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Energy dissipation in high temperature ceramic superconductors
Applied Superconductiviiy Vol. 3, No. 7-10, pp. 509-520, 1995 Copyright 8 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0964-1807(95)000860
Pergamon
0964-1087195 $9.50 + 0.00
ENERGY
DISSIPATION IN HIGH TEMPERATURE SUPERCONDUCTORS
M. CISZEK,? Interdisciplinary
A. M. CAMPBELL, Research
S. P ASHWORTH
Centre in Superconductivity, University Cambridge CB3 OHE, U.K.
CERAMIC
and B. A. GLOWACKQ of Cambridge,
Madingley
Road,
Abstract-Losses in high-T, superconductors are of vital importance in practical applications. The mechanisms are similar to those in low-T, materials but there are a number of complications. In this paper the loss mechanisms in different regimes are summarized and simple approximate expressions obtained. Losses due to the transport current are analysed with the theory of Norris for elliptical conductors. This predicts different loss voltages according to where the voltage contacts are placed on the conductor which has been confirmed experimentally. This poses difficulties in turning experimental data into reliable loss measurements, but a series of experiments with different contact layouts and phase setting techniques is found to give consistent results, in good agreement with theory.
1. INTRODUCTION
The practical application of superconductors, in, for example, power transmission cables, transformers, motors and other devices working under alternating current conditions, are limited by the magnitude of the electric currents which can be transported with a very low level of energy loss. The energy dissipation in type-II superconductors is connected to the intrinsic properties of these materials. Among them the most important are critical current density (pinning force), surface energy barriers for entry and exit of the magnetic flux, flux flow or flux creep, and the geometry. We also need to know the effect of combined fields and currents. Reviews of the losses in low-T, materials are given in, for example, Refs 1 and 2, and the same techniques can be applied to high-T, materials. Extra complications arise due to granularity and the presence of weak spots in most real conductors. D.C. measurements with a transport current show a measurable voltage near to critical current I,. This d.c. “resistance” is generated by steady state flux motion and contributes to losses with a.c. currents and magnetic fields, in addition to those losses arising from the changing flux within the superconductor. In high quality wires for transport currents less than 0.91, the “d.c.” component is negligible and the energy loss per cycle is frequency independent [3]. It is convenient to divide applications into two regimes: small field penetrations and large field penetrations. The division between these two regimes is the external field which just fills the cross-section with the critical current density. Two electrical a.c. loss measurement techniques, inductive and transport current, are common. In the first method the superconductor is placed in a varying external magnetic field and the currents induced in the sample (electric fields) are measured by means of a voltage pick-up coil wrapped around the specimen (contactless a.c. magnetic methods). These methods also allow us to measure the magnetic flux distribution within the sample [4]. In the case of transport methods an a.c. current flows through a superconductor, and the voltage drop along the sample is a measure of resistance, and thus of a dissipated energy (self-field losses). Magnetic methods permit us to measure a.c. losses in the inter-granular matrix of the ceramics (inter-granuiar losses), as well as in the individual grains (intra-granular losses). The transport loss measurements describe only the inter-granular losses (in the net of weak-linked Josephson junctions). Due to the very complex structure and morphology of the ceramic materials, on both a micro and a macro scale, distribution of the transport current is not homogenous and the current “meander”, taking paths t
On leave from Institute for Low Temperature and Structure Research, Polish Academy $ Also at the Department of Materials Science and Metallurgy, University of Cambridge, APSUP 3.7/N-K 509
of Science, Wroclaw, Poland. Cambridge CB2 342, U.K.
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with the locally highest critical parameters. As shown in [5], the local critical current density which depends on the local microstructure can be several times higher than the average one. 2.
LOSSES
DUE
TO EXTERNAL
MAGNETIC
FIELDS
Type-II superconductors subjected to a time-varying external magnetic field exhibit losses, mainly due to irreversible movement of flux vortices. Local dissipation per unit volume is generally expressed as J, x E, where J, is the critical current density and E = B x v is the electric field with the vortex velocity v. For small amplitudes bc of the external field, below the full penetration field b, (defined as the external applied field which just fills the superconductor cross-section with the critical current density) the loss is proportional to the surface area of the sample and varies as the cube of the amplitude. However an easier and more important regime is if the magnetic field penetrates to the centre of the superconductor (fl> 1, where /3 = b,/b,). In this case, if self fields are small compared with the external field, the rate of the change of flux density is everywhere the same as that at the surface. For a rectangular superconductor of width 2a and thickness d (see Fig. 1) the penetration of a field parallel to the long face (a) occurs at b, = paJ,d/2. The full penetration of a field directed perpendicular to the width of the sample (along d in Fig. 1) occurs at a field b, [6]: 6r = (poJcd/n)(z arc cot(z) + 0.5 ln(1 + z?)),
(1) where z = 2a/d. Since the field does not vary significantly across the sample we can assume constant J, and since the applied field is not significantly distorted we can assume that straight flux lines move towards the electric centre (Fig. 1). This defines the electric field, which rises linearly with x from zero at the electric centre, which is determined by symmetry. The electric field at a distance x from the centre of the slab is x dB/dt, so the local dissipation is J&E?/dt. Integrating this over the cross-section and a complete cycle of an a.c. signal with peak amplitude bO> b,, gives a loss per cycle Q in a perpendicular field bs: Q = 2JCbOa (J/m3).
(2) The same result can be obtained from the area of the hysteresis loop in the magnetization curve. The following conclusions can be drawn in this regime. First, the loss per unit volume is proportional to the width of the superconductor perpendicular to the field. Second, it is proportional to the critical current density. Third, the loss per cycle is independent of frequency. The loss for superconductors of various shapes is given in Table 1. If we pass a d.c. current equal to the critical value through the sample, the Bean model [7] tells us that the electric centre moves to the edge of the sample as in Fig. l(b). The flux has to move twice as far so the loss is now 4J,b,,a. In fact it can be shown that this expression is exact for any
-
E-O
E=O Fig. 1. Penetration of the magnetic flux lines into a rectangular superconductor. (a) Without d.c. transport current. The flux lines are not distorted and move at velocity proportional to x. (b) With d.c. transport current equal to Z, (the electric centre is now at the edge).
Energy dissipation in high-T, ceramic superconductors Table 1. Magnetic
losses per cycle for various
511
sample geometries
Sample geometry
B
PoJ,R
8>1
shape of the cross-section
[8] if a is taken as the maximum value of x which is inside the sample. This brings out the fact that the loss is rather independent of sample shape but depends mainly on the width perpendicular to the applied field. We can apply the same reasoning to find the loss in a silver sheath around a ceramic since the electric field will be the same as in the superconductor at the same value of x. The loss is obtained by averaging oE2 and is (7r/3)(~~&&2~) J/m3 Ag. If we have an alternating current with amplitude Ia = iI,, the loss is multiplied by a factor (1 + 3i2/8). The effect of the electric field in the superconductor due to the transport current has been ignored since it will be much smaller. For the result to be valid we must avoid the skin effect regime. This means that the area of silver must be small compared with 6’, where 6 is the skin depth of silver. If this does not hold the superconductor will be shielded from the field by eddy currents and can be ignored. If the condition is met we can then write the ratio R of the silver loss to the superconductor loss as: R = (7V3)(odl~2)(b,l&,),
(3)
where b, is the parallel penetration field [6]. At a frequency of 50 Hz the skin depth of silver at 77 K is about 3 mm and for our tapes ad/d2 is about 0.1. This means that the silver losses will begin to dominate the hysteresis losses at an amplitude of about 60 mT which is considerably smaller than the fields in transformers. These losses will dominate in magnets and transformers in a.c. conditions and the approximations used are rather accurate, although it will be necessary to use the local value of J, at each part of the winding.
3. TRANSPORT
CURRENT
LOSSES
Losses in cables are due to transport currents, which can never involve large penetrations since, when the critical state reaches the centre, we have reached the critical current. For low currents we can often use the expression for the surface loss in an applied field, inserting the local field generated by the current for an applied field. This will give an order of magnitude estimate even at the critical current. However the most important shapes of conductor have analytic solutions. The round wire problem was in fact the first application of the Bean model [8]. Solutions for an ellipse, a thin strip, asperities, and a gap between strips were obtained by Norris [lo] in a very comprehensive paper which applied conformal transformations to the critical state model for the first time. The elliptical wire is the easiest to deal with. The critical state penetrates as a series of concentric ellipses and the rather surprising result is that the loss per unit volume is independent of the aspect ratio. This is in marked contrast to the loss in an oscillating field which tends to infinity as the thickness goes to zero. It means that we can use the loss of a round wire for a wire of
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any aspect ratio, and that the loss is not very dependent on the wire shape. According to Ref. 10, for a circular or elliptical wire, self-field losses are given by: Q, =
and for a thin superconducting Q,
!$ [(I _ i)ln(l
-
i) +
v]
strip:
=&!$[(1 -
i)ln(l - i) + (1 + i)ln(l + i) - i*],
where i = Is/Z,. At the critical current the loss per cycle per unit length for an elliptical wire with a critical current Z, is 0.16 & and that for a thin rectangular film is 0.12 Z& so that the difference is not very great. However at low currents the behaviour is rather different. The ellipse follows the usual 1, dependence in common with most hysteretic systems which trace out a Rayleigh loop at low amplitudes. However in the film the losses are proportional to 1:. The losses depend rapidly on the transport current value. At 0.51, the loss in an ellipse is reduced by a factor of 16 compared with Z, and for a film it is 40. If losses are a significant problem there are clearly advantages to be gained by making a cable large enough to be operated well below its critical current. As in the magnetic case an alternating current in the superconductor is associated with an electric field at the surface which for thin cladding will cause an electric field in the silver of the same magnitude. Provided we are not in the skin effect regime we can assume that the magnetic field is that due to the superconductor alone and calculate the electric field from it. The expressions given in Ref. 10 for the elliptical conductor can be used to find the flux inside any point in the silver cladding and the electric field calculated from the rate of change of flux. This involves some rather heavy algebra but we can get a good estimate as follows. We divide the loss by the current to get the in-phase voltage. We then apply this voltage to the silver cladding and find the loss from Ohm’s law. At the critical current the loss for an elliptical wire is (o,&/(4rr)2) W/m so the rms electric field is o&/(2n2) V/m. The silver loss is therefore a(oZ.&/(27r2)) W/ m3 of Ag. The ratio of silver loss to superconductor loss is (2/7~~)(-4s/~~), w h ere A, is the cross-section area of silver, so that by avoiding the skin depth regime we automatically ensure that the silver loss is less than the superconductor loss. This underestimates the loss since the silver loss depends on the rms voltage, rather than the in-phase component. For planar geometry, which can be solved analytically, the error is a factor of four. 4.
COMBINED
FIELDS
AND
CURRENTS
Analytic solutions have been found for the case of an applied current and transverse field in thin films [ll, 121. There are a number of different cases, the one illustrated in Fig. 2 is where the applied field is less than the self field so that the effect is an asymmetry of the critical state on the two sides. Losses in this situation were not quoted but since one side gets smaller while the other gets bigger we would expect the effect of the external field to be second order in be/b,.
Fig. 2. (a) A field applied normal to a film increases the penetration on one side and decreases it on the other. (b) A parallel field does not affect edge penetration.
Energy dissipation
Fig. 3. A schematic representation
in high-T, ceramic superconductors
513
of the arrangement of eight strips around a former to form a cable.
Of more practical importance, since it is directly relevant to cables, is the effect of a field parallel to the film surface. Experimental results have shown that the longitudinal loss voltage in these circumstances is the same whether the applied field is d.c. or a.c. with the same peak amplitude [ 131. This means that the main effect of the field is the reduction in J, due to the increased total field, which is a reasonable result. A parallel field produces a low loss on its own and, if it is less than the parallel penetration field, adds little to the penetration at the edges, where the main loss due to the transport current is taking place. Therefore if J, were to remain constant the application of a parallel field should have little effect on the losses and there is little difference between an a.c. and d.c. field in this geometry.
5. CABLES
The main application of these results is to the calculation of a.c. losses in cables. These consist of tapes mounted on the outside of a cylindrical former as in Fig. 3. Since the critical current is reached when B reaches the penetration field these cables are never in the high penetration regime (unless there are several layers). The ideal geometry would be a single uniform layer of superconductor, which was how low-T, power cables were designed. If the superconducting layer is 2d thick, and the radius r, the loss can be written (2/3n)&,Z$3(d/r)). This is a lower limit to the loss. However mechanical constraints will require a significant separation. According to Ref. 10, the edge losses at a gap in a sheet go as the square of the gap size, and by the time the gap width is equal to the strip width they will be effectively behaving as independent strips. Hence a better expression for the loss is (Z~~Z,2/27r)(g/a)~,where g is half the gap width up to a maximum of a. The only effect of the radius on this expression is that a larger radius will reduce the surface field and so increase Z,. If several layers of tapes are used it will be important to decouple them with an insulating layer between each conducting layer. The direct measurement of losses from the voltage along a cable poses even more problems than that for a single conductor and a comparison of electrical measurements with thermal ones is needed.
6.
FILAMENTARY
MATERIAL
Multifilamentary high-T, materials are now becoming available, and are proving to have unexpected benefits in terms of resistance to mechanical deformation. However without twisting the wire there is no reduction in a.c. losses and there may be an increase [ 141. Figure 4 shows the field lines for an applied field and transport current for a monolithic and multifilamentary material. The flux lines must cross the superconductor and the normal material and for the case of the multifilamentary material we can just average the properties over the cross-section. It is clear that the wider the area over which we spread the superconductor, the more flux must cross the current and the higher the loss. The total loss of the conductor is proportional to the radius of the circle containing the filaments and subdivision within this area does not affect the losses, but the monofilament has a smaller radius and therefore has the lowest loss. This conclusion is based on the assumption that the flux moves across the filaments. Flux can reach the centre of the conductor by two routes. It can cut the superconducting filaments and move in from the edge, or it can diffuse between the filaments from the ends of the wire (Fig. 5). In the
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Fig. 4. If we spread out the superconductor
more flux is included and losses increase for both self and external fields.
Fig. 5. Flux can diffuse from the ends or cut through the sides.
latter case the filaments behave independently and the loss is proportional to the filament diameter rather than the conductor diameter. This clearly leads to a large reduction in losses. However the diffusion time constant for a kilometer of wire is about 20 yr, so measures must be taken to provide a shorter diffision path. This is done by twisting the wire so that flux rings can be nucleated at the filament crossovers and expand to the mid points between crossovers. The effective diffusion length is now the twist pitch. For twisting to be effective the pitch must be less than the skin depth, which is about 3 mm in pure silver at 77 IS and 50 Hz and a little longer for silver alloys. Making such a conductor is technically difficult, and in any case is only useful for external fields, it has no effect on self field losses, which require transp,osed filaments if they are to be decoupled. A twisted Ag/Bi-2223 conductor with critical current density of the order of 1.3 x 10 A/cm2 and twist pitches of 3.6 and 3.7 mm has been lately reported [ 151. It would appear that twisting high-T, conductors is not a high priority and the use of multifilamentary materials in cables must be justified on factors other than a.c. losses. However in magnets and transformers twisting will greatly reduce losses.
7.
LOSS
MEASUREMENTS
It is common to measure transport current losses by means of voltage taps on the conductor but it has recently been pointed out that this voltage is very dependent on the position of the contacts [16-181, and care must be taken to obtain a true measurement of the loss. The meaning of a
Energy dissipation
515
in high-T, ceramic superconductors
Fig. 6. A schematic representation of the measuring circuit containing an electrostatic voltmeter. The voltage contacts, on the edge of a strip in this example, form a loop which links with the magnetic field generated by the transport current.
voltage measurement is illustrated by reference to Fig. 6. Although the results are obviously independent of the nature of the meter the argument is clearest if we imagine an electrostatic voltmeter where the indicated voltage is the electric field E, between the plates. We now apply Faraday’s law of induction to a loop consisting of unit length along the centre of the wire, where the field is EC, the radial sections to the voltage leads themselves. The electric field E is zero along these sections. Then: E,, -EC = d(Oi + @&dt
or
E, = d(Q, + $)/‘dt
+ EC-
(6)
Hence what we measure on the voltmeter is the electric field along the centre line plus the flux change enclosed between this line and the leads to the meter, including the section within the superconductor, (I+ If we take our line integral along the surface instead of the centre line we see that an equivalent statement is that we measure the surface field plus the flux change between the surface and the meter, @i. If we are below 1, there is no electric field along the centre line so that the voltage is entirely attributable to the changing flux. If we have a d.c. current in a copper wire there is no change of flux and the voltage is the conventional resistive one. In general both components will contribute. The reason for the voltage differences is now clear. Figure 7(a) shows the flux around a strip carrying a current. There is far more flux cutting the edge contacts than the centre contacts and the voltage will be greater by a factor approximately equal to the aspect ratio of the conductor. To obtain a model-independent measurement of the loss, the loop containing the voltmeter should extend to a point well away from the conductor so that the lines of force are circular (Fig. 7b). The voltage will be the same for any point on the circumference so a single measurement will suffice, but there is the considerable disadvantage of a large inductive signal which will be difficult to balance out accurately.
Fig. 7. (a) The magnetic field lines round a strip carrying a current. (b) Only if the leads enclose a large loop is the voltage proportional to the loss.
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M. CISZEK et al. 8. EXPERIMENTAL
RESULTS
To obtain an unambiguous transport loss measurement we need to sample the flux over a distance much larger than the strip width. However for practical purposes quite a small loop will be adequate. We have measured the voltage from a series of contacts with different loop sizes as shown in Fig. 8. We used one of our typical Ag/BSCCO-2223 tape produced by powder-in-tube method. The ratio of ceramic to silver in the cross-section of the final tapes was about 1:4. The samples used were 27 mm long, with d.c. critical currents 1, in zero magnetic field at 77 K of 12.3 A (sample TS-12, overall cross-section 4 x 0.22 mm2, the thickness of the superconducting core in the central part of the tape was about 90 pm). The critical current (defined at 1 uV/cm criterion) corresponds to critical current densities of approximately 7 x lo3 A/cm2. The measurement set-up is shown in Fig. 9. For the a.c. transport measurements current leads were in the form of silver tapes of the same width as the sample which were directly soldered to the ends of the specimen. Measurements were carried out for frequencies from 40 to 300 Hz. A set of voltage leads were point soldered to the tape as shown schematically in Fig. 8. For magnetic loss measurements a single layer lo-turn pick-up coil. was wound along the sample. In this geometry the induced current flows along the tape as in the case of transport measurements. A liquid nitrogen cooled copper magnet was used to produce an a.c. magnetic field with a maximum amplitude of 140 mT. The lossy component U” (rms), of the voltage signal at the fundamental frequency, induced between the potential leads (or from the pick-up coil) was measured by using a dual phase lock-in amplifier. As the inductive component of the measured voltage in the case of large loops is more than two orders of magnitude higher than the loss signal, special precautions should be taken to set the phase and avoid common mode signals. A specially designed compensating (a) transport measurements Tl C
(b) magnetic measurements
Fig. 8. Sample arrangements for transport and magnetic loss measurements. (a) Voltage taps on the tape for measurement of transport losses. Potential leads E are on the edge of the tape, and C ones are on the central line of the tape. The distances ab, bc and cd are about 5 mm (similar dimensions have the central measurement loops). (b) Configuration of the sample and a.c. external magnetic field for inductive measurements. A single-layer pick-up coil is placed directly on to the surface of the tape in its central part.
Energy dissipation
in high-T, ceramic superconductors
517
J
-4 PC
5’ DVM A
Fig. 9. Block diagram of the a.c. transport current loss measurement set-up: S-sample tape, Ccompensating circuit, TR-isolating transformer (which also increases the current), CT-current transducer.
minitransformer was used as well as grounding only in one common point. Filters ensured that only the fundamental was measured; the harmonics do not contribute to the loss. All measurements were carried out at liquid nitrogen temperature. The sensitivity of our apparatus was better than 10v8 V and the voltage in each type of measurement was turned into an apparent loss using the usual expressions
where Qt is self-field loss, f is the frequency, ZmSis the transport current and VA is the loss voltage measured between potential leads (separation r>. In turn, the magnetic loss Qm, per surface area, per cycle, was calculated according to the equation
(8)
10s
P
1
/l&2.3
Al
10 4, (4
Fig. 10. The dependence of the loss voltage U,!&,on the amplitude of the a.c. transport current lo as measured for the different loops between the edge of the tape and potential leads. The inset shows the data for low currents in a linear scale.
518
M. CISZEK et al.
Fig. 11. Transport loss Qt per unit length and per cycle as a function of the current amplitude (i = lo/Z,) for different frequencies, as measured on the edge of the tape, and for larger potential wire loops. Solid and dashed lines are theoretical curves, for an ellipse and a strip, according to eqns (4) and (S), respectively.
where bc is the external field amplitude, N is the number of turns of the pick-up coil, S is the perimeter of the tape (about twice its length), and U,!&,is the loss voltage from the pick-up coil. Figure 10 shows the loss voltage as measured on the edge of the tape, for different loops between tape and the potential leads (see Fig. 8). There is a difference only for the smallest loop (taps no. 1, filled symbols). Due to the thickness of the silver this is already a loop about 0.3 mm thick. The data show that to measure correctly the loss voltage it is enough to increase the distance between sample surface and voltage wires to a few times the tape half width, in our case 5-6 mm separation was sufficient. Nearly the same data were obtained for taps placed in the central part of the sample, at least for currents greater than about 0.21,. The loss calculated using eqn (7) is shown in Fig. 11 and is in a very good agreement with the theoretical model for an elliptical or round wire. Compared to the theoretical prediction we have obtained very good agreement for the elliptical model (eqn 4). The strip model is a much less good approximation which is not surprising since most tape cross-sections show a tapered edge rather than a rectangular one. The loss varies as - Zo,4 in the range up to about 0.71,.
9.
GRANULARITY
At very low penetrations the grains can be treated as perfectly diamagnetic (Fig. 12). We can treat the system as a composite and average fields over many grains. B is the average flux density which, if J, were zero, would be less than an applied parallel field by something like the volume fraction of normal material. We can define a local H as the external field in equilibrium with the local value of B, just as we do in conventional superconductors, and a relative permeability, pr as B/p&t The relative permeability pr is equal to the volume fraction of normal material if the grains are long cylinders parallel to the field but in general the exact relationship will depend on the morphology and pL,needs to be measured experimentally. We have found a typical value of about 0.3. High-T, materials consist of grains with a high J,, the intragrain J,(intra), separated by weak links with a much weaker intergrain J,(inter). The detailed response of such a system to an alternating field is complex and detailed calculations will be found in Ref. 19. We need to introduce a third amplitude regime in which the grains as well as the weak links between them are penetrated by the external field. The same expressions apply to the grains as to the complete wire, although interactions between closely packed grains will make the results less accurate. We must add a loss of approximately 2BsJ,(intra) where s is the grain size. This result is not affected by a
Energy dissipation
in high-T, ceramic superconductors
519
Fig. 12. A composite of diamagnetic grains and weak links. Right side picture shows the penetration of the magnetic field Hand the magnetic induction B.
transport current, but since the only way of measuring Jc(intra) is from the magnetic hysteresis, we might as well use the measured magnetization curve to predict the loss in a.c. conditions. The fields which result in a cylinder parallel to an applied field are shown in Fig. 12 and the transport current is dH/d.x. Since for a given J, the value of B is lower by a factor pr compared with a homogeneous material, the effect of the diamagnetic grains is to reduce the losses by this factor. Clearly the assumption of perfect diamagnetism will not always be valid and there will be a surface loss on each grain. The importance of this will depend in detail on the shape of the grains, their size and the relative magnitudes of J,(inter) and J,(intra). This is something on which we need more information, but the good agreement of Fig. 11 suggests that, at least for transport currents, the grain surface losses can be ignored. The dependence of the magnetic a.c. losses Q,,, per unit surface area and per cycle, on the amplitude of the external magnetic field ba, for three different frequencies, is shown in Fig. 13. No remarkable differences are present for different frequencies of the field, which means that losses in the investigated sample are hysteretic in this frequency range. For the low bc the losses increase lo-’
F
a 2
G
1o-2 1o-a
E 5 v
lo-4
d
1o-s
% 9
10-a
.::
10-7
E B I
10-a
10-Q 3e-5
-
-2
3 2e-5 ‘2 le-5
-
Oe+O
b, (mT) Fig. 13. Magnetic loss Q,,, data vs external magnetic field amplitude be for Ag/BSCCO-2223 tape, for different frequencies. Only one peak at ba = b, = 5 mT in p” (b,) function is observed.
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M. CISZEK et al.
with a power close to four, whereas for higher magnetic fields this dependence is linear with bc. In the lower part of Fig. 13 the lossy component of the a.c. complex permeability is shown. Contrary to typical granular materials, e.g. sintered YBCO ceramics, we observe in our tape only one pronounced peak, at least in the available magnet amplitude range to 140 mT. The imaginary part of the complex permeability $’ is related to the losses via Q, = $‘7rb@/2~,,, where d is the thickness of the sample (d << length and width of the plate) and Qm is expressed as loss per surface unit. In the simplest case of the Bean model, for plate geometry (see Table l), taking into account the relation between loss and magnetic permeability $‘, the maximum in I” dependence takes place at b, = b, = 2/3pOJcd. For the peak which occurs at 5 mT the value of the critical current density is determined as about 6.6 x lo3 A/cm’, the value which differs only by 5% from that measured by four-point transport method. 10.
SUMMARY
In spite of the complexity of high-T, tapes we have an understanding of the losses in single strands which is probably adequate for the design of demonstration samples. The most accurate results are in the high penetration limit where the internal field is equal to the external field and the electric field is easily calculated. These losses will dominate the loss in a coil and will also probably be dominated by silver losses since these go as B* while the hysteresis losses go as B. Cables work in the low field limit where the loss is due to a transport current. The theory of Norris for elliptical conductors can be used here for the hysteresis loss of single strands. This loss is the same as that for round wire of the same critical current. Complete cables have a combination of pure transport current and an applied field which is a less certain situation but the loss of single strands with the appropriate J, should be a reasonable approximation. The determination of losses due to transport currents using the voltage generated gives results which depend strongly on the configuration of the measurement circuit. In particular the measured voltage varies by up to three orders of magnitude depending on the positioning of voltage contacts and the size of the “loop” linked with the changing magnetic field. Theoretically these measurement loops should extend to infinity to return the correct voltage, in practice experiments indicate that the results tend to a limit, in good agreement with theory, if the loops extend out to a few times the tape width. REFERENCES 1. M. N. Wilson, Superconducting Magnets, pp. 159-199. Oxford University Press, London (1989). 2. J. R. Clem, in Magnetic Susceptibility of Superconductors and Other Spin Systems (Edited by R. A. Hein et al.), pp. 177211. Plenum Press, New York (1991). 3. S. P. Ashworth, Physica C 229, 355 (1994). 4. A. M. Campbell, 1 Phys. C 2, 1492 (1969). 5. D. C. Larbalastier, X. Y. Cai, Y. Feng, H. Edelman, A. Umezawa, G. N. Riley Jr and W. L. Carter, Physica C 221,299 (1994). 6. A. M. Campbell, IEEE Trans. Appl. Supercond. 5, 682 (1995). 7. C. P. Bean, Rev Mod. Phys. 36, 31 (1964). 8. A. M. Campbell, Cryogenics 22, 3 (1978). 9. R. A. Kamper, Phys. Lett. 2, 290 (1962). 10. W. T. Norris, 1 Phys. D 3, 489 (1970). 11. E. H. Brandt and M. Indenbom, Phys. Rev. B. 48, 12893 (1993). 12. E. Zeldov, J. R. Clem, M. McElfresh and M. Darwin, Phys. Rev B 49, 9802 (1994). 13. M. Ciszek, B. A. Glowacki, S. P Ashworth, A. M. Campbell and J. E. Evetts, IEEE Trans. Appl. Supercond. 5, 709 (1995). 14. K. Kwasnitza, St. Clerc, Physica C 233, 423 (1994). 15. C. J. Christopherson and G. N. Riley Jr, Appl. Phys. Lett. 66, 2277 (1995). 16. T. Fukunaga, S. Maruyama and A. Oota, Advances in Superconductivity (Edited by T. Fujita and Y. Shiohara), p. 633. Springer, Tokyo (1994). 17. M. Ciszek, A. M. Campbell and B. A. Glowacki, Physica C 233, 203 (1994). 18. Y. Yang, T. Hughes, C. Bedux, D. M. Spiller, Z. Yi and R. G. Scurlock, IEEE lkans. Appl. Supercond. 5, 701 (1995). 19. K.-H. Miiller, in Magnetic Susceptibility of Superconductors and Other Spin Systems (Edited by R. A. Hein et al.), pp. 129250. Plenum Press, New York (1991).
Pergamon
0964-1087195 $9.50 + 0.00
ENERGY
DISSIPATION IN HIGH TEMPERATURE SUPERCONDUCTORS
M. CISZEK,? Interdisciplinary
A. M. CAMPBELL, Research
S. P ASHWORTH
Centre in Superconductivity, University Cambridge CB3 OHE, U.K.
CERAMIC
and B. A. GLOWACKQ of Cambridge,
Madingley
Road,
Abstract-Losses in high-T, superconductors are of vital importance in practical applications. The mechanisms are similar to those in low-T, materials but there are a number of complications. In this paper the loss mechanisms in different regimes are summarized and simple approximate expressions obtained. Losses due to the transport current are analysed with the theory of Norris for elliptical conductors. This predicts different loss voltages according to where the voltage contacts are placed on the conductor which has been confirmed experimentally. This poses difficulties in turning experimental data into reliable loss measurements, but a series of experiments with different contact layouts and phase setting techniques is found to give consistent results, in good agreement with theory.
1. INTRODUCTION
The practical application of superconductors, in, for example, power transmission cables, transformers, motors and other devices working under alternating current conditions, are limited by the magnitude of the electric currents which can be transported with a very low level of energy loss. The energy dissipation in type-II superconductors is connected to the intrinsic properties of these materials. Among them the most important are critical current density (pinning force), surface energy barriers for entry and exit of the magnetic flux, flux flow or flux creep, and the geometry. We also need to know the effect of combined fields and currents. Reviews of the losses in low-T, materials are given in, for example, Refs 1 and 2, and the same techniques can be applied to high-T, materials. Extra complications arise due to granularity and the presence of weak spots in most real conductors. D.C. measurements with a transport current show a measurable voltage near to critical current I,. This d.c. “resistance” is generated by steady state flux motion and contributes to losses with a.c. currents and magnetic fields, in addition to those losses arising from the changing flux within the superconductor. In high quality wires for transport currents less than 0.91, the “d.c.” component is negligible and the energy loss per cycle is frequency independent [3]. It is convenient to divide applications into two regimes: small field penetrations and large field penetrations. The division between these two regimes is the external field which just fills the cross-section with the critical current density. Two electrical a.c. loss measurement techniques, inductive and transport current, are common. In the first method the superconductor is placed in a varying external magnetic field and the currents induced in the sample (electric fields) are measured by means of a voltage pick-up coil wrapped around the specimen (contactless a.c. magnetic methods). These methods also allow us to measure the magnetic flux distribution within the sample [4]. In the case of transport methods an a.c. current flows through a superconductor, and the voltage drop along the sample is a measure of resistance, and thus of a dissipated energy (self-field losses). Magnetic methods permit us to measure a.c. losses in the inter-granular matrix of the ceramics (inter-granuiar losses), as well as in the individual grains (intra-granular losses). The transport loss measurements describe only the inter-granular losses (in the net of weak-linked Josephson junctions). Due to the very complex structure and morphology of the ceramic materials, on both a micro and a macro scale, distribution of the transport current is not homogenous and the current “meander”, taking paths t
On leave from Institute for Low Temperature and Structure Research, Polish Academy $ Also at the Department of Materials Science and Metallurgy, University of Cambridge, APSUP 3.7/N-K 509
of Science, Wroclaw, Poland. Cambridge CB2 342, U.K.
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with the locally highest critical parameters. As shown in [5], the local critical current density which depends on the local microstructure can be several times higher than the average one. 2.
LOSSES
DUE
TO EXTERNAL
MAGNETIC
FIELDS
Type-II superconductors subjected to a time-varying external magnetic field exhibit losses, mainly due to irreversible movement of flux vortices. Local dissipation per unit volume is generally expressed as J, x E, where J, is the critical current density and E = B x v is the electric field with the vortex velocity v. For small amplitudes bc of the external field, below the full penetration field b, (defined as the external applied field which just fills the superconductor cross-section with the critical current density) the loss is proportional to the surface area of the sample and varies as the cube of the amplitude. However an easier and more important regime is if the magnetic field penetrates to the centre of the superconductor (fl> 1, where /3 = b,/b,). In this case, if self fields are small compared with the external field, the rate of the change of flux density is everywhere the same as that at the surface. For a rectangular superconductor of width 2a and thickness d (see Fig. 1) the penetration of a field parallel to the long face (a) occurs at b, = paJ,d/2. The full penetration of a field directed perpendicular to the width of the sample (along d in Fig. 1) occurs at a field b, [6]: 6r = (poJcd/n)(z arc cot(z) + 0.5 ln(1 + z?)),
(1) where z = 2a/d. Since the field does not vary significantly across the sample we can assume constant J, and since the applied field is not significantly distorted we can assume that straight flux lines move towards the electric centre (Fig. 1). This defines the electric field, which rises linearly with x from zero at the electric centre, which is determined by symmetry. The electric field at a distance x from the centre of the slab is x dB/dt, so the local dissipation is J&E?/dt. Integrating this over the cross-section and a complete cycle of an a.c. signal with peak amplitude bO> b,, gives a loss per cycle Q in a perpendicular field bs: Q = 2JCbOa (J/m3).
(2) The same result can be obtained from the area of the hysteresis loop in the magnetization curve. The following conclusions can be drawn in this regime. First, the loss per unit volume is proportional to the width of the superconductor perpendicular to the field. Second, it is proportional to the critical current density. Third, the loss per cycle is independent of frequency. The loss for superconductors of various shapes is given in Table 1. If we pass a d.c. current equal to the critical value through the sample, the Bean model [7] tells us that the electric centre moves to the edge of the sample as in Fig. l(b). The flux has to move twice as far so the loss is now 4J,b,,a. In fact it can be shown that this expression is exact for any
-
E-O
E=O Fig. 1. Penetration of the magnetic flux lines into a rectangular superconductor. (a) Without d.c. transport current. The flux lines are not distorted and move at velocity proportional to x. (b) With d.c. transport current equal to Z, (the electric centre is now at the edge).
Energy dissipation in high-T, ceramic superconductors Table 1. Magnetic
losses per cycle for various
511
sample geometries
Sample geometry
B
PoJ,R
8>1
shape of the cross-section
[8] if a is taken as the maximum value of x which is inside the sample. This brings out the fact that the loss is rather independent of sample shape but depends mainly on the width perpendicular to the applied field. We can apply the same reasoning to find the loss in a silver sheath around a ceramic since the electric field will be the same as in the superconductor at the same value of x. The loss is obtained by averaging oE2 and is (7r/3)(~~&&2~) J/m3 Ag. If we have an alternating current with amplitude Ia = iI,, the loss is multiplied by a factor (1 + 3i2/8). The effect of the electric field in the superconductor due to the transport current has been ignored since it will be much smaller. For the result to be valid we must avoid the skin effect regime. This means that the area of silver must be small compared with 6’, where 6 is the skin depth of silver. If this does not hold the superconductor will be shielded from the field by eddy currents and can be ignored. If the condition is met we can then write the ratio R of the silver loss to the superconductor loss as: R = (7V3)(odl~2)(b,l&,),
(3)
where b, is the parallel penetration field [6]. At a frequency of 50 Hz the skin depth of silver at 77 K is about 3 mm and for our tapes ad/d2 is about 0.1. This means that the silver losses will begin to dominate the hysteresis losses at an amplitude of about 60 mT which is considerably smaller than the fields in transformers. These losses will dominate in magnets and transformers in a.c. conditions and the approximations used are rather accurate, although it will be necessary to use the local value of J, at each part of the winding.
3. TRANSPORT
CURRENT
LOSSES
Losses in cables are due to transport currents, which can never involve large penetrations since, when the critical state reaches the centre, we have reached the critical current. For low currents we can often use the expression for the surface loss in an applied field, inserting the local field generated by the current for an applied field. This will give an order of magnitude estimate even at the critical current. However the most important shapes of conductor have analytic solutions. The round wire problem was in fact the first application of the Bean model [8]. Solutions for an ellipse, a thin strip, asperities, and a gap between strips were obtained by Norris [lo] in a very comprehensive paper which applied conformal transformations to the critical state model for the first time. The elliptical wire is the easiest to deal with. The critical state penetrates as a series of concentric ellipses and the rather surprising result is that the loss per unit volume is independent of the aspect ratio. This is in marked contrast to the loss in an oscillating field which tends to infinity as the thickness goes to zero. It means that we can use the loss of a round wire for a wire of
M. CISZEK et al.
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any aspect ratio, and that the loss is not very dependent on the wire shape. According to Ref. 10, for a circular or elliptical wire, self-field losses are given by: Q, =
and for a thin superconducting Q,
!$ [(I _ i)ln(l
-
i) +
v]
strip:
=&!$[(1 -
i)ln(l - i) + (1 + i)ln(l + i) - i*],
where i = Is/Z,. At the critical current the loss per cycle per unit length for an elliptical wire with a critical current Z, is 0.16 & and that for a thin rectangular film is 0.12 Z& so that the difference is not very great. However at low currents the behaviour is rather different. The ellipse follows the usual 1, dependence in common with most hysteretic systems which trace out a Rayleigh loop at low amplitudes. However in the film the losses are proportional to 1:. The losses depend rapidly on the transport current value. At 0.51, the loss in an ellipse is reduced by a factor of 16 compared with Z, and for a film it is 40. If losses are a significant problem there are clearly advantages to be gained by making a cable large enough to be operated well below its critical current. As in the magnetic case an alternating current in the superconductor is associated with an electric field at the surface which for thin cladding will cause an electric field in the silver of the same magnitude. Provided we are not in the skin effect regime we can assume that the magnetic field is that due to the superconductor alone and calculate the electric field from it. The expressions given in Ref. 10 for the elliptical conductor can be used to find the flux inside any point in the silver cladding and the electric field calculated from the rate of change of flux. This involves some rather heavy algebra but we can get a good estimate as follows. We divide the loss by the current to get the in-phase voltage. We then apply this voltage to the silver cladding and find the loss from Ohm’s law. At the critical current the loss for an elliptical wire is (o,&/(4rr)2) W/m so the rms electric field is o&/(2n2) V/m. The silver loss is therefore a(oZ.&/(27r2)) W/ m3 of Ag. The ratio of silver loss to superconductor loss is (2/7~~)(-4s/~~), w h ere A, is the cross-section area of silver, so that by avoiding the skin depth regime we automatically ensure that the silver loss is less than the superconductor loss. This underestimates the loss since the silver loss depends on the rms voltage, rather than the in-phase component. For planar geometry, which can be solved analytically, the error is a factor of four. 4.
COMBINED
FIELDS
AND
CURRENTS
Analytic solutions have been found for the case of an applied current and transverse field in thin films [ll, 121. There are a number of different cases, the one illustrated in Fig. 2 is where the applied field is less than the self field so that the effect is an asymmetry of the critical state on the two sides. Losses in this situation were not quoted but since one side gets smaller while the other gets bigger we would expect the effect of the external field to be second order in be/b,.
Fig. 2. (a) A field applied normal to a film increases the penetration on one side and decreases it on the other. (b) A parallel field does not affect edge penetration.
Energy dissipation
Fig. 3. A schematic representation
in high-T, ceramic superconductors
513
of the arrangement of eight strips around a former to form a cable.
Of more practical importance, since it is directly relevant to cables, is the effect of a field parallel to the film surface. Experimental results have shown that the longitudinal loss voltage in these circumstances is the same whether the applied field is d.c. or a.c. with the same peak amplitude [ 131. This means that the main effect of the field is the reduction in J, due to the increased total field, which is a reasonable result. A parallel field produces a low loss on its own and, if it is less than the parallel penetration field, adds little to the penetration at the edges, where the main loss due to the transport current is taking place. Therefore if J, were to remain constant the application of a parallel field should have little effect on the losses and there is little difference between an a.c. and d.c. field in this geometry.
5. CABLES
The main application of these results is to the calculation of a.c. losses in cables. These consist of tapes mounted on the outside of a cylindrical former as in Fig. 3. Since the critical current is reached when B reaches the penetration field these cables are never in the high penetration regime (unless there are several layers). The ideal geometry would be a single uniform layer of superconductor, which was how low-T, power cables were designed. If the superconducting layer is 2d thick, and the radius r, the loss can be written (2/3n)&,Z$3(d/r)). This is a lower limit to the loss. However mechanical constraints will require a significant separation. According to Ref. 10, the edge losses at a gap in a sheet go as the square of the gap size, and by the time the gap width is equal to the strip width they will be effectively behaving as independent strips. Hence a better expression for the loss is (Z~~Z,2/27r)(g/a)~,where g is half the gap width up to a maximum of a. The only effect of the radius on this expression is that a larger radius will reduce the surface field and so increase Z,. If several layers of tapes are used it will be important to decouple them with an insulating layer between each conducting layer. The direct measurement of losses from the voltage along a cable poses even more problems than that for a single conductor and a comparison of electrical measurements with thermal ones is needed.
6.
FILAMENTARY
MATERIAL
Multifilamentary high-T, materials are now becoming available, and are proving to have unexpected benefits in terms of resistance to mechanical deformation. However without twisting the wire there is no reduction in a.c. losses and there may be an increase [ 141. Figure 4 shows the field lines for an applied field and transport current for a monolithic and multifilamentary material. The flux lines must cross the superconductor and the normal material and for the case of the multifilamentary material we can just average the properties over the cross-section. It is clear that the wider the area over which we spread the superconductor, the more flux must cross the current and the higher the loss. The total loss of the conductor is proportional to the radius of the circle containing the filaments and subdivision within this area does not affect the losses, but the monofilament has a smaller radius and therefore has the lowest loss. This conclusion is based on the assumption that the flux moves across the filaments. Flux can reach the centre of the conductor by two routes. It can cut the superconducting filaments and move in from the edge, or it can diffuse between the filaments from the ends of the wire (Fig. 5). In the
M. CISZEK et al.
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Fig. 4. If we spread out the superconductor
more flux is included and losses increase for both self and external fields.
Fig. 5. Flux can diffuse from the ends or cut through the sides.
latter case the filaments behave independently and the loss is proportional to the filament diameter rather than the conductor diameter. This clearly leads to a large reduction in losses. However the diffusion time constant for a kilometer of wire is about 20 yr, so measures must be taken to provide a shorter diffision path. This is done by twisting the wire so that flux rings can be nucleated at the filament crossovers and expand to the mid points between crossovers. The effective diffusion length is now the twist pitch. For twisting to be effective the pitch must be less than the skin depth, which is about 3 mm in pure silver at 77 IS and 50 Hz and a little longer for silver alloys. Making such a conductor is technically difficult, and in any case is only useful for external fields, it has no effect on self field losses, which require transp,osed filaments if they are to be decoupled. A twisted Ag/Bi-2223 conductor with critical current density of the order of 1.3 x 10 A/cm2 and twist pitches of 3.6 and 3.7 mm has been lately reported [ 151. It would appear that twisting high-T, conductors is not a high priority and the use of multifilamentary materials in cables must be justified on factors other than a.c. losses. However in magnets and transformers twisting will greatly reduce losses.
7.
LOSS
MEASUREMENTS
It is common to measure transport current losses by means of voltage taps on the conductor but it has recently been pointed out that this voltage is very dependent on the position of the contacts [16-181, and care must be taken to obtain a true measurement of the loss. The meaning of a
Energy dissipation
515
in high-T, ceramic superconductors
Fig. 6. A schematic representation of the measuring circuit containing an electrostatic voltmeter. The voltage contacts, on the edge of a strip in this example, form a loop which links with the magnetic field generated by the transport current.
voltage measurement is illustrated by reference to Fig. 6. Although the results are obviously independent of the nature of the meter the argument is clearest if we imagine an electrostatic voltmeter where the indicated voltage is the electric field E, between the plates. We now apply Faraday’s law of induction to a loop consisting of unit length along the centre of the wire, where the field is EC, the radial sections to the voltage leads themselves. The electric field E is zero along these sections. Then: E,, -EC = d(Oi + @&dt
or
E, = d(Q, + $)/‘dt
+ EC-
(6)
Hence what we measure on the voltmeter is the electric field along the centre line plus the flux change enclosed between this line and the leads to the meter, including the section within the superconductor, (I+ If we take our line integral along the surface instead of the centre line we see that an equivalent statement is that we measure the surface field plus the flux change between the surface and the meter, @i. If we are below 1, there is no electric field along the centre line so that the voltage is entirely attributable to the changing flux. If we have a d.c. current in a copper wire there is no change of flux and the voltage is the conventional resistive one. In general both components will contribute. The reason for the voltage differences is now clear. Figure 7(a) shows the flux around a strip carrying a current. There is far more flux cutting the edge contacts than the centre contacts and the voltage will be greater by a factor approximately equal to the aspect ratio of the conductor. To obtain a model-independent measurement of the loss, the loop containing the voltmeter should extend to a point well away from the conductor so that the lines of force are circular (Fig. 7b). The voltage will be the same for any point on the circumference so a single measurement will suffice, but there is the considerable disadvantage of a large inductive signal which will be difficult to balance out accurately.
Fig. 7. (a) The magnetic field lines round a strip carrying a current. (b) Only if the leads enclose a large loop is the voltage proportional to the loss.
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M. CISZEK et al. 8. EXPERIMENTAL
RESULTS
To obtain an unambiguous transport loss measurement we need to sample the flux over a distance much larger than the strip width. However for practical purposes quite a small loop will be adequate. We have measured the voltage from a series of contacts with different loop sizes as shown in Fig. 8. We used one of our typical Ag/BSCCO-2223 tape produced by powder-in-tube method. The ratio of ceramic to silver in the cross-section of the final tapes was about 1:4. The samples used were 27 mm long, with d.c. critical currents 1, in zero magnetic field at 77 K of 12.3 A (sample TS-12, overall cross-section 4 x 0.22 mm2, the thickness of the superconducting core in the central part of the tape was about 90 pm). The critical current (defined at 1 uV/cm criterion) corresponds to critical current densities of approximately 7 x lo3 A/cm2. The measurement set-up is shown in Fig. 9. For the a.c. transport measurements current leads were in the form of silver tapes of the same width as the sample which were directly soldered to the ends of the specimen. Measurements were carried out for frequencies from 40 to 300 Hz. A set of voltage leads were point soldered to the tape as shown schematically in Fig. 8. For magnetic loss measurements a single layer lo-turn pick-up coil. was wound along the sample. In this geometry the induced current flows along the tape as in the case of transport measurements. A liquid nitrogen cooled copper magnet was used to produce an a.c. magnetic field with a maximum amplitude of 140 mT. The lossy component U” (rms), of the voltage signal at the fundamental frequency, induced between the potential leads (or from the pick-up coil) was measured by using a dual phase lock-in amplifier. As the inductive component of the measured voltage in the case of large loops is more than two orders of magnitude higher than the loss signal, special precautions should be taken to set the phase and avoid common mode signals. A specially designed compensating (a) transport measurements Tl C
(b) magnetic measurements
Fig. 8. Sample arrangements for transport and magnetic loss measurements. (a) Voltage taps on the tape for measurement of transport losses. Potential leads E are on the edge of the tape, and C ones are on the central line of the tape. The distances ab, bc and cd are about 5 mm (similar dimensions have the central measurement loops). (b) Configuration of the sample and a.c. external magnetic field for inductive measurements. A single-layer pick-up coil is placed directly on to the surface of the tape in its central part.
Energy dissipation
in high-T, ceramic superconductors
517
J
-4 PC
5’ DVM A
Fig. 9. Block diagram of the a.c. transport current loss measurement set-up: S-sample tape, Ccompensating circuit, TR-isolating transformer (which also increases the current), CT-current transducer.
minitransformer was used as well as grounding only in one common point. Filters ensured that only the fundamental was measured; the harmonics do not contribute to the loss. All measurements were carried out at liquid nitrogen temperature. The sensitivity of our apparatus was better than 10v8 V and the voltage in each type of measurement was turned into an apparent loss using the usual expressions
where Qt is self-field loss, f is the frequency, ZmSis the transport current and VA is the loss voltage measured between potential leads (separation r>. In turn, the magnetic loss Qm, per surface area, per cycle, was calculated according to the equation
(8)
10s
P
1
/l&2.3
Al
10 4, (4
Fig. 10. The dependence of the loss voltage U,!&,on the amplitude of the a.c. transport current lo as measured for the different loops between the edge of the tape and potential leads. The inset shows the data for low currents in a linear scale.
518
M. CISZEK et al.
Fig. 11. Transport loss Qt per unit length and per cycle as a function of the current amplitude (i = lo/Z,) for different frequencies, as measured on the edge of the tape, and for larger potential wire loops. Solid and dashed lines are theoretical curves, for an ellipse and a strip, according to eqns (4) and (S), respectively.
where bc is the external field amplitude, N is the number of turns of the pick-up coil, S is the perimeter of the tape (about twice its length), and U,!&,is the loss voltage from the pick-up coil. Figure 10 shows the loss voltage as measured on the edge of the tape, for different loops between tape and the potential leads (see Fig. 8). There is a difference only for the smallest loop (taps no. 1, filled symbols). Due to the thickness of the silver this is already a loop about 0.3 mm thick. The data show that to measure correctly the loss voltage it is enough to increase the distance between sample surface and voltage wires to a few times the tape half width, in our case 5-6 mm separation was sufficient. Nearly the same data were obtained for taps placed in the central part of the sample, at least for currents greater than about 0.21,. The loss calculated using eqn (7) is shown in Fig. 11 and is in a very good agreement with the theoretical model for an elliptical or round wire. Compared to the theoretical prediction we have obtained very good agreement for the elliptical model (eqn 4). The strip model is a much less good approximation which is not surprising since most tape cross-sections show a tapered edge rather than a rectangular one. The loss varies as - Zo,4 in the range up to about 0.71,.
9.
GRANULARITY
At very low penetrations the grains can be treated as perfectly diamagnetic (Fig. 12). We can treat the system as a composite and average fields over many grains. B is the average flux density which, if J, were zero, would be less than an applied parallel field by something like the volume fraction of normal material. We can define a local H as the external field in equilibrium with the local value of B, just as we do in conventional superconductors, and a relative permeability, pr as B/p&t The relative permeability pr is equal to the volume fraction of normal material if the grains are long cylinders parallel to the field but in general the exact relationship will depend on the morphology and pL,needs to be measured experimentally. We have found a typical value of about 0.3. High-T, materials consist of grains with a high J,, the intragrain J,(intra), separated by weak links with a much weaker intergrain J,(inter). The detailed response of such a system to an alternating field is complex and detailed calculations will be found in Ref. 19. We need to introduce a third amplitude regime in which the grains as well as the weak links between them are penetrated by the external field. The same expressions apply to the grains as to the complete wire, although interactions between closely packed grains will make the results less accurate. We must add a loss of approximately 2BsJ,(intra) where s is the grain size. This result is not affected by a
Energy dissipation
in high-T, ceramic superconductors
519
Fig. 12. A composite of diamagnetic grains and weak links. Right side picture shows the penetration of the magnetic field Hand the magnetic induction B.
transport current, but since the only way of measuring Jc(intra) is from the magnetic hysteresis, we might as well use the measured magnetization curve to predict the loss in a.c. conditions. The fields which result in a cylinder parallel to an applied field are shown in Fig. 12 and the transport current is dH/d.x. Since for a given J, the value of B is lower by a factor pr compared with a homogeneous material, the effect of the diamagnetic grains is to reduce the losses by this factor. Clearly the assumption of perfect diamagnetism will not always be valid and there will be a surface loss on each grain. The importance of this will depend in detail on the shape of the grains, their size and the relative magnitudes of J,(inter) and J,(intra). This is something on which we need more information, but the good agreement of Fig. 11 suggests that, at least for transport currents, the grain surface losses can be ignored. The dependence of the magnetic a.c. losses Q,,, per unit surface area and per cycle, on the amplitude of the external magnetic field ba, for three different frequencies, is shown in Fig. 13. No remarkable differences are present for different frequencies of the field, which means that losses in the investigated sample are hysteretic in this frequency range. For the low bc the losses increase lo-’
F
a 2
G
1o-2 1o-a
E 5 v
lo-4
d
1o-s
% 9
10-a
.::
10-7
E B I
10-a
10-Q 3e-5
-
-2
3 2e-5 ‘2 le-5
-
Oe+O
b, (mT) Fig. 13. Magnetic loss Q,,, data vs external magnetic field amplitude be for Ag/BSCCO-2223 tape, for different frequencies. Only one peak at ba = b, = 5 mT in p” (b,) function is observed.
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with a power close to four, whereas for higher magnetic fields this dependence is linear with bc. In the lower part of Fig. 13 the lossy component of the a.c. complex permeability is shown. Contrary to typical granular materials, e.g. sintered YBCO ceramics, we observe in our tape only one pronounced peak, at least in the available magnet amplitude range to 140 mT. The imaginary part of the complex permeability $’ is related to the losses via Q, = $‘7rb@/2~,,, where d is the thickness of the sample (d << length and width of the plate) and Qm is expressed as loss per surface unit. In the simplest case of the Bean model, for plate geometry (see Table l), taking into account the relation between loss and magnetic permeability $‘, the maximum in I” dependence takes place at b, = b, = 2/3pOJcd. For the peak which occurs at 5 mT the value of the critical current density is determined as about 6.6 x lo3 A/cm’, the value which differs only by 5% from that measured by four-point transport method. 10.
SUMMARY
In spite of the complexity of high-T, tapes we have an understanding of the losses in single strands which is probably adequate for the design of demonstration samples. The most accurate results are in the high penetration limit where the internal field is equal to the external field and the electric field is easily calculated. These losses will dominate the loss in a coil and will also probably be dominated by silver losses since these go as B* while the hysteresis losses go as B. Cables work in the low field limit where the loss is due to a transport current. The theory of Norris for elliptical conductors can be used here for the hysteresis loss of single strands. This loss is the same as that for round wire of the same critical current. Complete cables have a combination of pure transport current and an applied field which is a less certain situation but the loss of single strands with the appropriate J, should be a reasonable approximation. The determination of losses due to transport currents using the voltage generated gives results which depend strongly on the configuration of the measurement circuit. In particular the measured voltage varies by up to three orders of magnitude depending on the positioning of voltage contacts and the size of the “loop” linked with the changing magnetic field. Theoretically these measurement loops should extend to infinity to return the correct voltage, in practice experiments indicate that the results tend to a limit, in good agreement with theory, if the loops extend out to a few times the tape width. REFERENCES 1. M. N. Wilson, Superconducting Magnets, pp. 159-199. Oxford University Press, London (1989). 2. J. R. Clem, in Magnetic Susceptibility of Superconductors and Other Spin Systems (Edited by R. A. Hein et al.), pp. 177211. Plenum Press, New York (1991). 3. S. P. Ashworth, Physica C 229, 355 (1994). 4. A. M. Campbell, 1 Phys. C 2, 1492 (1969). 5. D. C. Larbalastier, X. Y. Cai, Y. Feng, H. Edelman, A. Umezawa, G. N. Riley Jr and W. L. Carter, Physica C 221,299 (1994). 6. A. M. Campbell, IEEE Trans. Appl. Supercond. 5, 682 (1995). 7. C. P. Bean, Rev Mod. Phys. 36, 31 (1964). 8. A. M. Campbell, Cryogenics 22, 3 (1978). 9. R. A. Kamper, Phys. Lett. 2, 290 (1962). 10. W. T. Norris, 1 Phys. D 3, 489 (1970). 11. E. H. Brandt and M. Indenbom, Phys. Rev. B. 48, 12893 (1993). 12. E. Zeldov, J. R. Clem, M. McElfresh and M. Darwin, Phys. Rev B 49, 9802 (1994). 13. M. Ciszek, B. A. Glowacki, S. P Ashworth, A. M. Campbell and J. E. Evetts, IEEE Trans. Appl. Supercond. 5, 709 (1995). 14. K. Kwasnitza, St. Clerc, Physica C 233, 423 (1994). 15. C. J. Christopherson and G. N. Riley Jr, Appl. Phys. Lett. 66, 2277 (1995). 16. T. Fukunaga, S. Maruyama and A. Oota, Advances in Superconductivity (Edited by T. Fujita and Y. Shiohara), p. 633. Springer, Tokyo (1994). 17. M. Ciszek, A. M. Campbell and B. A. Glowacki, Physica C 233, 203 (1994). 18. Y. Yang, T. Hughes, C. Bedux, D. M. Spiller, Z. Yi and R. G. Scurlock, IEEE lkans. Appl. Supercond. 5, 701 (1995). 19. K.-H. Miiller, in Magnetic Susceptibility of Superconductors and Other Spin Systems (Edited by R. A. Hein et al.), pp. 129250. Plenum Press, New York (1991).