Self-field hysteresis loss in periodically arranged superconducting strips

Transcription

1 Ž Physica C Self-field hysteresis loss in periodically arranged superconducting strips K-H Muller ) CSIRO, DiÕision of Telecommunications and Industrial Physics, PO Box 18, Lindfield NSW 070, Australia Received 16 May 1997; accepted 3 July 1997 Abstract The magnetic field and current distributions inside superconducting strips arranged in a z-stack and an x-array are calculated analytically using the transformation method proposed by Mawatari for the case where a transport current is passed through the strips The magnetic field distributions are used to calculate the self-field ac hysteresis losses of both the z-stack and the x-array The self-field ac hysteresis loss of the z-stack is always larger than the loss of the x-array q 1997 Elsevier Science BV PACS: 855Kx; 8470qp; 7476Bz 1 Introduction The critical state model w1,x has been used extensively to describe the electrodynamics of type II superconductors and to calculate the ac hysteresis loss Calculations were first performed for the simple cases of long superconducting cylinders and infinite superconducting slabs in parallel magnetic fields The more complicated case of the critical state in a flat superconducting strip was first investigated by Swan wx 3 and Norris wx 4 High-temperature superconductors Ž HTS have renewed the interest in the critical state of flat superconductors with demagnetizing factors close to 1 w5,6 x Examples are HTS single crystals which are thin along the c-direction and ) Tel: q ; fax: q ; karl@dapcsiroau wide in the ab-directions used in experiments on flux vortex dynamics as well as Bi-3rAg tapes wx 7 and YBCOrNi Ž hastelloy thin-film flexible tapes wx 8 to be used in power applications In most applications, the tapes will be used in array or stack configurations were the strips Ž tapes couple electromagnetically The ac hysteresis loss of multiple strips is expected to be quite different from that of an isolated strip, and it is therefore important to investigate the behaviour of superconducting strips in array and stack configurations Mawatari w9,10x recently calculated the magnetic field distribution HŽ x and the current density distribution JŽ x Žx is the coordinate along the width of the strip for a z-stack and an x-array of strips where an ac magnetic field is applied perpendicular to the strips Mawatari wx 9 showed that, for both the x-array and the z-stack, the H J relation of Biot Savart s law can be transformed into the H J relation of an r97r$1700 q 1997 Elsevier Science BV All rights reserved Ž PII S

2 14 K-H MullerrPhysica C 89 ( 1997) isolated strip HŽ x and JŽ x for multiple strips can then be obtained from the HŽ x and JŽ x of the isolated strip by a simple transformation In the following paper we adopt the transformawx 9 to calculate tion method developed by Mawatari the magnetic field distribution HŽ x and the current density distribution JŽ x for an x-array and a z-stack for the important cases where dc or ac transport currents are flowing through the strips From HŽ x we calculate the self-field ac hysteresis losses of the z-stack and the x-array The self-field critical state in a periodic array of superconducting strips 1 The z-stack The magnetic field HŽ r at position r created by a current distribution Jr Ž is given by Biot Savart s law 1 ryr 3 HŽ r s H JŽ r = d r Ž 1 3 4p < ryr < Let us consider a single infinitely long superconducting strip of width a along the x-direction, thickness d along the z-direction Ž a 4 d, positioned at the coordinate origin In this case one obtains from Eq Ž 1 d a xyu HzŽ x,z sy H JŽ u du, p ya Ž xyu qz Ž where Ju Ž is the current density in the superconducting strip averaged over the thickness d of the strip, 1 dr H y d ydr JŽ x s J Ž x,z dz Ž 3 Let us now consider an infinite number of infinitely long strips periodically stacked above each other along the z-direction with a spacing D where D)d Such a z-stack of strips is shown in Fig 1 Fig 1 z-stack of infinitely long superconducting strips Ž l ` where d is the thickness of a strip, a the strip width and D the stacking spacing The z-component of the magnetic field, H Ž x z across the strip, is given by HzŽ x s d ` a xyu y Ý H JŽ u du p nsy` ya Ž xyu qž nd Ž 4 or w9,11x HzŽ x d a pž xyu sy JŽ u coth D D d u Ž 5 H ž / ya Introducing the variable transformations D p x D pu xs tanh ž /, us tanh p D p ž D/ D pa and as tanhž / Ž 6 p D Ž Ž Ž one obtains for Eq 5 using JusJyu d ã J Ž u H x sh zž zž x sy H du, Ž 7 p yã xyu where JusJ Ž Ž u Eq Ž 7 has the same form as Eq Ž for zs0 which describes H Ž x of a single strip The current z

3 K-H MullerrPhysica C 89 ( 1997) 15 and magnetic field distributions for a single strip have been discussed by Norris wx 4, Brandt et al wx 5 and Zeldov et al wx 6 The major problem solved by these authors was to find a current distribution which produces a zero z-component of the magnetic field inside the superconducting strip for < x< - b and a critical state magnetic field distribution for b-< x< - a, where b represents the distance of the flux front from the centre of the strip From their calculations it follows that for the z-stack Hz Ž x 0, < x< -b, 1r x x yh artanh, b- < x< d -a, ~ < x< s a 1r x a yh artanh, < x < d )a, < < x x Ž 8 where D p b bs tanhž / and HdsJcdrp Ž 9 p D Here, Jc is a field-independent critical current den- sity ŽThe more complicated case of a field-depenw1 x dent J has been discussed in Ref c For J Ž x one obtains from wx 5 J Ž x Jc b yx s~ 1r arctan, < x < -b, p a J, b- < x< c -a, 0, < x < )a Ž 10 Let us assume that after zero-field cooling a current Idc is applied to each strip where Idc - Ic and Ic is the critical current of a strip, Ics adj c Then a ã J Ž u Idc sdh JŽ u dusdh du ya ya 1y Ž p urd Ž 11 Using Eq Ž 7 and the fact that H Ž x syh Ž yx we obtain from Eq Ž 11 I sydh dc zž xsdrp Ž 1 Because xsdrp)a, we derive from the last line of Eq Ž 8 using Eqs Ž 9 and Ž 1 D cosh Ž p ard bs arcosh Ž 13 ž / p coshž p aidcrdic Fig Ž a shows the magnetic field distribution H Ž x rh ŽEqs Ž 8 and Ž 9 z d for different values of IdcrIc for a z-stack with Dra s 05 and for an isolated strip Ž Dra ` The field Hz increases with decreasing Dra In the limit D d 0 the magnetic field distribution H Ž x z is that of an infi- nite slab of width a ie HzŽ x rhd 0, < x< -b, p x y Ž < x<, b -< x< -a, s~ d < x< Ž 14 x pai dc y, < x < )a, < x < di c where b saž 1yI ri dc c Fig Ž b shows the current distribution JŽ x rj c ŽEq Ž 10 for different values of IdcrIc for a z-stack with Dra s 05 and for an isolated strip Ž Dra ` The smaller the vertical spacing D becomes, the more the current density profiles resemble that of an infinite slab of width a, ie 0, < x< JŽ x -b, ~ s 1, b -< x< -a, Ž 15 J c 0, < x < )a Self-field hysteresis loss of the z-stack When an ac current I Ž ac t sim cos v t is flowing through each strip, the magnetic field and current density distributions can be expressed in terms of H z of Eq Ž 7 and Ž 8 For the first half of the ac cycle Ž 0-t-prv, the z-component of the magnetic field distribution H Ž x,t is given by wx 5 z x H Ž x,t sh Ž x,j,i si zx z c dc m Ž yh x, J, I si yi Ž t Ž 16 z c dc m ac

4 16 K-H MullerrPhysica C 89 ( 1997) and the current density distribution J Ž x,t x is given by J Ž x,t sjž x,j,i si x c dc m Ž yj x, J, I si yi Ž t Ž 17 c dc m ac The self-field hysteresis loss Prl per strip per length l is defined as P v d prv s dt E x,t J x,t dx, 18 l p 0 0 a H H y Ž Ž Ž where E Ž x,t y is the electrical field inside a strip generated by the self-field magnetic flux which moves in and out during each ac cycle The integral over the time t accounts for the time averaging of the loss The electric field E is given by Faraday s law as y d x EyŽ x,t smo H HzxŽ u,t du Ž 19 dt 0 Using the fact that JŽ x sj E Ž x,t /0 one obtains y c for those x where P 4m o a x sy djcvh H HzxŽ u,ts0 du dx, l p b0 Ž b0 Ž Ž 0 where D coshž p ard bž 0 s arcosh p coshž p ai rdi Ž 1 ž / m c Ž Eq 0 can be simplified further by partial integration P 4m o a sy djcvh Ž xya HzxŽ x,ts0 dx l p b0 Ž Ž Ž Ž Ž Using Eq 6, Eq 8 and Eq 9 we obtain Fig Ž a z-stack magnetic field H rh and Ž z d b current density Jr Jc versus xra at different values of IdcrIc for Dras05 Ž solid lines and for an isolated strip Ž Dra ` Ž dashed lines Ic a s v Ž ayx H l p a b0 Ž ž Ž / =artanh tanh Ž p xrd ytanh p bž 0 rd tanh Ž p ard ytanh Ž p bž 0 rd d x Ž 3

5 K-H MullerrPhysica C 89 ( 1997) 17 Ž isolated strip Eq Ž 3 becomes the Norris formula wx 4, ie / ž / P m I I s vic 1y ln 1y l ž I I where o m m p c c ž / ž / ž / c c c Im Im Im q 1q ln 1q y, Ž 4 I I I Im 4 s v for I <I Ž 5 m c l 1p I c and P m o s Ž lny1 vic for ImsI c Ž 6 l p Fig 3 z-stack ac hysteresis loss Prl per strip per length versus ac current amplitude Im ric for different spacings Dra Here vrps50 Hz, as05 cm, Ic s100 A and ds1 mm Fig 3 shows the self-field hysteresis loss Prl of the z-stack as a function of ImrIc for different values of Dra using vrps50 Hz, as05 cm and Ic s100 A The loss per strip increases dramati- cally with decreasing spacing D as the magnetic field increases rapidly inside the strip For Dra ` In contrast, for small D Ž D-a the z-stack behaves similar to an infinite superconducting slab In the limit D d, the loss per strip corresponds to that of an infinite slab of width a which is, using Eq Ž 14, Eq Ž 16 and Eq Ž aim 3 s sv Ž 7 l 6p d I c In Fig 3, the curve for Drasdra can be obtained by either using Eq Ž 3 or Eq Ž 7 As can be see in Fig 3, Prl changes from a Im 4 behaviour at large Dra to a I 3 behaviour at small Dra m Ž Fig 4 x-array of infinitely long superconducting strips l ` The x-array periodicity is L

6 18 K-H MullerrPhysica C 89 ( 1997) 3 The x-array Let us consider an infinite number of long superconducting strips periodically placed next to each other along the x-direction at a spacing L where L) a Such an x-array is shown in Fig 4 From Eq Ž 1, assuming a4d, one finds d ` a JŽ u HzŽ x sy Ý H du, p ya xyž uqnl nsy` which alternatively can be expressed by w9,11x Ž 8 d a HzŽ x sy H JŽ u cotž p Ž xyu rl du L ya Ž 9 Introducing the variable transformations L p x L pu xs tan ž /, us tan p L p ž L/ L pa and as tanž / Ž 30 p L one obtains for Eq Ž 9 d ã J Ž u H x sh zž zž x sy H du, Ž 31 p yã xyu which is the same as Eq Ž 7 It is important to realize that the transformation D i LŽ where i s' y1 causes Eqs Ž 5 and Ž 6 to become Eqs Ž 9 and Ž 30 This transformation has also been used in Ref wx 9 From Eq Ž 13, with D i L, we obtain for the position b of the flux front in the strips L cos Ž p arl bs arccos Ž 3 p cosž p ai rli ž / dc c Fig 5Ž a shows the magnetic field distribution H Ž x rh ŽEqs Ž 8 and Ž 9 z d of the x-array for different currents IdcrIc with Lra s 5 as well as for the isolated strip Ž Lra ` The smaller the spacing L is, the weaker H Ž x z becomes, contrary to the z-stack behaviour Fig 5Ž b shows the current density distribution JŽ x rj ŽEq Ž 10 c for different values of IdcrIc for the x-array with Lras5 and Fig 5 Ž a x-array magnetic field H rh and Ž z d b current density Jr Jc versus xra at different values of IdcrIc for Lras5 Ž solid lines and for an isolated strip Ž Lra ` Ž dashed lines

7 K-H MullerrPhysica C 89 ( 1997) 19 where ž / m c L cosž p arl bž 0 s arccos Ž 34 p cosž p ai rli Fig 6 shows the self-field hysteresis loss as a function of ImrIc for different spacings Lra using vrps50 Hz, as05 cm and Ic s100 A As can be seen, in the limit L a the loss Prl goes to zero while for Lra ` Eq Ž 33 becomes the Norris formula wx 4 of Eq Ž 4 It is important to realize that, for L close to a, the loss which is due to the x-component of the magnetic field, H x, becomes more important than the loss caused by the perpendicular magnetic field Hz which has been considered so far The loss from the x-component of the magnetic field is dim 3 s v, Ž 35 l 4p a I c Fig 6 x-array ac hysteresis loss Prl per strip per length versus ac current amplitude Im ric for different spacings Dra Here vrps50 Hz, as05 cm and Ic s100 A The dashed curve represents Eq Ž 35 which is obtained from Eq Ž 7 by replacing d by a and a by dr Prl of Eq Ž 35 is shown as a dashed line in Fig 6 where vr ps50 Hz, as05 cm, Ic s100 A and ds1 mm The total loss cannot become smaller than the loss of Eq Ž 35 for an isolated strip Away from the edges, JŽ x becomes nearly constant, ie independent of x, when the spacing L is small In the limit L a one obtains JŽ x rj si ri c dc c 4 Self-field hysteresis loss of the x-array Using the transformation D i L, one obtains for the self-field hysteresis loss Prl per strip per length l of the x-array Ic a s v Ž ayx H l p a b0 Ž ž Ž / =artanh tan Ž p xrl ytan p bž 0 rl tan Ž p arl ytan Ž p bž 0 rl d x, Ž 33 3 Conclusion Our investigations show that in the case of a z-stack made of long superconducting strips of rectangular cross-section, the z-component of the magnetic field inside the strip increases when the spacing D is decreased For small D Ž D-a the z-stack behaves magnetically similar to a superconducting slab The z-stack self-field ac loss per strip is smallest for the isolated strip where Prl;Im 4 ric for Im-I c For Dsd the loss is maximal and corre- sponds to that of a slab of width a where Prl; Ž ardži 3 ri m c In the case of the x-array, the z-component of the magnetic field inside the strip decreases with decreasing spacing L In the limit L a the z-component of the magnetic field vanishes and the current density distribution JŽ x becomes constant where

8 130 K-H MullerrPhysica C 89 ( 1997) JŽ x rjcsidcri c The x-array self-field ac loss per strip is largest for the isolated strip and decreases with decreasing spacing L For L, a the loss from the x-component of the magnetic field, H x, becomes larger than the loss from the z-component The H -loss behaves like Prl; Ž draž I 3 ri x m c The self- field loss of the x-array is always smaller than the self-field loss of the z-stack The ratio between the minimum loss in the x-array, which occurs when Ls a, and the maximum loss in the z-stack, which occurs when Dsd, isž dra The behaviours of the x-array and the z-stack are reversed when the arrays are exposed to an ac magnetic field applied in the z-direction with zero applied currents wx 9 In this case the loss of the x-array is always larger than the loss of the z-stack After this manuscript was completed we became aware of a preprint by Mawatari w13 x, which applies the same formalism to the self-field case In contrast to his work our paper gives a more detailed derivation of the essential results and discusses the self-field ac loss in greater detail Acknowledgements The author gratefully acknowledges the support of MM Cables, Metal Manufactures Ltd, the Energy Research and Development Corporation and DIST References wx 1 CP Bean, Phys Rev Lett 8 Ž wx CP Bean, Rev Mod Phys 36 Ž wx 3 GW Swan, J Math Phys 9 Ž wx 4 WT Norris, J Phys D 3 Ž wx 5 EH Brandt, M Indenbom, Phys Rev B 48 Ž wx 6 E Zeldov, JR Clem, M McElfresh, M Darwin, Phys Rev B 49 Ž wx 7 K Heine, J Tenbrink, M Thoner, Appl Phys Lett 55 Ž wx 8 D Wu, SR Foltyn, PN Arendt, WR Blumenthal, IH Campbell, JD Cotton, JY Coulter, WL Hults, MP Maley, HF Safar, JL Smith, Appl Phys Lett 67 Ž wx 9 Y Mawatari, Phys Rev B 54 Ž w10x Y Mawatari, ASC 96 proceedings Ž in press w11x IS Gradshteyn, IM Ryzhik, Tables of Ingegrals, Series and Products, Academic Press, New York, 1980 w1x J McDonald, JR Clem, Phys Rev B 53 Ž w13x Y Mawatari, ISS 96 proceedings Ž in press