Excerpt from the Proceedings of the COMSOL Conference 2009 BangaloreSimulation of AC Loss in High Temperature SuperconductingCable using COMSOL MultiphysicsG. Konar * , R. K. Mandal, A. Chakrabarty, J. Das, N. ChakrabortyPower Engineering Department, Jadavpur UniversityBlock-LB, Plot-8, Saltlake, Kolkatafirstname.lastname@example.orgAbstract: In this paper simulation of thealternating current loss (AC loss) in a hightemperature superconducting tape and a HTScable having 4 layers of polygon conductors usingCOMSOL Multiphysics software has beenreported. The simulation results for the singlerectangular tape are verified with the Norrisequation for a HTS tape. The effect of increasednumber of layers on AC loss is also observedusing the simulation. All the simulations are doneusing two dimensional cross-section of HTS tapeand the multiple numbers of tapes are assembledto form the polygon conductor. Using A-Vformulation, the dynamic electromagneticanalysis has been done. For this, two dimensionalPDE based module of COMSOL Multiphysicshas been used with two dimensional geometrywith proper Dirichlet and Neumann boundaryconditions.Keywords: HTS tape, AC loss, polygonconductor cable, YBCO tape.carrying capacity HTS cables extend theoperating life of the existing high load corridors.There are no soil heating & no oil leak leading toenvironment friendly power transmission anddistribution. The most challenging task fordesigning an appropriate HTS cable is to calculateAC loss due to transport current (self field) and /or applied field, because it finally leads to thermalrunway of the cable. In this study, a single phaseHTS cable (figure 1) has been used. This HTScable has of 4 layers of polygon HTS conductorsin sub-domain C6, each assembled by HTS tapeshaving proper aspect ratio. For simulation of ACloss we are considering only this sub-domain. Thenumerical simulations for current distribution andthe AC loss are carried out for an isolated tape ofrectangular cross section and the results areverified by Norris’ theory . The simulation ofAC loss characteristics for the polygonconductors in 4 layers is also done and theinfluence of the number of layers on AC loss isobserved. All the simulations are performed usingthe COMSOL Multiphysics software.1. IntroductionUse of superconductors rather than conventionalconductors is a revolution of 21 st century. In orderto provide higher power capacity, higher currentdensity and lower impedance, HTS cables offersignificant benefits over conventional overheadand underground power cables. These cables areable to carry as much as 10 times more power &150 times more current than conventional cablewhich make them ideal solution for networkupgrades & urban profit projects. HTS cables alsoprovide a good solution to grid congestion. Hightemperature superconductors have transitiontemperature above Liquid Nitrogen temperatureof 77K. The best known high temperaturesuperconductors presently used for power cablesare Bismuth Strontium Calcium Copper Oxide(BSCCO) & Yttrium Barium Copper Oxide(YBCO). Other than higher current & powerFigure 1 Schematic diagram of single phase HTScable. For simulation of AC loss sub-domain C5 hasbeen used only.
2. Mathematical modelingIn this study the polygon conductors comprised ofmultiple HTS tapes are carrying the AC transportcurrents. They are also subjected to the ACmagnetic field produced by the transport currentsin the neighboring tapes. Each tape of the polygonconductor is multifilamentary and considered asbulk superconductor. Figure 2 illustrates theschematic diagram of an isolated tape and figure 3shows the schematic diagram of ¼th symmetry ofthe 4-layer polygon conductor cable. The detailspecifications of the cable are shown in table 1.For a single HTS tape (figure 2) an orthogonal coordinatesystem (x, y, z) has been opted where thez-axis coincides with the axis of the tape and thetransport current flows along the z-axis i.e. alongthe length of the tape. Considering a uniformelectromagnetic field in z direction, the electricfield and the current density have only the zcomponent. The basic electromagnetic equationfor the dynamic analysis is expressed as1 ∂A∇ ∇A+ σ = −σ∇V(1)μ 0∂twhere A is the magnetic vector potential due toself field and the external field produced due tothe currents through the neighboring tapes. ∇ Vis the voltage gradient. The electric field isexpressed as∂AE = − − ∇V(2)∂tThe E-j relationship as proposed by Rhyner  isexpressed asn−1⎛ J ⎞ JE = E ⎜ ⎟0(3)J⎝ c ⎠ JcHere n is the power exponent and the J c is thecritical current density at E 0 . Both of them aredependent on local magnetic field. E 0 has thevalue of 10 -4 V/m which is universally accepted. From equation (3) the non-linear conductivityσ is expressed as follows:1−nJσ = E n(4)Ec1ncBy numerically solving the equations (1) – (3)and using (4), the distribution of current densityand the electric field across the isolated tape crosssection can be evaluated. Then the AC losses perunit length are computed by equation (5) for anisolated tape. Then the AC loss is calculated forthe 4 layer polygon conductors.T1P = dtT∫ ∫0SJ.EdS3. Use of COMSOL Multiphysics(5)In the model navigator of COMSOL Multiphysicsthe PDE based mode has been chosen first. Themagnetic vector potential (A) is considered asdependent variable. A rectangular HTS strip isdrawn in 2D plane using draw menu. Properconstants for simulations (E 0 , J c , n, f etc) areentered in the option menu. Then the equationsfor current density, electric field, AC loss etc arespecified as scalar expressions in the same optionmenu. The expression of non-linear conductivity(4) is written in the sub-domain expression field.The expressions of A at the boundaries,determined from∇× A = B , are specified in theboundary expression. Then in the physics settingmenu the proper value/expressions of differentcoefficients of PDE (1) are set. Boundaryconditions are also specified at differentboundaries as Dirichlet or Neumann boundarycondition. Then the shape function and the initialvalues of the variable A are set in the element editfield and in the Init field respectively. After thatchoosing the free mesh parameters from meshmenu, meshing is done with maximum elementsize of 1e-2 in the superconducting sub-domain.Then the solution is done by clicking the solvemenu and setting proper solver parameters. Thesimulation is done for a single time period. Thepost processing menu has been used to get surfaceplots A, J, Q, E etc and other plots. Following thesame procedure the simulation o f AC loss hasbeen done for HTS wire with 4 layers of polygonconductors.Figure 2 Schematic diagram of an isolated tape of apolygon conductor, w =3.5 mm and d =200μm.
LayerNo.Table 1 Specification of each layer of HTS polygonconductor cableRadius(mm)Tapewidth(mm)Tapethickness(μm)TapenumberGap(mm)1 28 3.5 200 28 2.82 24 3.5 200 28 1.93 20 3.5 200 28 14 16 3.5 200 28 0.14. Numerical results and discussionThe AC losses of an isolated tape are simulatedusing COMSOL multiphysics software fordifferent values of transport currents (self field)and under influence of applied field. Externalapplied fields are due to transport currents in theneighboring tapes. Figure 4 shows the AC lossesin the isolated tape for different RMS values oftransport currents normalized by critical currentdensity. The AC losses are calculated for anaspect ratio of 17.5. For higher value of aspectratio the tape cross-section becomes almost onedimensional. To get better simulation results, herewe have considered two dimensional tape crosssectionswith the above aspect ratio. Thesimulation results are verified with the theoreticalvalues obtained from Norris equation of AC lossfor rectangular strip. As shown in Figure 4, theAC loss for an isolated tape has beentapes. These simulated loss values obtained byCOMSOL multiphysics match well with thetheoretical values of AC loss. They are almostproportional to the I 4 where I is the rms current.Next the AC loss calculation for a single layerpolygon conductor is done for J/J c = 0.7. Thisvalue is obtained in two ways. Firstly, simplymultiplying the AC loss value of an isolated tapewith the number of tapes in an individual layer.Secondly, the loss value has been simulated forthe layer with 1/4 th symmetry using COMSOLmultiphysics. A single layer polygon conductorcontains 28 numbers of tapes. For the first casethe obtained AC loss value is 3.64 W/m. The latermethod gives a lower value of AC loss (2.98W/m) which may be due to partial cancellation ofself fields produced by two adjacent tapes of thepolygon conductor. Next the numbers of layershave been increased to four . It is found thatthe values of AC losses for different currentdensities are more for increased numbers oflayers. For all these above calculations the n-value i.e. is the power index has been chosen tobe 50 which suits better for the YBCO tapes .10 2 Ac loss in isolated tapeTheoretical values of AC loss10 1Ac loss in 4 layer polygon conductor cable per phase10 010 -110 -210 -310 -410 -1 10 0Figure 3 Schematic diagram of 1/4 th symmetry of 4-layer HTS polygon conductor cable with suitable gapbetween two tapes; x and y axis are drawn in mm (1unit = 8mm).increased with the increased value of transportcurrent through itself and through the neighboringFigure 4 AC loss values for an isolated tape are plottedand compared with the theoretical values from Norrisequation. Also the plot of simulated AC loss values ofcable with 4 layers of HTS conductors per phase hasbeen shown.5. ConclusionThe AC loss simulation of assembled conductorof multiple thin HTS tapes in polygonarrangement has been done using COMSOLmultiphysics. The simulation results are verified
with the theoretical values. So the application ofthis software for AC loss calculation for largenumber of thin tapes in a polygon arrangementwith multiple numbers of layers is justified andhelps in the numerical analysis of loss withdifferent geometrical configurations. It could bebetter, if we can include some iteration forcalculating the nonlinear conductivity values inthe superconducting sub-domain using thissoftware, which might be given better matchingof AC loss values with the theoretical values.6. References W. T. Norris, Calculation of hysteresis lossesin hard superconductors carrying ac; Isolatedconductors and edges of thin sheets, J. Phys. D, 3,489-507 (1970). J. Rhyner, Vector potential theory of AClosses in superconductors, Physica C 377, 56- 66(2002). H.Noji, AC loss of a high-Tc superconductingpower-cable conductor, Supercond. Sci.Technol.10, 552 (1997). Noji H, Haji K and Hamada T Alternatingcurrent loss calculation in a high-Tcsuperconducting transmission cable consideringthe magnetic field distribution, Supercond. Sci.Technol.16, 14-18 (2003). N Chakraborty, A V Volkozub and A DCaplin, Bolometric measurement of AC loss inHTS tapes: a novel approach of microwattsensitivity Supercond. Sci. Technol. 13, 1062-1066(2000). S Chakraborty, N Chakraborty, A finitevolume based numerical simulation ofAnisotropic heat transfer behavior in HTS tapesdue to AC losses Proc. AppliedSuperconductivity 1931-1938 (2003) S. Fukui, R. Kojima, J. Ogawa, M.Yamaguchi, T. Sato, and O. Tsukamoto,“Numerical analysis of AC loss characteristics ofcable conductor assembled by HTS tapes inpolygonal arrangement,” IEEE Trans. AppliedSuperconductivity, 16(2), 143–146 (2006) www.comsol.com