Plasma and Fusion Research: Regular Articles

Volume 10, 3401059 (2015)

Numerical Investigation on Contactless Methods for Identifying
Defects in High-Temperature Superconducting Film∗)Teruou TAKAYAMA, Atsushi KAMITANI1) and Hiroaki NAKAMURA2)
1)

Department of Informatics, Yamagata University, Yonezawa 992-8510, Japan
Graduate School of Science and Engineering, Yamagata University, Yonezawa 992-8510, Japan
2)
National Institute for Fusion Science, Toki 509-5292, Japan(Received 25 November 2014 / Accepted 6 April 2015)

A numerical method has been developed for analyzing the shielding current density in a high-temperature
superconducting (HTS) film containing multiple cracks. By using the method, the inductive method and thescanning permanent magnet method for contactlessly measuring the critical current density has been successfully
reproduced, and the identifiability of the multiple cracks in the HTS film has been investigated. The results ofcomputations show that, when the cracks are separated from each other, the multiple cracks can be detected individually in the two types of the contactless method. On the other hand, if the crack is pretty close to the interface,
the multiple cracks is considered as a single crack. However, the crack position can be detected collectively.
Therefore, the contactless methods is useful method for detecting the multiple cracks in an HTS film.
c 2015 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: multiple crack, critical current density, finite element analysis, numerical simulation, hightemperature superconducting film
DOI: 10.1585/pfr.10.3401059

1. Introduction
A measurement of a critical current density jC isessential for designing high-temperature superconducting(HTS) devices such as a MRI, an HTS cable, and a linear
motor car. An Inductive method [1,2] for measuring a critical current density jC is employed as the crack detection in
an HTS sample. Kim experimentally investigated the detectability of a crack from observing the V3 -I0 curves [3].
He found that the threshold current of I0 decreases if thecoil approaches the crack. This result means that the crack
can be detected by the inductive method.
As an novel method, Ohshima et al. proposed a standard permanent magnet method [4]. This method is used
for a crack detection. In the method, while moving a permanent magnet above an HTS film, an electromagnetic
force Fz acting on the film is monitored. They foundthat when the magnet approached the crack, the maximum
repulsive force FM decreases. Consequently, the crack
can be detected by measuring FM . However, it has timeconsuming because to the evaluation of FM at each measurement point.
For the purpose of jC -measurements with a highspeed, Hattori et al. have recently proposed a scanning
permanent magnet method [5]. A magnet is placed in relation to a film surface at a constant distance between an
HTS surface and the magnet bottom, and it is moved in thedirection parallel to the surface. As a result, although theauthor’s e-mail: takayama@yz.yamagata-u.ac.jp∗) This article is based on the presentation at the 24th International Toki
Conference (ITC24).

Fig. 1 Time evolution of the magnetic flux lines and the shielding current density in an HTS film with a single crack
when a magnet is moving from the left edge to the rightedge. An important point is that you can see an animation on a PDF file by clicking this figure only when you
download and install the Adobe Acrobat Reader and theQuickTime on your computer.

jC -measurement speed is higher than the standard method,they have not yet been applied scanning method to thecrack detection.
In the previous study, we develop a numerical code foranalyzing the time evolution of a shielding current density
in an HTS film with a single crack [6–8]. By using thecode, we obtain the time evolution of the magnetic lines
and the shielding current density is illustrated for a typical case in Fig. 1. In this figure, the green lines and thearrows indicate the magnetic flux lines and the flow of ashielding current density, respectively. The blue plane, thegray cylinder and the black line show the HTS film, thepermanent magnet, and a single crack, respectively. In addition, we investigate the detectability of the scanning permanent magnet/inductive methods. The results of the com-3401059-1

c 2015 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Regular Articles

Volume 10, 3401059 (2015)

putations show that the single crack can be detected withhigh speed and high accuracy by combining the inductive
method and the scanning one [8].
We see from this figure that a shielding current density
j flows along the crack. Furthermore, j remains in the film
after passing through the magnet. The magnetic flux lines
are greatly changed under the influence of the crack andthe edge.
The purpose of the present study is to investigate theidentifiability of multiple cracks by the contactless methods by modifying an above mentioned numerical code.

2. Governing Equations
In the present study, we suppose that an HTS film contains m pieces of cracks. A rectangle cross-section Ω of thefilm has not only the outer boundary C0 but also the innerboundaries Ck (k = 1, 2, · · · , m). Moreover, n and t on Ck
are a normal unit vector and a tangential unit vector.
As usual, we use the Cartesian coordinate system
O : e x , ey , ez  (see Fig. 2): z-axis is parallel to the thickness direction. Also, it should be noted that the origin O
is chosen at the centroid of an HTS in the scanning permanent magnet method. On the other hand, O is taken at thecenter of an HTS upper surface in the inductive method. Interms of the coordinate system, the symmetry axis of themagnet and the coil can be expressed as (x, y) = (xA , yA ).
As is well known, a closely relationship between ashielding current density j an electric field E can be written
as the J-E constitutive equation N jj.
, E ( j) = EC
E = E (| j|)jC| j|Here, jC , EC , and N are the critical current density, thecritical electric field, and a positive value, respectively. Asa function E( j), we adopt the power law for characterizing
a HTS property.
Similar to the previous study, we use the thin-layer
approximation [9]: the thickness of an HTS film is thinenough so that the shielding current density j can hardly
flow in the thickness direction. From this approximation itFig. 2 A schematic view of a scanning permanent magnet
method.

is follows that j = (2/b)(∇S × ez ), where a scalar function
S (x, t) and its time evolution is governed by the following
integro-diﬀerential equation [9]:μ0 ∂t (ŴS ) + ez · (∇ × E) = −∂t B · ez .

(1)Here, the square bracket   denotes an average operator
over the thickness, and B is an applied magnetic field. The
explicit form of Ŵ is given in Ref. [9].
Applying the initial condition S = 0 at t = 0 to (1),
we assume the three boundary conditions:S = 0 on C0 ,
∂S /∂s = 0 on Ck , and hk (E) ≡ C E · tds = 0, where sk
is an arclength along Ck . By solving the initial-boundaryvalue problem of (1), we can determine the time evolutionof the shielding current density j in an HTS film with themultiple cracks.
As the discretization methods, we apply the finite
element method and the backward Euler method to theinitial-boundary-value problem of (1). The problem is reduced to the nonlinear boundary-value problem, and it istransformed by simultaneous linear equations by using theNewton method and virtual voltage method [10].
Incidentally, we adopt Hitachi SR16000 XM1 as thenumerical computations. In the computers, the matrix calculation for the numerical libraries (MATRIX/MPP) is implemented. We use the HDLGEM subroutine to solve simultaneous linear equations in the initial-boundary-value
problem of (1). As a result, the CPU time is about 92
times faster than before the implementation of the above
subroutine.
Throughout the present study, the physical and geometrical parameters of the HTS film are fixed as follows:
l = 30 mm, w = 12 mm, b = 1 μm, jC = 1 MA/cm2 , EC =1 mV/m, N = 20.

3. Identifiability of Multiple Cracks
by Using Contactless Methods
In this section, we numerically investigate the identifiability of the multiple cracks by using the two types of thecontactless methods for measuring a critical current density jC . In the present study, the number m of cracks are m= 3, and three cracks are denoted by C1 , C2 , and C3 (seeFig. 3). The shape of two cracks, C1 and C3 , is the line segment paralleled to the y-axis, and C2 is the vertical line tothe y-axis. The length of C1 , C2 , and C3 is fixed as Lc . InFig. 3 The three cracks, C1 , C2 , and C3 , containing an HTS film.

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addition, the center points (x, y) = (±xc , 0) and the origin
in the xy-plane, and the distance between cracks is denotedby dc (≡ 2xc ).

3.1

Scanning permanent magnet method

A cylindrical permanent magnet of radius R andheight H is located above an HTS film, and a distance
between a magnet bottom and a film surface is denotedby L. For characterizing the magnet strength, we use
the magnitude BF of a magnetic flux density at (x, y, z) =(xA , yA , b/2) when the magnet is not moving.
In order to numerically assess the identifiability of themultiple cracks, the movement of the magnet is controlled
by xA (t) = ±(vt − l/2) (≡ x± (t)), where v is the magnitudeof the scanning velocity. Specifically, the magnet is moved
toward the right edge of the film for xA (t) = x+ (t). On theother hand, it is moved toward the left edge of the film forxA (t) = x− (t).
The parameters in this subsection are fixed as follows:
R = 1 mm, H = 2 mm, BF = 0.1 T, v = 30 cm/s.
Although an electromagnetic force Fz acts on an HTS
film by moving the magnet, it is diﬃcult to detect the multiple cracks from the behavior of Fz . In order to identify
cracks numerically, we adopt a defect parameter(2)d ≡ sgn(ΔFz+ · ΔFz− ) |ΔFz+ · ΔFz− |.
Here, force deferences ΔFz± are determined form the diﬀerence between the electromagnetic forces Fz± with a crack
and Fz± without a crack. In addition, the superscripts (+)
and (-) denote the movement of the magnet for the casewith x+ (t) and x− (t), respectively.
In Fig. 4, we show the dependence of the defect parameter d on the magnet position xA . We see from this figure that the value of d almost vanishes near the film edge,
and we obtain d near the crack. For the case with yA =3 mm, d only appears near the crack C2 , whereas d aﬀect
the all cracks for yA = 0 mm. Here, the defect parameter dis useful for the crack detection.
Finally, let us investigate the distribution of d for twotypes of the crack position. In Figs. 5 (a) and (b), we show
the contour maps of the defect parameter d. We see fromFig. 5 (a) that three regions for d > 0 mN containing ineach crack can be obtained. In addition, it is found thatthe largest value of d becomes around the both ends of thecracks C1 and C3 . On the other hand, d does not increases
at the end of C2 . From Fig. 5 (b), since the defect parameter d is forced near the cracks, it is diﬃcult to identify
crack individually. However, it is possible to estimate theapproximate position and shape of the crack for this case.

3.2

Inductive method

A small M-turn coil of inner radius Rin , outer radiusR, and height H is located just above an HTS film and an
ac current I(t) = I0 sin 2π f t flows in the coil. In addition,
L denotes a distance between the coil bottom and the film

Fig. 4 Dependences of a defect parameter d on the magnet position xA for the case with dc = 5 mm.

Fig. 5 Contour maps of a defect parameter |d| for the case with(a) dc = 5 mm and (b) dc = 2 mm.

surface. As a result, a harmonic voltage Vn (n = 1, 2, · · · )is induced in the coil. A critical current density jC can be
estimated from a third-harmonic voltage V3 . In this subsection, we adopt the parameters as follows: Rin = 0.75 mm,
R = 1.25 mm, f = 1 kHz, M = 400.
In order to determine jC , it needs to evaluate a threshold current IT when a third-harmonic voltage V3 begins todevelop in the coil. To this end, we use the conventional
voltage criterion [2]V3 = 0.1 mV ⇔ I0 = IT∗ ,(3)because it is diﬃcult to determine IT accurately. Here, IT∗
denotes an estimated value of the coil current I0 .
Let us first investigate the influence of the threecracks, C1 , C2 , and C3 , on a third-harmonic voltage V3 .
In Fig. 6, we show V3 -I0 curves for various xA . From thisfigure, V3 gradually develops above the certain value ofI0 . Using the voltage criterion (3), we get IT∗ = 158.8 mA,
234.4 mA, and 261.1 mA for xA = 0 mm, 2 mm, and 4 mm,respectively.
Next, let us investigate the detectability of multiple

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collectively.

4. Conclusion
We have developed a numerical code analyzing thetime evolution of the shielding current density in an HTS
film containing the multiple cracks. By using the code,
we numerically investigate the identifiability of the threecracks in the HTS film by the inductive method and thescanning permanent magnet method. Conclusions obtained in the present study are summarized as follows:
1. In the two types of the contactless method for measuring a critical current density, when the cracks areseparated from each other, the multiple cracks canbe detected individually. In detecting the cracks, thescanning method is used for a defect parameter determined from an electromagnetic force acting on an
HTS film. In the inductive method, a distribution of athreshold current is adopted.
2. If the crack is pretty close to the interface, the multiple
cracks is considered as a single crack. However, thecrack position can be detected collectively.

Fig. 6 V3 -I0 curves for the case with yA = 0 mm and dc = 5 mm.

Acknowledgment

Fig. 7 Contour maps of the relative error e for the case with (a)dc = 5 mm and (b) dc = 2 mm.

cracks. To this end, a relative error is defined bye ≡ |ITA − IT∗ |/ITA ,(4)

as a measure of the crack detection, where ITA is an analytic value of the threshold current determined from theMawatari’s theory [2].
In Figs. 7 (a) and (b), we show the contour maps of therelative error e for two types of the crack position. We seefrom Fig. 7 (a) that the distribution of the error e is contained in a region for e > 0% except for a HTS film edge
and the both ends of cracks. Moreover, the value of e is e
= 0% near the both ends of each crack. This result meansthat the crack position can be detected by examining thedistribution of the relative error e. In Fig. 7 (b), although
the values of e vanish near the left end of the crack C1 andthe right end of C3 , e at the top end of C2 is about e = 35%.
Consequently, the multiple cracks is considered as a single
crack, if the distance between the cracks is pretty close tothe interface. However, the crack position can be detected

This work was supported in part by Japan Society for the Promotion of Science under a Grant-inAid for Scientific Research (C) (No. 24560321, No.
26520204) and under a Grant-in-Aid for Encouragement of Scientists (No. 26919011). A part of this work
was also carried out with the support and under theauspices of the NIFS Collaboration Research program(NIFS13KNTS025, NIFS13KNXN267). Moreover, thenumerical computations were carried out on Hitachi
SR16000 XM1 of the LHD Numerical Analysis Server inNIFS.
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