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Simulation of AC Loss in Superconducting Cable using COMSOL ...

Excerpt from the Proceed**in**gs **of** the **COMSOL** Conference 2009 Bangalore**Simulation** **of** **AC** **Loss** **in** High Temperature Superconduct**in**g**Cable** us**in**g **COMSOL** MultiphysicsG. Konar * , R. K. Mandal, A. Chakrabarty, J. Das, N. ChakrabortyPower Eng**in**eer**in**g Department, Jadavpur UniversityBlock-LB, Plot-8, Saltlake, Kolkata-700098*gargi_konar@yahoo.co.**in**Abstract: In this paper simulation **of** thealternat**in**g current loss (**AC** loss) **in** a hightemperature superconduct**in**g tape and a HTScable hav**in**g 4 layers **of** polygon conductors us**in**g**COMSOL** Multiphysics s**of**tware has beenreported. The simulation results for the s**in**glerectangular tape are verified with the Norrisequation for a HTS tape. The effect **of** **in**creasednumber **of** layers on **AC** loss is also observedus**in**g the simulation. All the simulations are doneus**in**g two dimensional cross-section **of** HTS tapeand the multiple numbers **of** tapes are assembledto form the polygon conductor. Us**in**g 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.carry**in**g capacity HTS cables extend theoperat**in**g life **of** the exist**in**g high load corridors.There are no soil heat**in**g & no oil leak lead**in**g toenvironment friendly power transmission anddistribution. The most challeng**in**g task fordesign**in**g an appropriate HTS cable is to calculate**AC** loss due to transport current (self field) and /or applied field, because it f**in**ally leads to thermalrunway **of** the cable. In this study, a s**in**gle phaseHTS cable (figure 1) has been used. This HTScable has **of** 4 layers **of** polygon HTS conductors**in** sub-doma**in** C6, each assembled by HTS tapeshav**in**g proper aspect ratio. For simulation **of** **AC**loss we are consider**in**g only this sub-doma**in**. Thenumerical simulations for current distribution andthe **AC** loss are carried out for an isolated tape **of**rectangular cross section and the results areverified by Norris’ theory [1]. The simulation **of****AC** loss characteristics for the polygonconductors **in** 4 layers is also done and the**in**fluence **of** the number **of** layers on **AC** loss isobserved. All the simulations are performed us**in**gthe **COMSOL** Multiphysics s**of**tware.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 **of**fersignificant 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 pr**of**it projects. HTS cables alsoprovide a good solution to grid congestion. Hightemperature superconductors have transitiontemperature above Liquid Nitrogen temperature**of** 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** s**in**gle phase HTScable. For simulation **of** **AC** loss sub-doma**in** C5 hasbeen used only.

2. Mathematical model**in**gIn this study the polygon conductors comprised **of**multiple HTS tapes are carry**in**g the **AC** transportcurrents. They are also subjected to the **AC**magnetic field produced by the transport currents**in** the neighbor**in**g 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 **of**the 4-layer polygon conductor cable. The detailspecifications **of** the cable are shown **in** table 1.For a s**in**gle HTS tape (figure 2) an orthogonal coord**in**atesystem (x, y, z) has been opted where thez-axis co**in**cides with the axis **of** the tape and thetransport current flows along the z-axis i.e. alongthe length **of** the tape. Consider**in**g 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 neighbor**in**g tapes. ∇ Vis the voltage gradient. The electric field isexpressed as∂AE = − − ∇V(2)∂tThe E-j relationship as proposed by Rhyner [2] 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[3]. From equation (3) the non-l**in**ear conductivityσ is expressed as follows:1−nJσ = E n(4)Ec1ncBy numerically solv**in**g the equations (1) – (3)and us**in**g (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 us**in**g 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-l**in**ear conductivity(4) is written **in** the sub-doma**in** expression field.The expressions **of** A at the boundaries,determ**in**ed from∇× A = B , are specified **in** theboundary expression. Then **in** the physics sett**in**gmenu 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 **in**itialvalues **of** the variable A are set **in** the element editfield and **in** the Init field respectively. After thatchoos**in**g the free mesh parameters from meshmenu, mesh**in**g is done with maximum elementsize **of** 1e-2 **in** the superconduct**in**g sub-doma**in**.Then the solution is done by click**in**g the solvemenu and sett**in**g proper solver parameters. Thesimulation is done for a s**in**gle time period. Thepost process**in**g menu has been used to get surfaceplots A, J, Q, E etc and other plots. Follow**in**g 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 simulatedus**in**g **COMSOL** multiphysics s**of**tware fordifferent values **of** transport currents (self field)and under **in**fluence **of** applied field. Externalapplied fields are due to transport currents **in** theneighbor**in**g tapes. Figure 4 shows the **AC** losses**in** the isolated tape for different RMS values **of**transport 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 obta**in**ed from Norris equation **of** **AC** lossfor rectangular strip. As shown **in** Figure 4, the**AC** loss for an isolated tape has beentapes. These simulated loss values obta**in**ed by**COMSOL** 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 s**in**gle layerpolygon conductor is done for J/J c = 0.7. Thisvalue is obta**in**ed **in** two ways. Firstly, simplymultiply**in**g the **AC** loss value **of** an isolated tapewith the number **of** tapes **in** an **in**dividual layer.Secondly, the loss value has been simulated forthe layer with 1/4 th symmetry us**in**g **COMSOL**multiphysics. A s**in**gle layer polygon conductorconta**in**s 28 numbers **of** tapes. For the first casethe obta**in**ed **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 **of**self fields produced by two adjacent tapes **of** thepolygon conductor. Next the numbers **of** layershave been **in**creased to four [4]. It is found thatthe values **of** **AC** losses for different currentdensities are more for **in**creased numbers **of**layers. For all these above calculations the n-value i.e. is the power **in**dex has been chosen tobe 50 which suits better for the YBCO tapes [7].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).**in**creased with the **in**creased value **of** transportcurrent through itself and through the neighbor**in**gFigure 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 **of**cable with 4 layers **of** HTS conductors per phase hasbeen shown.5. ConclusionThe **AC** loss simulation **of** assembled conductor**of** multiple th**in** HTS tapes **in** polygonarrangement has been done us**in**g **COMSOL**multiphysics. The simulation results are verified

with the theoretical values. So the application **of**this s**of**tware for **AC** loss calculation for largenumber **of** th**in** 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 **in**clude some iteration forcalculat**in**g the nonl**in**ear conductivity values **in**the superconduct**in**g sub-doma**in** us**in**g thiss**of**tware, which might be given better match**in**g**of** **AC** loss values with the theoretical values.6. References[1] W. T. Norris, Calculation **of** hysteresis losses**in** hard superconductors carry**in**g ac; Isolatedconductors and edges **of** th**in** sheets, J. Phys. D, 3,489-507 (1970).[2] J. Rhyner, Vector potential theory **of** **AC**losses **in** superconductors, Physica C 377, 56- 66(2002).[3] H.Noji, **AC** loss **of** a high-Tc superconduct**in**gpower-cable conductor, Supercond. Sci.Technol.10, 552 (1997).[4] Noji H, Haji K and Hamada T Alternat**in**gcurrent loss calculation **in** a high-Tcsuperconduct**in**g transmission cable consider**in**gthe magnetic field distribution, Supercond. Sci.Technol.16, 14-18 (2003).[5] N Chakraborty, A V Volkozub and A DCapl**in**, Bolometric measurement **of** **AC** loss **in**HTS tapes: a novel approach **of** microwattsensitivity Supercond. Sci. Technol. 13, 1062-1066(2000).[6] S Chakraborty, N Chakraborty, A f**in**itevolume based numerical simulation **of**Anisotropic heat transfer behavior **in** HTS tapesdue to **AC** losses Proc. AppliedSuperconductivity 1931-1938 (2003)[7] S. Fukui, R. Kojima, J. Ogawa, M.Yamaguchi, T. Sato, and O. Tsukamoto,“Numerical analysis **of** **AC** loss characteristics **of**cable conductor assembled by HTS tapes **in**polygonal arrangement,” IEEE Trans. AppliedSuperconductivity, 16(2), 143–146 (2006)[8] www.comsol.com