Numerical Investigation on Applicability of Scanning Permanent
Magnet Method to Crack Detection in High-Temperature
Superconducting FilmTeruou Takayama1 and Atsushi Kamitani2
1

1

Department of Informatics, Faculty of Engineering, Yamagata University, Japan2
Graduate School of Science and Engineering, Yamagata University, Japan

Permanentzr MagnetIntroduction

As is well known, a critical current density jC is one of
the most important parameters for engineering applicationof high-temperature superconductors (HTSs). Although
the standard four-probe method is generally used to measure jC , it may lead to degradation of HTS characteristics.
For this reason, some contactless methods such as the inductive method [1], the hall sensor method, and the hallprobe method have been proposed. These methods havebeen applied to the measurement of jC -distributions for alarge-area samples such as an HTS tape or wire.
As a novel contactless method, Ohshima et al. proposed
the standard permanent magnet method for measuring jC
in an HTS thin ﬁlm. While moving a permanent magnet
above an HTS ﬁlm, the electromagnetic force Fz acting on
the ﬁlm is measured. Consequently, they found that themaximum repulsive force FM is roughly proportional to jC .
This means that jC can be estimated from the measured
value of FM . Although the standard method is used forthe determination of jC -distributions or the detection of
any cracks in an HTS tape [2], it becomes time-consuming
due to the measurement of FM at each measurement point.
Recently, Ohshima et al. have proposed a modiﬁed permanent magnet method [3]. In the method, the magnet
is ﬁrst placed just above an HTS sample at the constantdistance between an HTS surface and the magnet bottom,
and it is subsequently moved in the direction parallel to thesurface. As a result, they found that a spatial distributionof jC can be obtained from a measured Fz -distribution. Inaddition, they have concluded that, from the standpointof the saving of the measurement time, this method is superior to the standard one. In the following, this method
is called the scanning permanent magnet method.
In the previous study, a numerical code was developed
for analyzing the time evolution of a shielding current density in an HTS ﬁlm [4]. By using the code, the standard permanent magnet method was reproduced for anHTS ﬁlm without any cracks. The results of computations
showed that, even if the symmetry axis of the magnet approaches the ﬁlm edge, the maximum repulsive force FM is
almost proportional to jC . From this result, we conclude
that the standard method is applicable to the measurement of jC even near the edge. However, any cracks were
not at all taken into consideration in the code.
The purpose of the present study is to develop a numerical code for analyzing the time evolution of a shielding
current density in an HTS ﬁlm with a crack, and to reproduce the scanning method by using the code. Moreover,mhmcLyb(xm, ym, b/2) OaHTS FilmxFig. 1. A schematic view of a scanning method.

we investigate whether or not the scanning method is applicable to the crack detection in an HTS ﬁlm.
2Governing equation and numerical methodIn Fig. 1, we show a schematic view of a scanning permanent magnet method. A cylindrical permanent magnetof radius rm and height hm is placed above a rectangleshaped HTS ﬁlm of width a, length c, and thickness b.
Also, a constant distance between a magnet bottom and aﬁlm surface is denoted by L.
Throughout the present study, we use the Cartesian coordinate system hO : ex , ey , ez i, where the z-axis is parallelto the thickness direction and the origin O is the centroidof the ﬁlm. In terms of the coordinate system, the symmetry axis of the permanent magnet can be expressed as(x, y) = (xm , ym ). For the purpose of characterizing themagnet strength, we use the magnitude BF of a magnetic
ﬂux density at (x, y, z) = (0, 0, b/2).
As usual, we assume the thin-layer approximation [5]:the thickness of an HTS ﬁlm is suﬃciency thin that ashielding current density can hardly ﬂow in the thickness
direction. The shielding current density j is closely related
to the electric ﬁeld E through the J-E constitutive equation: E = E(|j|)[j/|j|]. As the function E(j), we use thepower law: E(j) = EC [j/jC ]N , where EC is the criticalelectric ﬁeld.
Under the above assumptions, the shielding current density j can be written as j = (2/b)(∇S × ez ) and its time
evolution is governed by the following integro-diﬀerentialequation [5]:µ0 ∂t (Ŵ S) + ∂t h B · ez i + (∇ × E) · ez = 0,(1)

where h i is an average operator over the thickness, andB is an applied magnetic ﬂux density. In addition, Ŵ S is

Fig. 2. Spatial distribution of the electromagnetic force
Fz∗ for xc = −5 mm.
RR
deﬁned by Ŵ S ≡ Ω Q(|x−x0 |)S(x0 , t)d2 x0 +(2/b) S(x, t).
0Here, x and x are position vectors in the xy-plane. The
explicit form of Q(r) is described in [5].
The initial and boundary conditions to (1) are assumed
as follows:
R S = 0 at t = 0, S = 0 on C0 , ∂S/∂s = 0 on
C1 and C1 E · tds = 0. Here, C0 is the outer boundaryof the square cross-section of the ﬁlm, whereas C1 is thecrack surface. In addition, t and s denote a tangential unit
vector on C1 and an arclength along C1 , respectively.
By applying the ﬁnite element method and the backward Euler method to the initial-boundary-value problemof (1), the nonlinear boundary-value problem is obtainedat each time step. By using the Newton method, it can besolved easily. By means of the above method, a numerical code is developed for analyzing the time evolution of ashielding current density in an HTS ﬁlm with a crack.
Throughout the present study, the geometrical andphysical parameters are ﬁxed as follows: a = 30 mm,
c = 10 mm, b = 1 µm, rm = 2.5 mm, hm = 3 mm, ym= 0 mm, EC = 0.1 mV/m, N = 20, BF = 0.3 T, andL = 0.5 mm. Moreover, we assume that jC is uniform allover the ﬁlm: jC = 1.5 MA/cm2 . Incidentally, the magnet is moved from the left end of the ﬁlm to the right one
except for Fig. 4, and the magnet position is controlled asxm (t) = −a(2t/τ0∗ − 1)/2. Here, τ0∗ is ﬁxed as τ0∗ = 140 s.

3

Numerical results

We investigate whether or not the scanning method isapplicable to the crack detection in an HTS ﬁlm. Throughout the present study, the crack is assumed to be parallel toy-axis and its shape is assumed to be a line segment. The
center position of the crack is denoted by (x, y) = (xc , 0)in xy-plate, and the crack size is given by Lc .
Let us ﬁrst investigate the inﬂuence of the crack sizeon the Fz∗ -distribution. In Fig. 2, we show the spatial
distribution of the electromagnetic force Fz∗ . We see fromthis ﬁgure that, for the case with Lc = 0 mm, Fz∗ becomesalmost constant for −7.5 mm . xm . 7.5 mm. On theother hand, a widely diﬀerent behavior of Fz∗ is observed
for the ﬁlm containing a crack: Fz∗ becomes an attractive force Fa for xc > xm , whereas it becomes a repulsive
force Fr for xc < xm . In addition, the magnitudes of Fa
and Fr increase with the crack size Lc . This means thatthe crack position and size can be estimated from the Fz∗ distribution.
Next, let us investigate the detectable range of a crack.
In Fig. 3, we show the Fz∗ -distribution for various crackpositions xc ’s. We see from this ﬁgure that, for xm ≤ −7.5

Fig. 3. Spatial distribution of the electromagnetic force
Fz∗ for Lc = 0.83 mm.

mm, Fa cannot be observed because of the edge eﬀect. Inconclusion, it is found that, when the crack is located nearthe ﬁlm edge, the crack position cannot be estimated fromFa and Fr .
Finally, let us investigate how the Fz∗ -distribution is affected by changing the scanning direction of the magnet.
For the case where the magnet is moved from the rightend to the left one, Fz∗ is calculated as a function of xm
and is depicted in Fig. 4. In this ﬁgure, two curves ofFz∗ intersect at the crack position xc . From this result, wecan conclude that, if two Fz∗ -distributions are obtained byusing two scanning directions of the magnet, the crack position can be accurately determined from the intersectionof two curves for Fz∗ (xm ).
References[1] J. H. Claassen et al., A contactless method for measurement of the critical current density and critical temperature of superconductiong ﬁlms, Rev. Sci. Instrum., vol. 62,no. 4, pp. 996-1004, 1991.
[2] S. Ohshima et al., Detection of critical current distribution of YBCO-coated conductors using permanent magnet
method, IEEE Trans. Appl. Supercond., vol. 21, no. 3, pp.
3385-3388, 2011.
[3] S. Hattori et al., Detection of smaller Jc region and damage
in YBCO coated conductors by using permanent magnet
method, Physica C, vol. 471, pp. 1033-1035, 2011.
[4] T. Takayama et al., Numerical investigation of inductive
and permanent-magnet methods to measure critical current density of HTS thin ﬁlms, IEEE Trans. Appl. Supercond., vol. 21, pp. 3360-3363, 2011.
[5] A. Kamitani et al., Magnetic shielding analysis of axisymmetric HTS plates in mixed state, IEEE Trans. Electron,
vol. E82-C, pp. 766-773, 1999.

Fig. 4. Spatial distribution of the electromagnetic force
Fz∗ for Lc = 0.83 mm and xc = −5 mm. Here, two typesof arrows indicate the scanning direction of the magnet.