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FORCE-FREE COIL PRINCIPLES APPLIED TOHIGH-TEMPERATURE SUPERCONDUCTINC MATERIALS

H.P. Furth, S.C. Jardin,Princeton Plasma Physics Laboratory

Princeton, New Jersey 08544and

D.B. MontgomeryMassachusetts Institute of TechnologyPlasma Fusion CenterCambridge, Massachusetts 02319

P P P L 2 5 0 7D E 8 8 0 0 9 3 4 3

ABSTRACT

Force-free magnetic-field configurations , where the current flowsparallel to the magnetic field vector, have the potential to raise thecritical, magnetic field and current-density limits for high-temperaturesuperconductors.

WtMD I S T R I B U T I O N OF J f S S E S H E K T R U H U K I T H

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In a force-free magnetic-field configuration, the current densit y flowsalong the magnetic field (J = J . . , so that the Lorentz force 3 x 5v a n i s h e s . A simple illust ration is provided by the force-fre e cable ofF i g .1. The helical winding pattern generates a B z~field component on theinterior of the cabl e, thus permitting 1 to flow parallel to B, instead ofperpendicular to 5, as would be the case in a conventional cable. Theexterior magnetic field (BQ) is the same for force-free and conventionalcables carryingthe same total current I z >

Force-free cables and coil s* have two advantages that should beimportant in superconduc tor applications involving high magne tic fields andcurrent densities: (1) thevanishingof the overall Lor entz force exertedonthe conductor greatly relaxestheconditionson themechanical strengthof them a t e r i a l; (2) the vanishing of the Lorentz force on the current-carryingelectrons within the conductor is favorable to the achievement of highercritical current densitiesJ__.-,. th anare likelyto be reached w ith currents

i crit 'flowing perpendicularto themagnetic field(J - _ : . . 58In the case of conventional hard superconductors, where comparative

measurements have been madefor J..c r ^ c and J c r i t o n e * i n a s Jn c r i t c o b etypically larger by an order of m a g n i t u d e . In the case of the new h i g h -temperature superconductors, anisotropics in critical current density havebeen reported for different orientation sof 6 relativeto the crystal planesof the s a m p l e 7' ~ but these measurements were made by inducing closedcurrent loops within the sample and, therefore, they refer ocly to J c r i t(The measuredJ c r j r ^ o r a current loop closed within the favorable crystalplanehas been found to be higher than the J c r ; c r a current loop thatpassesin and out of the favorable pl ane.) To measureJ.c r ; t the inductivemethod is not applicable: one must pass an externally generated current

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through the sample, in the direction along 5, and measur e the voltage dropwithin the sample.

The advantages of force-free configurations are partly off-set by theirgreater geomet ric complexity, and especially by the need for a helical windingpatte rn, as illustrated in Fig. 1. To estimat e the potent ial-be nefit of theforce-fre e design a pproach , we write r * J. c r i t / J , c r t and assumeB J. c _ : t * const, (as was found in Ref. 5 ) , up to some critical fieldB - t , whe re J.. drops to zero. In the simple geometry of Fig . i, the

force-fre e condition is expressed by

B 2 d B 2/dr + Bg/r dC rB eJ/d r = 0 (1)

Among the many possible solutions of Eq. (1 ), we select the parti cular onethat corresponds to a solid cylindrical force-free windi ng with E J =const. The solution is shown in Fig. 2. In comparison with a conventionalcable of the same radius a, unifor m current densit y J , and the -ante totallongitudinal current Iz(a)> we find that for the force-free cable, therequired product E J is 2.2 times great er. The factor by whic h I,(a) can behigher in the force-free case is therefore 0.45 r. (By comp aris on, themaximum factor by which I z(a ) could be increased in a conventional-type cableby optimizing J(r) according to B J = const, is limited to1.33.}

Z ZAn approximately force-f ree coil of large El/a (ratio of major to minor

radi us) can be realized by bending the cable of Fig s. 1 and 2 into a closedloop. Coils of this type are potentially useful for inductive energy storageor , conceivably, to form part of the guide field for a high- energy partic le

accelerator.

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For purposes of producing maximal magnetic field strength in a smallexperimental volum e, a lou-H/a coil of the type shown in Fig. 3 isappro priat e. The coil has nested toroidal winding surfaces that coincide withthe magnetic surf aces . It generates a poloidal field on its exterior , like aconventional coil , and also generates a toroidal-field component on itsinterior. The helical pitch af Che windings is varied from surface tosurface, so that J points along B an all surfaces. To be truly force- free,the coil must be supported by an external magnet ic field from a conventional

force-bearing coil but the maximum pre ssure B /8ir appearing at thesupporting coil can be kept relatively small. A force-free toroid of Low ft/acan be used to ach iev e typical fie ld- enh anc eme nt fact ors R"-'< = 2. 5- 3. 0, wher eR* is defined as the ratio of central ma gnetic field strength to fieldstrength at the supporting coil, normalized to che same ratio for thesupporting coil alone. Refere nce 4 finds that there is a trade-of f between R*and the maximum diamete r of the useful force -free coil bore that can beachieved with a supporting coil of given si2e.

For the coil design of Fig. 3, the mathemat ical appro ach was to solve7 x 5 = u 0 3 , J = A.B, \ = I|I , wh er e \|> is the no rm al ize d po lo id al ma gn et icflux, which has a null at the coil surf ace. The figure uses radial and axialcoordinates x and z, normalized to give r * r at x ~ 1. In Fig . 3a, we seethe poloidal flux surfaces and the variation of B z( x ) / B 2 s(o ) along themidplane (z = 0 ) , where B s refers to the field of the supporting coilalone . Figure 3b gives the contours of constant toroidal current density Jand the magnitu de of J normalized relative to the total toroidal currentI = Sr B s (o ) (in units of HA, m, T ) on the midplarte. Figure 3c shows thecontours and normalized magnitud e of xB_. This particular coil design givesR* = 2.7, along with a high degree of magneti c-field uniformity (3 B /3x 2)/B

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- 0.1 within a moderate- sized coil bore . (The external force- bearing coil isidealized as Che single-layer solenoid at x = 1 in Fig. 3a.)

Experiments with low-pressure plasmas have demonstrated the formation andstability of virtually force-free configuration s ' much like those of Fi gs . 1and 3. By analogy, a tendency of 3 to flow along 5 within superconductingsurfaces would favor the spontaneous emerg ence of a force-free pattern ofcurrent flow thus simplifying the construc tion of force-free cables andcoils.

ACKNOWLEDGMENTThis work supported by U.S . Department of Energy Contract

N o . DE-ACO2-76CHO3073.

REFERENCES1 R. Lust and A. Schluter, Z. Astro phys. 34, 263 (1954).

2 U.P. Furth, M.A. Levine, and R.W. Waniek, Rev. Sci. Inst rum. 28_ 949(1957).

3 H.P. Furth and M.A. Levine, J. Appl . Phys. 33, 747 (1962).

H.P. Furth and S.C. Jardin, Princeton Plasma Physics Labor atory ReportPPPL-2465 (1987).

5 D. 8. Montgomery and W. Sampson, App l. Phy s. Lett. 6, 108 (1965).

G.D . Cody and G.W. Culle n, RC A Rev . 2J5, 466 (1964).

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S.T. Sekula, R.W. Boom, and C.J. Bergeron, Appl. Phy s. Lett. 2 1021963).

a C.J. Bergeron, App l. Phys. Lett. 3 63 1963).

9 V. Hidaka ec al. , Jpn. J. Appl. Phys. 26_, 4 ) , L377 198 7) .

0 T.R. Dinger ec al ., Phys . Rev. Lett. 58_, 2687 19 87 ).

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FIGURE CAPTIONS

Fig. 1. Force-free cables use heLical windings of varying picch.

Fig. 2. For the condition BJ = const., a particular force-free solution isobtained in cylindrical geometry.

Fig. 3. Insertion of a force-free coil into the force-bearing solenoid atx = I serves to triple Che field st rength B at x = 0, z = 0.

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s8?A0053

F i g

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1.0

0 . 5

1 r 1

Bfi \ BiCO \BzC =i . 6 B g f a .B0

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0J 0o

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