Reduction of low frequency ac losses in coaxial cables of type II superconductors by asteady bias currentM. A. R. LeBlanc, Daniel S. M. Cameron, David LeBlanc, and Jinglei Meng Citation: Journal of Applied Physics 79, 334 (1996); doi: 10.1063/1.360835 View online: http://dx.doi.org/10.1063/1.360835 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/79/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Hysteretic losses of a typeII superconductor in parallel ac and dc magnetic fields of comparable magnitude J. Appl. Phys. 50, 3531 (1979); 10.1063/1.326350 Theory of ac losses in typeII superconductors with a fielddependent surface barrier J. Appl. Phys. 50, 3518 (1979); 10.1063/1.326349 Reduction of the ac losses of multifilament superconductors by the use of low twist rates J. Appl. Phys. 47, 5038 (1976); 10.1063/1.322462 Temperature dependence of ac loss in type−II superconductors J. Appl. Phys. 46, 426 (1975); 10.1063/1.321354 ac Losses in Superconductors J. Appl. Phys. 39, 2538 (1968); 10.1063/1.1656612

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Reduction of low frequency ac losses in coaxial cables of type IIsuperconductors by a steady bias current

M. A. R. LeBlanc, Daniel S. M. Cameron, David LeBlanc, and Jinglei MengUniversity of Ottawa, Faculty of Science, Physics, 150 Louis Pasteur, Ottawa ON K1N 6N5, Canada

~Received 17 July 1995; accepted for publication 25 September 1995!

Hysteresis losses,Wac, in the core of a monolithic coaxial cable carrying an alternating current offixed amplitudeI ac are predicted to trace a valley as a steady bias currentI bias is superimposed onI ac, when~a! the critical current densityj c diminishes with increasing magnetic fieldH, and/or~b!a Meissner currentI M or a surface barrier currentI SB opposing flux entry play a role. The predictedI bias,min where the valley minimum occurs and the value ofWac at the minima are displayed forvarious I M>0 and I SB>0 when j c5a ~Bean! and j c5a/H ~Kim approximation!. © 1996American Institute of Physics.@S0021-8979~96!09901-7#

I. INTRODUCTION

Many workers1–7 have observed thatWac, the dissipa-tion of energy in type II superconductors subjected to a lowfrequency alternating magnetic field,H total(t) of fixed ampli-tudeHac, diminishes and traverses a minimum when a sta-tionary bias magnetic fieldHbias is superimposed collinearlyuponHac. Consequently,Wac plotted versusHbias with Hackept constant traces a valley which has been labeled theClem valley.1 This interesting and possibly useful feature hasbeen investigated in some detail in situations where the totalexternally applied magnetic field,H total5Hbias1Hac f~t!, wasdirected along the axis of long cylinders and parallel to thelength of wide ribbons. Heref (t) is an arbitrary smoothfunction of time varying between11 and21. The majorfeatures of the observations have been well accounted for inthe framework of the critical state model also taking intoaccount contributions fromDH in andDHex, surface barriersto flux entry and exit and the role of the lossless diamagneticMeissner currentI M .

1,3,4,7–10

The prominent properties of the ac loss valley of specialexperimental and theoretical interest are the following:

~i! Hbias,min, the ‘‘location’’ or ‘‘position’’ of the mini-mum for the selected amplitude,

~ii ! Wmin~Hbias,Hac!/W~0,Hac!, the ‘‘depth’’ of the valleymeasured with respect to the losses in zero bias at thecorresponding amplitude, and

~iii ! the evolution of the two above quantities as a functionof Hac.

In this article, we develop detailed quantitative predic-tions for an ac loss valley in a solid core coaxial cable oftype II superconductors in zero externally applied magneticfield when a low frequency alternating currentI acf (t) ismade to flow along the cable and is superimposed on asteady bias currentI bias, hence,I (t)5I bias1I acf (t). Here I acis the amplitude of the alternating current. We restrict ouranalysis to situations whereImax5 I bias1 I ac is smaller thanI C , the critical current of the cable, hence we do not considerthe flux flow regime.11 We map out the shape of the valleyand examine:

~i! the migration ofI biaswhere the minimum appears andof

~ii ! the evolution ofWmin~I bias,I ac!/W~0,I ac!, the depth ofthe minimum as a function ofI ac, I C .

We also investigate the effect on these two salient propertiesof:

~i! a Meissner surface currentI M and of~ii ! a field independent surface barrier currentI SB.

We study the influence ofI M and I SB as their magnituderelative to the bulk critical currentI cb is varied over a widerange.

In this survey we focus on two simple well known re-gimes for the dependence of the bulk critical current densityj c on the magnetic flux densityB, namely the Bean–Londonapproximation,12,13 j c5a, and the Kim–Anderson ap-proximation,14 j c5m0a/B. Herea is a temperature depen-dent pinning parameter characterizing the superconductingsolid core of the cable.

II. GENERAL FRAMEWORK OF THE INVESTIGATION

We calculate the energy dissipation in a monolithic, ho-mogeneous and isotropic central core of a coaxial cable inzero externally applied magnetic field in the low frequencyregime where viscosity effects in the displacement of the fluxlines can be neglected. Consequently, the succession of con-figurations of magnetic flux densityBf(r ), theB profiles,are taken to correspond to quasi-static critical states pre-scribed by

¹3H56 zj c@T,Hf~r !#. ~1!

For an infinitely long cylindrical core Eq.~1! reads

Hf

r1dHf

dr56 j c@T,Hf~r !#. ~2!

We takeBf(r ) 5 m0Hf(r ), wherem0 5 4p(1027)T/m A, hence ignore equilibrium diamagnetism in the bulkof the specimen.

We focus on the Bean–London and Kim–Anderson ap-proximations for the dependence ofj c on B since:

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~i! these two simple prescriptions bracket a wide spec-trum of physical situations and

~ii ! lead to analytic expressions not only for the sequenceof Bf profiles but also for Bf& 5 *0

RBfdr/R, their

spatial averages as the conduction currentI carried bythe coaxial cable is varied.

Exploiting the closed form formulae developed in theappendix we calculate curves of^Bf& as I is made to varyover a full cycle between, Imax5I bias1I ac,I c andImin5I bias2I ac. The energy dissipation,Wcycle, per unit sur-face of cable core per cycle is then obtained by numericalevaluation of the area enclosed by the hysteresis loops tracedout in this manner, hence by numerical integration of therelationships

Wcycle5R R Hf~R!d^Bf&

52R R ^Bf&dHf~R!

51

2p R Id^Bf&

521

2p R ^Bf&dI51

2pR R V~ t !I ~ t !dt, ~3!

which follow from the Poynting formulation,**(E3H)• dS dt for the net energy flow per cycle across the surfaceS of radiusR of the core. HereHf(R) 5 I /2pR, and,¹3E5 2]B/]t, reads ]Ez /]r 5 ]Bf /]t, hence, E(R)5R]^Bf&/]t.

We visualize that the total conduction currentI carriedby the cable core comprises two contributions:

~i! A netcurrentI b flowing in the bulk of the core with acritical current densityj c[Bf(r ),T] governed by Eq.~2! and

~ii ! one of two kinds of surface currents whose nature andbehavior we now describe.

We refer the reader to Figs. 1 and 2 for aid in followingour description of these properties.

~i! A Meissner surface screening current, denotedI M ,whose magnitude and direction are dictated by and in equi-librium with the external surface field,Hf(R)5 I /2pR. TheMeissner current obeys the linear relation,I M 5 I M85 Hf(R)2pR in the range, 0, Hf(R) 5 Hc1 . Hence herethe total currentI5I M and, thenet bulk current I b50 al-though the latter may consist of counterflowing concentricannular current zones. For simplicity8 we take I M to be aconstant,I M 5 Hc1/2pR hence independent of the magnitudeof Hf(R) whenHf(R)>Hc1 .

~ii ! A hysteretic surface barrier current, denotedI SB,which exhibits the following dependence on the history ofHf(R). This surface current opposes flux entry into the corewhen Hf(R) ~hence I ! are increasing in magnitude untilI SB8 attains a maximum value, denotedI SB. Again forsimplicity8 we takeI SB to be a constant, hence independentof the magnitude ofHf(R). Also in accord with precedent8

and some observations15–23 we assume that no surface bar-

rier current exists to oppose flux exit afterI SBhas been madeto vanish by the dimution ofI hence by a correspondingdecline in the magnitude ofHf(R).

Figures 1 and 2 illustrate the variety of hysteresis loopsfor ^Bf& vs I which ensue from the properties chosen for thedependence ofI M and I SB on the external field as describedin the foregoing. In these sketches the horizontal segmentsportray the essential feature that the configuration ofBf(r )remains fixed, henceBf& remains constant whenI is madeto vary over the specified ranges.

Sequences ofBf(r ) profiles generated asI b , the bulkcurrent, is varied fromI b,max to 2I b,min are displayed sche-matically in Fig. 3.

The maximum lossless steady current which the cablecore can support is

I c,max5I cb1I M , ~4a!

or

FIG. 1. Schematics of the evolution of hysteresis loops of^Bf&, the spatialaverage of the azimuthal magnetic flux density as the conduction currentI isvaried between1Imax and2Imin . Here uIminu is decreased by the superpo-sition of a static positive bias currentI bias. For simplicity in the display ofthe family of hysteresis loops,Imax is kept fixed, hence hereI ac, the ampli-tude of the alternating current is diminishing whenI bias is augmented sinceImax5I bias1I ac. It is visualized that in~a! a Meissner current of maximummagnitudeI M is present while in~b!, a surface barrier currentI SB opposingflux entry plays a role.I SBdecreases to zero at the extremes of the hysteresisloops asuI u is diminished over the corresponding range. For simplicityI SB istaken independent of the magnitude of the surface field whenImax.I SB anduIminu.uI SBu.

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I c,max5I cb1I SB, ~4b!

where the maximum lossless critical current which the‘‘bulk’’ of the core can carry reads

I cb52pE0

R

j c@Bf~r !#r dr , ~5!

with j c[Bf(r )] flowing in one direction onlythrough theentire cross section of the cable core.

III. RESULTS AND DISCUSSION

We examine the behavior predicted forWac vs I bias forfour sets of conditions:~a! the Bean–Londonj c5a approximation with

~i! a Meissner currentI M>0, or~ii ! a surface barrier currentI SB>0, and

~b! the Kim–Andersonj c5a/H approximation with

~i! I M>0 or~ii ! I SB>0.

For these four sets of conditions we investigate situationswhereI ac: I M andI ac: I SB.

A. j c5 a, IM50, ISB50

London,13 Hancox,15 and Norris16 focusing on the spe-cial casewherej c 5 a, I M 5 0, I SB5 0, anduImaxu 5u Iminu,henceI bias50 and the swing ofI is symmetric, developed aclosed form expression forWac vs I ac given below.

It is a somewhat more tedious but generally straightfor-ward exercise to show that for the ‘‘displaced’’ hysteresisloops whereI biasÞ0, henceuImaxu Þu Iminu and the swing ofIis asymmetric,

Wac

m0I c2 5F22S Imax2Imin

2I cbD G S Imax2Imin

2I cbD

12F12S Imax2Imin2I cb

D G ln F12S Imax2Imin2I cb

D G ,~6!

which is equivalent to the London, Hancox, and Norris for-mula since

I ac5Imax2Imin

2.

Wilson24 compares

~i! full-wave ~i.e., symmetric! oscillations where,Imax5u Iminu, henceI bias50, with

FIG. 2. Complements Fig. 1. HereI ac is kept fixed as a bias current,I bias5 Imax2 I ac is increased.Imin 5 I bias2 I ac is indicated by the dots at the leftend of the horizontal segments. Whenj c5a ~Bean approximation!, the twouppermost hysteresis loops embrace equal areas in~a! when Imin>0 and in~b! when Imin>IM . Consequently hereWac vs I bias for fixed I ac traces aplateau.

FIG. 3. Schematics of the sequences of quasi-staticBf(r ) profiles generatedasI b , the net current carried by the bulk of the cable core varies fromI b,maxto 2I b,min in ~a! and returns toI b,max in ~b!. The numbers along the curvesindicate the three different categories of segments ofBf profiles which areencountered and described by analytic expressions in the appendix.

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~ii ! half-wave oscillations where Imin50, henceImax52I ac, and,I bias5I ac.

He then states that ‘‘when the transport current does not re-verse but oscillates from zero to a maximum in one directiononly, the losses are considerably reduced.’’ This is mislead-ing, since in the half-wave case the amplitude of the currentoscillations is taken ashalf of that in the corresponding full-wave case. Further, he displays graphs of a quantity labeledthe loss factor which omit the prefactor, (I ac/I c)

2, hence donot give a true picture of the variations of the total and rela-tive losses with amplitude. Sheahen25 reproduces this sectionfrom Wilson24 and states: ‘‘The case of fully reversing cur-rent shows greater losses than does the case of unidirectionaloscillations from zero.’’ Again, the crucial feature that, thisarises because the amplitude in the former situation is twicethat of the corresponding case in the latter, is not mentioned.

The ‘‘building blocks’’ for the development of Eq.~6!are given in the appendix and are exploited for the calcula-tion of Wac in more complicated situations which we nowaddress.

B. Iac < IM and Iac < ISB with j c 5 a and j c 5 a/H

Figure 4 schematically displays the evolution ofWac vsI bias for various fixedI ac for the circumstances where

~i! the fixed I ac< I SB, and~ii ! the fixed I ac< I M .

Here j c may be constant or decline withH increasing.Three regions may appear in the evolution ofWac vs

I bias. These are labeleda, b, andg.In regiona, no losses occur since no migration of flux

lines takes place because

Imax5I bias1I ac,I SB,

hence

I bias,I SB2I ac,

or

Imax5I bias1I ac,I M ,

hence

I bias,I M2I ac.

Region b appears when the augmentation ofI bias with afixed I ac, I SB ~or I ac, I M! causesImax to exceed I SB ~orI M!. Now migration of flux into and out of the specimentakes place during part of the hysteresis cycle as displayed inFigs. 2~a! and 2~b!.

From inspection of Fig. 2~a!, we note that the hysteresisloops with a surface barrierI SB possess equal upper andlower horizontal segments whenImin 5 I bias2 I ac> 0, henceI bias> I ac. With j c5a, the hysteresis loops for a fixedI acareidentical and thus embrace equal areas asI bias is augmentedover the range

I ac<I bias<~ I cb1I SB2I ac!. ~7!

Consequently,Wac vs I bias traces a plateau in regiong. How-ever, with j c5a/H and similar dependences ofj c onH,Wac

does not trace a plateau but displays a less rapid rate ofincrease versusI bias in regiong compared to that in regionb.Here, the latter spans the range

I SB,Imax5I bias1I ac,2I ac, ~8a!

hence

~ I SB2I ac!<I bias<I ac. ~8b!

We stress that our calculations address only the situa-tions whereImax, I c,max @see Eq.~4!#.

We note in Fig. 4~a! that regionb shrinks while regiongexpands asI ac is made smaller butI SB is kept fixed.

Consequently, both regionsb and g vanish and no aclosses arise when

I ac<I SB2, ~9!

although here

I SB,Imax,I c,max. ~10!

FIG. 4. Schematics of the evolution of the ac losses vsI bias for various fixedamplitudesI ac< I SB in ~a! or I ac< I M in ~b!. All currents are normalized toI cb , the critical current of the bulk of the core. The dots indicate the limitsof validity of our model since here,Imax 5 I bias 1 I ac 5 I cb 1 I SB , or I cb1 I M . The plateaus arise whenj c5a ~Bean approximation! whereas thedashed segments will occur whenj c diminishes with increasing flux density.For purpose of illustration we choseI SB/I cb51.4 and I M/I cb50.7 in ~a!and ~b!.

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In these circumstances, azimuthal flux penetrates into thebulk of the core whenI is impressed for the first time andImaxexceedsI SB. TheBf profile, however, remains frozen inthe configuration established atI5Imax during the subse-quent swings ofI of total magnitude, 2I ac, I SB.

Regionb terminates without regiong arising when theaugmentation ofI bias ~with fixed I ac, I SB! causesImax to at-tainI c,max5I cb1 I SB, while,Imin5I bias2 I ac, remainsnegative.

From inspection of Fig. 2~b! we note that the hysteresisloops trace no horizontal segments when,Imin 5 I bias 2 I ac> I M , hence whenI bias > (I M 1 I ac). Consequently, withj c5a and fixedI ac, the family of hysteresis loops generatedasI biasis varied over the range, (I M 1 I ac) < I bias< (I cb1 I M2 I ac), embrace identical areas@see Fig. 2~b!#. ThusWac vsI bias traces a plateau over this range@regiong in Fig. 4~b!#.However for j c5a/H and similar dependences ofj c on H,Wac vs I bias does not trace a plateau but displays a less rapidrate of increase in regiong than in regionb.

The latter spans the range, (I M 2 I ac) < I bias< (I M 1 I ac).Consequently,when(I M 1 I ac)> (I cb1 I M 2 I ac), hence,2I ac> I cb , regiong will not be encountered. Again we note theconstraint that the onset of flux flow is outside the realm ofour model.

C. j c5 a, Iac> IM>0

As we have already seen, the presence of a Meissnercurrent or of a surface barrier drastically alters the ac lossbehavior. Now we note that for a fixed amplitudeI ac . I M. 0,Wac displays a decrease to a minimum and can trace avalley asI bias is augmented. The evolution ofWac vs I bias isdisplayed in Fig. 5~a! for several fixed amplitudes and a fixedI M . For easy comparison, each curve ofWac(I bias,I ac,I M! vsI bias is normalized to the corresponding loss per cycle at zerobias,Wac(0,I ac,I M).

The salient features of Fig. 5 can be understood frominspection of pertinent families of hysteresis loops withoutrecourse to computations. We have already explored the ori-gin of regionsb and g in the preceding section. Again wenote that the plateau does not appear if (I M 1 I ac) > (I cb1 I M 2 I ac), henceI ac> I cb/2. Region b is also suppressedif ( I ac2 I M) > @ I cb 2 (I ac2 I M)#, hence if, I ac> (I cb/2)1I M .

The appearance of a region whereWac declines~regiona8 of Fig. 5! can be understood from careful consideration ofthe evolution of the hysteresis loops displayed in Fig. 1~a! asI bias is augmented but withI ac kept fixed. We note that theinitial application ofI bias causes the open space between thehorizontal segments of the hysteresis loops to shrink inheight. It is the diminution of the area enclosed by this por-tion of the hysteresis loops which plays the dominant role inthe descent ofWac in regiona8.

A valley, hence a minimum, will be traced byWac vs I biaswhen the increase ofI biascan cause this open space to vanishas sketched in the uppermost hysteresis loop of Fig. 1~a! andlowermost of Fig. 2~b!. This configuration of the hysteresisloops is attained asI bias is augmented when

Imin5I bias2I ac52I M , ~11a!

hence

I bias,min5I ac2I M . ~11b!

If, howeverI ac> @(I cb/2) 1 I M#, hence neither regionbnor g appear and only regiona8 is encountered, the mini-mum value ofWacwill not correspond to a disappearance butbe associated only to a shrinkage of the open space betweenthe horizontal segments.

Two important features for possible applications arereadily identified in Fig. 5:

~i! The magnitude ofI biaswhen the minimum inWac oc-curs. Let I bias,min denote this quantity which, as wehave seen, is a function ofI ac and I M .

~ii ! The ‘‘depth’’ of the valley which we measure by tak-ing the ratio ofWac at the minimum to the corre-

FIG. 5. Illustrates the evolution ofWac vs I bias calculated usingj c5a ~theBean approximation! for various fixedI ac . I M or I ac . I SB . The losses ineach case are normalized to the corresponding zero bias value. The dotsindicate the limits of our model since here,Imax5 I bias1 I ac5 I cb 1 I M , or,I cb 1 I SB . A further increase ofI bias will bring about viscous~velocity ef-fects! losses which are not taken into account in our model. For the purposeof illustration we choseI M/I cb50.2 in ~a! and I SB/I cb50.2 in ~b!. Thenumbers along the curves indicateI ac normalized toI cb . We stress thatWac

vs I bias for fixed I ac will trace horizontal lines whenI M50, andI SB50 indisagreement with misleading statements in the literature~see Refs. 26 and27!. ~The losses in each case are normalized to the corresponding zero biasvalue!.

338 J. Appl. Phys., Vol. 79, No. 1, 1 January 1996 LeBlanc et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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sponding value at zero bias,@i.e., Wac 5 (I bias,min,I ac,I M)/Wac(0,I ac,I M)]. LetWac,mindenote this quan-tity.

Figure 6 displays the evolution ofI bias,minvs I ac for vari-ous fixedI M . The numbers given correspond to the quanti-ties, I bias,min, I ac, andI M normalized with respect toI cb . Thegraphs in this figure ensue both from detailed computationsand from the analysis presented above. We note that Fig. 6simply displays the relationships we have already identified,namely,

I bias,min5I ac2I M , ~12a!

valid whenI M< I ac< @ I M 1 (I cb/2)#

I bias,min5I cb2~ I ac2I M !, ~12b!

valid when@ I M1 (I cb/2)# < I ac< (I M1 I cb).Figure 7~a! displays the evolution of the ‘‘depth’’ of the

valleyWac,min vs I ac for various fixedI M . The formulae ex-ploited in our computations of the contours of the varioushysteresis loops are illustrated in the appendix. A change inslope of the curves ofWac,min vs I ac occurs atI ac 5 I M1 (I cb/2) because here regionb vanishes@see Fig. 5~a!#. Werecall that whenI ac. @ I M 1 (I cb/2)#, the open space betweenthe horizontal segments of the hysteresis loops remains finite@see Fig. 1~a!#. Consequently the behavior ofWac,min in thelatter range ofI ac is qualitatively and quantitatively differentfrom that in the range,I ac, @ I M 1 (I cb/2)#, where this openspace has vanished whenWac5 Wac,min.

The extrapolation ofWac,minto zero in the limit whereI acapproachesI M is noteworthy. We will see below that when a

surface barrier is postulated,Wac,min at the correspondinglimit extrapolates to finite values which are far from negli-gible.

D. j c5 a, Iac> ISB>0

A family of curves ofWac vs I bias for variousI ac and afixed I SB is schematically illustrated in Fig. 5~b! and is seento be similar to that for the preceding situation although theshapes of the hysteresis loops are quite different~see Fig. 1!.The origin and boundaries of regionsb andg have alreadybeen examined in Sec. III B.

Regiona8 now appears because the application ofI biascauses the open space in the hysteresis loops which isbounded by the two adjacent horizontal segments and theiropposite contours to diminish in area@see Fig. 1~b!#. Thearea enclosed by the hysteresis loops attains a minimum@seeuppermost hysteresis loop of Fig. 1~b! and lowermost of Fig.2~a!# when

FIG. 6. Displays the evolution ofI bias,min the value ofI bias where the valleyminimum occurs for a chosen fixedI ac ~see Fig. 5!. HereI bias,minis plotted vsthe correspondingI ac for variousI M/I cb ~or I SB/I cb! indicated by the num-bers along the curves. For the left hand side of the family of curves,I bias,min5 I ac2 I M , or, I bias,min5 I ac2 I SB . For the right hand sides,I bias,min5 I cb2 (I ac2 I M),or I bias,min5 I cb2 (I ac2 I SB).

FIG. 7. Complements Fig. 6 and displays the corresponding variations ofthe depths of the ac loss valleys~see Fig. 5! calculated for various selectedI M in ~a! andI SB in ~b! indicated by the numbers along the curves~normal-ized to I cb!. Note that the normalizedWmin extrapolate to zero in~a! butexhibit the same finite ‘‘threshold’’ value'0.667 in~b!.

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Imin5I bias2I ac52I SB, ~13a!

hence

I bias,min5I ac2I SB. ~13b!

If ( I ac2 I SB)> @ I cb2 (I ac2 I SB)# @seeFig.5~b!#, hence

I ac>~ I cb/2!1I SB, ~14!

neither regionb nor regiong are encountered andWac de-clines to its smallest value, whenImax 5 I bias 1 I ac 5 I cb1 I SB, hence in these circumstances

I bias,min5I cb2~ I ac2I SB!. ~15!

Figure 6 displays the evolution ofI bias,minvs I ac for vari-ous I SB which follows from Eqs. 13~b! and 15 and emergesfrom the detailed computations of families of curves ofWacvs I bias for variousI ac and I SB as illustrated in Fig. 5~b!.

The evolution of the ‘‘depth’’ of the valley defined as

Wac,min8 [Wac~ I bias,min,I ac,I SB!

Wac~0,I ac,I SB!~16!

vs I ac for variousI SB is displayed in Fig. 7~b!. A change inslope of the curves ofWac,min8 vs I ac occurs atI ac 5 I SB1 (I cb/2) because here regionb vanishes~see Fig. 5!. WhenI ac . I SB 1 (I cb/2), the adjacent horizontal segments of thehysteresis loops do not make contact@see Fig. 1~b!#. Conse-quently the behavior ofWac,min8 in the latter range ofI ac isquite different from that in the range,I ac , I SB 1 (I cb/2)where these horizontal segments have established contact.

We wish to stress thatWac,min8 extrapolates to a signifi-cant finite value~e.g., 0.667! in the limit where I ac ap-proachesI SB. This feature is in sharp contrast with the be-havior ofWac,min in the Meissner case which extrapolates tozero in the limit whereI ac approachesI M @see Fig. 7~a!#.Related to this feature is the observation thatWac,min is al-ways smaller than the correspondingWac,min8 @compare Figs.7~a! and 7~b!#. In other words, the valleys forWac vs I bias forvarious fixedI ac are always deeper whenI M plays a role thanthe corresponding valleys whereI SB is introduced ~seeFig. 5!.

E. j c5a

H, IM50, ISB50

The evolution ofWac vs I bias where j c5a/H, is illus-trated in Fig. 8 for the situations where neither a Meissnercurrent nor a surface barrier are made to play a role. We notefrom examination of the family of curves that the location ofthe valley minimum and the depth of the valley both evolveas I ac is augmented. The detailed dependence of these twoquantities onI ac is displayed in Figs. 9 and 10. We note thatthe relationship betweenI bias,minandI ac is not linear and thatthe slopeDI bias,min/DI ac < 0.67. Of special interest for pos-sible applications is the feature that the depth of the valleynormalized to the corresponding zero bias is nearly constantand'0.4 over the pertinent rangeI ac < 0.63I cb . When I ac. 0.63I cb , the onset of the flux flow regime dictates thatI bias,min5 I cb 2 I ac since the threshold for this regime is at-tained whenImax5 I bias1 I ac5 I cb .

F. j c5 a/H, IM>0, ISB50

Typical valleys ofWac vs I bias for a fixedI M and variousI ac are displayed in Fig. 11~a!. A comparison with Fig. 8

FIG. 8. Illustrates the evolution ofWac vs I bias calculated usingj c5a/H~Kim approximation! for various fixedI ac/I cb indicated along the curves.Here no Meissner or surface barrier current are present.

FIG. 9. Displays the evolution ofI bias,minvs I ac/I cb for variousI M/I cb>0 in~a! and I SB/I cb>0 in ~b! calculated usingj c5a/H ~see Figs. 8 and 11 andcompare this figure with Fig. 6!.

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shows that the presence ofI M does not appreciably affect thestructure of the valleys. The evolution ofI bias,min and of thenormalized depths of the valleys,Wac,minvs I ac are displayedin Figs. 9~a! and 10~a!.

As in the previous case, we observe that the variation ofI bias,minwith I ac on the left of the curves is not linear and thatthe slopes although steeper than in the case whereI M50,remain less than unity. Of special interest for possible appli-cation is the feature thatWac,min falls appreciably below theI M50 curve@see Fig. 10~a!#.

G. j c5 a/H, ISB>0, IM50

Typical valleys ofWac vs I bias for a fixedI SB and variousI ac are displayed in Fig. 11~b!. A comparison with Fig. 8shows that the presence ofI SB does not appreciably modifythe structure of the valleys. The evolutionI bias,minand of thenormalized depth of the valleys,Wac,min8 vs I ac are displayedin Figs. 9~b! and 10~b!. Again we note that the variation ofI bias,minwith I ac on the left of the curves is not linear and theslopes are smaller than unity. The feature that the depths ofthe valleys at intermediate amplitudes remain of the order of0.5 over a broad range of amplitudes is of technical interest.

IV. COMMENT ON ac LOSSES IN THE OUTERCYLINDER

The study of the ac losses in the outer cylinder~sheath!of the coaxial cable involves the introduction of an additionalparameter, namely,Ri /R, the ratio of the inner radius of thesheath to the radius of the core. We visualize that the crosssection of the sheath is appreciably greater than that of thecore. Consequently the current in that component will remainfar below critical even if a critical value is attained in thecore. Thus the outer radius of the sheath will play no role inan analysis of the ac losses whenI b < I cb .

The behavior of the ac losses expected for the sheath canbe estimated from that predicted for the core taking the fol-lowing features into account.

SinceHC1 dictates the maximum Meissner currentI M ,and a surface barrier field,Hbarrier, determines the surfacebarrier current, then assigning these same properties to thesheath we write, from Ampere’s law,

FIG. 10. Complements Fig. 9 by displaying the corresponding variations ofthe ‘‘depths’’ of the ac loss valleys~see Figs. 8 and 11! for variousI M/I cb>0 in ~a! and I SB/I cb>0 in ~b!. Compare this with Fig. 7.

FIG. 11. Illustrates the evolution ofWac vs I bias calculated usingj c5a/Hfor various fixedI ac/I cb when ~a! a Meissner currentI M/I cb50.4 plays arole, and~b! a surface barrier currentI SB/I cb50.4 is taken into account. The‘‘location’’ and depths of the valleys as a function ofI ac for severalI M andI SB are presented in Figs. 9 and 10.

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HC15I M2pR

5I M8

2pRi, Hbarrier5

I SB2pR

5I SB8

2pRi,

~17a!

hence,

I M8 5I MSRi

R D ,and ~17b!

I SB8 5I SBSRi

R D .We let a prime denote a quantity for the sheath.

By conservation of current

I5I b1I M5I b81I M8 5I b81I MSRi

R D , ~18a!

hence

I b85I b2I MF SRi

R D21G ~18b!

and similarly,

I5I b1I SB5I b81I SB8 5I b81I SBSRi

R D , ~18c!

hence

I b85I b2I SBF SRi

R D21G . ~18d!

Consequently, the horizontal segments of the hysteresisloops ~see Fig. 1! are lengthened by the factorRi /R accord-ing to Eq. ~17! and the curved contours are, from Eq.~18!,correspondingly shortened. The enclosed areas are therebydiminished.

Since the area embraced by the hysteresis loop measuresthe loss per unit surface, the factorRi /R must also be takeninto account in comparing the total losses in the sheath rela-tive to that in the core. Nevertheless in view of the foregoingwe expect the total ac losses in the sheath to be smaller thanthat in the core whenI M or I SBplay a role. However with theparameters set so that the core losses fit to a valley minimumit is unlikely that the sheath losses will also correspond to avalley minimum.

V. SUMMARY AND CONCLUSION

We have examined the effect of a steady bias current,I bias, on the energy dissipation in a coaxial cable carrying analternating current of amplitudeI ac within the framework ofthe quasi-static critical state, hence when the maximum cur-rent is less than critical.

We have also investigated the effect of a Meissner cur-rent I M and of a surface barrier currentI SBon the ac losses inthese circumstances exploiting simple but standard depen-dences ofI M and I SB on the surface magnetic field and itshistory.

In our calculations we have focused on two simple de-pendences ofj c on H, namely j c5a and j c5a/H sincethese lead to closed form expressions for theB profiles and

their spatial averages. In all the situations we have examinedexcept the case wherej c5a, I M50, andI SB50, a valley ispredicted in the graph ofWac, the energy dissipation percycle vsI bias with I ac kept fixed.

The evolution of the depth of the valley and of its posi-tion (I bias,min! have been mapped out as a function ofI ac forseveral values ofI M and I SB relative toI cb , the critical cur-rent for the bulk of the central core of the coaxial cable.

Our results and earlier calculations1,8 show that a valleyin the ac losses appears whenj c diminishes with increasingH. Our exploratory investigations with standard functions ofthe form

j c5a

~H1H0!n , j c5 j c0 e

2H/H0, ~19!

~wheren.0, H0 and j c0 are parameters of the specimen!confirm this behavior. The choice of function and parametersaffect the depth of the valley and the value ofI bias,min/I cb . Animportant feature which emerges from this study is that themore rapid the decline ofj c with H in the range 0, H, I cb/2pR, the deeper the valley.

When I M and I SB play no role, the behavior we predictalso applies to a cable constructed usingstraight tightlypacked filaments of arbitrary cross sections.

APPENDIX

Maxwell’s equation,¹3H5j and the critical state as-sumption thatj 5 6 zj c@Hf(r )#, or zero reads

1

r

d

dr~rHf!56 j c@Hf~r !# ~A1!

for an infinitely long cylinder in zero externally applied mag-netic field.

The sequences of quasi-staticBf profiles encountered asthe transport current,I b , carried by the bulk of the coaxialcable core is varied fromI b,max to 2I b,min and returns toI b,max are sketched in Fig. 3.

Three basic categories of segments ofBf(r ) profiles canbe identified in Fig. 3 where these are labeled 1, 2, and 3. Incategory 1,Bf(r ) increasesin magnitudeversusr , whereasBf(r ) decreasesin magnitudeversusr for categories 2 and3. The difference between the latter is that category 3 has anouter boundary whereBf(r )50. For each category,Bf(r )may be1 or 2.

The pertinent variables areI b , the net current flowing inthe bulk of the cable core andBS , the corresponding mag-netic flux density just inside the surface of the core. By Am-pere’s law,

HS5I b

2pR, ~A2!

and we takeBS5 m0HS .It is convenient to normalizeI b with respect toI cb , the

critical current whenj c[Hf(r )] fills the bulk of the core andflows in one direction only. Let

342 J. Appl. Phys., Vol. 79, No. 1, 1 January 1996 LeBlanc et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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i5I bI cb

~A3!

and

hn~r !5Hf~r !

~ I cb/2pR!, ~A4!

where the subscriptn51, 2, or 3 denotes the category of theBf(r )5m0Hf(r ) profile.

The normalized spatial average of the magnetic flux per-meating a segment of theBf profile is obtained from

^hn&51

REr ir0hn~r !dr, ~A5!

where r i and r 0 denote the appropriate boundaries of thesegment under consideration. These boundaries are deter-mined byI b,max, I b,min and the evolution ofI b between theselimits. The prescription thatBf(r ) must be continuous,hence the various segments must ‘‘match’’ at common inter-faces, dictates the relationships ofr 0 and r i with I b,max,I b,min, andI b .

The spatial average of aBf(r ) profile consisting of ajuxtaposition of several categories of segments reads

^h~r !&5(n

^hn&. ~A6!

We address the case wherej c5a ~Bean approximation!and, j c5a/H ~Kim approximation! separately.

1. j c5a (Bean approximation)

Integration of Eq.~A1! leads to

RHf~R!2rHf~r !56 j cR2

2 F12S rRD 2G ~A7!

with

I cb5 j cpR2, ~A8!

hence

h1~r !56F i21

~r /R!1S rRD G , ~A9!

h2~r !56F i11

~r /R!2S rRD G , ~A10!

h3~r !56F2 i11

~r /R!2S rRD G . ~A11!

We note that hereh3(r ) 5 2h1(r ). The spatial integralsof Eqs.~A9!, ~A10!, and~A11! are well known and need notbe given here. By way of illustration we give^h(r )& for themiddle Bf(r ) configuration of Fig. 3~b! which consists ofthe following juxtaposition of segment categories:1 h1(r ), 1 h3(r ), 2 h1(r ), 2 h3(r ) and1 h1(r ) ~threeconcentric annular current zones!

^h~r !&51S 12 imax2 D ln ~12 imax!2F12S imin1 imax

2 D G

3 ln F12S imin1 imax2 D G1F12S i1 imin

2 D G3 ln 1F2S i1 imin

2 D G1i

2. ~A12!

We note that the ‘‘previous’’h(r )& wherei is negative as itswings from2 imin to imax corresponds to Eq.~A12! where2 i replaces1 i .

2. j c5a/H (Kim approximation)

Equation~A1! now leads to

E ~rHf!d~rHf!56aE r 2dr, ~A13!

which yields

h1~r !561

~r /R!@2~12 i 2!1~r /R!3#1/2, ~A14!

h2~r !561

~r /R!@1~11 i 2!2~r /R!3#1/2, ~A15!

h3~r !561

~r /R!@1~12 i 2!1~r /R!3#1/2. ~A16!

The spatial average for each category of profile segmentreads

^h1~r !&5E h1d~r /R!

52

3 F @~r /R!32~12 i 2!#1/2

2~12 i 2!1/2 tan21 S ~r /R!32~12 i 2!

12 i 2 D 1/2G , ~A17!

^h2~r !&5E h2d~r /R!

51

3 F2@~11 i 2!2~r /R!3#1/21~11 i 2!1/2

3 ln S ~11 i 2!1/22@~11 i 2!1/22~r /R!3#1/2

~11 i 2!1/21@~11 i 2!1/22~r /R!3#1/2D G ,~A18!

^h3~r !&5E h3d~r /R!

51

3 F2@~12 i 2!2~r /R!3#1/21~12 i 2!1/2

3 ln~12 i 2!1/22@~12 i 2!1/22~r /R!3#1/2

~12 i 2!1/21@~12 i 2!1/22~r /R!3#1/2G . ~A19!

Again, by way of illustration we give h(r )& for themiddleBf(r ) configuration of Fig. 3~b!

^h~r !&522

3~12 imax

2 !1/2 tan21 S imax2 2 imin

2

2~12 imax2 !

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21

3~12 imin

2 !1/2 lnS ~12 imin2 !1/22@~ imax

2 2 imin2 !/2#

~12 imin2 !1/21@~ imax

2 2 imin2 !/2#

D12

3~12 imin

2 !1/2 tan21S imin2 2 i 2

2~12 imin2 !

D 1/211

3~12 i 2!1/2 lnS ~12 i 2!1/22@~ imin

2 2 i 2!/2#1/2

~12 i 2!1/21@~ imin2 2 i 2!/2#1/2

D12

3 F i2~12 i 2!1/2 tan21S i

~12 i 2!1/2D G . ~A20!

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