AC losses in high-temperature superconductor BSCCO hollow cylinders with induced current

Physica C 319 Ž1999. 238–248

AC losses in high-temperature superconductor BSCCO hollow cylinders with induced current V. Meerovich ) , V. Sokolovsky, S. Goren, G. Jung Physics Department, Ben-Gurion UniÕersity of the NegeÕ, P.O. Box 653, 84105 Beersheba, Israel Received 10 November 1998; received in revised form 15 April 1999; accepted 26 April 1999

Abstract The issue of the correlation of AC loss data obtained in different experimental configurations for high-temperature superconductor ŽHTSC. samples of different topology is investigated using a practically important example of a hollow cylinder with induced current. The basic configuration under consideration is a transformer device where the current in a BSCCO cylinder is induced by the magnetic flux from a coil positioned inside the cylinder. Using a novel contactless method, the AC losses in the cylinder are measured. It is shown that Bean’s critical state model is unsuitable for explanation of the experimental data. Moreover, in the case of complete penetration, the AC loss values appear to be quite different for a slab, a hollow cylinder in uniform external magnetic field and a cylinder in the investigated configuration. Thus, neither the critical state model nor the experimental data obtained on samples of simple geometry can be used to determine AC losses in HTSC hollow cylinders. A method for the AC loss evaluation in HTSC cylinders that is based on using the real E–J characteristic and the mathematical model of a transformer is proposed. It provides a good agreement of experimental and calculated values of AC losses. q 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: AC losses; High-Tc superconductors; Critical state

1. Introduction The problem of AC loss evaluation is an important issue for the development of superconducting devices for power applications at alternating currents. AC losses in low-temperature superconductors have traditionally been calculated within the framework of Bean’s critical state model ŽCSM. w1x based on the concept of the critical current density. This approach is based on the stepwise dependence of the electric field E on the current density J that is )

Corresponding author. Tel.: q972-7-647-2458; fax: q972-7647-2903.

typical for low-temperature superconductors. The validity of this approach for a low-temperature superconducting sample of any shape and size, carrying the transport current or effected by external magnetic fields, was confirmed by numerous experiments. Therefore, the experimental data obtained for small samples of simple geometry have provided the comparison with calculation. Using these data, AC losses in superconducting elements of full-scale devices were evaluated. In contrast, in high-temperature superconductor ŽHTSC. materials, the observed E–J curves show noticeably non-linear behavior over the wide range of current densities w2x. The question of legitimacy of

0921-4534r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 3 0 1 - 9

V. MeeroÕich et al.r Physica C 319 (1999) 238–248

using the CSM for the description of electromagnetic properties of HTSC materials was discussed in many papers w3–6x. Many experiments showed deviations from this model w4,5x. Nevertheless, a number of authors have reported good agreement of experimental values of low frequency AC losses and susceptibility with ones calculated in the framework of the CSM w3,6x. Most of the researchers investigating the applicability of the CSM for various HTSC materials relate the criteria of this applicability with the shape of E–J characteristic. In the previous papers w7,8x, we have shown that the applicability of the CSM is determined not only by the characteristic of a superconductor but also by the parameters of applied magnetic field as well. We obtained a simple criterion of the CSM validity for the AC loss calculation in a slab placed in a uniform magnetic field w9x. Because of the simplicity of the comparison of experimental results with theoretical models, slab samples or other simple topologies have been used by the majority of the researchers for investigations of AC losses. There is a question as to how the results that were obtained for samples of simple geometry and small sizes can be applied to samples of more complicated shapes and larger sizes. The purpose of this paper is to investigate AC losses in a two-linked configuration — a hollow cylinder with the current induced by external magnetic field. The validity of the classical CSM for samples of this topology will be discussed. This configuration is of practical importance because of the application for superconducting magnetic shields w10x and inductive current limiting devices w11x.

239

2. Experimental 2.1. Experimental configuration The experimental study of AC losses is usually carried out on samples placed in external uniform magnetic field w4x. The basic distinguished feature of the configuration that is investigated in this paper is the placement of the magnetic field source inside the bore of a superconducting cylinder. The experimental model, shown schematically in Fig. 1, constitutes an open core transformer and is similar to the models of an inductive fault current limiter previously investigated by us w11,12x. A 400-turn primary coil, 36 mm in height and 22 mm in diameter, made out of a 0.3 mm cooper wire is inserted tightly into a secondary one-turn superconducting coil formed by melt cast 2212 BSCCO cylinder. The primary coil and cylinder are centered on a 10 = 12 mm2 cross-section ferromagnetic core. The model was immersed in liquid nitrogen at the temperature of 77 K. The measured inductance of the primary coil in the model without the superconducting cylinder was 2.6 = 10y2 H. Our experimental setup ŽFig. 1. has been described in detail in Refs. w11,12x. In the current experiments, we have supplemented the standard arrangement with a Hall probe and equipment for magnetic field measurements. To prevent heating of the superconducting cylinder, an electronic switch was incorporated into the test circuit. The switch enables measurements in a short-run regime at high currents above 3.2 A to last about 0.05 s. The impedance of our test circuit in this regime was

Fig. 1. Experimental model and setup.

240

V. MeeroÕich et al.r Physica C 319 (1999) 238–248

about 4 V. It was sufficient to provide a quick attenuation of the transient component of the current. 2.2. BSCCO samples The BSCCO cylinders were fabricated using the melt cast technology developed by Hoechst w13,14x. The melt cast process relies on casting a homogeneous melt of the starting materials in moulds of the required shape and size. The hot melt is cast into rotating moulds, where it is evenly distributed on the inner side of the walls. After solidification of the melt, elements undergo a suitable heat treatment in order to obtain proper superconducting properties. As obtained casts were machined into 30 mm height, 35 mm inside diameter and 5.1 mm wall thickness cylinders. Characterization of the specimens by DC four-point method gave the value of 570 Arcm2 for the critical current density Žfrom 1 mVrcm criterion. and the critical temperature of 94 K. DC current– voltage characteristic and critical current vs. magnetic field have been studied in detail and described elsewhere w14x. Using the same experimental configuration as shown in Fig. 1, we have studied the response of the cylinders to a step of the current in the primary coil w15x. The decay of the induced current in the cylinder was well fitted to the logarithmic expression describing the classical flux creep in hard superconductors. This typical behavior was observed in the wide range of electric field intensities from 10y9 to 3 = 10y3 Vrcm. The dynamic E–J characteristic of the cylinder ŽFig. 2. was determined from the relation between the maximum electric field and the maximum current induced in the cylinder by a step of magnetic field. The E–J curves are characterized by a dissipation onset visible even well below the field intensity of 1 mVrcm. The smooth form of the characteristic extends into the entire range of electric field intensities investigated. For current densities above about 600 Arcm2 the dependence is well fitted by the power law with the exponent equal to 7. 2.3. Procedure of AC loss determination Inductive methods of measurements of electromagnetic properties of superconducting samples are

Fig. 2. E – J characteristic of HTSC BSCCO cylinder.

widely used by many researchers. A typical inductive setup consists of a driving coil producing magnetic field and pick-up coils that enables one to detect the changes in the field imposed by the presence of the sample under test w5x. Our novel procedure of AC loss determination in a hollow cylinder relies on measuring, with the help of Hall probe technique, the magnetic flux density Bm in the ferromagnetic core, on the cylinder axis, as a function of the instantaneous current i c in the primary coil. The procedure can be described as following. A general expression for AC loss power in the superconductor can be written as Ps

1

T

H U i dt, T 0 s s

Ž 1.

where T is the period of AC current, Us is the voltage drop across the secondary coil, and i s is the current in the superconductor. Let us express the losses as a function of the magnetic field Bm and the primary current i c . The magnetic field in the device results from the superposition of the magnetic field Bc produced by the primary coil current i c and the field Bs due to the current i s circulating in the superconducting cylinder. In the range of fields used in the experiment, the ferromagnetic core has linear magnetic characteristics and, therefore, all magnetic fields are proportional to currents that produce them. The proportionality coefficient k b between the magnetic field Bc and the primary current i c can be obtained from an

V. MeeroÕich et al.r Physica C 319 (1999) 238–248

241

2.4. Measurement results

Fig. 3. AC losses in hollow cylinder as a function of current in primary coil: experimental points and CSM calculation for various cases: slab Žcurve 1., hollow cylinder in uniform magnetic field Žcurve 2., hollow cylinder in our experimental model Žcurve 3.. Frequency 50 Hz, current is given in RMS values.

experiment without a superconducting cylinder. The cylinder current is determined as i s s w Ž Bm y i c k b . rk b ,

Ž 2.

where w is the number of turns of the primary coil. The voltage drop Us is determined from the equation of a two-coil transformer:

ž

L d is w

2

dt

/

q Us w s yM

d ic dt

,

Fig. 3 shows the dependence of the loss power in the cylinder on the RMS current in the primary coil at the frequency of 50 Hz. The points related to low currents and shown by cross symbols in the figure were obtained in prolonged AC regime Žunder steady-state conditions.. The points marked by squares were obtained in the short-run regime of the test circuit, in which high AC currents were applied to the primary coil for several periods only. For example, when the current in the primary coil is 8.5 A rms , the losses are equal to 150 W and can increase the temperature by not more than 0.5 K during the measuring time of 0.05 s Žunder the assumption of homogeneous heating of the cylinder.. The negligible heating is confirmed by the fact that the loop ‘‘current in the cylinder vs. current in the coil’’ does not change during several AC cycles ŽFig. 4.. In the case of marked heating, the amplitude of the current in the cylinder is bound to decrease from period to period resulting in the narrowing of the loop. As one can see from Fig. 3, the observed dependence of AC losses vs. the primary current has a smooth character with gradually increasing slope. At low currents below about 2.5 A, the experimental points apparently follow the cubic law. Above 6 A, the dependence becomes practically linear.

Ž 3.

where L is the self-inductance of the primary coil, M s Ž 1 y kXbrk b . L is the mutual inductance of the coil and cylinder, kXb is the proportionality coefficient between Bm and i c at low currents when the losses in the cylinder are negligible. The expression for the power of AC losses that contains only measured quantities can be obtained by multiplying Eq. Ž3. by i s and using Eq. Ž2.: Ps

1

M

T

H U i d t s y k T EB T 0 s s

md ic .

Ž 4.

b

Note that expression Ž4. gives the total loss power in the device including hysteresis losses in the ferromagnetic core. To determine the losses in the ferromagnetic core, experiments with the model without a cylinder were carried out. These losses were found to be only a few percents of measured total losses.

Fig. 4. Current in hollow cylinder of the experimental model as a function of current in primary coil Žinstantaneous values..

242

V. MeeroÕich et al.r Physica C 319 (1999) 238–248

3. Calculation in the framework of Bean’s CSM A number of papers reported the experimental and theoretical results confirming the applicability of the CSM for HTSC samples w3,6,16x. The measurements of AC losses in small BSCCO bars and tubes prepared using the same technology as our cylinders were reported in Ref. w6x. Good agreement with the CSM was obtained: AC losses are proportional to the cube of the amplitude of applied magnetic field in the case of incomplete penetration and linearly increase with the field above the complete penetration field. Let us start with the treatment in the framework of the CSM assuming that the critical current density Jc is independent of magnetic field. This treatment has a two-fold purpose. First, we want to determine the differences of the case under consideration in the comparison with other simple configurations. Second, our purpose is to check the applicability of the CSM for the AC loss calculation in hollow cylinders possessing the E–J characteristic described by the plot in Fig. 2. 3.1. Slab and hollow cylinder in uniform magnetic field Let us compare our experimental configuration with a slab and a cylinder in uniform sinusoidal magnetic field He s yH0 cos v t directed along to the axis of the sample. These are the typical configurations used in the majority of the papers devoted to

the magnetic properties of superconductors. For the simplification of the problem, the slab and cylinder are assumed to be of infinite length. Fig. 5 shows the evolution of the distribution of magnetic field in a slab and a hollow cylinder when the external field He is increased from zero. The external magnetic field penetrates both sides of the slab and only the outside of the cylinder. The approach to AC loss calculation in the framework of the CSM was developed in many papers w17,18x. It is based on Maxwell’s equations and the assumptions that the current density is equal to "Jc or zero and the magnetic induction B and magnetic field intensity H are linearly related: B s m 0 H Ž m 0 is the magnetic permeability of vacuum.. In the case of incomplete penetration Žthe magnetic field amplitude H0 is less than the complete penetration field Hp s Jc D, where D is the halfthickness of the slab and the thickness of the cylinder wall., the only difference in the calculation procedure for a slab and a hollow cylinder is taking into account the curvature of the cylinder wall. The loss density per period and per surface unit is determined using the usual CSM procedure w17,18x as 2 m0 3 for a slab: p s H , Ž 5. 3 Jc 0 for a hollow cylinder: p s

2 m0 3 Jc

ž

H03 1 y

H0 2 Jc R

/

,

Ž 6. where R is the outside radius of the cylinder.

Fig. 5. Evolution of CSM profiles of magnetic field for a slab Ža. and a hollow cylinder Žb. in uniform magnetic field.

V. MeeroÕich et al.r Physica C 319 (1999) 238–248

The behavior is substantially different for the case of the complete penetration of the magnetic field, H0 ) Hp . Consider one half-period of AC cycle where the external magnetic field He increases from yH0 to qH0 . For the part of the cycle when He - yH0 q 2 Hp , the process of the field penetration is similar to the incomplete penetration case. When the instantaneous value of the external magnetic field exceeds the value yH0 q 2 Hp , the electric field E remains equal to zero in the center of the slab. In contrast, the electric field at the inner surface of the cylinder is determined from the Faraday law: E Ž R1 . s y

m 0 R1 d 2

dt

Ž He y Hp .

Ž 7.

where R 1 s R y D is the inside radius of the hollow cylinder. The distribution of the electric field in the cylinder at R 1 - r - R is described by the expression: EŽ r . s y

m 0 r d He 2

dt

.

Ž 8.

The loss density per period in the case of the complete penetration is: for a slab: p s

2 m 0 Hp2 Jc

Ž H0 y 23 Hp . ,

Ž 9.

for a hollow cylinder: ps

2 m 0 Hp2 Jc q

Ž H0 y 23 Hp .

1y

R 12 Jc

Ž H 0 y Hp .

R

Hp Ž H0 y 23 Hp .

.

2 D 3 R

ž

H0 y 34 Hp H0 y 32 Hp

/ Ž 10 .

The first terms in expressions for AC losses in a cylinder ŽEqs. Ž6. and Ž10.. are identical to the corresponding expressions for AC losses in a slab. The second negative terms in the brackets ŽEqs. Ž6. and Ž10.. reflect the influence of the wall curvature and are negligible at R 4 D. The last term in Eq. Ž10. is associated with the change of the magnetic flux in the bore of the cylinder and increases with the cylinder’s radius. Note that this part of the losses exists also in rings, cylinders and other closed loops made from type-I superconductors. Buchhold w19x

243

was the first to study these AC losses. In addition to his result, expression Ž10. includes the terms related to the penetration of magnetic field into a type-II superconducting material of finite thickness. In a cylinder of large diameter, the part of AC losses associated with the cylinder bore becomes dominant. This part is also increased by insertion of a ferromagnetic core into the cylinder bore. 3.2. Hollow cylinder as a secondary coil of a transformer Now let us consider our experimental configuration ŽFig. 1.. The device constitutes a two-coil transformer where the current in the cylinder is induced by the magnetic flux of the primary coil. Therefore, one can use the standard methods and approximations accepted in the theory of transformers w20,21x. The basic assumptions that are usually used at the calculation of magnetic field in transformers are: - the magnetic system is linear; - all the magnetic flux lines in the air gap between two concentric coils of a transformer are straight lines which parallel to the coil axes; - magnetic field produced by the current in a coil outside the coil is negligible in the comparison with the field inside the coil. For the case of the performance of the secondary coil in the form of a superconducting cylinder, the second assumption is valid even at large air gaps because the superconductor compensates in part the magnetic field component perpendicular to its surface. Under the assumptions above, we can use the approximation of an infinitely long system for the AC loss calculation in our experimental configuration. In the framework of the CSM, the magnetic field profiles in the model is shown in Fig. 6. The magnetic field penetrates the cylinder from the inside. Below the complete penetration field, the induced current in the cylinder keeps the value of the total magnetic flux inside the cylinder constant. As the magnetic field penetrates the whole of the wall thickness, the current in the cylinder achieves the critical value. Further increase of the current in the primary coil results only in the increase of the magnetic field inside of the primary coil while the cylinder current does not change any more. Magnetic field at the

244

V. MeeroÕich et al.r Physica C 319 (1999) 238–248

Fig. 6. Evolution of CSM profiles of magnetic field for a hollow cylinder in experimental model shown in Fig. 1.

internal surface of the cylinder remains constant and is equal to Jc D. This last point distinguishes principally our model from the simple cases that were considered above where the magnetic field at the superconductor surface depends on external magnetic field. The Maxwell equations for electric and magnetic fields in the cylinder wall in the cylindrical coordinates have the form:

the cylinder and the magnetic field Hs produced by the cylinder current: m 0 m 1 R 1 d Ž Hc q Hs . E Ž R1 . s y Ž 13 . 2 dt where Hc is the magnetic field produced in the ferromagnetic core by the current in the primary coil; m 1 is the relative permeability of the ferromagnetic core. Expression Ž13. is taken as one of the boundary conditions at the solution of Eq. Ž12.. We start with the case of the incomplete penetration of the magnetic field into the cylinder wall. Let us consider only one-half of the period of a magnetic field, e.g., between the minimum Hs s yHs0 to the maximum Hs s Hs0 . According to the CSM, for t s 0, the current density equals to yJc for all the points reached by the penetrating magnetic field. This initial profile of the magnetic field is marked as ‘‘1’’ in Fig. 7. As the magnetic field increases, the profile changes and starts to contain a region with current density qJc . Let us denote by r 1 the point at which current density reverses its sign. We have Jc for r F r 1 and yJc for r ) r 1. The continuity of the electric field at the point r 1 is the second boundary condition in our problem. The solution of the Maxwell equations has the following form: yHs 0 q Jc Ž r y R 1 . if r ) r 1 Hs Ž 14 . Hs y Jc Ž r y r 1 . if r F r 1

½

° E s~ m ¢2

0

if r ) r 1

0 d Hs dt

ž

r 12 r

yr

/

if r F r 1

Ž 15 .

where r 1 s R 1 q Ž Hs0 q Hs .rŽ2 Jc ..

EH s yJc , Er 1 E Ž rE . EH s ym 0 . r Er Et

Ž 11 . Ž 12 .

The electric field at the internal surface of the cylinder is determined by the value of the total magnetic flux in the cylinder bore. Neglecting the portion of the magnetic flux in the air gaps in the comparison with the flux in the ferromagnetic core, we obtain from the Faraday law the relationship between the electric field E at the internal surface of

Fig. 7. Profiles of AC magnetic field in wall of hollow cylinder. The magnetic field produced by current in the cylinder increases from y Hs0 Žline 1. to q Hs0 Žline 3..

V. MeeroÕich et al.r Physica C 319 (1999) 238–248

From Eqs. Ž13. and Ž15., we obtain the equation for determining the magnetic field at the internal surface of the cylinder:

ž

d2 2dq

R1

/

d Hs dt

s ym 1 R 1

d Ž Hc q Hs . dt

Ž 16 .

where d s r 1 y R 1 F H0rJc , and the maximum of d is D. Because d is a function of Hs , Eq. Ž16. is a non-linear equation for the magnetic field at the cylinder surface. If Ž2 D q D 2rR1 . ; m 1 R 1 , the magnetic field Hs can differ noticeably from the magnetic field Hc even in the case of incomplete penetration. This difference is due to hysteresis losses and is a characteristic feature of our configuration. For our experimental model, 2 D f R 1 but, for an open core transformer, m 1 is determined as the effective permeability to be about 10. Therefore, we assume here that Hs s yHc and the loss density per period is ps

2 m 0 Hs30 3 Jc

ž

1q

Hs 0 2 R 1 Jc

/

.

Ž 17 .

For our model Hs 0 s k c wIc 0rh s

Ž 18 .

where Ic0 is the amplitude of the AC current in the primary coil; h s is the height of the cylinder; k c s Ž1 y kXbrk b . has a meaning of the coupling coefficient of the primary and secondary Žcylinder. coils Žsee notations to expression Ž3... In the case of complete penetration, the magnetic field at the inner surface of the cylinder increases during a half of the period from yHp to Hp . At the minimum of the primary current, the magnetic field Hs is equal to yHp and, with increasing current, achieves the maximum value Hp and then remains constant despite a further increase of the current in the primary coil. The electric field at the inner surface of the cylinder is determined by Eq. Ž13., where Hs is constant. The loss density per period is ps

2 m 0 Hp3 3 Jc

ž

D 1q

2 R1

/

245

and the bore. To calculate the total losses in a sample, the surface density must be multiplied by the area of inner surface of the cylinder in our model, outer surface of the cylinder in uniform magnetic field and twice surface of the slab. 3.3. Comparison of calculation and experimental results The results of AC loss calculation in the framework of the classical CSM for the three cases above are shown in Fig. 3 together with the experimental points. As the reference for this comparison, we took the relationship ŽEq. Ž18.. between the current in the primary coil of our experimental model and the magnetic field produced by this current. The critical current density for all three calculated cases was taken to be 600 Arcm2 . Below the magnetic field of the complete penetration corresponding to the current of about 1.75 A rms in the primary coil, the experimental points apparently follow the CSM predictions. The difference between the calculated curves below the point corresponding to 1.75 A rms is only due to the wall curvature of the cylinder and decreases as this point is approached. The more detailed analysis of the experimental data at low currents shows that the deviations from the CSM exist even in this range of the current. One of these deviations is the reduction of the AC loss per period with frequency ŽFig. 8..

q m 0 m 1 R 1 Hp Ž Hc 0 y Hp . .

Ž 19 . As in the previous case, the expression contains the terms associated with the cylindrical geometry

Fig. 8. Frequency dependence of AC losses per period in a hollow cylinder.

246

V. MeeroÕich et al.r Physica C 319 (1999) 238–248

This behavior was reported by many researchers w4,5,22x and predicted in our theoretical paper w9x devoted to the analysis of AC losses in the framework of the extended CSM. The difference between the experimental data and the calculation results becomes crucial at the currents higher than 1.75 A rms when the magnetic field penetrates the sample completely. The slope of the experimental curve continues to increase up to a current of 6 A rms while the losses calculated according to the CSM increase linearly. The slopes of the lines describing the losses for different configurations are quite different. The explanation of this difference lies in various values of the electric field induced in the sample of different geometry and tested in various conditions. For a slab or small-size samples, the electric field induced by AC magnetic fields is relatively small and the current density is close to the critical value. This is a reason why the CSM applicability was observed up to fields significantly exceeding the complete penetration field w6x. In contrast to the slab, the electric field in the wall of a hollow cylinder depends on the total magnetic flux in its bore and sharply increases after the penetration. Because of strongly increasing electric field, the current density in a superconductor can be markedly more than the critical one. Thus, we cannot obtain satisfactory predictions of the losses for our experimental configuration from the CSM. To calculate the AC losses, especially in the case of the complete penetration, one has to solve Maxwell’s equations while taking into account the real E–J characteristic. Numerical solutions for a number of simple configurations were obtained w5,22,23x. However, the case considered is much more complicated because of finite dimensions of the cylinder and the complex boundary condition ŽEq. Ž13... Below, we suggest a simplified method for the evaluation of AC losses in the cylinder in our experimental configuration.

form E s E0 Ž JrJ0 . n , we have proposed w9x to define an effective critical current density as Jcef s

ž

m 0 v Hs20 J0n 3p E0

1r Ž nq1 .

/

Ž 20 .

where J0 and E0 are values which generally depend on the magnetic field and the temperature. Further, these values are assumed to be constants as determined from the E–J plot. Note that even in this case, the effective current becomes to be dependent on the frequency and magnetic field amplitude. Below the complete penetration field determined by the relationship Hp s Jcef Ž Hp . D, we calculate the losses applying the CSM result ŽEq. Ž17... For magnetic field amplitudes above Hp , the total losses are determined as the sum of two contributions. The first is calculated in the framework of the CSM with Hs0 s Hp and Jc s Jcef Ž Hp .. The second contribution, for instantaneous values of magnetic field above Hp , is calculated using an approach based on the equation of a two-coil transformer ŽEq. Ž3.. and the relationship UsŽ i s . given by the E–J characteristic presented in Fig. 2. The carefully measured inductance of the model and the coupling coefficient between the primary coil and the cylinder integrally account for the finite dimensions of the elements of the experimental model. Note that the calculated results are very sensitive to the variation of these

4. Evaluation of AC losses from E–J characteristic To extend the applicability range of the CSM to HTSC with a power law E–J characteristic in the

Fig. 9. Comparison of the experimental data and the calculation from E – J characteristic.

V. MeeroÕich et al.r Physica C 319 (1999) 238–248

247

5. Conclusion

Fig. 10. Influence of inductance of the magnetic system on AC losses: Žcurve 1. Ls 5.2=10y2 H; Žcurve 2. Ls 2.6=10y2 H; Žcurve 3. Ls1.3=10y2 H.

parameters. Also, the results depend significantly on the exponent in the power law approximation of E–J characteristic. The AC losses calculated within this approach with the exponent equal to 7 are shown in Fig. 9. Use of the CSM with the effective critical density extends the applicability of this model to the higher current of 2.7 A rms . Moreover, this model now describes the frequency dependence of AC losses ŽFig. 8.. The deviations at higher currents may be explained by the fact that our simplified theoretical model that is based on the equation of a transformer does not take into account the real distributions of the electric field and current density in the cylinder. Finally, let us analyze the influence of the parameters of the magnetic system on AC losses. Fig. 10 shows how the losses change with the variation of the inductance of the system resulted, for example, from the change of effective permeability m 1. Below the complete penetration field, the losses are calculated using the CSM expressions that do not include inductance. Therefore, the portions of the curves in this range coincide. At the complete penetration, the losses depend also on the magnetic flux in the bore and, hence, on the inductance. However, the difference becomes pronounced only at currents above 4 A rms when the voltage drop across the superconductor achieves a marked value.

It was shown that above the magnetic field of the complete penetration the AC losses are mainly determined by the sample shape and by the configuration of magnetic system and differ significantly from the values predicted by the CSM. This can explain the discrepancies regarding to the CSM applicability obtained in experiments on samples of various geometrical forms and under various conditions. For small bars in low-frequency uniform magnetic field, AC losses can be described in the framework of the CSM approximation up to high fields several times exceeding the complete penetration field. However, neither the CSM nor the data obtained on samples of simple geometry can be used to calculate AC losses in hollow cylinders. The proposed approach based on using the real E–J characteristic and mathematical model of a transformer provides a good agreement of experimental and calculated values of AC losses in a hollow cylinder.

Acknowledgements This research was supported by the Israeli Science Foundation founded by the Israeli Academy of Sciences and Humanities, and the G.I.F., German–Israeli Foundation for Scientific Research and Development.

References w1x w2x w3x w4x w5x w6x w7x w8x

w9x

C.P. Bean, Phys. Rev. Lett. 8 Ž1962. 250. K. Yamafuji, T. Wakuda, T. Kiss, Cryogenics 37 Ž1997. 421. L. Dresner, Appl. Supercond. 4 Ž1996. 167. H. Jiang, C.P. Bean, Appl. Supercond. 2 Ž1994. 689. T. Wakuda, T. Nakano, M. Iwakuma, K. Funaki, M. Takeo, K. Yamada, S. Yasuhara, Cryogenics 37 Ž1997. 381. E. Beghin, J. Bock, G. Duperray, D. Legat, P.F. Herrmann, Appl. Supercond. 3 Ž1995. 339. V. Sokolovsky, V. Meerovich, M. Sinder, Appl. Supercond. 4 Ž1996. 625. V. Meerovich, M. Sinder, V. Sokolovsky, S. Goren, G. Jung, G.E. Shter, G.S. Grader, Supercond. Sci. Technol. 9 Ž1996. 1042. V. Sokolovsky, V. Meerovich, S. Goren, G. Jung, Physica C 306 Ž1998. 154.

248

V. MeeroÕich et al.r Physica C 319 (1999) 238–248

w10x M.J. Hurben, O.G. Symko, W.J. Yeh, S. Kulkarni, M. Novak, IEEE Trans. Magn. 27 Ž1991. 1874. w11x V. Sokolovsky, V. Meerovich, G. Grader, G. Shter, Physica C 209 Ž1993. 277. w12x V. Meerovich, V. Sokolovsky, S. Goren, G. Jung, IEEE Trans. Appl. Supercond. 5 Ž1995. 1044. w13x S. Elschner, J. Bock, G. Brommer, P.F. Herrmann, IEEE Trans. Magn. 32 Ž1996. 2724. w14x J. Bock, S. Elschner, P.F. Herrmann, B. Rudolf, Proc. of European Applied Superconductivity Conference, Edinburgh, 3–6 July 1995, Inst. Phys. Conf. Ser. No. 148, 1995, p. 67. w15x V. Meerovich, V. Sokolovsky, J. Bock, S. Gauss, S. Goren, G. Jung, Proc. of European Applied Superconductivity Conference, The Netherlands, 30 June–3 July, Inst. Phys. Conf. Ser. No. 158, 1997, p. 1179.

w16x K.-H. Muller, IEEE Trans. Magn. 27 Ž1991. 2174. w17x H. Brechna, Superconducting Magnetic Systems, SpringerVerlag, Berlin, 1973. w18x I.A. Glebov, C. Laverik, V.N. Shakhtarin, Electrophysical Problems of Superconductivity, Nauka, Leningrad, 1980. w19x T.A. Buchhold, Cryogenics 3 Ž1963. 141. w20x L.V. Leites, Electromagnetic Calculations of Transformers and Reactors, Energija, Moscow, 1981. w21x P.M. Tikchomirov, Calculation of Transformers, Energoatomizdat, Moscow, 1986. w22x N. Amemiya, K. Miyamoto, N. Banno, O. Tsukamoto, IEEE Trans. Appl. Supercond. 7 Ž1997. 2110. w23x M.J. Qin, X.X. Yao, Physica C 272 Ž1996. 142.