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Mechanism of decay of trapped magnetic field in HTS bulk caused by application of AC magnetic field
Journal of Materials Processing Technology 161 (2005) 52–57
Mechanism of decay of trapped magnetic field in HTS bulk caused by application of AC magnetic field O. Tsukamotoa,∗ , K. Yamagishia , J. Ogawaa , M. Murakamib , M. Tomitab b
a Yokohama National University, 79-5 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan Superconductivity Research Laboratory, ISTEC, 1-16-25 Shibaura, Minato-ku, Tokyo 105-0023, Japan
Abstract In our previous work, it was observed that trapped magnetic field in an high-temperature superconductor (HTS) bulk was decayed and even erased by application of AC external field whose amplitude was much smaller than the peak value of the trapped magnetic field. Therefore, knowledge on the mechanism of the decay of the trapped magnetic field is important to design the machines and to develop a method to suppress the decay. This work studies a mechanism of the decay due to application of AC magnetic field by a numerical analysis and experiments. An analytical model is proposed to explain the mechanism based on the thermal effect due to AC losses in the bulk. In the model it is assumed that the AC loss characteristic follows the Bean model, which was experimentally demonstrated also in our previous work. We conducted experiments in which temperature rise of the bulk was measured by thermocouples and the decay of the trapped magnetic was observed by a hall probe. Results numerically calculated from the analytical model well agreed with those obtained from the experiments and we consider it was clarified that the decay of the trapped field was caused by the thermal effect due to AC loss. Once the mechanism has been clarified, methods to suppress the decay can be easily developed. © 2004 Elsevier B.V. All rights reserved. Keywords: HTS bulk; Trapped magnetic field; AC loss; HTS motor
1. Introduction and background Recent progresses of high-temperature superconductor (HTS) bulk technology have made various applications realistic. Especially, compact and highly efficient electric motors, magnetic levitation systems and flywheel energy storage systems are most promising applications [1–3]. The HTS bulks in these applications are exposed to magnetic field perturbation. The perturbation causes AC losses in the bulks and decay of the magnetic field trapped in the bulk. According to the Bean model, the reduction of the trapped field does not depend on the frequency, and the tapped field is decreased at the first cycle of the AC external field by the amplitude of the external field and, after this first reduction, the trapped field is unchanged, if the bulk temperature is constant. In a previous work, we measured the magnetic field distributions on the surface of the bulk with the trapped field after applying AC magnetic field. It was observed that
amount of the reduction of the trapped field was dependent on the frequency of the AC external field and that the trapped field was kept decreasing even to zero during the application of the relatively high external field [4]. This result is different from that is predicted from the simple Bean model of constant bulk temperature. To apply HTS bulks to electric machines knowledge on the mechanism of the decay of the trapped magnetic field is important to design the machines and to develop the countermeasures against the decay. In this work, mechanism of the decay of the trapped magnetic field is studied from the standpoint of thermal effect due to AC losses in the bulk and the validity of the mechanism is assessed by comparison of analytical and experimental results.
2. Analysis 2.1. Behavior of trapped magnetic field
∗
Corresponding author. Tel.: +81 45 339 4124; fax: +81 45 338 1157. E-mail address: [email protected] (O. Tsukamoto).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.07.009
Behavior of the trapped magnetic field in an HTS bulk subject to external AC magnetic field is analyzed here as-
O. Tsukamoto et al. / Journal of Materials Processing Technology 161 (2005) 52–57
53
magnetic field. The depth of the penetration of the external magnetic field rm is given by the following equation: rm = r0
Bm Bmp
(1)
where Bmp is the full penetration magnetic field and equal to µ0 Jc r0 . Fig. 1(c) shows the areas where the AC external magnetic field penetrates and AC shielding currents flow. AC losses are dissipated in these penetration areas. When the AC external field is reduced gradually to zero, the magnetic field and shielding currents in the penetration area become zero as shown in Fig. 1(d). If it is assumed that the bulk temperature is constant during the application of the AC external magnetic field, the reduction of the trapped magnetic field occurs at the first cycle of the AC external field and does not depend on the frequency nor on the duration time of the AC external field. However, experimental results were different from those theoretical predictions [4]. The AC losses caused by the magnetic flux movements in the penetration areas raise the bulk temperature, which causes the reduction of Jc and the increase of rm . If rm exceeds r0 then the trapped magnetic field disappears. Therefore, it is considered that influence of the AC external magnetic field on the trapped magnetic field is due to the temperature rise of bulk caused by AC losses. 2.2. Temperature rise of bulk caused by AC losses
Fig. 1. Distributions of magnetic field and current density in the cylindrical bulk subject to AC external field of amplitude Bm parallel to the cylinder axis. Bm is lower than the initial peak of the trapped magnetic field Bp0 : (a) initial state; (b) after one cycle application of AC external magnetic field; (c) during application of AC external magnetic field; (d) after gradual decrease of AC external magnetic field to zero.
suming the Bean model. The bulk is assumed to be a cylinder of infinite length for the simplicity of the analysis. Fig. 1 shows distributions of the trapped magnetic field B and current density J in an HTS cylinder of radius r0 subject to AC magnetic field of amplitude Bm parallel to the cylinder axis. Fig. 1 is for the case that Bm is lower than the peak of the initial trapped magnetic field Bp0 . Jc is the critical current density of the bulk. In the initial state, the magnetic field is trapped in the bulk by the superconducting current as shown in Fig. 1(a). Applying the external magnetic field it starts to penetrate into the cylinder from the surface, and shielding current whose density Jc is induced in the bulk. At the end of one cycle application of the AC external magnetic field, the peak of the trapped magnetic field is reduced by the magnitude Bm and the shielding current induced by the external field is hold in the bulk as shown in Fig. 1(b). If the temperature of the bulk and Jc are not changed, the distributions of the magnetic field and current density are the same as of Fig. 1(b) even after multiple cycles application of the AC
In the experiment described in the next section, we used a bulk of short-cylinder shape. Therefore, in the following analysis, the temperature rise of a short-cylinder bulk is studied. It has been shown that the AC loss characteristics of a short-cylinder YBCO bulk were well described by a cylinder model of infinite length following the Bean model [5]. Losses in a superconductor cylinder subject to AC external magnetic field of amplitude Bm parallel to the cylinder axis are given by the following equations [6]: Q=
2 2Bm µ0
2 2Bm Q= µ0
2β β2 − 3 3
2 1 − 3β 3β2
for β < 1
(2)
for β > 1
(3)
where β=
Bm Bmp
(4)
where Q (J/m3 per cycle) is the loss per cycle per unit volume. When β ≥ 1, the external magnetic field fully penetrates into the bulk (rm > r0 ) and the trapped magnetic field disappears when the application of the external field is terminated.
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O. Tsukamoto et al. / Journal of Materials Processing Technology 161 (2005) 52–57
It is assumed in the analysis that dependence of Jc on the bulk temperature T is given by Jc (T ) =
Jc0 (Tc − T ) (Tc − T0 )
(5)
where T0 is the coolant temperature, Jc0 the critical current density at T = T0 and Tc the critical temperature of the bulk. The thermal equilibrium equation of a short-cylinder bulk of radius r0 and thickness d subject to AC external field is expressed as follows, assuming the bulk temperature is uniform: Cp
dθ(t) + λθ(t) = P(t) dt
(6)
where θ(t) = T − T0 ,
P(t) = Qfv
where Cp (J/K) and λ (W/K) are the heat capacity of the bulk and the heat transfer coefficient from the bulk surface to the coolant, respectively, f is the frequency of AC external magnetic field and v = πr0 2 d is the volume of the bulk. The peak value of the trapped magnetic field Bp (T) is given by µ0 Jc (T )(r0 − rm ) for β < 1 Bp (T ) = (7) 0 for β ≥ 1 As T is increased by the AC loss, Jc decreases and rm increases. Thus, Bp decreases and when β exceeds 1, Bp becomes zero. From the above equations, time evolutions of T, β and Bp can be calculated by a numerical analysis.
Fig. 2. Trapped magnetic field distribution of the YBCO bulk: (a) sample arrangement; (b) relative positions of the thermocouples TC1–TC5 to the trapped magnetic field pattern.
was covered with a polystyrene foam layer of 5 mm thickness to thermally insulate the thermocouples from the liquid nitrogen (Fig. 3(a)). The side surface of the bulk directly contacted to liquid nitrogen. Fig. 3(b) shows relative positions of the thermocouples to the distribution of the trapped magnetic field. The bulk sample was placed in a bore of a copper coil that applied AC magnetic field to the bulk parallel to the axis of the short-cylinder bulk. The bulk and the copper coil were put in a liquid nitrogen bath (T0 = 77.3 K). During the application of the AC magnetic field, the temperature could not be measured because large inductive noise caused by the AC field
3. Experiment To investigate relation between temperature rise and behavior of trapped magnetic field in a bulk subject to AC external field, we performed an experiment using a short-cylinder YBCO bulk. 3.1. Experimental setup The bulk used in the experiment was epoxy impregnated YBCO bulk of 45 mm diameter and 20 mm thickness. The bulk was magnetized by the field cooling method applying DC external magnetic field parallel to the bulk axis. The maximum peak value of the trapped magnetic field was 0.31 T on the bulk surface. Magnetic field distribution was observed by a hall probe. Fig. 2 shows the magnetic field distribution of 0.5 mm above the bulk surface just after the magnetization. Temperature distributions of the bulk were measured by five Cr versus constantan thermocouples TC1–TC5 attached to the bulk surface. The sample arrangement and the positions of the thermocouples are shown in Fig. 3(a) and (b), respectively. The bulk was glued to the GFRP sample holder by vacuum grease and the bulk surface with thermocouples
Fig. 3. Experimental arrangement of the YBCO sample. (a) During application of AC external magnetic field. At t = 0 s the external field of Bm = 0.1 T (61 Hz) was applied. (b) During cooling process. At t = 0 s the application of the external field was finished.
O. Tsukamoto et al. / Journal of Materials Processing Technology 161 (2005) 52–57
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was mixed in the signals from the thermocouples. Therefore, the AC current was applied for a certain time and the copper coil current was shut off to measure the signals from the thermocouples. Change of the bulk temperature was negligibly small in several seconds. The experiment to investigate temperature rises was conducted for the bulk without trapped magnetic field. 3.2. Experimental results Fig. 4(a) and (b) shows temperature distributions measured by the thermocouples TC1–TC5 for various duration time of application of the 61 Hz AC external field of Bm = 0.1 T and during the cooling process after the application of the external field finished. The temperature distribution was almost uniform, though above 88 K it became rather obvious that the temperature in the peripheral area was lower than that in the central area, which was because the cooling in the peripheral was better. Fig. 5 shows average temperature on the bulk surface T versus duration time of the application of AC external magnetic field td for various values of Bm . In the graph analytical results are also shown that are explained in the next section. For the application of AC magnetic field of Bm = 0.1 T for more than 100 s, the bulk temperature went up to close to Tc which was about 91 K. In other cases of Bm = 0.08, 0.06, 0.04 T, the bulk temperatures reached steady-state values which were lower than Tc .
Fig. 4. Time-varying temperature distribution measured by thermocouples during application of AC and cooling process.
Fig. 5. Average temperature on the bulk surface T vs. duration time of 61 Hz AC magnetic fields td for various amplitudes of Bm . Lines in the graph were analytically calculated. (a) Before application of AC field; (b) Bm = 0.08 T, td = 240 s; (c) Bm = 0.1 T, td = 60 s; (d) Bm = 0.1 T, td =80 s; (e) Bm = 0.1 T, td = 100 s.
The trapped magnetic field distributions were measured on the surface of the polystyrene foam insulation layer (5 mm above the bulk surface) applying the AC magnetic field of a constant amplitude Bm for a certain time td and decreasing the amplitude to zero gradually in 1 s. Fig. 6 shows various
Fig. 6. Patterns of the trapped magnetic field in the bulk after the application of the AC external magnetic field. (a) Time evolutions of β; (b) time evolutions of calculated and experimental Bp normalized by Bp0 .
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O. Tsukamoto et al. / Journal of Materials Processing Technology 161 (2005) 52–57
patterns of the trapped magnetic field. Fig. 6(a) is the pattern before the application of the AC external field. Fig. 6(b) is the pattern after the 240 s application of Bm = 0.08 T. In this case the bulk temperature reached a steady-state 82.9 K and the peak trapped magnetic field was decreased by 30% from the value before the application of the AC magnetic field. Fig. 6(c)–(e) are after the 60, 80 and 100 s applications of Bm = 0.1 T, respectively. In the case of Fig. 6(c), the bulk temperature reached 84.4 K and the peak of the trapped magnetic field was decreased by 41% from the initial state. In the case of Fig. 6(d), the bulk temperature reached 86.6 K and the trapped magnetic field almost disappeared. In the case of Fig. 6(d), the bulk temperature reached 89.9 K and the trapped magnetic field completely disappeared. It is obvious from these results that the reduction and disappearance of the trapped magnetic field were caused by the temperature rise due to the AC losses and that the trapped magnetic field disappeared only when the bulk temperature rose close to Tc and the AC external magnetic field fully penetrated into the bulk. It should be noted that the disappearance of the trapped field is not caused by the normal transition of the bulk. If the bulk becomes normal, there is no energy enough to raise the bulk temperature, because the AC losses are generated only in the superconducting state bulk. It was experimentally demonstrated that eddy current losses in the normal state bulk caused by the AC external magnetic field was negligibly small compared with the AC losses in the superconducting state bulk. Therefore, the bulk temperature does not exceed Tc even in the case of the trapped magnetic field disappears. Fig. 7. Calculated time evolutions of β and Bp .
4. Comparison of the experimental and analytical results From Eqs. (1)–(7), the time evolutions of T, Bp and β were numerically calculated. The parameters used in the calculation are listed in Table 1. The time evolutions of T are shown in Fig. 5. Jc0 was estimated from the measured AC loss value and Eq. (2). λ was estimated from measured steady-state temperature of the bulk and the calculated AC loss value. τ was experimentally determined from the time evolution of T in the cooling process. As seen in Fig. 5, the calculated time evolutions of T agree well with the experimental results. Fig. 7(a) and (b) shows the numerically calculated time evolutions of β and ≡p for various values of Bm . As seen in Fig. 7, when t > 93 s for Bm = 0.1 T, β exceeds 1 and Bp becomes 0. On Table 1 Parameters used in the calculation Bulk size
Ø 45 mm, 19.5 mm thick
Jc0 ( A/m2 at 77.3 K) λ (W/K) Cp (J/K) τ = Cp /λ (s) Tc (K)
1.61 × 107 0.60 30.0 50 91.0
the other hand, for t < 93 s for Bm = 0.1 T, β is less than 1 and Bp remains. It is obvious from Fig. 7 for Bm = 0.08 T that β stays below 1 and that Bp remains even in the steady state. Experimental values of Bp in Fig. 6(b)–(e) are plotted in Fig. 7(b) normalized by the value of Bp0 in Fig. 6(a). Bp in Fig. 6(c)–(e) are close to the calculated values, though there is some discrepancy for the case of Fig. 6(b). These analytical results well explain the experimental results.
5. Concluding remarks As is described above, the phenomena of the reduction and disappearance of the trapped magnetic field by application of AC external magnetic field can be explained by thermal effect caused by the AC losses. In the thermal analysis in this paper the AC losses were calculated based on the Bean model assuming that the bulk used in the experiment could be simulated as a cylinder of infinite axis length, though it was a short cylinder. Results calculated from this analytical model well agreed with the experimental results and its validity was demonstrated.
O. Tsukamoto et al. / Journal of Materials Processing Technology 161 (2005) 52–57
Thus, we consider that mechanism of the influence of the AC external magnetic field on the trapped magnet field in the bulk was clarified. Once the mechanism has been clarified, various methods to suppress the decay can be developed. We are now developing the methods and assessing their effectiveness.
References [1] B. Oswald, M. Krone, M. Soll, J. Oswald, K.J. Best, W. Gawalek, L. Kovalev, Superconducting reluctance motors with YBCO bulk material, IEEE Trans. Appl. Supercond. 9 (1999) 1201–1204.
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[2] H. Hayashi, H. Ueda, M. Tsuda, A. Ishiyama, A new type of activeMaglev system using YBCO bulk and multiple electromagnets, IEEE Trans. Appl. Supercond. 12 (2002) 907–910. [3] K. Nagaya, et al., Study on high temperature superconducting magnetic bearing for 10 kW h flywheel energy storage system, IEEE Trans. Appl. Supercond. 11 (2001) 1649–1652. [4] J. Ogawa, M. Iwamoto, O. Tsukamoto, M. Murakami, M. Tomita, Interaction between trapped magnetic field and AC loss in HTS bulk, Physica C 372–376 (2002) 1754–1757. [5] J. Ogawa, M. Iwamoto, K. Yamagishi, O. Tsukamoto, M. Murakami, M. Tomita, Influence of AC external magnetic field perturbation of trapped magnetic field in HTS bulk, Physica C 386 (2003) 26– 30. [6] M. Wilson, Superconducting Magnet, Oxford Science Publications, 1983.
Mechanism of decay of trapped magnetic field in HTS bulk caused by application of AC magnetic field O. Tsukamotoa,∗ , K. Yamagishia , J. Ogawaa , M. Murakamib , M. Tomitab b
a Yokohama National University, 79-5 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan Superconductivity Research Laboratory, ISTEC, 1-16-25 Shibaura, Minato-ku, Tokyo 105-0023, Japan
Abstract In our previous work, it was observed that trapped magnetic field in an high-temperature superconductor (HTS) bulk was decayed and even erased by application of AC external field whose amplitude was much smaller than the peak value of the trapped magnetic field. Therefore, knowledge on the mechanism of the decay of the trapped magnetic field is important to design the machines and to develop a method to suppress the decay. This work studies a mechanism of the decay due to application of AC magnetic field by a numerical analysis and experiments. An analytical model is proposed to explain the mechanism based on the thermal effect due to AC losses in the bulk. In the model it is assumed that the AC loss characteristic follows the Bean model, which was experimentally demonstrated also in our previous work. We conducted experiments in which temperature rise of the bulk was measured by thermocouples and the decay of the trapped magnetic was observed by a hall probe. Results numerically calculated from the analytical model well agreed with those obtained from the experiments and we consider it was clarified that the decay of the trapped field was caused by the thermal effect due to AC loss. Once the mechanism has been clarified, methods to suppress the decay can be easily developed. © 2004 Elsevier B.V. All rights reserved. Keywords: HTS bulk; Trapped magnetic field; AC loss; HTS motor
1. Introduction and background Recent progresses of high-temperature superconductor (HTS) bulk technology have made various applications realistic. Especially, compact and highly efficient electric motors, magnetic levitation systems and flywheel energy storage systems are most promising applications [1–3]. The HTS bulks in these applications are exposed to magnetic field perturbation. The perturbation causes AC losses in the bulks and decay of the magnetic field trapped in the bulk. According to the Bean model, the reduction of the trapped field does not depend on the frequency, and the tapped field is decreased at the first cycle of the AC external field by the amplitude of the external field and, after this first reduction, the trapped field is unchanged, if the bulk temperature is constant. In a previous work, we measured the magnetic field distributions on the surface of the bulk with the trapped field after applying AC magnetic field. It was observed that
amount of the reduction of the trapped field was dependent on the frequency of the AC external field and that the trapped field was kept decreasing even to zero during the application of the relatively high external field [4]. This result is different from that is predicted from the simple Bean model of constant bulk temperature. To apply HTS bulks to electric machines knowledge on the mechanism of the decay of the trapped magnetic field is important to design the machines and to develop the countermeasures against the decay. In this work, mechanism of the decay of the trapped magnetic field is studied from the standpoint of thermal effect due to AC losses in the bulk and the validity of the mechanism is assessed by comparison of analytical and experimental results.
2. Analysis 2.1. Behavior of trapped magnetic field
∗
Corresponding author. Tel.: +81 45 339 4124; fax: +81 45 338 1157. E-mail address: [email protected] (O. Tsukamoto).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.07.009
Behavior of the trapped magnetic field in an HTS bulk subject to external AC magnetic field is analyzed here as-
O. Tsukamoto et al. / Journal of Materials Processing Technology 161 (2005) 52–57
53
magnetic field. The depth of the penetration of the external magnetic field rm is given by the following equation: rm = r0
Bm Bmp
(1)
where Bmp is the full penetration magnetic field and equal to µ0 Jc r0 . Fig. 1(c) shows the areas where the AC external magnetic field penetrates and AC shielding currents flow. AC losses are dissipated in these penetration areas. When the AC external field is reduced gradually to zero, the magnetic field and shielding currents in the penetration area become zero as shown in Fig. 1(d). If it is assumed that the bulk temperature is constant during the application of the AC external magnetic field, the reduction of the trapped magnetic field occurs at the first cycle of the AC external field and does not depend on the frequency nor on the duration time of the AC external field. However, experimental results were different from those theoretical predictions [4]. The AC losses caused by the magnetic flux movements in the penetration areas raise the bulk temperature, which causes the reduction of Jc and the increase of rm . If rm exceeds r0 then the trapped magnetic field disappears. Therefore, it is considered that influence of the AC external magnetic field on the trapped magnetic field is due to the temperature rise of bulk caused by AC losses. 2.2. Temperature rise of bulk caused by AC losses
Fig. 1. Distributions of magnetic field and current density in the cylindrical bulk subject to AC external field of amplitude Bm parallel to the cylinder axis. Bm is lower than the initial peak of the trapped magnetic field Bp0 : (a) initial state; (b) after one cycle application of AC external magnetic field; (c) during application of AC external magnetic field; (d) after gradual decrease of AC external magnetic field to zero.
suming the Bean model. The bulk is assumed to be a cylinder of infinite length for the simplicity of the analysis. Fig. 1 shows distributions of the trapped magnetic field B and current density J in an HTS cylinder of radius r0 subject to AC magnetic field of amplitude Bm parallel to the cylinder axis. Fig. 1 is for the case that Bm is lower than the peak of the initial trapped magnetic field Bp0 . Jc is the critical current density of the bulk. In the initial state, the magnetic field is trapped in the bulk by the superconducting current as shown in Fig. 1(a). Applying the external magnetic field it starts to penetrate into the cylinder from the surface, and shielding current whose density Jc is induced in the bulk. At the end of one cycle application of the AC external magnetic field, the peak of the trapped magnetic field is reduced by the magnitude Bm and the shielding current induced by the external field is hold in the bulk as shown in Fig. 1(b). If the temperature of the bulk and Jc are not changed, the distributions of the magnetic field and current density are the same as of Fig. 1(b) even after multiple cycles application of the AC
In the experiment described in the next section, we used a bulk of short-cylinder shape. Therefore, in the following analysis, the temperature rise of a short-cylinder bulk is studied. It has been shown that the AC loss characteristics of a short-cylinder YBCO bulk were well described by a cylinder model of infinite length following the Bean model [5]. Losses in a superconductor cylinder subject to AC external magnetic field of amplitude Bm parallel to the cylinder axis are given by the following equations [6]: Q=
2 2Bm µ0
2 2Bm Q= µ0
2β β2 − 3 3
2 1 − 3β 3β2
for β < 1
(2)
for β > 1
(3)
where β=
Bm Bmp
(4)
where Q (J/m3 per cycle) is the loss per cycle per unit volume. When β ≥ 1, the external magnetic field fully penetrates into the bulk (rm > r0 ) and the trapped magnetic field disappears when the application of the external field is terminated.
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O. Tsukamoto et al. / Journal of Materials Processing Technology 161 (2005) 52–57
It is assumed in the analysis that dependence of Jc on the bulk temperature T is given by Jc (T ) =
Jc0 (Tc − T ) (Tc − T0 )
(5)
where T0 is the coolant temperature, Jc0 the critical current density at T = T0 and Tc the critical temperature of the bulk. The thermal equilibrium equation of a short-cylinder bulk of radius r0 and thickness d subject to AC external field is expressed as follows, assuming the bulk temperature is uniform: Cp
dθ(t) + λθ(t) = P(t) dt
(6)
where θ(t) = T − T0 ,
P(t) = Qfv
where Cp (J/K) and λ (W/K) are the heat capacity of the bulk and the heat transfer coefficient from the bulk surface to the coolant, respectively, f is the frequency of AC external magnetic field and v = πr0 2 d is the volume of the bulk. The peak value of the trapped magnetic field Bp (T) is given by µ0 Jc (T )(r0 − rm ) for β < 1 Bp (T ) = (7) 0 for β ≥ 1 As T is increased by the AC loss, Jc decreases and rm increases. Thus, Bp decreases and when β exceeds 1, Bp becomes zero. From the above equations, time evolutions of T, β and Bp can be calculated by a numerical analysis.
Fig. 2. Trapped magnetic field distribution of the YBCO bulk: (a) sample arrangement; (b) relative positions of the thermocouples TC1–TC5 to the trapped magnetic field pattern.
was covered with a polystyrene foam layer of 5 mm thickness to thermally insulate the thermocouples from the liquid nitrogen (Fig. 3(a)). The side surface of the bulk directly contacted to liquid nitrogen. Fig. 3(b) shows relative positions of the thermocouples to the distribution of the trapped magnetic field. The bulk sample was placed in a bore of a copper coil that applied AC magnetic field to the bulk parallel to the axis of the short-cylinder bulk. The bulk and the copper coil were put in a liquid nitrogen bath (T0 = 77.3 K). During the application of the AC magnetic field, the temperature could not be measured because large inductive noise caused by the AC field
3. Experiment To investigate relation between temperature rise and behavior of trapped magnetic field in a bulk subject to AC external field, we performed an experiment using a short-cylinder YBCO bulk. 3.1. Experimental setup The bulk used in the experiment was epoxy impregnated YBCO bulk of 45 mm diameter and 20 mm thickness. The bulk was magnetized by the field cooling method applying DC external magnetic field parallel to the bulk axis. The maximum peak value of the trapped magnetic field was 0.31 T on the bulk surface. Magnetic field distribution was observed by a hall probe. Fig. 2 shows the magnetic field distribution of 0.5 mm above the bulk surface just after the magnetization. Temperature distributions of the bulk were measured by five Cr versus constantan thermocouples TC1–TC5 attached to the bulk surface. The sample arrangement and the positions of the thermocouples are shown in Fig. 3(a) and (b), respectively. The bulk was glued to the GFRP sample holder by vacuum grease and the bulk surface with thermocouples
Fig. 3. Experimental arrangement of the YBCO sample. (a) During application of AC external magnetic field. At t = 0 s the external field of Bm = 0.1 T (61 Hz) was applied. (b) During cooling process. At t = 0 s the application of the external field was finished.
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was mixed in the signals from the thermocouples. Therefore, the AC current was applied for a certain time and the copper coil current was shut off to measure the signals from the thermocouples. Change of the bulk temperature was negligibly small in several seconds. The experiment to investigate temperature rises was conducted for the bulk without trapped magnetic field. 3.2. Experimental results Fig. 4(a) and (b) shows temperature distributions measured by the thermocouples TC1–TC5 for various duration time of application of the 61 Hz AC external field of Bm = 0.1 T and during the cooling process after the application of the external field finished. The temperature distribution was almost uniform, though above 88 K it became rather obvious that the temperature in the peripheral area was lower than that in the central area, which was because the cooling in the peripheral was better. Fig. 5 shows average temperature on the bulk surface T versus duration time of the application of AC external magnetic field td for various values of Bm . In the graph analytical results are also shown that are explained in the next section. For the application of AC magnetic field of Bm = 0.1 T for more than 100 s, the bulk temperature went up to close to Tc which was about 91 K. In other cases of Bm = 0.08, 0.06, 0.04 T, the bulk temperatures reached steady-state values which were lower than Tc .
Fig. 4. Time-varying temperature distribution measured by thermocouples during application of AC and cooling process.
Fig. 5. Average temperature on the bulk surface T vs. duration time of 61 Hz AC magnetic fields td for various amplitudes of Bm . Lines in the graph were analytically calculated. (a) Before application of AC field; (b) Bm = 0.08 T, td = 240 s; (c) Bm = 0.1 T, td = 60 s; (d) Bm = 0.1 T, td =80 s; (e) Bm = 0.1 T, td = 100 s.
The trapped magnetic field distributions were measured on the surface of the polystyrene foam insulation layer (5 mm above the bulk surface) applying the AC magnetic field of a constant amplitude Bm for a certain time td and decreasing the amplitude to zero gradually in 1 s. Fig. 6 shows various
Fig. 6. Patterns of the trapped magnetic field in the bulk after the application of the AC external magnetic field. (a) Time evolutions of β; (b) time evolutions of calculated and experimental Bp normalized by Bp0 .
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patterns of the trapped magnetic field. Fig. 6(a) is the pattern before the application of the AC external field. Fig. 6(b) is the pattern after the 240 s application of Bm = 0.08 T. In this case the bulk temperature reached a steady-state 82.9 K and the peak trapped magnetic field was decreased by 30% from the value before the application of the AC magnetic field. Fig. 6(c)–(e) are after the 60, 80 and 100 s applications of Bm = 0.1 T, respectively. In the case of Fig. 6(c), the bulk temperature reached 84.4 K and the peak of the trapped magnetic field was decreased by 41% from the initial state. In the case of Fig. 6(d), the bulk temperature reached 86.6 K and the trapped magnetic field almost disappeared. In the case of Fig. 6(d), the bulk temperature reached 89.9 K and the trapped magnetic field completely disappeared. It is obvious from these results that the reduction and disappearance of the trapped magnetic field were caused by the temperature rise due to the AC losses and that the trapped magnetic field disappeared only when the bulk temperature rose close to Tc and the AC external magnetic field fully penetrated into the bulk. It should be noted that the disappearance of the trapped field is not caused by the normal transition of the bulk. If the bulk becomes normal, there is no energy enough to raise the bulk temperature, because the AC losses are generated only in the superconducting state bulk. It was experimentally demonstrated that eddy current losses in the normal state bulk caused by the AC external magnetic field was negligibly small compared with the AC losses in the superconducting state bulk. Therefore, the bulk temperature does not exceed Tc even in the case of the trapped magnetic field disappears. Fig. 7. Calculated time evolutions of β and Bp .
4. Comparison of the experimental and analytical results From Eqs. (1)–(7), the time evolutions of T, Bp and β were numerically calculated. The parameters used in the calculation are listed in Table 1. The time evolutions of T are shown in Fig. 5. Jc0 was estimated from the measured AC loss value and Eq. (2). λ was estimated from measured steady-state temperature of the bulk and the calculated AC loss value. τ was experimentally determined from the time evolution of T in the cooling process. As seen in Fig. 5, the calculated time evolutions of T agree well with the experimental results. Fig. 7(a) and (b) shows the numerically calculated time evolutions of β and ≡p for various values of Bm . As seen in Fig. 7, when t > 93 s for Bm = 0.1 T, β exceeds 1 and Bp becomes 0. On Table 1 Parameters used in the calculation Bulk size
Ø 45 mm, 19.5 mm thick
Jc0 ( A/m2 at 77.3 K) λ (W/K) Cp (J/K) τ = Cp /λ (s) Tc (K)
1.61 × 107 0.60 30.0 50 91.0
the other hand, for t < 93 s for Bm = 0.1 T, β is less than 1 and Bp remains. It is obvious from Fig. 7 for Bm = 0.08 T that β stays below 1 and that Bp remains even in the steady state. Experimental values of Bp in Fig. 6(b)–(e) are plotted in Fig. 7(b) normalized by the value of Bp0 in Fig. 6(a). Bp in Fig. 6(c)–(e) are close to the calculated values, though there is some discrepancy for the case of Fig. 6(b). These analytical results well explain the experimental results.
5. Concluding remarks As is described above, the phenomena of the reduction and disappearance of the trapped magnetic field by application of AC external magnetic field can be explained by thermal effect caused by the AC losses. In the thermal analysis in this paper the AC losses were calculated based on the Bean model assuming that the bulk used in the experiment could be simulated as a cylinder of infinite axis length, though it was a short cylinder. Results calculated from this analytical model well agreed with the experimental results and its validity was demonstrated.
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Thus, we consider that mechanism of the influence of the AC external magnetic field on the trapped magnet field in the bulk was clarified. Once the mechanism has been clarified, various methods to suppress the decay can be developed. We are now developing the methods and assessing their effectiveness.
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