AC MAGNETIZATION AND LOSSES OF STACKED HTS TAPES IN EXTERNAL MAGNETIC FIELD PERPENDICULAR TO THEIR WIDE FACES

AC MAGNETIZATION AND LOSSES OF STACKED HTS TAPES IN EXTERNAL MAGNETIC FIELD PERPENDICULAR TO THEIR WIDE FACES K. Kajikawa, Y. Fukuda, H. Moriyama, T. Hayashi, M. Iwakuma and K. Funaki Department of Electrical and Electronic Systems Engineering, Kyushu University ABSTRACT The AC magnetization and losses in stacked Bi-2223 Agsheathed tapes are experimentally and numerically evaluated in an external magnetic field perpendicular to their wide faces. The sample consists of 1, 3 or 1 pieces of non-twisted Bi-2223 multifilamentary tapes, which are vertically stacked. The magnetization in the external AC field, whose frequency is varied from 1 to 6 Hz, is measured with a saddle-shaped pickup coil. The measured AC magnetization and losses are compared not only with the theoretical curves based on the critical state model but also with the numerically calculated results which are derived using the voltage-current characteristics represented by the power law. 1. INTRODUCTION When high-t c superconducting (HTS) magnets for pulsed and AC uses are designed and fabricated, it is important to estimate AC losses of windings especially near the ends of the coils. This is because the HTS wires with high current capacity are tape-shaped and the AC losses cannot be ignored around the ends due to the magnetic fields perpendicular to the wide faces of the tapes [1-3]. The external magnetic field perpendicular to the wide faces is generally called a perpendicular magnetic field, and the AC loss dissipated in this type of external field is designated a perpendicular-field loss. Although there are a lot of articles [4-7] on the perpendicular-field losses in a single tape, the superconducting tapes are usually stacked face-to-face to enlarge the current capacity in the actual devices, and thus the exact evaluation of the perpendicular-field losses is desired in such actual electromagnetic configuration. The present authors [8] have experimentally evaluated the influence of stacks on the perpendicular-field losses observed in non-twisted Bi-2223 multifilamentary tapes. As a result, it was found that the AC losses decrease with increasing the number of tapes and asymptotically approach to a theoretical curve predicted with the critical state model in a superconducting infinite-slab with the thickness equal to the width of filamentary region in the tape. This can be understood by the demagnetization effect that the magnetic fields applied to a superconducting wire under consideration are reduced with increasing the stack numbers due to the magnetization in the other wires. The similar investigation has been also carried out by Suenaga et al. [9] and Oomen et al. [1]. Up to now, there are some reports [11-13] on a collec- tive interaction of turns in superconducting coils wound with metallic superconducting wires. Zenkevitch et al. [11] have measured hysteresis losses of solenoidal sample coils consisting of monofilamentary wires, and quantitatively explained the effect of magnetic field generated by the magnetization of other turns on the AC losses with a theoretical expression. Sumiyoshi et al. [12] have derived the magneticfield distribution and effective demagnetization factor in single-layer coils wound with multifilamentary wires, and investigated the influence of distance between adjacent turns on coupling losses in relatively small amplitudes of external magnetic fields. Furthermore, the authors [13] recently found that hysteresis losses in multifilamentary wires are determined by maximum magnetic fields applied to their filamentary regions utilizing effective demagnetization factors in double-layer non-inductive coils obtained by the theoretical consideration. In all the cases, the investigations were made in NbTi round wires. On the other hand, the influence of magnetic interaction between wires have been scarcely discussed in tape-shaped wires that are typical as HTS wires. In this paper, the influence of stack numbers on the perpendicular-field losses is studied in Bi-2223 Ag-sheathed tapes from a viewpoint of the magnetic interaction between them. The frequency dependence of AC magnetization in the stacks is also investigated in addition to the losses. Furthermore, the numerical simulation of electromagnetic-field distribution in a superconducting infinite-slab is carried out with the voltage-current characteristics represented by the power law, and the results are compared with the measured AC magnetization and losses. 2. EXPERIMENTS The specification of the sample wire is listed in Table 1. The sample wire is a non-twisted tape with 55 filaments of Bi- 2223, which was produced by the powder-in-tube technique. The superconductors in the sample tape are surrounded by pure silver. The thickness and width of the sample tape are.25 and 2.89 mm, respectively. In order to estimate an effective demagnetization factor below, it is assumed that the profiles of the cross sections of whole tape and filamentary region can be expressed by the equation, p q w( f) w( f) w( f ) w( f ) x / a + y/ b =1 where a w(f), b w(f), p w(f) and q w(f) are the geometrical parameters for the coordinate system in which the major and minor axes are parallel to the x- and y-directions, respectively. (1)

Table 1. Specification of non-twisted Bi-2223 Ag-sheathed tape. Width (and thickness) 2.89 mm (.25 mm) Number of filaments 55 Twist pitch Infinite Material (and ratio) of matrix Pure silver (2.8) Cross-sectional area Wire, A w.581 mm 2 Filamentary region, A f.41 mm 2 Silver matrix, A Ag.428 mm 2 Critical current (A) 2 15 1 5 critical current n-value.2.4.6.8.1 External DC magnetic field (T) Fig. 1. Critical currents and n-values of sample wire. 2 15 1 5 n-value In the case of the present wire, these are 2a w = 2.89 (mm), 2b w =.25 (mm), p w = 3, q w = 2 and a f =.95a w, b f =.8b w, p f = 6, q f = 2, where the subscripts, w and f, represent the quantities for the whole tape and the filamentary region, respectively. The surface of the sample tape is electrically insulated. Fig. 1 shows the dependence of critical currents I c and n-values in the sample wire on the external DC magnetic field. The measurements were carried out in liquid nitrogen by mean of the usual four-probe method, and the external fields were applied perpendicular to the wide faces of the tape. The critical currents I c were determined by the electric-field criterion of 1 µv/m, and the n-values were obtained from the slopes at I c on the measured voltage-current characteristics. The solid and dotted lines in Fig. 1 represent approximated curves of the critical currents I c and n-values in the background fields from.1 to.1 T, respectively. Three kinds of samples, whose numbers of stacked tapes were 1, 3 and 1, were prepared in this paper. The lengths of all the sample tapes were about 39 mm. The external AC magnetic field was applied perpendicular to the wide faces of the tapes in the sample bundles with an excitation magnet, and the magnetization of the samples was observed with a saddle-shaped pickup coil located around them. The AC losses of the samples can be obtained by evaluating the area of the magnetization curve. The detailed description of the experimental setup was already presented in [8]. The measurements were carried out at the frequency of 1-6 Hz. Fig. 2 shows the dependence of the perpendicular-field losses measured at 1 Hz in liquid nitrogen on the amplitude of external magnetic field. The vertical axis in this figure is plotted as the AC loss per unit volume of the filamentary regions in the tapes per a cycle of the external field. It can be found in Fig. 2 that the perpendicular-field losses decrease with increasing the number of tapes in the stacks in the range of small field amplitude. For larger amplitude, on the other hand, the AC losses are scarcely dependent on the stack numbers. Fig. 3 shows the experimental results of AC losses in the 1-tape stack for different frequencies. The typical example of the magnetization curve is also given in Fig. 4. One can see in Figs. 3 and 4 that the AC losses and magnetization of the stacked tapes strongly depend on the frequency in the perpendicular magnetic field. 1 5 1 4 1 3 1 2 1 1 experimental results (1 tape) experimental results (3 tapes) experimental results (1 tapes) theoretical curve (1 tape) theoretical curve (3 tapes) theoretical curve (1 tapes) 77 K, 1 Hz 1 1-3 1-2 1-1 Amplitude of external magnetic field, µ H m (T) Fig. 2. Dependence of perpendicular-field losses at 1 Hz for different tape stacks on amplitude of external magnetic field. 1 5 1 4 1 3 1 2 1 1 experimental results (1Hz) experimental results (1Hz) experimental results (6Hz) 77 K, 1 tapes 1 1-3 1-2 1-1 Amplitude of external magnetic field, µ H m (T) Fig. 3. Dependence of perpendicular-field losses in 1-tape stack for different frequencies on amplitude of external magnetic field.

Negative magnetization, µ M (T).8.4 -.4 µ H m experimental results (1Hz) experimental results (3Hz) experimental results (1Hz) experimental results (3Hz) experimental results (6Hz) =.1 (T) 77 K, 1 tapes -.8 -.12 -.8 -.4.4.8.12 External magnetic field, µ H e (T) Fig. 4. Experimental results of AC magnetization in 1-tape stack for different frequencies in external-field amplitude of.1 T. 3. EFFECTIVE DEMAGNETIZATION FACTORS AND AC LOSS EVALUATION In order to evaluate the influence of the magnetic interaction between superconducting tapes theoretically, let us estimate effective demagnetization factors which depend on the stack number of the tapes. Since the multifilamentary tapes used in the experiments have no twisting of filaments, the whole of the filamentary region behaves like a superconductor in the external transverse magnetic field. In this paper, it is assumed that the filamentary region in the tape can be regarded as a uniform magnetic substance. Such an assumption may be usable for the case that the external magnetic field is much smaller than the full penetration field or that there are a lot of filaments in twisted multifilamentary wires. In order to consider only the geometrical effect on the AC loss, however, let us neglect the influences of the distribution of currents and magnetic fields inside the tapes. Under the situation of the tapes stacked at intervals of 2d in the y-direction, whose filamentary regions are regarded as magnetic substances in the form of an elliptic cylinder with an infinitely long axis in the z-direction, it is assumed that all the filamentary regions in the tapes are uniformly magnetized into M in the same direction as the external magnetic field H e applied in the y-direction. In this case, if the contribution of magnetization from each tape in a stack is superposed and averaged, the uniform magnetic flux density B iy in the y-direction inside the filamentary region is given by [14] B = µ H + µ b a b M µ + + k 2ab + e iy e 2 2u j = 1i= 1, i j c 1 M (2) where µ is the magnetic permeability of the free space, k the number of the stacked tapes, a and b (b < a) represent the major and minor semi-axes on the cross section of the k k ( ) elliptic cylinder in the x- and y-direction, and u = ln[(y /c) + {(y /c) 2 + 1} 1/2 ] with y = 2 j i d and c = (a 2 b 2 ) 1/2. Here, the magnetic field at the center of the j-th wire under consideration is used as the contribution due to the magnetization of the other i-th wire. In Eq. (2), the second term on the right-hand side represents the inside magnetic field generated by the magnetization of the wire under consideration itself, whereas the third one is the contribution due to the magnetization of the other wires. When the effective demagnetization factor N e is defined by Hs = He NeM (3) with the magnetic field H s applied to the filamentary region, the inside magnetic flux density B iy is generally given by Biy = µ ( Hs + M) = µ He + µ ( 1 Ne) M (4) By comparing Eq. (2) with (4), the expression of the factor N e can be obtained as follows: k k a ab Ne = 1 (5) a + b k u j i i j c ( 2 2 2 = 1 = 1, 1 + e ) Table 2 shows the effective demagnetization factors N e calculated with Eq. (5) for the samples used in the experiments, in which the distance 2d between the adjacent tapes was.21 mm. As seen in Table 2, the increase in the stack number k decreases the factor N e. Next, let us evaluate maximum magnetic fields applied to the filamentary regions in the tapes. Since the magnetic field applied to the filamentary region is generally expressed by Eq. (3), the maximum H sm should be given by [ ] max H H t N M t = () () sm e e As seen in Eq. (6), the maximum fields H sm can be evaluated with the magnetization curves observed in the experiments and the effective demagnetization factors N e listed in Table 2. The experimental results of the perpendicular-field losses in Fig. 2 are replotted in Fig. 5 as a function of the maximum magnetic field H sm applied to the filamentary region. It is found in Fig. 5 that the AC losses as a function of H sm scarcely depend on the number of tapes in the stacks, and that they are plotted on a master curve. The replot of Fig. 3 is also given in Fig. 6, where the theoretical curve for an infinite-slab with the width of 2a f is added as a solid line on the basis of the critical state model [15]. It is found in Fig. 6 that the perpendicular-field losses as a function of the maximum applied field H sm have the strong dependence on frequency, and that they differ from the prediction of the critical state model. Table 2. Effective demagnetization factors of stacked sample tapes. number of tapes, k effective demagnetization factor, N e 1.944 3.846 1.633 (6)

1 5 1 4 1 3 1 2 1 1 experimental results (1 tape) experimental results (3 tapes) experimental results (1 tapes) numerical results for slab 77 K, 1 Hz 1 1-3 1-2 1-1 Maximum of applied magnetic field, µ H sm (T) Fig. 5. Dependence of perpendicular-field losses at 1 Hz for different tape stacks on maximum of applied magnetic field. 1 5 1 4 1 3 1 2 1 1 experimental results (1Hz) experimental results (1Hz) experimental results (6Hz) numerical results (1Hz) numerical results (1Hz) numerical results (6Hz) Irie-Yamafuji model 77 K, 1 tapes 1 1-3 1-2 1-1 Maximum of applied magnetic field, µ H sm (T) Fig. 6. Dependence of perpendicular-field losses in 1-tape stack for different frequencies on maximum of applied magnetic field. The solid line shows the theoretical curve based on the critical state model, whereas the broken lines are the numerical results for the slab whose transport property is represented by the power law. 4. NUMERICAL EVALUATION OF AC MAGNETIZATION AND LOSSES In order to calculate the losses of the stacked tapes in the perpendicular magnetic field, the electromagnetic-field distribution is numerically evaluated in a superconducting slab with infinitely wide surfaces, whose demagnetization factor is equal to zero. If the x-axis is perpendicular to the surfaces of the slab of width D and the external AC magnetic field H e is applied in the y-direction, the electromagnetic quantities in the slab satisfy one-dimensional Maxwell s equations, Ext, Bxt, (7) x t 1 Bxt (, ) = Jxt (, ) (8) µ x where B is the local magnetic flux density in the y-direction, E and J are the local electric field and current density in the z-direction, respectively. In Eq. (8), the term for a displacement current is ignored as usual. It is also assumed that the E-J characteristics of the slab are expressed by the n-value model, ( ) = ( ) Ext E Jxt nb ( ) (, ) (, )= c (9) λjc( B) where J c is the critical current density in the superconductor, λ the volume fraction of superconductor in the filamentary region, and E c the electric-field criterion of the average critical current density λj c. In carrying out the numerical calculations, the solid and dotted lines in Fig. 1 were used for the critical currents I c and n-values of the sample wire. The average critical current density λj c was given by converting the critical current I c into the current density in the filamentary region as λj c (B) = I c (B) / A f with the cross-sectional area of filamentary region, A f. The distribution of B, E and J in the slab is numerically calculated by means of the finite difference method with Eqs. (7)-(9) under the boundary conditions of B(, t) = B(D, t) = µ H e (t). By using the obtained distribution of magnetic flux density B, the AC magnetization M per unit volume of the filamentary region can be derived by 1 D Mt ()= (1) D Bxtdx (, ) He() t µ Therefore, the AC loss W per unit volume per a cycle is finally given by W = µ MdH e (11) The solid line in Fig. 5 and the broken lines in Fig. 6 show the numerical results of AC losses in the slab. Here, it is assumed that the width D of the slab is equal to the major axis of the filamentary region, 2a f. As shown in Figs. 5 and 6, the experimental results of AC losses are well reproduced by the numerical simulation. Fig. 7 show the comparison between the experimental and numerically calculated magnetization curves at the different frequencies in the external-field amplitude of.1 T. The horizontal axes in Fig. 7 are represented by the magnetic field H s applied to the filamentary region for the experimental results in the stacked tapes. The theoretical curves obtained with the Irie-Yamafuji model [15] are also drawn by dashed lines as a reference. When the frequencies are below 1 Hz, the numerical results have a good agreement with the experimental ones. In the cases of 3 and 6 Hz, on the other hand, the measured magnetization is rather larger than the results of numerical simulation. This may be due to the effect of eddy currents induced in a silver matrix, which

Negative magnetization, µm (T).6 (a) 1Hz -.6.6 (c) 3Hz experimental results numerical results Irie-Yamafuji model (b) 1Hz (d) 6Hz -.6 -.12.12 -.12.12 Applied magnetic field, µ H s (T) Fig. 7. Comparison between experimental and numerical results of AC magnetization in external magnetic fields with the amplitude of.1 T and the different frequencies. The theoretical curves based on the critical state model are also drawn by dashed lines. The experimental results of AC losses are 4.7, 6.6, 8. and 1.5 kj/ m 3 at 1, 1, 3 and 6 Hz, while the numerical results are 5.1, 6.7, 7.3 and 8.1 kj/m 3, respectively. is not taken into account above. Let us estimate roughly the frequency dependence of the eddy-current losses. When the bundle of 1 tapes is regarded as an effective slab with infinitely wide surfaces, the width of slab, D e, becomes 2.77 mm. In this case, the eddy-current loss W e per unit volume of the filamentary region per a cycle of the external magnetic field with the amplitude of H m is expressed by [16] W A ( De ) ( De ) e = Ag 2 πδ sinh / δ sin / δ µ Hm (12) Af D e cosh ( De / δ)+ cos ( De / δ) where A Ag is the cross-sectional area of silver matrix, and δ the skin depth given by δ = [ρ / (µ πf)] 1/2 with the resistivity ρ of silver and the frequency f. If the resistivity ρ of silver matrix is equal to 3 1 9 Ωm, the eddy-current losses W e in the amplitude H m of.1 T are calculated as.44,.44, 1.3 and 2.6 kj/m 3 at 1, 1, 3 and 6 Hz, respectively. Since these values seem to account for the discrepancies between the experimental and numerical results in Figs. 6 and 7, it is concluded that the eddy-current losses cannot be ignored in the perpendicular-field losses near the commercial frequency. 5. PREDICTION OF PERPENDICULAR-FIELD LOSSES In this section, let us predict the perpendicular-field losses in a stack with any number of tapes using the AC loss property in the superconducting slab obtained by the numerical simulation. Although the maximum magnetic field H sm applied to the filamentary region can be generally expressed by Eq. (6), H sm is simplified as Hsm = Hm NeMm (13) where M m is the magnetization per unit volume of the fila- Negative magnetization, µ M (T).2.15.1.5 µ H m µ H sm Eq.(13) for 1 tape.2.4.6.8.1 Maximum of applied magnetic field, µ H sm (T) Fig. 8. Initial-magnetization characteristics of superconducting slab. mentary region at H e = H m. The solid line in Fig. 8 represents the numerical results of the magnetization M m of the slab at the peak of the external magnetic field, H m (= H sm ), whereas the dotted line is given as an example using Eq. (13) for the external-field amplitude of.4 T and the effective demagnetization factor of.944 in a single tape. As indicated in Fig. 8, the intersection of the dotted line and the horizontal axis is the amplitude H m of external field. On the other hand, the intersection of the dotted and solid lines gives the maximum of applied field, H sm. In this way, H m and H sm can be related with each other in any number of tapes. Since the slope of the dotted line in Fig. 8 does not vary for constant factor N e, H sm becomes larger than H m in the case of small field amplitude. In very large amplitude, on the other hand, H sm and H m are nearly equal to each other. The AC losses obtained by means of this technique are drawn as curves in Fig. 2. It can be found in Fig. 2 that the experimental results in the bundles with the finite number of tapes are almost reproduced by the theoretical evaluation described here. 6. CONCLUSIONS The AC magnetization and losses of stacked Bi-2223 Ag-sheathed tapes without twisting of the filaments were experimentally and numerically evaluated in an external magnetic field perpendicular to their wide faces. It was confirmed that the perpendicular-field losses decrease with increasing the number of tapes in the stacks in the range of small field amplitude. In the larger amplitude, on the other hand, the losses scarcely depended on the stack number. By using the theoretical expression of effective demagnetization factors, the perpendicular-field losses were plotted on a master curve for the maximum magnetic field applied to the filamentary region in the tape. Next, the numerical calculation for the superconducting infinite-slab, including the voltage-current characteristics represented by the power law, reproduced the experimental results of the magnetization

curves well at frequencies below 1 Hz. It was also suggested that the eddy-current losses in the silver matrix have to be taken into account for the exact evaluation of the perpendicular-field losses when the external magnetic field with the commercial frequency is applied to the tapes. Furthermore, by utilizing the magnetic property calculated numerically in the slab, the experimental results of the perpendicular-field losses in the stacks with the finite number of tapes were almost reproduced as a function of the amplitude of the external magnetic field. The filamentary regions in the non-twisted multifilamentary tapes focused on in this paper behave like isolated superconductors for the application of the external transverse magnetic field, and therefore the hysteresis losses become a main component in the dissipation. The further discussion may be needed for recently developed wires, in which the electromagnetic coupling between filaments is suppressed to achieve lower AC losses, such as twisted wires, and wires with oxide barriers around the filaments. 7. ACKNOWLEDGMENT The authors would like to thank Showa Electric Wire & Cable Co., Ltd. for the preparation of the sample wires. 8. REFERENCES [1] K. Funaki, M. Iwakuma, K. Kajikawa, M. Takeo, J. Suehiro, M. Hara, K. Yamafuji, M. Konno, Y. Kasagawa, K. Okubo, Y. Yasukawa, S. Nose, M. Ueyama, K. Hayashi, and K. Sato, Development of a 5 kva-class oxide-superconducting power transformer operated at liquid-nitrogen temperature, Cryogenics, Vol. 38, No. 2, pp. 211-22, 1998.2. [2] M. Iwakuma, K. Funaki, K. Kajikawa, H. Kanetaka, H. Hayashi, K. Tsutsumi, A. Tomioka, M. Konno, and S. Nose, Ac loss and temperature distribution in a cryocooler-cooled 1T pulse coil with an interlayer-transposed parallel conductor, Adv. Supercond. XI, Vol. 2, pp. 963-966, 1999. [3] M. Iwakuma, K. Funaki, K. Kajikawa, H. Tanaka, T. Bohno, A. Tomioka, H. Yamada, S. Nose, M. Konno, Y. Yagi, H. Maruyama, T. Ogata, S. Yoshida, K. Ohashi, K. Tsutsumi, and K. Honda, Ac loss properties of a 1MVA single-phase HTS power transformer, IEEE Trans. Appl. Supercond., Vol. 11, No. 1, pp. 1482-1485, 21.3. [4] K. Miyamoto, N. Amemiya, N. Banno, M. Torii, E. Hatasa, E. Mizushima, T. Nakagawa, H. Mukai, and K. Ohmatsu, Measurement and FEM analysis of magnetization loss in HTS tapes, IEEE Trans. Appl. Supercond., Vol. 9, No. 2, pp. 77-773, 1999.6. [5] J.J. Rabbers, O. van der Meer, W.F.A. Klein Zeggelink, O.A. Shevchenko, B. ten Haken, and H.H.J. ten Kate, Magnetisation loss of BSCCO/Ag tape in uni-directional and rotating magnetic field, Physica C, Vol. 325, Nos. 1-2, pp. 1-7, 1999.11. [6] E. Martínez, Y. Yang, C. Beduz, and Y.B. Huang, Experimental study of loss mechanisms of AgAu/PbBi-2223 tapes with twisted filaments under perpendicular AC magnetic fields at power frequencies, Physica C, Vol. 331, Nos. 3-4, pp. 216-226, 2.5. [7] P. Fabbricatore, S. Farinon, F. Gömöry, and S. Innocenti, Ac losses in multifilamentary high-t c tapes due to a perpendicular ac magnetic field, Supercond. Sci. Technol., Vol. 13, No. 9, pp. 1327-1337, 2.9. [8] M. Iwakuma, M. Okabe, K. Kajikawa, K. Funaki, M. Konno, S. Nose, M. Ueyama, K. Hayashi, and K. Sato, Ac losses in Bi2223 multifilamentary wires exposed to an magnetic field perpendicular to the wide surface, Adv. Supercond. X, Vol. 2, pp. 833-836, 1998. [9] M. Suenaga, T. Chiba, S.P. Ashworth, D.O. Welch, and T.G. Holesinger, Alternating current losses in stacked Bi 2 Sr 2 Ca 2 Cu 3 O 1 /Ag tapes in perpendicular magnetic fields, J. Appl. Phys., Vol. 88, No. 5, pp. 279-2717, 2.9. [1] M.P. Oomen, J.J. Rabbers, B. ten Haken, J. Rieger, and M. Leghissa, Magnetisation loss in stacks of high-t c superconducting tapes in a perpendicular magnetic field, Physica C, Vol. 361, No. 2, pp. 144-148, 21.9. [11] V.B. Zenkevitch, V.V. Zheltov, and A.S. Romanyuk, Hysteresis losses in superconductors of round cross-section with collective interaction, Cryogenics, Vol. 18, No. 2, pp. 93-99, 1978.2. [12] F. Sumiyoshi, F. Irie, and K. Yoshida, The effect of demagnetization on the eddy-current loss in a single-layered multifilamentary superconducting coil, J. Appl. Phys., Vol. 51, No. 7, pp. 387-3811, 198.7. [13] K. Kajikawa, A. Takenaka, M. Iwakuma, and K. Funaki, Influences of collective interaction of turns on transverse-field losses in superconducting multifilamentary wires, Inst. Phys. Conf. Ser. No 167, Vol. 1, pp. 931-934, 2. [14] K. Kajikawa, M. Nishimura, H. Moriyama, M. Iwakuma, and K. Funaki, Quantitative evaluation of perpendicular-field losses in stacked oxide superconducting tape-shaped wires, Trans. IEE Japan, Vol. 121-B, No. 1, pp. 1283-1289, 21.1, in Japanese. [15] F. Irie and K. Yamafuji, Theory of flux motion in non-ideal type-ii superconductors, J. Phys. Soc. Jpn., Vol. 23, No. 2, pp. 255-268, 1967.8. [16] K. Funaki and F. Sumiyoshi, Multifilamentary wires and conductors, Sangyo Tosho, Tokyo, 1995, pp. 125-129, in Japanese.