3-D Numerical Simulations of Twisted Stacked Tape Cables
Philipp A. C. Krüger, Victor M. R. Zermeño, Makoto Takayasu, Francesco Grilli
11 Philipp A. C. Kr¨uger, Victor M. R. Zerme˜no, Makoto Takayasu, and Francesco Grilli
Abstract —Different magnet applications require compact highcurrent cables. Among the proposed solutions, the TwistedStacked Tape Cable (TSTC) is easy to manufacture and has veryhigh tape length usage efficiency. In this kind of cables the tapesare closely packed, so that their electromagnetic interaction isvery strong and determines the overall performance of the cable.Numerical models are necessary tools to precisely evaluate thisinteraction and to predict the cable’s behavior, e.g. in terms ofeffective critical current and magnetization currents. For thispurpose, we developed a fully three-dimensional model of aTSTC, which not only takes into account the twisted geometryof these cables, but is also able to account for the contactresistances of the current terminations. The latter can haveprofound influence on the way the current is partitioned amongthe tapes, especially on short laboratory prototypes.In this paper, we first use the numerical model to computethe critical current and the magnetization AC loss of a twistedtape, showing the differences with the case of a straight tape.Then, we use it to calculate the current distribution in a TSTCcable, comparing the results with those experimentally obtainedon a cable composed of four straight stacked tapes. The resultsshow the ability of the model to simulate twisted conductors andconstitutes a first step toward the simulation of TSTC in high-field magnet applications. The presented modeling approach isnot restricted to the TSTC geometry, but may be used for anycable configuration with periodical translational symmetry.
Index Terms —3-D FEM, Twisted Stacked Tape Cables, AClosses.
I. I
NTRODUCTION D IFFERENT cable designs are currently being consideredfor fusion magnet applications using HTS coated con-ductors [1]. In all these designs, the tapes are very compactlyarranged, in a geometrical configuration involving either twist-ing, transposition or helical winding. Their electromagneticinteraction is very strong, and it is therefore desirable to havenumerical tools able to predict the performance of such cables,for example the calculation of the effective critical current, thecurrent distribution between the tapes, and their losses whensubjected to varying currents and fields. Such tools must beable to calculate the local components of the magnetic field:in this way, the typically anisotropic transport properties ofHTS tapes can be properly taken into account, as it has beendone by a variety of 2-D models developed by several authorsin the past years (see, for example, [2], [3], [4], [5], [6]).
P. A. C. Kr¨uger, V. Zerme˜no, and F. Grilli are with the Karlsruhe Instituteof Technology, Karlsruhe Germany. M. Takayasu is with the MassachusettsInstitute of Technology, Cambridge, MA, USA. Corresponding author’s e-mail: [email protected] work was partly supported by the Helmholtz Association (Grant VH-NG- 617).Manuscript received August 10, 2014.
In this paper, we developed a full 3-D model for twistedsuperconductors based on the popular H -formulation of eddycurrents problems implemented in the software package Com-sol [7], [8], which has been recently extended to 3-D [9],[10], [11]. In addition to considering twisted structures andincluding the field dependence of J c , the model is able tosimulate the contact resistances of the electrical terminations,while keeping the possibility of simulating only a periodic cell(e.g. one twist pitch) of the superconducting parts.We use this model to simulate the electromagnetic behaviorof a twisted tape and of a Twisted-Stacked-Tape-Cable in avariety of working conditions, emphasizing the effects of thetwisted geometry on the tape’s and cable’s performances. Wealso use it to study the current distribution between tapes ina straight stacked-tape cable manufactured at MIT [12], forwhich experimental data are available.In this work, we use critical current density values, J c ,typical for coated conductors at
77 K , also for the purpose ofvalidations against available experimental data. The appliedbackground fields are rather small, in the range of 5-
20 mT .The main goal of this paper is to show the feasibility offull 3-D electromagnetic calculations involving twisted coatedconductors and assemblies thereof. The paper can thereforebe seen as a first stepping stone toward the simulation ofspecific applications of these conductors (e.g. magnets forfusion or accelerator applications) characterized by muchhigher currents and background fields as well as by differentangular dependences of J c .II. S INGLE T WISTED T APE
First we simulated a single twisted coated conductor tape, wide, with a twist pitch of 40 cm. The superconductoris modeled with a power-law resistivity which includes thedependence of the critical current density on the componentsof the magnetic flux density parallel and perpendicular to theflat face of the tape [13], [14]: J c ( B (cid:107) , B ⊥ ) = J c (cid:104) (cid:113) ( kB (cid:107) ) + B ⊥ /B c (cid:105) b . (1)Here we considered values typical of the characteriza-tion of HTS coated conductors at
77 K : J c = . − , B c = . , k =0.275, b =0.6, power index n =21.In the finite-element (FE) model, the geometry is builtby first drawing the transversal cross-section of the tape( xy plane) and then by extruding it in several steps alongthe tape direction ( z direction). This allows fixing the meshof the cross-section and sweeping it along the direction of a r X i v : . [ c ond - m a t . s up r- c on ] M a y imefieldcurrent t H t I t MAX a b c
Fig. 1. Magnetic field and current ramps applied for the determination of thetape’s or cable’s critical current in the presence of a background field. Thecircles represent the instants when the current density distributions of Fig. 3are taken. − − − − − − − current I (A) e l ec t r i c fi e l d E ( V / m ) twisted slow ramptwisted fast rampstraight slow B k straight slow B ⊥ Fig. 2. E − I characteristic of a twisted tape with a background field,compared to those of an equivalent straight tape experiencing a purelyperpendicular or parallel background field. The curve obtained with “fast”field and current ramps is also shown (circle symbols). extrusion [15]. The local parallel and perpendicular fieldcomponents appearing in (1) need to be locally evaluated atevery point of the tape geometry. For the general case ofsimultaneous presence of transport current and external field, itis necessary to simulate one full twist pitch. Other situations(e.g. external field only) may allow reducing the simulatedlength to half or a quarter of the pitch length, A. E − I Characteristics with Background Field
For our first tests, we calculated the voltage-current char-acteristics of a twisted tape in a uniform magnetic field of
20 mT applied along the y -direction. For this purpose, theexternal field is increased from zero to the maximum valuewith a linear ramp of t H = ; then it is left constant, and,starting at t I =
10 s , the current is ramped up to
100 A witha rate of
10 A s − ). The applied field and current rampsare schematically shown in Fig. 1 and the resulting E − I characteristic is given by the continuous line in Fig. 2 (“twisted Fig. 3. Distribution of the current density component J z (normalized to thelocal J c ) at different time instants of the field and current ramps (see circlesin Fig. 1). Some portions of the twisted tape are removed, in order to makethe internal current distribution at the different places visible. For readabilitypurposes, a thick ( µ m ) tape is shown here, but similar distributions werefound for a tape as thin as µ m . slow ramp” in the legend), where E is the volumetric averageof E z . For the twisted tape, the critical current (determinedwhen the electric field reaches × − V m − ) is
66 A . Inthe case of a straight tape (with the same J c ( B ) properties)subjected to a purely perpendicular or parallel field, the criticalcurrent is 63.5 and . , respectively, as a consequence of astronger or weaker reduction of J c ( B ) according to (1). Thetwisted tape experiences both parallel and perpendicular localcomponents, so its critical current is between the two cases ofthe straight tape in perpendicular and parallel applied field.Besides the effects related to the presence of a twistedgeometry discussed just above, a closer look at the currentdensity distribution inside the tape at different instants dur-ing the field and current ramps reveals a general importantconsequence of the use of a power-law (PL) in transientelectromagnetic models – not particularly related to twistedgeometries: current relaxation. During the initial field ramp( t < t H ), the current density has a pattern very similar tothat described by the critical state model (CSM): at the edgesof the tape there are regions carrying the maximum possiblecurrent (in the framework of the CSM, J/J c = 1 , here thecurrent density is slightly over-critical because of the PL) –see Fig. 3(a). As the field is held constant ( t H < t < t I ),the current density relaxes to sub-critical values (Fig. 3(b)).Finally, as the new current kicks in ( t > t I ), the regions with c (cid:13) igher J c values change: note that the current distributionis no longer symmetric because of the superposition of thetransport current to the pre-existing magnetization currents(Fig. 3(c)). Whether this relaxation behavior is realistic or justa consequence of the use of the PL relationship is not clear.In [16], the authors compare models based on the PL andon the CSM to experiments for the calculation of AC ripplelosses and argue that the behavior observed in experiments isin better agreement with the predictions based on the CSM, asopposed to the PL model. However, they also say that furtherinvestigations are necessary on the topic. With the PL, a criticalstate-like behavior can be obtained by quickly ramping currentand fields (e.g. with t H = t I = .
01 s and t MAX = .
02 s – seeFig. 1 for reference). In Fig. 1, the circle symbols (“twistedfast ramp” in the legend) represent the E − I characteristicobtained with such fast ramp. In that case, the obtained E − I curve is different from the one that can be obtained in realexperiments (compare also with section IV and with the resultsreported in the appendix of [14]), but the correct value of thecritical current can be obtained by extrapolating the upper-right part of the curve down to the critical electric field of × − V m − . The advantage of doing so is that ‘fast’ramps are much faster to simulate in transient electromagneticproblems. The reason is that when using the H -formulationand the PL relationship, the zero divergence of the magneticflux density is only imposed by means of the time derivativeterm in the resulting non-linear diffusion equation. This isexplained in detail in [17] and in section 2 of [18]. For lowfrequencies, this time derivative term becomes numericallyless significant. Overall, this increases the workload on thesolver which requires more iterations for reaching a solutionin accordance with the target tolerance values. B. Magnetization AC Losses
As a second step, we calculated the magnetization AC lossescaused by a background magnetic field oriented along the y direction. The field has a varying orientation with respect to thetwisted tape, whose behavior varies from ‘strip-like’ (when thefield is perpendicular to the flat face of the tape) to ‘slab-like’(when the field is parallel). This is clearly visible in Fig. 4:shown are the current density distribution (along z ) inducedby a field of
20 mT of amplitude applied in the y -directionand the corresponding power density, which is much higher inthe regions of the tape characterized by a strip-like behavior.The power density for different field amplitudes is displayedin Fig. 5: as for the critical currents (Fig. 2), the power densityof the twisted tape lie between that of a similar straight tapeexperiencing fully perpendicular and fully parallel magneticfield. The main difference with the critical current case dis-cussed above is that the AC loss dynamics are very differentin strip-like [19], [20] and slab-like [21] configurations, whichresults in power densities orders of magnitude apart, as shownin the figure. The losses of a straight tape in perpendicularand parallel field can be calculated with simple 2-D models.However, they constitute just two far limit values and cannotbe used for a realistic evaluation of the dissipation in a twistedtape. This is a clear example of the usefulness of a numericalmodel able to calculate the losses of twisted tapes. Fig. 4. Instantaneous power density in a twisted tape subjected to anoscillating (frequency
50 Hz ) magnetic field of
20 mT . The correspondingcurrent density distribution is shown in the inset. The figures refer to theinstant of the peak of the field. A C p o w e r d e n s i t y ( W / m ) twisted parallel perpendicular Fig. 5. AC power density for the case of a tape subjected to a backgroundfield (at
50 Hz ) of different amplitudes. Comparison between a twisted tapeand a straight tape experiencing only a perpendicular or parallel field.
III. T
WISTED S TACKED -T APE C ABLE
We considered a twisted stacked-tape cable composed offour tapes, subjected to a current ramp in background field.In order to construct the geometry and the mesh, we followedthe same procedure as in the case of the single twisted tape:the transversal cross section is first built in a 2-D workplane and successively extruded along the cable direction witha multi-step process. The current is imposed to the wholesuperconductor cross-section and let free to distribute in thedifferent tapes. Additionally, we imposed that the currententering each tape’s cross section is the same at the twoends of the simulated geometry (one twist pitch). The resultsof the current distribution between the tapes are shown inFig. 6. At the beginning, most of the current flows in themost external tapes. When they reach the current capacity, c (cid:13)
50 100 150 200 250 300020406080 total current in the cable (A) c u rr e n t i n i nd i v i du a l t a p e s ( A ) tape 1tape 2tape 3tape 412341, 4 2, 3 Fig. 6. Distribution of the current in the 4-tape TSTC with an exemplaryimage of the utilized mesh. c u rr e n t i n i nd i v i du a l t a p e s ( A ) tape 1tape 2tape 3tape 412341 23 4 Fig. 7. Measured (symbols) and calculated (lines) distribution of the currentin a stacked-tape cable composed of four tapes. The continuous and dashedlines represent the results of simulation with and without angular dependenceof J c , respectively. significant current starts to flow in the internal tapes as well.For very high total current, all the tapes carry almost thesame current: however, the innermost tapes carry a little morecurrent than the outermost ones, most probably because theyhave a slightly higher critical current as a consequence of thesmaller experienced magnetic field. Figure 6 also shows themesh for a portion of the TSTC.IV. I NFLUENCE OF C ONTACT R ESISTANCES
In addition to being able to simulate twisted tapes, ourmodel is able to take into account the contact resistance offeredby the electrical terminations through which the current is injected. This is done by placing the resistances in a simulationdomain disconnected from that of the tapes and by transferringthe current from a contact resistance to the correspondingtape by means of integral constraints. The advantage of thisapproach with respect to simulating the full geometry is that,for the simulation of the superconducting tapes, one can takefull advantage of the existing periodicity. So, for example, onecan still simulate just a twist pitch of superconducting cableand adjust the contact resistance to take into account the realcable length. Technical details for the implementation of thisapproach can be found in [15].Here we used the model to calculate the current distributionin a straight cable composed of four stacked tapes, for whichexperimental data are available [12]. The results are shown inFig. 7. The current distribution is different from that presentedin Fig. 6 in two points: 1) The initial distribution of the currentis governed by the contact resistance of each tape; 2) In thehigh current regime, the currents carried by the tapes aredifferent: in particular, tape number 4 carries substantially lesscurrent than the others because it has a much lower criticalcurrent than the others.One similarity with the data of Fig. 6 is that in some tapesthe current does not increase monotonically, but it shows aplateau or even a decrease. This is probably an effect ofthe generated self-field: as the current carried by the cableincreases, so does the generated self-field, which in turnsdecreases the maximum current a given tape can carry. Itis worth noting that this effect is not present if the J c ( B ) dependence of the superconductor is not taken into account(dashed lines in the figure).More details can be found in a recent publication ofours [22], where other techniques to take the presence ofcontact resistances into account are presented. The currentapproach, however, is the only one that allows simulatingcontact resistances and complex 3-D geometries (such asTSTC) within the same model.V. C ONCLUSION
In this paper we presented preliminary simulation resultsof individual and stacked twisted HTS tapes, both in DCand AC conditions. For the evaluation of the critical currentof the tapes or cables in background magnetic field, theissue on the type of field/current ramps to apply with a PLmodel is discussed. Fast ramps, although not describing thebehavior occurring in reality, can provide a fast calculationof the effective I c . The 3-D model offers the possibilityof including the presence of contact resistances, which cansignificantly influence the current distribution between thetapes. The simulation domain of the contact resistance isdisconnected from that of the superconductor devices, whichallows the simulation of translational geometries without theintroduction of end-effect perturbations. The model presentedhere has all the necessary features to handle simulation ofHTS devices employing twisted conductors with complexangular J c ( B ) dependence. In future work it will be used toinvestigate the electromagnetic behavior of TSTC conductorsexperiencing the conditions typical of large-scale applications,such as fusion or accelerator magnets. c (cid:13) EFERENCES[1] W. H. Fietz, C. Barth, S. Drotziger, W. Goldacker, R. Heller, S. I.Schlachter, and K.-P. Weiss, “Prospects of High Temperature Supercon-ductors for fusion magnets and power applications,”
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