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Hysteresis loss and the voltage–current relation in BSCCO tape superconductors
Physica C 401 (2004) 165–170 www.elsevier.com/locate/physc
Hysteresis loss and the voltage–current relation in BSCCO tape superconductors J.J. Rabbers *, B. ten Haken, H.H.J. ten Kate Faculty of Science and Technology, Low Temperature Division, University of Twente, P.O. Box 217, Enschede 7500 AE, The Netherlands
Abstract Hysteresis loss is an important factor when the performance of superconducting electric power devices is evaluated. The non-ideal voltage–current relation in Bi2 Sr2 Ca2 Cu3 Ox /Ag conductors is one of the reasons that critical state based loss relations do not accurately predict the AC loss in these conductors. In this paper the influence of the magnetic field dependent voltage–current relation on the hysteresis loss is discussed. An analytical high-field approach is used to demonstrate the effect of the finite steepness of the voltage–current relation on the induced current. Besides, a numerical technique is used to calculate also for applied magnetic fields below the penetration field the influence of both a field dependent critical current density and a field dependent steepness of the voltage–current relation. Both the magnitude of the hysteresis loss and the field dependence of the loss are influenced by the voltage–current relation. Especially for small applied fields the field dependence of the loss deviates from the cubic dependence that is predicted by the critical state model based relations. Results of calculations are compared with measured data. An intrinsic critical current density vs. magnetic field relation is determined in order to obtain agreement between measurements and calculations. 2003 Elsevier B.V. All rights reserved. PACS: 74.72.)h; 74.25.ha; 74.60.Jg Keywords: BSCCO/Ag tapes; AC loss; E–J relation; Hysteresis loss
1. Introduction Since the beginning of AC loss research on high-Tc superconductors it was clear that these new conductors differ on several points from the low-Tc conductors [1]. First of all, the filamentary core of Bi2223/Ag conductors has a rectangular or elliptical cross-section, where the low-Tc conductors are usually available in the form of (round) wires.
*
Corresponding author. Tel.: +31-53-489-4839/3889; fax: +31-53-489-1099. E-mail address: [email protected] (J.J. Rabbers).
The influence of the shape of the conductor crosssection on the hysteresis loss is treated elsewhere [2]. The superconducting to normal transition (voltage–current relation) in BSCCO superconductors differs from the idealised critical state description that is used to model low-Tc conductors. BSCCO tapes have a voltage–current relation that is much less steep (low n-value) than the one for classical superconductors. At liquid nitrogen temperature, the Bi2223/Ag conductors are used in a field range up to several tenths of a Tesla. In this field range the critical current density and the n-value decrease
0921-4534/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2003.09.030
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drastically, especially for magnetic field perpendicular to the wide face of the tape conductor. In this paper the influence of the magnetic field dependent voltage–current relation on the hysteresis loss in BSCCO tape conductors is investigated. Analytical approaches, which are possible in some cases, are discussed. The results of numerical calculations are shown for cases where no analytical models are available. It is shown that an ÔintrinsicÕ voltage–current relation is necessary to describe the measured hysteresis loss accurately.
2. The voltage–current relation and AC loss When a superconductor is exposed to an alternating magnetic field dB=dt, an electrical field E is induced according to FaradayÕs law (r E ¼ dB=dt) (see Fig. 1). The magnitude of the induced current depends on the (local) relation between E and J (referred to as the voltage–current relation). In analytical models for the hysteresis loss, which are based on the critical state model the voltage–current relation is infinitely steep. In the 1=n power law description, J ¼ Jc ðE=Ec Þ , which is generally used to model the voltage–current relation of superconductors, this means an infinite index n (the n-value). In that case, the magnitude of the induced current is þJc or Jc and does not depend on the electric field. However, in Bi2223/ Ag superconductors the n-value varies between 10 and 20 and also decreases when magnetic field is present. This means that the induced current de1=n pends on the electrical field. The factor ðE=Ec Þ in the power law is the ratio between the induced current density J and the critical current density Jc ,
Fig. 1. Schematic picture of a tape conductor and the definition of the coordinate system and the magnetic field directions.
which is usually defined as the current density at E ¼ Ec ¼ 104 V/m. Since E varies in the conductor cross-section also the induced current is not constant over the conductor cross-section. Furthermore, the induced electric field depends on the dB=dt so that also the induced current varies during a period of a sinusoidal field change or as a function the frequency of the magnetic field. Furthermore, the critical current density Jc and the n-value decrease when magnetic field is present, especially for field perpendicular to the wide face of the tape conductor. In fact, the induced current density, instead of the critical current density is the quantity that matters when the loss is calculated. The influence of magnetic field on the critical current density and the n-value is referred to as the magnetic field dependent voltage–current relation. This relation determines the induced current and thus the hysteresis loss. The typical dimensions of a BSCCO tape parallel to the applied magnetic field are 0.3 mm (2b) for parallel applied magnetic field (Bk ) and 4 mm (2a) for perpendicular field (B? ). For magnetic field amplitudes up to 0.1 T with a frequency of 50 Hz the electrical field, in a one-dimensional simplification; Ez ¼ y dBx =dt, is up to 5 · 103 V/m for parallel field and up to 6 · 102 V/m for perpendicular field. In Fig. 2 the relation between the ratio J =Jc and the electrical field is shown for various values of the n-value. The induced current density can be
Fig. 2. The power law E–J relation in a double logarithmic graph for various values of the n-value.
J.J. Rabbers et al. / Physica C 401 (2004) 165–170
either lower or higher than the critical current density, depending on the induced electrical field.
3. Analytical models In general the calculation of the hysteresis loss in superconductors cannot be performed analytically. Only for the infinite slab in parallel applied magnetic field an analytical solution of MaxwellÕs equations can be obtained when the critical state model is assumed. However, when the applied magnetic field is much larger than the field of full penetration, the magnetic field in the conductor is (almost) equal to the applied field and the hysteresis loss can be calculated analytically for various conductor cross-sections, see e.g. [3], even for a power law voltage–current relation with finite nvalue and field dependent critical current density. The hysteresis loss for a conductor with rectangular cross-section (width 2a), critical current density Jc;0 at zero field in perpendicular applied magnetic field with amplitude Ba and frequency f is: 1=n 2aJc;0 Ba B0 Ba þ B0 2pfaBa Qm ¼ log : ð1Þ B0 Ec p 1 þ 2n1 Ba The magnetic field dependence of the critical current is modelled with the Kim model with a characteristic field B0 The term in square brackets displays the E=Ec ratio. The larger n, the closer this ratio is to one and the closer the induced current is to the critical current. The part before the brackets is a cross-sectional dependent term [3]. The proportionality of the loss with the magnetic field is slightly higher than in the critical state model. For conductors with an n-value in the range of 10–20: Qm / Ba1:1–1:05 vs. Qm / Ba for n ¼ 1 (critical state). The field proportionality increases because E and thus the induced J increases with increasing magnetic field. This causes an extra increase of the hysteresis loss because B / J for constant field for Ba Bp . The hysteresis loss also depends on frequency. Compared to the critical state model the loss increases with a factor of 1.5 for an n-value of 10 and a frequency of 50 Hz.
167
Dresner investigated the influence of the magnetic field dependence on the hysteresis loss in an infinite slab with dimensional analysis [4]. The calculations show how the field proportionality of the hysteresis loss, also for field amplitudes smaller than the penetration field changes with n-value and field dependent critical current density. The field proportionality of the magnetisation loss for low values of the magnetic field varies from 2.7 to 2.9 for n ¼ 5–20 [4], slightly less than the cubic dependence in the critical state approach. The field proportionality decreases because the induced current increases with increasing magnetic field and the hysteresis loss below penetration field, at constant field, is proportional to 1=J . The slope of the loss relation below penetration field becomes more than cubical when the critical current depends on magnetic field. The slope depends on the relation between critical current and magnetic field. The induced current decreases when the field increases, which leads to an extra increase of the loss on top of the cubic field dependence. Unfortunately no analytical solutions with magnetic field dependent voltage–current relation for the hysteresis loss for magnetic field amplitude below the penetration field are known yet. Numerical calculations have to be performed. Results are shown in Section 4.
4. Numerical modelling The hysteresis loss is calculated numerically by solving the Poisson equation for the magnetic vector potential [5], including the field dependence of the critical current density and the n-value as described in [6]. All calculations are performed for a conductor with a rectangular cross-section of 3.6 mm · 0.28 mm and a magnetic field frequency of 48 Hz. In Fig. 3 hysteresis loss for various n-values for a conductor with rectangular cross-section in perpendicular applied magnetic field is shown. Critical current density and n-value do not depend on magnetic field. The slope of the loss as a function of applied field for small applied magnetic field decreases as the n-value decreases, as explained in the former
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Fig. 3. Numerically calculated magnetisation loss as a function of applied magnetic field for various n-values.
section. Above the penetration field the proportionality of the loss on the applied field increases for decreasing n-value, as predicted by dimensional analysis [4] and Eq. (1) [3]. In Fig. 4 the influence of both a magnetic field dependent critical current density and n-value on the magnetisation loss is shown (solid line). As a comparison the loss for constant critical current density and constant, but finite, n-value is also shown (dotted and dash-dotted line).
Fig. 4. Numerically calculated magnetisation loss with field dependent current–voltage relation as a function of applied magnetic field perpendicular and parallel to the tape. Comparison with constant Jc and n-value.
The relation between critical current density and magnetic field, and n-value and magnetic field is modelled with a two current path model [7]. The quantities Ic ð0Þ and nð0Þ refer to the Ic and n-value at zero applied magnetic field. Only the influence of local perpendicular magnetic field is taken into account. Even in the case of a tape conductor in parallel applied magnetic field the influence of this component is dominant. The screening currents produce locally at the edges of the tape a perpendicular field component, which decreases the critical current density much more than the applied parallel magnetic field. The influence of the magnetic field dependent voltage–current relation on the magnetisation loss is in both parallel and perpendicular magnetic field an increase of the slope of the loss relation for small applied fields. The increase is even more pronounced in parallel magnetic field than in perpendicular field. Above the penetration field the slope of the loss relation changes hardly, in both field orientations. As a comparison the loss relation in perpendicular magnetic field with the same critical current (140 A) and n-value as in the calculation with field dependent voltage–current relation is shown. The effect of the higher critical current and n-value is a lower loss for small fields since Qm is proportional to 1=J . Also the penetration field is higher because J is higher and therefore the loss for magnetic field above full penetration also. A striking result is that the hysteresis loss in perpendicular magnetic field for a conductor with constant critical current of 50 A and n-value of 15 is almost equal to the hysteresis loss of the conductor with field dependent critical current and nvalue (Ic ð0Þ ¼ 140, nð0Þ ¼ 15). This is caused by the fact that first of all the critical current in perpendicular field decreases and at the same time also the n-value decreases which causes an extra decrease of the induced current for low applied field, i.e. E < Ec (see Fig. 2).
5. Comparison with measurements In Fig. 5 the measured magnetisation loss in a BSCCO tape is shown in perpendicular and par-
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169
Fig. 6. Critical current as a function of magnetic field. Comparison between the measured values and the ÔintrinsicÕ relation. Fig. 5. Magnetisation loss as a function of applied magnetic field parallel and perpendicular to the wide face of the tape. Comparison between the experimental data, an analytical model and numerical calculations.
allel magnetic field (symbols). The experiments are performed at 77 K on a multifilament Bi2223 tape with untwisted filaments and a silver matrix. The outer dimensions are 4.1 mm · 0.27 mm. The critical current, determined at E ¼ 104 V/m is 128 A. The frequency of the applied magnetic field is 48 Hz. The loss calculated with the analytical model for a conductor with elliptical cross-section [8] is also shown (solid lines). For perpendicular magnetic field there is good agreement between the measured loss and the model, except for the lowest field range. In parallel applied magnetic there is a larger deviation between the analytical model and the measurements. Especially the field dependence of the measured loss for small applied fields deviates from the field (cubic) dependence predicted by the analytical model. Numerical calculations are performed with a model [5,6] with magnetic field dependent voltage– current relation (dotted lines). In order to obtain good agreement with the measured data the Jc ðBÞ dependence obtained from a DC Ic measurement had to be modified. As a starting point the measured critical current as a function of applied constant magnetic field is used, symbols in Fig. 6.
Close to zero applied field, there is the self-field due to the DC transport current of the conductor. This magnetic field limits the critical current. However, in a magnetisation loss experiment at applied fields smaller or not much larger than the self-field in the DC measurement the critical current density is larger than the self-field critical current density. In order to obtain the magnetisation loss shown by the dotted lines in Fig. 5, a sharp increase of the Jc for B < 0:01 T is needed (see Fig. 6). This is an ÔintrinsicÕ critical current relation, which cannot be measured in a DC Ic experiment. The height and width of the peak in the curve for low field is determined iteratively, the deviation between the measured loss and the numerically calculated loss is minimised. The height of the peak can be understood when the loss in parallel applied field is considered. The analytical model with a constant Ic of 128 A predicts a loss for small fields that is about one order of magnitude higher than the measured loss. Since the loss below the penetration field is proportional to 1=J at constant field the (averaged) induced current from zero field to applied field must be up to one order of magnitude higher than the current used in the analytical model. The sharp peak in the low field regime of the critical current vs. field relation is responsible for the increase of the slope of the loss relation for low fields, as explained before. The sharper the peak the higher the slope [4].
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6. Conclusions
Acknowledgements
The magnetic field dependency of the voltage– current relation (critical current density and nvalue) changes the hysteresis loss when compared to the critical state model based loss calculations. The influence of a low n-value on the hysteresis loss is a slight decrease of the field proportionality below the penetration field and a slight increase of the slope above penetration field. A sharp peak in the magnetic field dependence of the critical current towards zero field causes a more than cubic field proportionality of the hystresis loss relation. In order to describe the measured hysteresis loss of a BSCCO tape conductor accurately, a critical current relation that deviates from the one measured in a DC Ic experiment is needed. The more than cubic field proportionality for low fields is correctly described with an ÔintrinsicÕ critical current relation that has a sharp peak for fields smaller than the self-field in the DC Ic experiment.
This research is supported by the Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs.
References [1] K. Kwasnitza, St. Clerc, Adv. Cryog. Eng. 40 (1994) 53. [2] J.J. Rabbers, B. ten Haken, H.H.J. ten Kate, Modelling Hysteresis Loss of BSCCO/Ag Tape Superconductors, The Influence of Conductor Cross-section, in press. [3] N. Banno, N. Amemiya, Cryogenics 39 (1999) 99. [4] L. Dresner, Appl. Supercond. 4 (1996) 167. [5] E.H. Brandt, Phys. Rev. 54 (1996) 4246. [6] T. Yazawa, J.J. Rabbers, B. ten Haken, H.H.J. ten Kate, H. Maeda, J. Appl. Phys. 84 (1998) 5652. [7] D.C. van der Laan, H.J.N. van Eck, B. ten Haken, J. Schwartz, H.H.J. ten Kate, IEEE Trans. Appl. Supercond. 11 (2001) 3345. [8] B. ten Haken, J.J. Rabbers, H.H.J. ten Kate, Physica C 377 (2002) 156.
Hysteresis loss and the voltage–current relation in BSCCO tape superconductors J.J. Rabbers *, B. ten Haken, H.H.J. ten Kate Faculty of Science and Technology, Low Temperature Division, University of Twente, P.O. Box 217, Enschede 7500 AE, The Netherlands
Abstract Hysteresis loss is an important factor when the performance of superconducting electric power devices is evaluated. The non-ideal voltage–current relation in Bi2 Sr2 Ca2 Cu3 Ox /Ag conductors is one of the reasons that critical state based loss relations do not accurately predict the AC loss in these conductors. In this paper the influence of the magnetic field dependent voltage–current relation on the hysteresis loss is discussed. An analytical high-field approach is used to demonstrate the effect of the finite steepness of the voltage–current relation on the induced current. Besides, a numerical technique is used to calculate also for applied magnetic fields below the penetration field the influence of both a field dependent critical current density and a field dependent steepness of the voltage–current relation. Both the magnitude of the hysteresis loss and the field dependence of the loss are influenced by the voltage–current relation. Especially for small applied fields the field dependence of the loss deviates from the cubic dependence that is predicted by the critical state model based relations. Results of calculations are compared with measured data. An intrinsic critical current density vs. magnetic field relation is determined in order to obtain agreement between measurements and calculations. 2003 Elsevier B.V. All rights reserved. PACS: 74.72.)h; 74.25.ha; 74.60.Jg Keywords: BSCCO/Ag tapes; AC loss; E–J relation; Hysteresis loss
1. Introduction Since the beginning of AC loss research on high-Tc superconductors it was clear that these new conductors differ on several points from the low-Tc conductors [1]. First of all, the filamentary core of Bi2223/Ag conductors has a rectangular or elliptical cross-section, where the low-Tc conductors are usually available in the form of (round) wires.
*
Corresponding author. Tel.: +31-53-489-4839/3889; fax: +31-53-489-1099. E-mail address: [email protected] (J.J. Rabbers).
The influence of the shape of the conductor crosssection on the hysteresis loss is treated elsewhere [2]. The superconducting to normal transition (voltage–current relation) in BSCCO superconductors differs from the idealised critical state description that is used to model low-Tc conductors. BSCCO tapes have a voltage–current relation that is much less steep (low n-value) than the one for classical superconductors. At liquid nitrogen temperature, the Bi2223/Ag conductors are used in a field range up to several tenths of a Tesla. In this field range the critical current density and the n-value decrease
0921-4534/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2003.09.030
166
J.J. Rabbers et al. / Physica C 401 (2004) 165–170
drastically, especially for magnetic field perpendicular to the wide face of the tape conductor. In this paper the influence of the magnetic field dependent voltage–current relation on the hysteresis loss in BSCCO tape conductors is investigated. Analytical approaches, which are possible in some cases, are discussed. The results of numerical calculations are shown for cases where no analytical models are available. It is shown that an ÔintrinsicÕ voltage–current relation is necessary to describe the measured hysteresis loss accurately.
2. The voltage–current relation and AC loss When a superconductor is exposed to an alternating magnetic field dB=dt, an electrical field E is induced according to FaradayÕs law (r E ¼ dB=dt) (see Fig. 1). The magnitude of the induced current depends on the (local) relation between E and J (referred to as the voltage–current relation). In analytical models for the hysteresis loss, which are based on the critical state model the voltage–current relation is infinitely steep. In the 1=n power law description, J ¼ Jc ðE=Ec Þ , which is generally used to model the voltage–current relation of superconductors, this means an infinite index n (the n-value). In that case, the magnitude of the induced current is þJc or Jc and does not depend on the electric field. However, in Bi2223/ Ag superconductors the n-value varies between 10 and 20 and also decreases when magnetic field is present. This means that the induced current de1=n pends on the electrical field. The factor ðE=Ec Þ in the power law is the ratio between the induced current density J and the critical current density Jc ,
Fig. 1. Schematic picture of a tape conductor and the definition of the coordinate system and the magnetic field directions.
which is usually defined as the current density at E ¼ Ec ¼ 104 V/m. Since E varies in the conductor cross-section also the induced current is not constant over the conductor cross-section. Furthermore, the induced electric field depends on the dB=dt so that also the induced current varies during a period of a sinusoidal field change or as a function the frequency of the magnetic field. Furthermore, the critical current density Jc and the n-value decrease when magnetic field is present, especially for field perpendicular to the wide face of the tape conductor. In fact, the induced current density, instead of the critical current density is the quantity that matters when the loss is calculated. The influence of magnetic field on the critical current density and the n-value is referred to as the magnetic field dependent voltage–current relation. This relation determines the induced current and thus the hysteresis loss. The typical dimensions of a BSCCO tape parallel to the applied magnetic field are 0.3 mm (2b) for parallel applied magnetic field (Bk ) and 4 mm (2a) for perpendicular field (B? ). For magnetic field amplitudes up to 0.1 T with a frequency of 50 Hz the electrical field, in a one-dimensional simplification; Ez ¼ y dBx =dt, is up to 5 · 103 V/m for parallel field and up to 6 · 102 V/m for perpendicular field. In Fig. 2 the relation between the ratio J =Jc and the electrical field is shown for various values of the n-value. The induced current density can be
Fig. 2. The power law E–J relation in a double logarithmic graph for various values of the n-value.
J.J. Rabbers et al. / Physica C 401 (2004) 165–170
either lower or higher than the critical current density, depending on the induced electrical field.
3. Analytical models In general the calculation of the hysteresis loss in superconductors cannot be performed analytically. Only for the infinite slab in parallel applied magnetic field an analytical solution of MaxwellÕs equations can be obtained when the critical state model is assumed. However, when the applied magnetic field is much larger than the field of full penetration, the magnetic field in the conductor is (almost) equal to the applied field and the hysteresis loss can be calculated analytically for various conductor cross-sections, see e.g. [3], even for a power law voltage–current relation with finite nvalue and field dependent critical current density. The hysteresis loss for a conductor with rectangular cross-section (width 2a), critical current density Jc;0 at zero field in perpendicular applied magnetic field with amplitude Ba and frequency f is: 1=n 2aJc;0 Ba B0 Ba þ B0 2pfaBa Qm ¼ log : ð1Þ B0 Ec p 1 þ 2n1 Ba The magnetic field dependence of the critical current is modelled with the Kim model with a characteristic field B0 The term in square brackets displays the E=Ec ratio. The larger n, the closer this ratio is to one and the closer the induced current is to the critical current. The part before the brackets is a cross-sectional dependent term [3]. The proportionality of the loss with the magnetic field is slightly higher than in the critical state model. For conductors with an n-value in the range of 10–20: Qm / Ba1:1–1:05 vs. Qm / Ba for n ¼ 1 (critical state). The field proportionality increases because E and thus the induced J increases with increasing magnetic field. This causes an extra increase of the hysteresis loss because B / J for constant field for Ba Bp . The hysteresis loss also depends on frequency. Compared to the critical state model the loss increases with a factor of 1.5 for an n-value of 10 and a frequency of 50 Hz.
167
Dresner investigated the influence of the magnetic field dependence on the hysteresis loss in an infinite slab with dimensional analysis [4]. The calculations show how the field proportionality of the hysteresis loss, also for field amplitudes smaller than the penetration field changes with n-value and field dependent critical current density. The field proportionality of the magnetisation loss for low values of the magnetic field varies from 2.7 to 2.9 for n ¼ 5–20 [4], slightly less than the cubic dependence in the critical state approach. The field proportionality decreases because the induced current increases with increasing magnetic field and the hysteresis loss below penetration field, at constant field, is proportional to 1=J . The slope of the loss relation below penetration field becomes more than cubical when the critical current depends on magnetic field. The slope depends on the relation between critical current and magnetic field. The induced current decreases when the field increases, which leads to an extra increase of the loss on top of the cubic field dependence. Unfortunately no analytical solutions with magnetic field dependent voltage–current relation for the hysteresis loss for magnetic field amplitude below the penetration field are known yet. Numerical calculations have to be performed. Results are shown in Section 4.
4. Numerical modelling The hysteresis loss is calculated numerically by solving the Poisson equation for the magnetic vector potential [5], including the field dependence of the critical current density and the n-value as described in [6]. All calculations are performed for a conductor with a rectangular cross-section of 3.6 mm · 0.28 mm and a magnetic field frequency of 48 Hz. In Fig. 3 hysteresis loss for various n-values for a conductor with rectangular cross-section in perpendicular applied magnetic field is shown. Critical current density and n-value do not depend on magnetic field. The slope of the loss as a function of applied field for small applied magnetic field decreases as the n-value decreases, as explained in the former
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Fig. 3. Numerically calculated magnetisation loss as a function of applied magnetic field for various n-values.
section. Above the penetration field the proportionality of the loss on the applied field increases for decreasing n-value, as predicted by dimensional analysis [4] and Eq. (1) [3]. In Fig. 4 the influence of both a magnetic field dependent critical current density and n-value on the magnetisation loss is shown (solid line). As a comparison the loss for constant critical current density and constant, but finite, n-value is also shown (dotted and dash-dotted line).
Fig. 4. Numerically calculated magnetisation loss with field dependent current–voltage relation as a function of applied magnetic field perpendicular and parallel to the tape. Comparison with constant Jc and n-value.
The relation between critical current density and magnetic field, and n-value and magnetic field is modelled with a two current path model [7]. The quantities Ic ð0Þ and nð0Þ refer to the Ic and n-value at zero applied magnetic field. Only the influence of local perpendicular magnetic field is taken into account. Even in the case of a tape conductor in parallel applied magnetic field the influence of this component is dominant. The screening currents produce locally at the edges of the tape a perpendicular field component, which decreases the critical current density much more than the applied parallel magnetic field. The influence of the magnetic field dependent voltage–current relation on the magnetisation loss is in both parallel and perpendicular magnetic field an increase of the slope of the loss relation for small applied fields. The increase is even more pronounced in parallel magnetic field than in perpendicular field. Above the penetration field the slope of the loss relation changes hardly, in both field orientations. As a comparison the loss relation in perpendicular magnetic field with the same critical current (140 A) and n-value as in the calculation with field dependent voltage–current relation is shown. The effect of the higher critical current and n-value is a lower loss for small fields since Qm is proportional to 1=J . Also the penetration field is higher because J is higher and therefore the loss for magnetic field above full penetration also. A striking result is that the hysteresis loss in perpendicular magnetic field for a conductor with constant critical current of 50 A and n-value of 15 is almost equal to the hysteresis loss of the conductor with field dependent critical current and nvalue (Ic ð0Þ ¼ 140, nð0Þ ¼ 15). This is caused by the fact that first of all the critical current in perpendicular field decreases and at the same time also the n-value decreases which causes an extra decrease of the induced current for low applied field, i.e. E < Ec (see Fig. 2).
5. Comparison with measurements In Fig. 5 the measured magnetisation loss in a BSCCO tape is shown in perpendicular and par-
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169
Fig. 6. Critical current as a function of magnetic field. Comparison between the measured values and the ÔintrinsicÕ relation. Fig. 5. Magnetisation loss as a function of applied magnetic field parallel and perpendicular to the wide face of the tape. Comparison between the experimental data, an analytical model and numerical calculations.
allel magnetic field (symbols). The experiments are performed at 77 K on a multifilament Bi2223 tape with untwisted filaments and a silver matrix. The outer dimensions are 4.1 mm · 0.27 mm. The critical current, determined at E ¼ 104 V/m is 128 A. The frequency of the applied magnetic field is 48 Hz. The loss calculated with the analytical model for a conductor with elliptical cross-section [8] is also shown (solid lines). For perpendicular magnetic field there is good agreement between the measured loss and the model, except for the lowest field range. In parallel applied magnetic there is a larger deviation between the analytical model and the measurements. Especially the field dependence of the measured loss for small applied fields deviates from the field (cubic) dependence predicted by the analytical model. Numerical calculations are performed with a model [5,6] with magnetic field dependent voltage– current relation (dotted lines). In order to obtain good agreement with the measured data the Jc ðBÞ dependence obtained from a DC Ic measurement had to be modified. As a starting point the measured critical current as a function of applied constant magnetic field is used, symbols in Fig. 6.
Close to zero applied field, there is the self-field due to the DC transport current of the conductor. This magnetic field limits the critical current. However, in a magnetisation loss experiment at applied fields smaller or not much larger than the self-field in the DC measurement the critical current density is larger than the self-field critical current density. In order to obtain the magnetisation loss shown by the dotted lines in Fig. 5, a sharp increase of the Jc for B < 0:01 T is needed (see Fig. 6). This is an ÔintrinsicÕ critical current relation, which cannot be measured in a DC Ic experiment. The height and width of the peak in the curve for low field is determined iteratively, the deviation between the measured loss and the numerically calculated loss is minimised. The height of the peak can be understood when the loss in parallel applied field is considered. The analytical model with a constant Ic of 128 A predicts a loss for small fields that is about one order of magnitude higher than the measured loss. Since the loss below the penetration field is proportional to 1=J at constant field the (averaged) induced current from zero field to applied field must be up to one order of magnitude higher than the current used in the analytical model. The sharp peak in the low field regime of the critical current vs. field relation is responsible for the increase of the slope of the loss relation for low fields, as explained before. The sharper the peak the higher the slope [4].
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6. Conclusions
Acknowledgements
The magnetic field dependency of the voltage– current relation (critical current density and nvalue) changes the hysteresis loss when compared to the critical state model based loss calculations. The influence of a low n-value on the hysteresis loss is a slight decrease of the field proportionality below the penetration field and a slight increase of the slope above penetration field. A sharp peak in the magnetic field dependence of the critical current towards zero field causes a more than cubic field proportionality of the hystresis loss relation. In order to describe the measured hysteresis loss of a BSCCO tape conductor accurately, a critical current relation that deviates from the one measured in a DC Ic experiment is needed. The more than cubic field proportionality for low fields is correctly described with an ÔintrinsicÕ critical current relation that has a sharp peak for fields smaller than the self-field in the DC Ic experiment.
This research is supported by the Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs.
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