defined by IT b C j n ds. Here, the integration is carried out along an arbitrary curve C connecting the outer boundary C0 with the inner boundary C1 (see Fig. 1). The time dependence of IT is determined by means of the two methods and the results of computations are depicted in Fig. 2. For the case with 0.1 < mod (ft, 1) < 0.3 or 0.6 < mod (ft, 1) < 0.8, there is a significant difference between the toroidal currents determined by the two methods. This result suggests that an accurate solution cannot be obtained by means of the conventional method. In fact, j-distributions by the two methods show slightly different patterns especially near both ends of the crack (see Figs. 3(a) and 3(b)).
3. Numerical simulation of inductive method
S ¼ 0 on C 0 ; @S ¼ 0 on C 1 ; @s I
c½E
169
E t ds ¼ 0;
C1
where s is an arclength. Note that c[E] = 0 is the integral form of Faraday’s law on the crack surface C1. By solving (1) together with the initial and boundary conditions, we can determine the time evolution of the shielding current density j. Throughout the present study, the parameters are assumed as follows: a = 20 mm, b = 600 nm, jC = 1 MA/cm2, EC = 1 mV/m, N = 20, (xc, yc) = (0 mm, 0 mm), f = 1 kHz, M = 400, Rin = 1.5 mm, R = 2.5 mm, H = 2 mm, and L = 0.5 mm.
On the basis of the virtual voltage method, a numerical code has been developed for analyzing the time evolution of the shielding current density. By executing the code, the threshold current IT can be easily evaluated. In this section, we numerically investigate the following two problems: How is the inductive method affected by a crack? Is the inductive method applicable to the crack detection?
2.2. Virtual voltage method Throughout this section, the superscript (n) denotes a value at time t = nDt, where Dt is a time step size. If the initial-boundaryvalue problem of (1) is discretized with the backward Euler method, S(n) becomes a solution S of the following nonlinear boundaryvalue problem:
c S þ Dt ðr EÞ ez u ¼ 0 in X; GðSÞ l0 W
ð2Þ
S 2 HðXÞ;
ð3Þ
c½E ¼ 0;
ð4Þ
Fig. 2. The time dependence of the toroidal current IT for the case with I0 = 400 mA, Lc = 16 mm and (xA, yA) = (0 mm, 0 mm).
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Fig. 3. Spatial distributions of the shielding current density j at time t = 1.2/f for the case with I0 = 400 mA, Lc = 16 mm and (xA, yA) = (0 mm, 0 mm). Here, the j-distribution is determined by using either (a) the conventional method or (b) the virtual voltage method. In this figure and Fig. 4, cracks are denoted by thick line segments.
Fig. 4. Spatial distributions of the shielding current density j at time t = 1.2/f for the case with I0 = IT and (xA, yA) = (0 mm, 0 mm). Here, (a) Lc = 2 mm and (b) Lc = 5 mm.
3.1. Influence of crack on inductive method First, let us investigate the influence of the crack size Lc on the inductive method for the case with (xA, yA) = (xc, yc). To this end, jdistributions are numerically determined for various values of Lc and are depicted in Figs. 4(a) and 4(b). For the case with Lc = 2 mm, the j-distribution is little affected by the crack and is almost axisymmetric. In contrast, for the case with Lc = 5 mm, the j-distribution is strongly influenced by the crack so that its axisymmetry is lost considerably. These results indicate that, for the case with Lc J 3 mm (= 2Rin), the axisymmetric j-distribution will be deformed. Hence, the accuracy of the inductive method is expected to be degraded for this case. In order to quantitatively investigate the accuracy degradation due to a crack, we determine the dependence of the accuracy on the crack size. As a measure of the accuracy, the following relative error is adopted: jjC jC j=jC . Here, jC denotes an approximate value of jC that is estimated from IT. The relative error is evaluated as a function of the crack size Lc and is depicted in Fig. 5. For the case with Lc [ 2Rin, the relative error takes an almost constant value and, hence, it hardly depends on Lc. In contrast, for the case with Lc J 2Rin, an increase in Lc will raise drastically. Apparently, Lc > 2Rin is the necessary
Fig. 5. Dependence of the relative error (xA, yA) = (0 mm, 0 mm).
on the crack size Lc for the case with
and sufficient condition for a crack overlapping the projection of the coil onto the film surface (the shaded regions in Figs. 4(a) and 4(b)). Therefore, the above results suggest that, if the crack overlaps the projection of the coil, the accuracy of the inductive method is remarkably degraded. Hereafter, the projection of the coil onto the film surface is called a shadow region of the coil and is denoted by Ds(xA, yA). Specifically, Ds(xA, yA) is given by
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part of a crack. This property is applied to the crack detection in Section 3.2. 3.2. Crack detection using inductive method
Fig. 6. Dependence of the relative error Lc = 5 mm and yA = 0 mm.
on the coil position xA for the case with
In general, jC is unknown before the measurement by the inductive method and, hence, cannot be evaluated in the actual exper iment. In order to approximately calculate ; jC is first evaluated as a function of (xA, yA) and, subsequently, its maximum over X is adopted as jC. By means of this method, we can determine the approximate distribution of that is employed in the crack detection. Figs. 5 and 6 also indicate that, if 6 8% is satisfied, any part of a crack is not contained in the shadow region Ds. On the basis of this tendency, we can approximately determine the position and the shape of a crack. The method for determining the region Dc, that contains a crack, is composed of the following three steps: (1) The relative error is evaluated as a function of coil position (xA, yA) and, subsequently, the region S = {(xA, yA) 2 Uj (xA, yA) < 8%} is determined numerically. Here, U is a subset of X. (2) When (xA, yA) moves in S, all points in Ds(xA, yA) constitute the region T that never contains a crack. The region T is determined by using
T¼
[
Ds ðxA ; yA Þ:
ðxA ;yA Þ2S
(3) The region Dc is determined as Dc = U \ Tc. Here, Tc denotes a complement of T. By means of the above three steps, the region Dc is numerically determined in the 1st quadrant and is depicted in Fig. 7. The contours of the relative error are also shown in Fig. 7. Since the crack is assumed to be a line segment, x = 0 mm and jyj 6 2.5 mm, it is completely contained in Dc. In addition, the area of Dc is extremely small as compared with that of X. From these results, we conclude that a crack can be accurately detected by using the inductive method. 4. Conclusion
Fig. 7. The region Dc and contours of for the case with Lc = 5 mm. Here, the region Dc is denoted by a shaded region.
n o Ds ðxA ; yA Þ ¼ ðx; yÞj R2in < ðx xA Þ2 þ ðy yA Þ2 < R2 : Next, we investigate the effect of the crack position on the inductive method. To this end, the dependence of the relative error on the crack position is numerically determined for yA = yc and the results of computations are depicted in Fig. 6. For the case with 2.5 mm 6 xA 6 7 mm, the relative error remains almost constant and it takes 8% or less. In contrast, for the case with 0 mm 6 xA < 2.5 mm, it drastically increases with a decrease in xA. Since the crack position is fixed as xc = 0 mm, the inequality, 0 mm 6 xA < 2.5 mm, is equivalent to jxA xcj < R. On the other hand, jxA xcj < R is the necessary and sufficient condition for a crack overlapping the shadow region Ds. Hence, the above result also implies that, if the crack overlaps Ds, the accuracy of the inductive method is remarkably degraded. From the above results, we can conclude that, if IT changes significantly from its ambient value, a shadow region Ds contains a
We have investigated the performance the virtual voltage method numerically. As a result, it turns out that the virtual voltage method is a powerful tool for accurately analyzing the timedependent shielding current density in an HTS film containing cracks. By using the virtual voltage method, we have investigated the inductive method numerically. Conclusions obtained in the present study are summarized as follows. When a crack overlaps the projection of a coil onto the film surface, the accuracy of the inductive method is remarkably degraded. This accuracy degradation is attributable to the deformation of the axisymmetric j-distribution. The shape and the position of a crack can be accurately determined by means of the inductive method.
Acknowledgments This work was supported in part by Japan Society for the Promotion of Science under a Grant-in-Aid for Scientific Research ((B) No. 22360042, (C) No. 24560321). A part of this work was also carried out with the support and under the auspices of the NIFS
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Collaboration KNXN237).
Research
program
(NIFS11KNTS011,
NIFS12
Appendix A. Virtual voltage method In the virtual voltage method, a virtual voltage /V is applied along the crack surface C1 so that N[E] = 0 can be satisfied. Specifically, the boundary condition (4) is replaced with the following two conditions:
c½E ¼ /V ; N½E ¼ 0;
ðA:1Þ ðA:2Þ
where /V is an unknown constant. The resulting nonlinear boundary-value problem is composed of (2), (3), (A.1) and (A.2). In the present study, the nonlinear problem is solved for S and /V with the Newton method. In this appendix, the method for solving the nonlinear problem is explained in detail. If the Newton method is applied to the above nonlinear problem, the solutions, S and /V, are iteratively determined by using the following two steps. (1) The linear boundary-value problem:
dG ¼ G½S;
ðA:3Þ
dS 2 HðXÞ;
ðA:4Þ
c½dE d/V ¼ ðc½E /V Þ;
ðA:5Þ
N½dE ¼ N½E;
ðA:6Þ
is solved to get the corrections, dS and d/V. (2) The solutions, S and /V, are updated by means of S :¼ S + dS and /V :¼ /V + d/V. Here, dG and dE are Fréchet derivatives of G(S) and E, respectively, and their explicit forms are given by
c dS þ Dt ðr dEÞ ez ; dG ¼ l0 W 2 dE ¼ PðjÞ ðrdS ez Þ: b In addition, P(j) is a 2nd-order tensor defined by
PðjÞ
d EðjjjÞ j j EðjjjÞ I; þ djjj jjj jjj jjj
where I denotes an identity tensor. The above two steps are repeated until both kdSk/kSk and jd/Vj/j/Vj become negligibly small. Incidentally, in solving the linear boundary-value problem, (A.5) and (A.6) are treated as a natural boundary condition and an essential boundary condition, respectively. In other words, the weak form equivalent to (A.3) and (A.5) is first derived and it is subsequently solved with the essential boundary conditions, (A.4) and (A.6). In the numerical code, the finite element method is applied to the discretization of the weak form and the essential boundary conditions. References [1] J.H. Claassen, M.E. Reeves, R.J. Soulen Jr, A contactless method for measurement of the critical current density and critical temperature of superconducting films, Rev. Sci. Instrum. 62 (1991) 996–1004. [2] Y. Mawatari, H. Yamasaki, Y. Nakagawa, Critical current density and thirdharmonic voltage in superconducting films, Appl. Phys. Lett. 81 (2002) 2424– 2426. [3] A. Kamitani, T. Takayama, Numerical simulation of shielding current density in high-temperature superconducting film: influence of film edge on permanent magnet method, IEEE Trans. Magn. 48 (2012) 727–730. [4] A. Kamitani, T. Takayama, A. Saitoh, H. Nakamura, Accurate and stable numerical method for analyzing shielding current density in hightemperature superconducting film containing cracks, Plasma Fusion Res. 7 (2012) 2405024. [5] S. Ohshima, K. Takeishi, A. Saito, M. Mukaida, Y. Takano, T. Nakamura, I. Suzuki, M. Yokoo, A simple measurement technique for critical current density by using a permanent magnet, IEEE Trans. Appl. Supercond. 15 (2005) 2911– 2914. [6] A. Kamitani, T. Takayama, A. Tanaka, S. Ikuno, Numerical simulation of inductive method for determining spatial distribution of critical current density, Physica C 470 (2010) 1189–1192. [7] Y.B. Jung, S.J. Salon, Calculation of ac losses in bulk HTS rods from the combined action of transport current and external field, IEEE Trans. Magn. 43 (2007) 1397–1400. [8] R. Fresa, G. Rubinacci, S. Ventre, F. Villone, W. Zamboni, Fast solution of a 3-D integral model for the analysis of ITER superconducting coils, IEEE Trans. Magn. 45 (2009) 988–991. [9] G.P. Lousberg, J.F. Fagnard, M. Ausloos, P. Vanderbemden, B. Vanderheyden, Numerical study of the shielding properties of macroscopic hybrid ferromagnetic/superconductor hollow cylinders, IEEE Trans. Appl. Supercond. 20 (2010) 33–41. [10] R. Brambilla, F. Grilli, L. Martini, Integral equations for computing AC losses of radially and polygonally arranged HTS thin tapes, IEEE Trans. Appl. Supercond. 22 (2012) 8401006. [11] Y. Yoshida, M. Uesaka, K. Miya, Magnetic-field and force analysis of high-Tc superconductor with flux-flow and creep, IEEE Trans. Magn. 30 (1994) 3503– 3506.