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Ag superconducting tape exposed to applied field with arbitrary angle
Cryogenics 42 (2002) 771–778 www.elsevier.com/locate/cryogenics
Magnetisation loss of BSCCO/Ag superconducting tape exposed to applied field with arbitrary angle J.J. Rabbers *, O. van der Meer, B. ten Haken, H.H.J. ten Kate Faculty of Applied Physics, Low Temperature Division, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands Received 15 July 2002; accepted 10 October 2002
Abstract Bi2 Sr2 Ca2 Cu3 Ox /Ag tape superconductors are used in applications like power cables and transformers. In these applications the superconductor is exposed to an alternating magnetic field that has different orientations with respect to the tape surface. In this paper the angle dependency of the magnetisation loss is considered from two points of view. First the measurement technique with pickup coils is analysed theoretically. Measured magnetisation loss in uni-directional magnetic field with various orientations and rotating magnetic field are compared. When the orientation is changed from perpendicular (0°) to parallel (90°) applied magnetic fields, the contribution of the perpendicular field component to the magnetisation loss is dominant up to 60°. A new model to describe the angle dependency of the magnetisation loss, based on the measured loss in perpendicular and parallel magnetic field is developed. Deviations between models and the measured loss are explained with the help of the theoretical analysis of the measurement technique. The new model is not only applicable for the magnetisation loss but also for other AC loss components. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: BSCCO/Ag tapes; Magnetisation loss; Rotating field
1. Introduction When Bi2 Sr2 Ca2 Cu3 Ox /Ag tapes are used in electric power applications the magnetisation loss contributes considerably to the total AC loss [1], i.e. magnetisation loss plus transport current loss. Towards the edges of a solenoidal coil the orientation of the magnetic field with respect to the tape surface changes from parallel to perpendicular. In order to calculate the AC loss in these kind of devices, it is necessary to know the magnetisation loss not only in the most often measured limiting cases of magnetic field parallel and perpendicular to the wide side of the tape. Also the intermediate angles should be considered. In this paper a theoretical evaluation of the measurement method of magnetisation loss in arbitrary uni-directional field and rotating magnetic field is given. Measurement results for a variety of magnetic field amplitudes and orientations are presented. A new approximation to estimate the magnetisation loss for orientations between parallel and
*
Corresponding author. Tel.: +31-53-489-4839/3889; fax: +31-53489-1099. E-mail address: [email protected] (J.J. Rabbers).
perpendicular magnetic field is developed. The deviations between the measured loss and the models are explained with the help of the theoretical analysis of the measurement technique.
2. Theory Magnetisation loss measurements on tape superconductors with applied magnetic field under various angles are already published in several papers [2–4]. In these measurements, the orientation of the tape superconductor in the magnetic field is changed and the pickup coils sense the magnetic moment of the sample in the direction of the applied magnetic field. There is no doubt that this is the correct way to measure the loss in this case. Here, the analysis that leads to this conclusion is shown in order to get a better understanding of the magnetisation loss as a function of the orientation of the applied magnetic field. Later on the analysis is used to explain differences between models and the measured loss. Furthermore, the derivation shows how the magnetisation loss in a magnetic field that rotates around the sample can be measured. Although the derivation is
0011-2275/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 1 - 2 2 7 5 ( 0 2 ) 0 0 1 4 5 - 5
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straightforward, it is shown here in some detail because no references for this specific case are known. A more general description of magnetic loss measurement techniques can be found in [5]. Consider a tape conductor with homogeneous properties and infinite dimension in the longitudinal direction, see Fig. 1. The directions ? and k are defined in the figure and the longitudinal direction is perpendicular to the tape cross-section. The magnetic field B is applied perpendicular to the longitudinal direction. The applied magnetic field can be written as: BðtÞ ¼ Bk ðtÞek þ B? ðtÞe? ; ð1Þ where (in the case of a sinusoidal varying magnetic field): B? ðtÞ ¼ Ba cosðhÞ sinð2pftÞ; ð2Þ Bk ðtÞ ¼ Ba sinðhÞ sinð2pft þ uÞ:
ð3Þ
For u ¼ 0° the applied field is unidirectional. The two extreme situations h ¼ 0° and h ¼ 90° are referred to as perpendicular (?) and parallel (k) respectively. When u 6¼ 0° the magnetic field vector B rotates around the tape. The case where h ¼ 45° and u ¼ 90° describes a circular rotation. Thep rotating magnetic field is characterised with Ba ¼ ðB2a;? þ B2a;k Þ and h ¼ atan= ðBa;k =Ba;? Þ, the unidirectional field is characterised with Ba and h. The resulting magnetic moment generated by the screening currents in the superconductor is written in a similar way: mðBÞ ¼ mk ðBÞek þ m? ðBÞe? :
ð4Þ
Due to the dot product only the equally directed components in Eq. (6) give non-zero contributions: dqðtÞ ¼ ½Bk ðtÞm_ k ðBÞ þ B? ðtÞm_ ? ðBÞ dt ¼ ½dQk ðtÞ þ dQ? ðtÞ dt:
It is important to realise that the magnetic moment is always a function of the applied magnetic field vector B, and not only the component parallel to the considered magnetic field. The component of B perpendicular to the considered magnetic moment does not contribute to the magnetisation loss, but it influences the properties (critical current) of the conductor and the induced current patterns can interact. This will be demonstrated later on. Continue with the parallel component, Q per field cycle in J/m3 (for the perpendicular component the calculation is identical): I 1 Bk ðtÞm_ k ðBÞ dt; ð8Þ Qk ðBÞ ¼ V field cycle where V is the sample volume. The components m_ k (and m_ ? ) of the time derivative of m can be written as a Fourier series: m_ k ðtÞ ¼
1 X
an sinð2npft þ uÞ þ bn cosð2npft þ uÞ:
Combining Eq. (8) and Eq. (9) yields: Z 1 1=f Qk ðBÞ ¼ Ba sinðhÞ sinð2pft þ uÞ V 0 X ½an ðBÞ sinð2npft þ uÞ
n
þ bn ðBÞ cosð2npft þ uÞ dt: ð5Þ
Combining Eq. (1) and the time derivative of Eq. (4) with Eq. (5) yields: dqðtÞ ¼ ½Bk ðtÞek þ B? ðtÞe? ½m_ k ðBÞek þ m_ ? ðBÞe? dt: ð6Þ
ð9Þ
n¼1
The magnetisation loss, in J, is calculated with: _ ðBÞdt: dqðtÞ ¼ BðtÞ dmðBÞ ¼ BðtÞ m
ð7Þ
ð10Þ
Only the a1 term in the Fourier series gives non-zero result after integration over an entire magnetic field cycle: Z 1 1=f Qk ðBÞ ¼ Ba sinðhÞ sinð2pft þ uÞa1 ðBÞ V 0
sinð2pft þ uÞdt: ð11Þ Calculating the integral and adding the loss component of the perpendicular field components yields the magnetisation loss (c1 is the first Fourier term in the perpendicular loss contribution, similar to Eq. (9)): QðBÞ ¼
Fig. 1. The tape cross-section and definition of the magnetic field directions k and the ? and the angle h.
1 Ba sinðhÞa1 ðBÞ 1 Ba cosðhÞc1 ðBÞ þ : V 2f V 2f
ð12Þ
The last equation shows that the loss in perpendicular and parallel direction have to be summed in order to obtain the magnetisation loss in uni-directional or rotating magnetic field. Only components of the magnetisation that have the same direction as the applied field component contribute to the loss, but their magnitude is a function of the field vector B. In the next section the
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practical implementation of the loss measurement is discussed.
3. Measurement setup The magnetisation loss in unidirectional magnetic field is measured in a dipole magnet that generates a magnetic field in de y-direction, see Fig. 2. The normal on the wide face of the tape sample makes an angle a with the y-axis. In that case only the magnetic moment in the y-direction (the direction parallel to the applied magnetic field) contributes to the loss, as shown before. A set of pickup coils that sense the magnetic moment in the y-direction is used to measure the loss. The magnetisation loss in a rotating magnetic field is measured differently. With a two-dipole magnet system, magnetic fields are generated perpendicular to each other, see Fig. 2. In this case the orientation a of the sample is either 0° or 90°. The pick-up coils sense the magnetic moment of the sample in the y-direction which is the perpendicular (?) or the parallel (k) direction respectively, depending on the orientation of the sample in the pick-up coils. The measurement of the magnetisation loss in rotating magnetic field is performed in two steps [2]. The magnetisation of the sample in the parallel and the perpendicular direction has to be measured and since the set-up has only one set of pick-up coils (for the ydirection) two measurements are necessary. First the magnetisation of the sample in perpendicular direction is measured, while the field vector B rotates around the sample. Then the orientation of the sample is changed 90° so the pick-up coils sense the magnetisation of the sample in parallel direction. But then also the orientation of the magnetic field has to be changed 90° and the magnetisation in the parallel direction is measured, while the field vector B rotates around the sample. As discussed before the two contributions are summed in order to obtain the total magnetisation loss in a rotating magnetic field.
Fig. 2. Cross-section of the measurement setup with dipole magnets, pick-up coils, sample (magnified) and the definition of the k-, ?, x- and y-direction.
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The signal from the pick-up coils is measured with a lock-in amplifier. The magnetic moment is given by: m_ ðBÞ ¼ CUpick-up , where C is the calibration constant of the pick-up coils. The procedure to obtain the calibration constant is described in detail in [6]. The lock-in amplifier displays the root mean square (rms) valuepof the signal a1 C 1 sinð2pft), which is equal to a1 =Cp 2, and of the signal c1 C 1 sinð2pft) (equal to c1 =C 2). These measured values are denoted with Uk;meas and U?;meas respectively. The rms-values of the magnetic p field components B k and B? are equal to Ba sinðaÞ= 2 and p Ba cosðaÞ= 2. These values are denoted with Bk;meas and B?;meas . Combining this with Eq. (12) gives for rotating magnetic field: Q¼
1 Bk;meas CUk;meas 1 B?;meas CU?;meas þ : V V f f
ð13Þ
The loss in uni-directional field is given by: Q¼
1 By;meas CUy;meas ; V f
ð14Þ
where the quantities By;meas and Uy;meas correspond to the quantities in Eq. (13). Details of the measurement set-up and the sample preparation are described extensively in [6]. All measurements presented in this paper are performed in liquid nitrogen at atmospheric pressure (77 K) on a Bi-2223 tape (3:7 mm 0:26 mm) with 55 untwisted filaments, a pure silver matrix and sheath and a critical current of 55 A (E ¼ 104 V/m, self-field).
4. Results The magnetisation loss of multifilamentary BSCCO/ Ag superconducting tapes consists in general of hysteresis loss and coupling current loss. In a magnetisation experiment both contributions are measured. However, in multifilamentary conductors with untwisted filaments, like the one studied in this paper, the bundle of filaments behaves like one big filament and the loss is hysteretic in nature. The magnetisation measurements presented in this paper are thus hysteresis loss measurements. The magnetisation loss in parallel magnetic field, perpendicular magnetic field and various orientations in between is shown in Fig. 3. The loss is plotted as the dimensionless loss function (C ¼ l0 Q=2B2a ). Qualitatively the shape of the loss functions for the different angles is comparable. The loss in perpendicular field (a ¼ 0°) is about one order of magnitude larger than the loss in parallel field (a ¼ 90°). From a ¼ 0° to a ¼ 45° the loss decreases relatively slowly, after that it decreases rapidly until a ¼ 80°. Further on to a ¼ 90° it again decreases slowly.
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Fig. 3. Loss functions as a function of applied magnetic field for various orientations of the uni-directional magnetic field, the symbols represent the measured data the lines are a guide to the eye (f ¼ 48:2 Hz, T ¼ 77 K).
The maximum of the loss function does not appear at the same field amplitude for all angles. Two effects play a role. First, the maximum appears on different positions for a tape in parallel field (ÔinfiniteÕ slab, Bmax ¼ 4=3Bp [7]) and a tape in perpendicular field (Bmax < Bp ) [8]. Secondly, the critical current in perpendicular field decreases more rapidly than the critical current in parallel field, resulting in a lower field of full penetration. The loss functions for a ¼ 0° to a ¼ 60° show a curvature (deviation from the straight line in the double logarithmic plot) for magnetic fields around 1 mT. Since the ÔbumpÕ is moving to lower field amplitude for increasing a, it probably appears below the lowest measured field amplitudes for a ¼ 70° to a ¼ 90°. A similar effect is observed in other BSCCO/Ag tapes [9]. The effect was attributed to the existence of two magnetic sub-systems in the tape: bulk and grains. This results in a loss function that is the sum of the functions of the bulk and the grains, with different maximum (position and height). In Fig. 4 the loss in uni-directional field (same data as in Fig. 3) is plotted versus the perpendicular component of the applied magnetic field. For angles a up to 60°, the magnetisation loss is practically completely determined by the magnetic field perpendicular to the tape. For angles closer to 90° the loss is larger than the loss contribution of the perpendicular magnetic field. For increasing magnetic field this difference becomes smaller. When the magnetic field angle a is greater than 60° the loss caused by the parallel component of the magnetic field becomes significant. The magnetisation loss for rotating magnetic field is shown in Fig. 5 for various values of h (see Fig. 1) and u ¼ 90°. In the graph also the measured loss in uni-
Fig. 4. Magnetisation loss as a function of the magnetic field component perpendicular to the wide face of the tape conductor, the symbols represent the measured data (f ¼ 48:2 Hz, T ¼ 77 K).
Fig. 5. Loss functions for rotating magnetic field (filled circles) with various ratios of perpendicular and parallel field amplitudes, the contribution of the perpendicular and parallel (open symbols) components and a comparison with the loss in unidirctional field (solid lines) (f ¼ 48:2 Hz, T ¼ 77 K).
directional field with the same value of h ( ¼ a in that case) is shown. The loss in rotating magnetic field is described well by the lines [2]. For values of h up to 70°, the contribution of the perpendicular magnetic field component (see Eq. (12)) to the magnetisation loss is dominant. This is illustrated with the perpendicular loss component for h ¼ 70° in Fig. 5 that is indicated with h ¼ 70°, ?. The perpendicular component for h ¼ 20° and 45°, which is not shown in the graph, practically coincides with the total magnetisation loss in rotating magnetic field. For h ¼ 80° the situation is different.
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Here the contribution of the parallel magnetic field component to the magnetisation loss becomes also important, it is even larger than the perpendicular component. The two components are both shown, indicated with h ¼ 80°; ?, and h ¼ 80°; k.
5. Models for the angle dependency In the former section it was already shown that the magnetisation loss in unidirectional magnetic field depends strongly on the magnetic field component perpendicular to the wide face of the conductor. In this section different methods to describe the magnetisation loss as a function of the angle between the tape and the magnetic field are shown. The difference between the equations and the measurement results is explained with the help of the equations that are derived in the section theory. In [4] and [10] similar analytical models for the magnetisation loss as a function of the orientation of the applied magnetic field are derived for field amplitudes far above the penetration field. However a comparison with the measurements in [4] shows that the model predicts the loss correctly for parallel and perpendicular field but not for orientations in between. An overestimation of the loss of about two times is observed. The model in [10] is not compared with measured data. No analytical models for the angle dependency of the magnetisation loss below and around the penetration field are known at present. In order to describe the magnetisation loss for arbitrary orientation, the measured data in parallel and perpendicular field is used to calculate the loss. Although a BSCCO/Ag superconducting tape is a multifilamentary conductor with filaments that consist of multiple grains, the tape is considered as a continuum. In the first place the filaments are untwisted and thus electromagnetically coupled in a 0.1 m tape with an applied field of 0.5–100 mT with a frequency of 50 Hz. Furthermore the contribution of the individual grains to the magnetisation loss is negligible [11]. In [3] a method is proposed to calculate the magnetisation loss as a function of the angle a between the magnetic field and the normal on the wide face of the conductor. In a lower limit for the loss (Ql ) only the magnetic field component perpendicular to the tape, Ba cos a (similar to Fig. 4) is accounted for: Ql ðBa ; aÞ ¼ Q? ðBa cos aÞ:
Qu ðBa ; aÞ ¼ Q? ðBa cos aÞ þ Qk ðBa sin aÞ:
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ð16Þ
In Fig. 6 the loss function, C ¼ l0 Q=2B2a is shown for perpendicular and parallel magnetic field and an orientation close to each of them (a ¼ 30° and a ¼ 70° respectively). Also the loss calculated with Eqs. (15) and (16) is shown for the two field orientations. Close to perpendicular magnetic field (a ¼ 30°) the calculated upper and lower limit for the loss are identical because the contribution from the parallel magnetic field component is negligible. The calculated line describes the measured points very well. The situation for a ¼ 70° is different. Because the loss contribution of the parallel magnetic field component becomes significant in Eq. (16), there is a clear difference between the calculated upper and lower limit for the magnetisation loss. When the model line is compared with the measured values different regimes can be distinguished. Below the maximum of the loss function the measured loss is larger than the calculated loss although the calculated upper limit (Qu ) deviates less from the measured data than the calculated lower limit (Ql ). Obvious, not only the perpendicular component of the magnetic field contributes to the magnetisation loss. The contribution from the parallel component of the magnetic field, used in Eq. (16), is calculated for the case that there is only magnetic field applied parallel to the conductor. In the case that a < 90° there is also a perpendicular magnetic field component that influences the critical current of the superconductor which in turn results in an increase of the magnetisation loss below the penetration field. So the second term of Eq. (16) gives a contribution that is too small.
ð15Þ
Close to parallel magnetic field (a ¼ 90°) the contribution from the parallel field component becomes significant. Assume that both components do not interact and can coexist in the conductor so they can be summed in order to obtain an upper limit (Qu ) for the magnetisation loss:
Fig. 6. Loss functions for uni-directional field with various orientation a. The symbols represent measured points. The lines represent the upper (Qu ) and lower (Ql ) limit for the loss calculated with Eqs. (15) and (16), the symbols represent the measured data (f ¼ 48:2 Hz, T ¼ 77 K).
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When the magnetic field increases after the maximum the calculated upper limit for the magnetisation loss for a ¼ 70° is larger than the measured value. This is a typical behaviour of a loss relation calculated with a too large Ic because in that case the penetration field is too large and the magnetisation loss above the penetration field is overestimated. Furthermore, a summation of the loss caused by the perpendicular and parallel field component leads to an overestimation of the loss when the conductor is completely filled with screening currents due to the field in one direction leaving no space for the other direction. However, the calculated lower limit for the loss approaches the measured values for increasing magnetic field. For a magnetic field well above the penetration field the induced currents from the parallel and perpendicular field components can of course not coexist. The electric field induced by the perpendicular magnetic field component is larger than the induced electric field from the parallel field component so screening and thus loss in this direction will be dominant [3]. The angle dependency of the magnetisation loss can be described also with another method. This method uses the local field dependency of the magnetisation loss relation in perpendicular applied magnetic field and the ratio between the loss in parallel and perpendicular applied magnetic field. In fact, the idea behind this method is the same as the idea behind the method described before (Eqs. (15) and (16)), but the equations that are obtained give a better insight in the angle dependent behaviour of the loss. The method is not only applicable for the magnetisation loss as a function of the orientation of the magnetic field but also for other loss components, e.g. the transport current loss in AC magnetic field with various orientations [12] or the Ôdynamic resistance lossÕ [13]. Starting at perpendicular magnetic field (a ¼ 0°) an increase of the angle a means that the magnetic field component perpendicular to the wide face of the conductor, B? ðaÞ changes as Ba cos a. For the magnetisation loss well below the penetration field a QðBa Þ / B3a dependence is observed. This means for the angle dependency: QðaÞ / cos3 a, since the magnetisation loss depends completely on the perpendicular component of the magnetic field when a < 60°. Well above the penetration field the loss depends linearly on the applied magnetic field. This results in: QðaÞ / cos a. In general QðBa Þ / BacðBa Þ and thus QðaÞ / coscðBa;? Þ ðaÞ, where cðBa;? Þ is the field dependent slope of the magnetisation loss in a double logarithmic plot of the magnetisation loss versus the applied magnetic field given by: cðBa;? Þ ¼
Ba;? dQðBa;? Þ : QðBa;? Þ dBa;?
ð17Þ
The magnetisation loss as a function of applied magnetic field perpendicular to the wide face of the tape
can be described with an analytical model, see e.g. [8], in order to calculate Eq. (17). Another possibility is to model the measured data, e.g. with [14]: QðBa;? Þ ¼
aBca bBa : aBca þ bBa
ð18Þ
The normalised magnetisation loss as a function of angle a is modelled as: QðaÞ ¼ Qð0°Þ
Qð90°Þ Qð90°Þ 1 ; coscðBa;? Þ a þ Qð0°Þ Qð0°Þ
ð19Þ
with cðBa;? Þ given by Eq. (17) and an appropriate description of the magnetisation loss relation in perpendicular magnetic field. The last term in the equation, Qð90°Þ=Qð0°Þ forces the relation to the correct value for parallel magnetic field. In fact only the change in the loss due to the change in perpendicular field component is taken into account. In [15] a similar approach with Q / a sinn ðaÞ þ b cosn ðaÞ is followed. A field dependent exponent for the sine and cosine function is introduced, modelled with a function that goes to infinity for B ! 0. The link with the slope of the loss relation is not made. A theoretical value of n ¼ 2 that is mentioned in [15] is valid for coupling current loss as a function of the orientation of the applied magnetic field [16] and not for the hysteresis loss that is considered. In Fig. 7 the normalised magnetisation loss as a function of the orientation a of the magnetic field is shown for various amplitudes of the applied magnetic field. The solid lines show the angle dependent loss calculated with Eq. (19). Also the angle depen-
Fig. 7. The normalised magnetisation loss as a function of the orientation of the magnetic field for various values of the amplitude Ba of the applied magnetic field. The dotted lines are a guide to the eye. The solid lines represent Eqs. (16) and (19) (f ¼ 48:2 Hz, T ¼ 77 K).
J.J. Rabbers et al. / Cryogenics 42 (2002) 771–778
dency of the loss calculated with Eq. (16) is shown is Fig. 7. For the lowest magnetic field amplitude (0.003 T), Eqs. (16) and (19) yield the same result. For the two higher field amplitudes, the loss calculated with Eq. (16) tends to values that are too high. The higher the field amplitude the larger the difference between the measured loss and the calculated loss. As mentioned before, this model assumes an independent behaviour of the loss caused by the perpendicular and the parallel field component that justifies summing of the loss. When the amplitude of the magnetic field becomes higher, this assumption becomes less valid and the difference with the measured loss is higher. On the other hand the values that are calculated with Eq. (19) tend to a too low estimate of the loss, especially when the angle a reaches 90°. This is explained by the fact that the contribution of the parallel field component is not accounted for accurately, only the extreme case a ¼ 90° is treated correctly. The method described here is also applicable for other AC loss components. When the AC loss due to the perpendicular field component is dominating and the loss dependence is Q / BcðBÞ , then the loss as a function of angle a follows a coscðBÞ a dependence.
6. Conclusions The magnetisation loss of tape superconductors as a function of the orientation of the applied magnetic field is investigated. Different models to describe the angle dependent behaviour of the loss are considered. The measurement technique with pickup coils in unidirectional and rotating magnetic field is studied theoretically. The analysis is also used to explain the difference between the magnetisation loss as a function of the orientation of the magnetic field and models to describe this dependency. The results of magnetisation loss measurements in uni-directional with various orientations and rotating magnetic field are presented. The magnetisation loss in a rotating magnetic field is described quite well with the loss in a uni-directional field. For orientations from perpendicular (0°) up to 60°, the perpendicular component of the magnetic field dominates the magnetisation loss. Different models to describe the magnetisation loss as a function of the orientation of the applied magnetic field from the measured loss in perpendicular and parallel magnetic field are considered. Both models do not take into account only the magnetisation loss caused by the perpendicular component of the magnetic field. Also the contribution of the parallel field component is taken into account. Depending on the field
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orientation and the magnetic field amplitude, the loss is either underestimated or overestimated. The assumption that the loss due to the perpendicular and the parallel field component can be treated independently is not correct. In order to increase the accuracy of the relations, the description of the loss caused by the parallel magnetic field component has to be improved. The new proposed method to describe the loss as a function of the orientation of the magnetic field is not only applicable for the magnetisation loss but also for other loss components.
Acknowledgements 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] Rabbers JJ, van der Laan DC, ten Haken B, ten Kate HHJ. Magnetisation and transport current loss of BSCCO/Ag tape in external AC magnetic field carrying an AC transport current. IEEE Trans Appl Supercond 1999;9:1185. [2] Rabbers JJ, van der Meer O, Klein Zeggelink WFA, Shevchenko OA, ten Haken B, ten Kate HHJ. Magnetisation loss of BSCCO/ Ag tape in uni-directional and rotating magnetic field. Physica C 1999;325:1. [3] Oomen MP, Rieger J, Leghissa M, ten Kate HHJ. Field-angle dependence of AC losses in multifilamentary high-Tc superconducting tapes. Appl Phys Lett 1997;70:3038. [4] Banno N, Ammemiya N, Mihoichi A, Ciszek M, Mukai H, Ohmatsu K. Penetration loss in BSCCO tape without transport current. IEEE Trans Appl Supercond 1999;9:2565. [5] Seeber B. Handbook of applied superconductivity. Bristol, United Kingdom: Institute of physics publishing; 1998. [6] Rabbers JJ, tenHaken B, tenKate HHJ. Advanced AC loss measurement methods for high-temperature superconducting tapes. Rev Sci Instr 2001;72:2365. [7] Wilson MN. Superconducting magnets. Oxford: Clarendon Press; 1983. [8] ten Haken B, Rabbers JJ, ten Kate HHJ. Magnetization and AC loss in a superconductor with an elliptical cross-section and arbitrary aspect ratio. Physica C 2002;277:156. [9] Kwasnitza K, Clerc St. AC losses of superconducting high-Tc multifilament Bi-2223/Ag sheathed tapes in perpendicular magnetic field. Physica C 1999;223:423. [10] Lethonen J, Paasi J, Pittel J, Kovac P. Magnetisation losses in magnets wound from anisotropic HTS tape conductors. Physica C 2000;336:261. [11] M€ uller K-H, Andrikidis C, Liu HK, Dou SX. AC hysteresis losses in multifilamentary Pb–Bi–Sr–Ca–Cu–O/Ag tapes. Physica C 1995;247:74. [12] Rabbers JJ, ten Haken B, ten Kate HHJ. Transport current loss of BSCCO/Ag tape in different orientations of the external AC magnetic field. IEEE Trans Appl Supercond 1999;9:801. [13] Ciszek M, Knoopers HG, Rabbers JJ, ten Haken B, ten Kate HHJ. Angular dependence of the dynamic resistance and its
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relation to the AC transport current loss in Bi-2223/Ag tape superconductors. Supercond Sci Technol 2002;15:1275. [14] Rabbers JJ, ten Haken B, Shevchenko OA, ten Kate HHJ. An engineering formula to describe the AC loss of BSCCO/Ag tape. IEEE Trans Appl Supercond 2001;11:2623.
[15] Wolfbrandt A, Magnusson N, H€ ornfeldt S. AC losses in BSCCO/Ag tape carrying AC transport currents in AC magnetic fields applied in different orientations. IEEE Trans Appl Supercond 2001;11:4123. [16] Carr WJ. AC loss and macroscopic theory of superconductors. New York: Gordon and Breach; 1983. p. 126.
Magnetisation loss of BSCCO/Ag superconducting tape exposed to applied field with arbitrary angle J.J. Rabbers *, O. van der Meer, B. ten Haken, H.H.J. ten Kate Faculty of Applied Physics, Low Temperature Division, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands Received 15 July 2002; accepted 10 October 2002
Abstract Bi2 Sr2 Ca2 Cu3 Ox /Ag tape superconductors are used in applications like power cables and transformers. In these applications the superconductor is exposed to an alternating magnetic field that has different orientations with respect to the tape surface. In this paper the angle dependency of the magnetisation loss is considered from two points of view. First the measurement technique with pickup coils is analysed theoretically. Measured magnetisation loss in uni-directional magnetic field with various orientations and rotating magnetic field are compared. When the orientation is changed from perpendicular (0°) to parallel (90°) applied magnetic fields, the contribution of the perpendicular field component to the magnetisation loss is dominant up to 60°. A new model to describe the angle dependency of the magnetisation loss, based on the measured loss in perpendicular and parallel magnetic field is developed. Deviations between models and the measured loss are explained with the help of the theoretical analysis of the measurement technique. The new model is not only applicable for the magnetisation loss but also for other AC loss components. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: BSCCO/Ag tapes; Magnetisation loss; Rotating field
1. Introduction When Bi2 Sr2 Ca2 Cu3 Ox /Ag tapes are used in electric power applications the magnetisation loss contributes considerably to the total AC loss [1], i.e. magnetisation loss plus transport current loss. Towards the edges of a solenoidal coil the orientation of the magnetic field with respect to the tape surface changes from parallel to perpendicular. In order to calculate the AC loss in these kind of devices, it is necessary to know the magnetisation loss not only in the most often measured limiting cases of magnetic field parallel and perpendicular to the wide side of the tape. Also the intermediate angles should be considered. In this paper a theoretical evaluation of the measurement method of magnetisation loss in arbitrary uni-directional field and rotating magnetic field is given. Measurement results for a variety of magnetic field amplitudes and orientations are presented. A new approximation to estimate the magnetisation loss for orientations between parallel and
*
Corresponding author. Tel.: +31-53-489-4839/3889; fax: +31-53489-1099. E-mail address: [email protected] (J.J. Rabbers).
perpendicular magnetic field is developed. The deviations between the measured loss and the models are explained with the help of the theoretical analysis of the measurement technique.
2. Theory Magnetisation loss measurements on tape superconductors with applied magnetic field under various angles are already published in several papers [2–4]. In these measurements, the orientation of the tape superconductor in the magnetic field is changed and the pickup coils sense the magnetic moment of the sample in the direction of the applied magnetic field. There is no doubt that this is the correct way to measure the loss in this case. Here, the analysis that leads to this conclusion is shown in order to get a better understanding of the magnetisation loss as a function of the orientation of the applied magnetic field. Later on the analysis is used to explain differences between models and the measured loss. Furthermore, the derivation shows how the magnetisation loss in a magnetic field that rotates around the sample can be measured. Although the derivation is
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straightforward, it is shown here in some detail because no references for this specific case are known. A more general description of magnetic loss measurement techniques can be found in [5]. Consider a tape conductor with homogeneous properties and infinite dimension in the longitudinal direction, see Fig. 1. The directions ? and k are defined in the figure and the longitudinal direction is perpendicular to the tape cross-section. The magnetic field B is applied perpendicular to the longitudinal direction. The applied magnetic field can be written as: BðtÞ ¼ Bk ðtÞek þ B? ðtÞe? ; ð1Þ where (in the case of a sinusoidal varying magnetic field): B? ðtÞ ¼ Ba cosðhÞ sinð2pftÞ; ð2Þ Bk ðtÞ ¼ Ba sinðhÞ sinð2pft þ uÞ:
ð3Þ
For u ¼ 0° the applied field is unidirectional. The two extreme situations h ¼ 0° and h ¼ 90° are referred to as perpendicular (?) and parallel (k) respectively. When u 6¼ 0° the magnetic field vector B rotates around the tape. The case where h ¼ 45° and u ¼ 90° describes a circular rotation. Thep rotating magnetic field is characterised with Ba ¼ ðB2a;? þ B2a;k Þ and h ¼ atan= ðBa;k =Ba;? Þ, the unidirectional field is characterised with Ba and h. The resulting magnetic moment generated by the screening currents in the superconductor is written in a similar way: mðBÞ ¼ mk ðBÞek þ m? ðBÞe? :
ð4Þ
Due to the dot product only the equally directed components in Eq. (6) give non-zero contributions: dqðtÞ ¼ ½Bk ðtÞm_ k ðBÞ þ B? ðtÞm_ ? ðBÞ dt ¼ ½dQk ðtÞ þ dQ? ðtÞ dt:
It is important to realise that the magnetic moment is always a function of the applied magnetic field vector B, and not only the component parallel to the considered magnetic field. The component of B perpendicular to the considered magnetic moment does not contribute to the magnetisation loss, but it influences the properties (critical current) of the conductor and the induced current patterns can interact. This will be demonstrated later on. Continue with the parallel component, Q per field cycle in J/m3 (for the perpendicular component the calculation is identical): I 1 Bk ðtÞm_ k ðBÞ dt; ð8Þ Qk ðBÞ ¼ V field cycle where V is the sample volume. The components m_ k (and m_ ? ) of the time derivative of m can be written as a Fourier series: m_ k ðtÞ ¼
1 X
an sinð2npft þ uÞ þ bn cosð2npft þ uÞ:
Combining Eq. (8) and Eq. (9) yields: Z 1 1=f Qk ðBÞ ¼ Ba sinðhÞ sinð2pft þ uÞ V 0 X ½an ðBÞ sinð2npft þ uÞ
n
þ bn ðBÞ cosð2npft þ uÞ dt: ð5Þ
Combining Eq. (1) and the time derivative of Eq. (4) with Eq. (5) yields: dqðtÞ ¼ ½Bk ðtÞek þ B? ðtÞe? ½m_ k ðBÞek þ m_ ? ðBÞe? dt: ð6Þ
ð9Þ
n¼1
The magnetisation loss, in J, is calculated with: _ ðBÞdt: dqðtÞ ¼ BðtÞ dmðBÞ ¼ BðtÞ m
ð7Þ
ð10Þ
Only the a1 term in the Fourier series gives non-zero result after integration over an entire magnetic field cycle: Z 1 1=f Qk ðBÞ ¼ Ba sinðhÞ sinð2pft þ uÞa1 ðBÞ V 0
sinð2pft þ uÞdt: ð11Þ Calculating the integral and adding the loss component of the perpendicular field components yields the magnetisation loss (c1 is the first Fourier term in the perpendicular loss contribution, similar to Eq. (9)): QðBÞ ¼
Fig. 1. The tape cross-section and definition of the magnetic field directions k and the ? and the angle h.
1 Ba sinðhÞa1 ðBÞ 1 Ba cosðhÞc1 ðBÞ þ : V 2f V 2f
ð12Þ
The last equation shows that the loss in perpendicular and parallel direction have to be summed in order to obtain the magnetisation loss in uni-directional or rotating magnetic field. Only components of the magnetisation that have the same direction as the applied field component contribute to the loss, but their magnitude is a function of the field vector B. In the next section the
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practical implementation of the loss measurement is discussed.
3. Measurement setup The magnetisation loss in unidirectional magnetic field is measured in a dipole magnet that generates a magnetic field in de y-direction, see Fig. 2. The normal on the wide face of the tape sample makes an angle a with the y-axis. In that case only the magnetic moment in the y-direction (the direction parallel to the applied magnetic field) contributes to the loss, as shown before. A set of pickup coils that sense the magnetic moment in the y-direction is used to measure the loss. The magnetisation loss in a rotating magnetic field is measured differently. With a two-dipole magnet system, magnetic fields are generated perpendicular to each other, see Fig. 2. In this case the orientation a of the sample is either 0° or 90°. The pick-up coils sense the magnetic moment of the sample in the y-direction which is the perpendicular (?) or the parallel (k) direction respectively, depending on the orientation of the sample in the pick-up coils. The measurement of the magnetisation loss in rotating magnetic field is performed in two steps [2]. The magnetisation of the sample in the parallel and the perpendicular direction has to be measured and since the set-up has only one set of pick-up coils (for the ydirection) two measurements are necessary. First the magnetisation of the sample in perpendicular direction is measured, while the field vector B rotates around the sample. Then the orientation of the sample is changed 90° so the pick-up coils sense the magnetisation of the sample in parallel direction. But then also the orientation of the magnetic field has to be changed 90° and the magnetisation in the parallel direction is measured, while the field vector B rotates around the sample. As discussed before the two contributions are summed in order to obtain the total magnetisation loss in a rotating magnetic field.
Fig. 2. Cross-section of the measurement setup with dipole magnets, pick-up coils, sample (magnified) and the definition of the k-, ?, x- and y-direction.
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The signal from the pick-up coils is measured with a lock-in amplifier. The magnetic moment is given by: m_ ðBÞ ¼ CUpick-up , where C is the calibration constant of the pick-up coils. The procedure to obtain the calibration constant is described in detail in [6]. The lock-in amplifier displays the root mean square (rms) valuepof the signal a1 C 1 sinð2pft), which is equal to a1 =Cp 2, and of the signal c1 C 1 sinð2pft) (equal to c1 =C 2). These measured values are denoted with Uk;meas and U?;meas respectively. The rms-values of the magnetic p field components B k and B? are equal to Ba sinðaÞ= 2 and p Ba cosðaÞ= 2. These values are denoted with Bk;meas and B?;meas . Combining this with Eq. (12) gives for rotating magnetic field: Q¼
1 Bk;meas CUk;meas 1 B?;meas CU?;meas þ : V V f f
ð13Þ
The loss in uni-directional field is given by: Q¼
1 By;meas CUy;meas ; V f
ð14Þ
where the quantities By;meas and Uy;meas correspond to the quantities in Eq. (13). Details of the measurement set-up and the sample preparation are described extensively in [6]. All measurements presented in this paper are performed in liquid nitrogen at atmospheric pressure (77 K) on a Bi-2223 tape (3:7 mm 0:26 mm) with 55 untwisted filaments, a pure silver matrix and sheath and a critical current of 55 A (E ¼ 104 V/m, self-field).
4. Results The magnetisation loss of multifilamentary BSCCO/ Ag superconducting tapes consists in general of hysteresis loss and coupling current loss. In a magnetisation experiment both contributions are measured. However, in multifilamentary conductors with untwisted filaments, like the one studied in this paper, the bundle of filaments behaves like one big filament and the loss is hysteretic in nature. The magnetisation measurements presented in this paper are thus hysteresis loss measurements. The magnetisation loss in parallel magnetic field, perpendicular magnetic field and various orientations in between is shown in Fig. 3. The loss is plotted as the dimensionless loss function (C ¼ l0 Q=2B2a ). Qualitatively the shape of the loss functions for the different angles is comparable. The loss in perpendicular field (a ¼ 0°) is about one order of magnitude larger than the loss in parallel field (a ¼ 90°). From a ¼ 0° to a ¼ 45° the loss decreases relatively slowly, after that it decreases rapidly until a ¼ 80°. Further on to a ¼ 90° it again decreases slowly.
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Fig. 3. Loss functions as a function of applied magnetic field for various orientations of the uni-directional magnetic field, the symbols represent the measured data the lines are a guide to the eye (f ¼ 48:2 Hz, T ¼ 77 K).
The maximum of the loss function does not appear at the same field amplitude for all angles. Two effects play a role. First, the maximum appears on different positions for a tape in parallel field (ÔinfiniteÕ slab, Bmax ¼ 4=3Bp [7]) and a tape in perpendicular field (Bmax < Bp ) [8]. Secondly, the critical current in perpendicular field decreases more rapidly than the critical current in parallel field, resulting in a lower field of full penetration. The loss functions for a ¼ 0° to a ¼ 60° show a curvature (deviation from the straight line in the double logarithmic plot) for magnetic fields around 1 mT. Since the ÔbumpÕ is moving to lower field amplitude for increasing a, it probably appears below the lowest measured field amplitudes for a ¼ 70° to a ¼ 90°. A similar effect is observed in other BSCCO/Ag tapes [9]. The effect was attributed to the existence of two magnetic sub-systems in the tape: bulk and grains. This results in a loss function that is the sum of the functions of the bulk and the grains, with different maximum (position and height). In Fig. 4 the loss in uni-directional field (same data as in Fig. 3) is plotted versus the perpendicular component of the applied magnetic field. For angles a up to 60°, the magnetisation loss is practically completely determined by the magnetic field perpendicular to the tape. For angles closer to 90° the loss is larger than the loss contribution of the perpendicular magnetic field. For increasing magnetic field this difference becomes smaller. When the magnetic field angle a is greater than 60° the loss caused by the parallel component of the magnetic field becomes significant. The magnetisation loss for rotating magnetic field is shown in Fig. 5 for various values of h (see Fig. 1) and u ¼ 90°. In the graph also the measured loss in uni-
Fig. 4. Magnetisation loss as a function of the magnetic field component perpendicular to the wide face of the tape conductor, the symbols represent the measured data (f ¼ 48:2 Hz, T ¼ 77 K).
Fig. 5. Loss functions for rotating magnetic field (filled circles) with various ratios of perpendicular and parallel field amplitudes, the contribution of the perpendicular and parallel (open symbols) components and a comparison with the loss in unidirctional field (solid lines) (f ¼ 48:2 Hz, T ¼ 77 K).
directional field with the same value of h ( ¼ a in that case) is shown. The loss in rotating magnetic field is described well by the lines [2]. For values of h up to 70°, the contribution of the perpendicular magnetic field component (see Eq. (12)) to the magnetisation loss is dominant. This is illustrated with the perpendicular loss component for h ¼ 70° in Fig. 5 that is indicated with h ¼ 70°, ?. The perpendicular component for h ¼ 20° and 45°, which is not shown in the graph, practically coincides with the total magnetisation loss in rotating magnetic field. For h ¼ 80° the situation is different.
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Here the contribution of the parallel magnetic field component to the magnetisation loss becomes also important, it is even larger than the perpendicular component. The two components are both shown, indicated with h ¼ 80°; ?, and h ¼ 80°; k.
5. Models for the angle dependency In the former section it was already shown that the magnetisation loss in unidirectional magnetic field depends strongly on the magnetic field component perpendicular to the wide face of the conductor. In this section different methods to describe the magnetisation loss as a function of the angle between the tape and the magnetic field are shown. The difference between the equations and the measurement results is explained with the help of the equations that are derived in the section theory. In [4] and [10] similar analytical models for the magnetisation loss as a function of the orientation of the applied magnetic field are derived for field amplitudes far above the penetration field. However a comparison with the measurements in [4] shows that the model predicts the loss correctly for parallel and perpendicular field but not for orientations in between. An overestimation of the loss of about two times is observed. The model in [10] is not compared with measured data. No analytical models for the angle dependency of the magnetisation loss below and around the penetration field are known at present. In order to describe the magnetisation loss for arbitrary orientation, the measured data in parallel and perpendicular field is used to calculate the loss. Although a BSCCO/Ag superconducting tape is a multifilamentary conductor with filaments that consist of multiple grains, the tape is considered as a continuum. In the first place the filaments are untwisted and thus electromagnetically coupled in a 0.1 m tape with an applied field of 0.5–100 mT with a frequency of 50 Hz. Furthermore the contribution of the individual grains to the magnetisation loss is negligible [11]. In [3] a method is proposed to calculate the magnetisation loss as a function of the angle a between the magnetic field and the normal on the wide face of the conductor. In a lower limit for the loss (Ql ) only the magnetic field component perpendicular to the tape, Ba cos a (similar to Fig. 4) is accounted for: Ql ðBa ; aÞ ¼ Q? ðBa cos aÞ:
Qu ðBa ; aÞ ¼ Q? ðBa cos aÞ þ Qk ðBa sin aÞ:
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ð16Þ
In Fig. 6 the loss function, C ¼ l0 Q=2B2a is shown for perpendicular and parallel magnetic field and an orientation close to each of them (a ¼ 30° and a ¼ 70° respectively). Also the loss calculated with Eqs. (15) and (16) is shown for the two field orientations. Close to perpendicular magnetic field (a ¼ 30°) the calculated upper and lower limit for the loss are identical because the contribution from the parallel magnetic field component is negligible. The calculated line describes the measured points very well. The situation for a ¼ 70° is different. Because the loss contribution of the parallel magnetic field component becomes significant in Eq. (16), there is a clear difference between the calculated upper and lower limit for the magnetisation loss. When the model line is compared with the measured values different regimes can be distinguished. Below the maximum of the loss function the measured loss is larger than the calculated loss although the calculated upper limit (Qu ) deviates less from the measured data than the calculated lower limit (Ql ). Obvious, not only the perpendicular component of the magnetic field contributes to the magnetisation loss. The contribution from the parallel component of the magnetic field, used in Eq. (16), is calculated for the case that there is only magnetic field applied parallel to the conductor. In the case that a < 90° there is also a perpendicular magnetic field component that influences the critical current of the superconductor which in turn results in an increase of the magnetisation loss below the penetration field. So the second term of Eq. (16) gives a contribution that is too small.
ð15Þ
Close to parallel magnetic field (a ¼ 90°) the contribution from the parallel field component becomes significant. Assume that both components do not interact and can coexist in the conductor so they can be summed in order to obtain an upper limit (Qu ) for the magnetisation loss:
Fig. 6. Loss functions for uni-directional field with various orientation a. The symbols represent measured points. The lines represent the upper (Qu ) and lower (Ql ) limit for the loss calculated with Eqs. (15) and (16), the symbols represent the measured data (f ¼ 48:2 Hz, T ¼ 77 K).
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When the magnetic field increases after the maximum the calculated upper limit for the magnetisation loss for a ¼ 70° is larger than the measured value. This is a typical behaviour of a loss relation calculated with a too large Ic because in that case the penetration field is too large and the magnetisation loss above the penetration field is overestimated. Furthermore, a summation of the loss caused by the perpendicular and parallel field component leads to an overestimation of the loss when the conductor is completely filled with screening currents due to the field in one direction leaving no space for the other direction. However, the calculated lower limit for the loss approaches the measured values for increasing magnetic field. For a magnetic field well above the penetration field the induced currents from the parallel and perpendicular field components can of course not coexist. The electric field induced by the perpendicular magnetic field component is larger than the induced electric field from the parallel field component so screening and thus loss in this direction will be dominant [3]. The angle dependency of the magnetisation loss can be described also with another method. This method uses the local field dependency of the magnetisation loss relation in perpendicular applied magnetic field and the ratio between the loss in parallel and perpendicular applied magnetic field. In fact, the idea behind this method is the same as the idea behind the method described before (Eqs. (15) and (16)), but the equations that are obtained give a better insight in the angle dependent behaviour of the loss. The method is not only applicable for the magnetisation loss as a function of the orientation of the magnetic field but also for other loss components, e.g. the transport current loss in AC magnetic field with various orientations [12] or the Ôdynamic resistance lossÕ [13]. Starting at perpendicular magnetic field (a ¼ 0°) an increase of the angle a means that the magnetic field component perpendicular to the wide face of the conductor, B? ðaÞ changes as Ba cos a. For the magnetisation loss well below the penetration field a QðBa Þ / B3a dependence is observed. This means for the angle dependency: QðaÞ / cos3 a, since the magnetisation loss depends completely on the perpendicular component of the magnetic field when a < 60°. Well above the penetration field the loss depends linearly on the applied magnetic field. This results in: QðaÞ / cos a. In general QðBa Þ / BacðBa Þ and thus QðaÞ / coscðBa;? Þ ðaÞ, where cðBa;? Þ is the field dependent slope of the magnetisation loss in a double logarithmic plot of the magnetisation loss versus the applied magnetic field given by: cðBa;? Þ ¼
Ba;? dQðBa;? Þ : QðBa;? Þ dBa;?
ð17Þ
The magnetisation loss as a function of applied magnetic field perpendicular to the wide face of the tape
can be described with an analytical model, see e.g. [8], in order to calculate Eq. (17). Another possibility is to model the measured data, e.g. with [14]: QðBa;? Þ ¼
aBca bBa : aBca þ bBa
ð18Þ
The normalised magnetisation loss as a function of angle a is modelled as: QðaÞ ¼ Qð0°Þ
Qð90°Þ Qð90°Þ 1 ; coscðBa;? Þ a þ Qð0°Þ Qð0°Þ
ð19Þ
with cðBa;? Þ given by Eq. (17) and an appropriate description of the magnetisation loss relation in perpendicular magnetic field. The last term in the equation, Qð90°Þ=Qð0°Þ forces the relation to the correct value for parallel magnetic field. In fact only the change in the loss due to the change in perpendicular field component is taken into account. In [15] a similar approach with Q / a sinn ðaÞ þ b cosn ðaÞ is followed. A field dependent exponent for the sine and cosine function is introduced, modelled with a function that goes to infinity for B ! 0. The link with the slope of the loss relation is not made. A theoretical value of n ¼ 2 that is mentioned in [15] is valid for coupling current loss as a function of the orientation of the applied magnetic field [16] and not for the hysteresis loss that is considered. In Fig. 7 the normalised magnetisation loss as a function of the orientation a of the magnetic field is shown for various amplitudes of the applied magnetic field. The solid lines show the angle dependent loss calculated with Eq. (19). Also the angle depen-
Fig. 7. The normalised magnetisation loss as a function of the orientation of the magnetic field for various values of the amplitude Ba of the applied magnetic field. The dotted lines are a guide to the eye. The solid lines represent Eqs. (16) and (19) (f ¼ 48:2 Hz, T ¼ 77 K).
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dency of the loss calculated with Eq. (16) is shown is Fig. 7. For the lowest magnetic field amplitude (0.003 T), Eqs. (16) and (19) yield the same result. For the two higher field amplitudes, the loss calculated with Eq. (16) tends to values that are too high. The higher the field amplitude the larger the difference between the measured loss and the calculated loss. As mentioned before, this model assumes an independent behaviour of the loss caused by the perpendicular and the parallel field component that justifies summing of the loss. When the amplitude of the magnetic field becomes higher, this assumption becomes less valid and the difference with the measured loss is higher. On the other hand the values that are calculated with Eq. (19) tend to a too low estimate of the loss, especially when the angle a reaches 90°. This is explained by the fact that the contribution of the parallel field component is not accounted for accurately, only the extreme case a ¼ 90° is treated correctly. The method described here is also applicable for other AC loss components. When the AC loss due to the perpendicular field component is dominating and the loss dependence is Q / BcðBÞ , then the loss as a function of angle a follows a coscðBÞ a dependence.
6. Conclusions The magnetisation loss of tape superconductors as a function of the orientation of the applied magnetic field is investigated. Different models to describe the angle dependent behaviour of the loss are considered. The measurement technique with pickup coils in unidirectional and rotating magnetic field is studied theoretically. The analysis is also used to explain the difference between the magnetisation loss as a function of the orientation of the magnetic field and models to describe this dependency. The results of magnetisation loss measurements in uni-directional with various orientations and rotating magnetic field are presented. The magnetisation loss in a rotating magnetic field is described quite well with the loss in a uni-directional field. For orientations from perpendicular (0°) up to 60°, the perpendicular component of the magnetic field dominates the magnetisation loss. Different models to describe the magnetisation loss as a function of the orientation of the applied magnetic field from the measured loss in perpendicular and parallel magnetic field are considered. Both models do not take into account only the magnetisation loss caused by the perpendicular component of the magnetic field. Also the contribution of the parallel field component is taken into account. Depending on the field
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orientation and the magnetic field amplitude, the loss is either underestimated or overestimated. The assumption that the loss due to the perpendicular and the parallel field component can be treated independently is not correct. In order to increase the accuracy of the relations, the description of the loss caused by the parallel magnetic field component has to be improved. The new proposed method to describe the loss as a function of the orientation of the magnetic field is not only applicable for the magnetisation loss but also for other loss components.
Acknowledgements 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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[15] Wolfbrandt A, Magnusson N, H€ ornfeldt S. AC losses in BSCCO/Ag tape carrying AC transport currents in AC magnetic fields applied in different orientations. IEEE Trans Appl Supercond 2001;11:4123. [16] Carr WJ. AC loss and macroscopic theory of superconductors. New York: Gordon and Breach; 1983. p. 126.