AC Loss in Fine Superconducting Filaments under Reversible Motion of Flux Lines Edmund Soji Otabe, Teruo Matsushita Abstract_ The magnetic flux distribution and the energy loss density under an external AC magnetic field were theo- retically analyzed for a fine superconducting filament. The Campbell model for the force-displacement relation and the continuity equation for flux lines were solved numerically. It was shown that the AC loss takes a much smaller value due to the reversible motion of flux lines than predicted by the critical state model. Keywords_ reversible flux motion, reduction of hysteresis loss, Bi-2223 silver sheathed tape I.Introduction URTHER reduction in the AC loss is a key issue for F applying oxide superconductors to practical AC appli- cations. It has been shown that the energy lossFdensityicang. 1. Variation in * *the flux distribution in a superconducting slab be reduced from the prediction of the criticalfstatermodelom the initial state. due to the reversible flux motion at 77.3 K even in an oxide superconductor when the filament thickness isTreducedhtoe current density, J, i* *s related to the magnetic flux den- the order of micrometer [1]-[3]. There is nosrigorousitheo-ty by retical work to be compared with the experimental results, 1 @B since the existing theoretical treatment has been done only __~0__@x= J: * * (2) the case where a small AC magnetic field is superposed on a large DC magnetic field [4]. JdisiequalstopJclinathecinitialesta* *te,mandechangesnwithttheof flux lines as [5] In this work, the reversible flux motion is theoretically ~ ` '~ analyzed for the case where only an AC magnetic field is -u_ applied on a thin superconductor. The magnetic field de-J = -Jc 1 - 2 exp 2di; * * (3) pendence of the critical current density is also taken into account. The theoretical result is compared withwthehex-ere diis the interactio* *n distance representing the half perimental result on a Bi-2223 multifilamentarystapeiwithze of the pinning pote* *ntial. At high fields, diis given fine filaments [2]. as af=i, where afis the flux line s* *pacing and i is a con- stant ranged from 4 to 2ss dependin* *g on the kind of pinning II.Theory centers. In the low field limit the* * above expression of didi- verges. In order to avoid such an u* *nphysical situation, it For simplicity, it is assumed that the AC magneticifields assumed that the upp* *er limit of diis given by 1/10 of Bm cos!t is applied along the z-axis parallelstopaneinfinitecimen size d. For t* *he magnetic field dependence of the superconducting slab occupying 0 x d. Fromcsymme-ritical current density, J* *c, modified Kim's model [2] is as- try we have to consider only one half, 0 xsumd=2.eItdis: ` * * ' assumed, for simplicity, that the flux distribution inJinitial(B) = __ff0_ 1 - * *|B|_;(4) state, Bini(!t = -ss) is in the critical state as showncin |B| + fi * * Bi Fig. 1. When the AC magnetic field is increased,wthehfluxere ff0, fi are parame* *ters and Biis the irreversibility distribution varies and the corresponding displacementfofield at which Jcreache* *s zero. The boundary conditions are flux lines, u(x), can be obtained from the continuityBequa-(0) = Bm cos!t at th* *e surface (x = 0) and dB=dx = ~0Jc tion for flux lines: at the center of the slab (x = d). * *The magnetization, M, @_(Bu) = -(B - B ): (1) is calculated from the spatial aver* *age of the magnetic flux @x ini densitydoverutherslabifornaghalfwcy* *clehfromithecinitialhstatethe surface field varies from -B * * m to Bm. For E. S. Otabe is with Kyushu Institute of Technology,s680-4,implicity, the minor* * curve of M in the next half cycle is Kawazu, Iizuka 820-8502 Japan (Telephone: +81-948-29-7683,ae-ssumed to be symme* *try to that in the previous half cycle mail: otabe@cse.kyutech.ac.jp) so that the minor magnetization cur* *ve closes. Then, the T. Matsushita is with Kyushu Institute of Technology, 680-4, Kawazu, Iizuka 820-8502 Japan (Telephone: +81-948-29-7663,ee-nergy loss density* * is approximately calculated from the mail: matusita@cse.kyutech.ac.jp) area of the closed magnetization cu* *rve. Fig. 2. Calculation result of the magnetic fluxFdistributioniingthe. 3. Compa* *rison of several calculated results of the energy loss superconducting slab. density. Closed symbols are exper* *imental results [2]. TABLE I irreversible. It is to be noted tha* *t W is saturated at large The parameters used in the theoretical calculation.Bm, since Bm becomes large* *r than Bi. On the other hand, the reversible * *flux motion becomes _ff0_____fi_____Bi_____d_______i______ significant when i is small, result* *ing in smaller W than the 3 x 1072:5 x 10-20.3 3:0 x 10-60.4-6 prediction of completely irreversib* *le modified Kim's model at medium AC magnetic field amplitu* *des. However, i = 0:4 obtained so as to fit with the * *measurement is too small in comparison with the theoreticall* *y predicted value. The III.Result and Discussion reason is still open. The calculated result of the variation in theTmagnetichfluxe calculation has n* *ot yet completed for smaller AC distribution in the superconducting slab formBma=g5nmTeistic field amplitudes, * *since the assumption that the shown in Fig. 2. The parameters used in the calculationinitial magnetic flux di* *stribution is in the critical state is are listed in Table I, which are determined sonasototobtainfulfilled. Therefor* *e, it is necessary to calculate the a good agreement for the major magnetizationscurvetbe-eady-state distribution a* *fter sufficient cycles started from tween experiment on a Bi-2223 multifilamentaryttapehate virgin state. 77.3 K and theory [2]. The effect of reversible flux motion is significant when the variation in the external magnetic References field from the initial state is small. On the[other1hand,]theE.S. Otabe, H. Mat* *suoka, T. Matsushita, J. Fujikami and K. flux motion is almost irreversible with a changeOinhJmfromatsu,R"Possibilityeof* *vReductioneinrHysteresissLossiDuebtole Fluxoid Motion in a Superconducting Bi* *-@ Jcto -Jcin the almost whole region of the filamentfwhenilamentary Wire," Jpn. J* *. Appl. Phys., vol. 37, pp. L382-L385, the variation in the external magnetic field is1large.998. In Fig. 3, the calculated results of the energy[loss2den-]E.S.LOtabe,oT.sMatsu* *shita,sJ.iFujikami,nK.aOhmatsu,S"Hysteresisuperconducting Bi-2223 Tape sity, W, are shown as a function of the AC magneticwfieldith Fine Filaments", I* *EEE Transaction on Applied Supercon- amplitude, Bm, and compared with the measured[resultductivity,3Vol.]9,ENo..2S(A* *SC'98),.pp.O2569-2572,t1999.abe, T. Matsushita, J. Fujikami, K. Ohmatsu, "Rev* *e@ of Bi-2223 multifilamentary tape at 77.3 K. InFthelfigure,uxoid Motion in a Sup* *erconducting Bi-2223 Tape with Fine the solid, chained and doted lines represent theFresultsiforlaments", Supercond* *uctor Science and Technology, Vol. 12, pp. i = 0:4, i = 6:28 and the prediction of the irreversible1112-1115,[1999.4]T. Ma* *tsushita, E. S. Otabe, B. Ni, "Effect of reversible fluxoid modified Kim's model, respectively. These threemcalcu-otion on AC susceptibili* *ty of high temperature superconduc- lated results agree with the measured result representedtors", Physica C, vol. * *182, pp. 95-102, 1991. by symbols at large Bm, where the flux motion[is5almost]A.M.FCampbell,r"TheeRes* *ponseqofuPinnedeFluxnVorticesctoyLow-Fields," J. Phys. C, vol. 2, pp. 1492-15* *0@