Modeling of High Temperature Superconducting Tapes, Arrays and AC Cables Using COMSOL

Transcription

1 Except fom the Poceedings of the COMSOL Confeence 2010 Pais Modeling of High Tempeatue Supeconducting Tapes, Aays and AC Cables Using COMSOL Oleg Chevtchenko * Technical Univesity of Delft, The Nethelands *Coesponding autho: EWI, TU Delft, Postbus 5031, 2600 GA Delft, NL; O.Chevtchenko@tudelft.nl Abstact: In this pape we pesent a set of numeical models ceated with COMSOL Multiphysics. The set includes quantitative models of a coated conducto tape, an aay of such tapes and a high tempeatue supeconducting cable. Simila models wee ceated in the past. An advantage of ou appoach is in additional tanspaency and taceability fo a use, these ae povided on one hand with existing COMSOL example and on the othe hand via step by step validation of each model with tusted expeimental and theoetical data. Keywods: High tempeatue supeconducting tapes, aays and cables, ac losses, non-lineaity. 1. Intoduction High tempeatue supeconducting tapes of the second geneation (coated YBCO tapes) and devices using the tapes show a tend to penetate makets. Steps towads commecialization ae made as the pice of YBCO tapes goes down and othe obstacles ae esolved. One of the emaining technical obstacles is eduction of the losses, especially fo AC applications such as cables. Numeical simulation using FEM becomes a poven tool to calculate AC losses in HTS tapes. The simulation is not tivial as the tapes show non linea esistivity, lage aspect atio and models of HTS tapes ae geneally not pat of a standad FEM package. Building models fom scatch equies time and involves isks. The majo isk is the typo. When ceating complex models, typos can aise. It is theefoe good pactice to use validated paths. The COMSOL Multiphysics modeling envionment contains a numbe of pedefined models that povide such path fo futhe manipulation. 2. Govening equations Seveal fomulations commonly used to solve Maxwell equations with 2D numeical models of supeconductos ae listed in [1, 2]. To illustate ou appoach, we use hee the H -field fomulation, see Eq. (1): µ H / t + ρ H = 0; (1) J = H; ρ = ρ( J). The fomulation is well documented, othe advantages of this fomulation (such as use of the edge elements) ae explained in [1]. Obviously, othe fomulations can be teated the same way. Fo COMSOL modeling Eq. (1) pesents a challenge as it involves coss poducts, which ae not yet pesent in the geneal fom PDE of COMSOL. They can howeve be implemented though the weak fom and using existing example of COMSOL as explained in the next sections. 3. Method Fo ou pupose (to model a supeconducto coss-section with minimum changes in COMSOL) an example exists in COMSOL: AC/DC module; Quasi-Statics, Magnetic; in-plane induction cuents vecto potential. In this example Eq. (2) is the diving equation: 1 1 e σ d A/ t + d( µ 0 µ A) = dj (2) B = A ; σ (E σ = ) ; E = A t Φ. One can solve Eq. (1), by solving Eq. (2) with a pope substitution of the vaiables and letting d 1 e and J 0. Tables 1, 2 povide the needed substitution fo the H -field fomulation. Table 1: Substitution of vaiables fo H -fomulation Oiginal desciption Paamete Substitute fo HTS Paamete Magnetic vecto A x A y Magnetic field H x H y potential Conductivity σ=1/ρ pemeability µ 0 *µ Relative µ conductivity 1/ρ pemeability Absolute pemeability µ 0 1

2 Figue 1. Sub-domain setting of the paametes in GUI of COMSOL Multiphysics 3.5a. In COMSOL Multiphysics 3.5a GUI the substitutions appea as shown in Fig. 1. In the backgound, COMSOL computes a numbe of useful vaiables, these can be found in the Equation System. Two most impotant of them ae listed in Table 2. Table 2: Two useful vaiables computed by COMSOL Paamete HTS substitute Paamete Cul A z_emqap - Cul H z B z_emqap Cuent density J z The extenal (tanspot) cuent is set as a vaiable I 1. The actual cuent density is integated ove the HTS domain. This scala value Iext is then balanced with the peset cuent in this point: I 1 -I ext =0. The location of this point is ielevant. It can be completely outside the active geomety. In the COMSOL GUI it appeas as shown in Fig. 2. The next step is to actually model a HTS tape. Resistivity ρ of a HTS tape is descibed by the so-called powe law, see Eq. (3). n Ec J z ρ ( J z) =. (3) Jc Jc When modeling a supeconducto placed in a non-conducting envionment, the high gadients can cause non-convegence. This poblem is solved by setting an ambient esistivity of seveal odes highe than that of the HTS tape. Similaly, the esistivity of HTS is kept above Ωm. Theefoe Eq. (3) is implemented into the COMSOL example using Eq. (4). 1+ n E B c z _ emqap 18 Rho = Rhosuco = * + 10, J c J c Rho = Rhoai =10 5. (4) Figue 2. Point settings in the Equation System. When needed, extenal (e.g. pependicula to the tape boad face) magnetic field can be applied by pope selection of the bounday conditions (e.g. by setting in Physics, Bounday settings, Bounday condition to: Magnetic potential and poviding to the Quantity A 0 a cetain value o expession). Despite the open chaacte of COMSOL, not all vaiables can be diectly accessed at pesent. A few backgound calculations ae pefomed fo vecto edge elements. Some specific bounday conditions howeve typically ely on these vaiables. In such cases wokaounds may be needed as explained below.

3 4. Application examples Stip geomety. In all cases below a filament o a tape with ectangula coss-section (a stip) is used as shown in Fig. 3. The stip is infinite in the z-diection. The extenal magnetic field is applied in the y-diection, the cuents flow in the z-diection. In pactice, most of YBCO tapes ae 2 to 10 mm wide, thei thickness t (in y-diection) vaies between 50 and 100 µm. The supeconducting YBCO laye is typically 1 µm thick. The supeconducto aspect atio is lage in a tape and it ceates a poblem fo meshing. The filaments ae typically less than 1 mm wide. Fo example, a stiated 12 mm wide tape used below consists of ten 0.84 mm wide filaments stacked togethe in x-diection and sepaated fom each othe with 0.4 mm wide gaps. Whee suitable, the symmeties as descibed in [1] ae used to speed up the computation. Othe specifications of the model stips ae listed in Table 3. y z -w/2 w/2 Figue 3. Geomety of the YBCO stip. Table 3: Specifications of the stips used in this pape Sample n. 1 2a 2b 3 W, [mm] t, [µm] J c 10-8 [A/m 2 ] N Meshing of vey thin and wide objects is a poblem in COMSOL. The aspect atio of the supeconducting laye can be as high as In this study we assume a supeconducting laye being 50 µm thick (except fo the stip n. 2b, Table 3) and use a tiangula mesh with a efinement, see example in Fig. 4. The AC losses (in W/m) in a peiod T ae computed ove the HTS tape domain as: 1 T P = dt o J 2 z ρ ( J z ) ds. (5) T S x 4.1 HTS stip in extenal pependicula magnetic field Figue 4. Example of a mesh fo 12 mm-wide YBCO tape (only pat of the actual tape coss-section centeed aound zeo is shown in the figue). Specifications of the two modeled stips n. 1 and 2 ae listed in Table 3, example of the mesh is shown in Fig. 4. Magnetic field is applied by the pope selection of bounday conditions (by selecting Physics, Bounday settings, setting Bounday condition to: Magnetic potential and poviding to Quantity A 0 a cetain value o expession, in ou case: A y0 sin(2π f t)), with f being a fequency. In Fig. 5 calculated AC losses fo a sample stip n. 1 and fo ten sepaate stips n. 2a, Table 3 (the computed points ae shown by the cicles and by the tiangles espectively) ae compaed to the theoy of Bandt [3] (shown by the dashed lines) and a good ageement is found, the eos ae within a few pecent ove the entie ange. AC loss, J/m 1E+0 1E-1 1E-2 1E-3 AC loss fo thin YBCO stip(s) at 50, 100 and 200 Hz 12 mm -wide tape 1E-4 ten 0.84 mm-wide filaments, sepaated with 0.4 mm gaps 1E-5 ten 0.84 mm-wide filaments sepaated with infinite gaps 1E Extenal pependicula magnetic field, H a, T ms Figue 5. AC losses of YBCO stipes: 12 mm wide tape; ten sepaate filaments; a finite aay of ten filaments sepaated by gaps of 0.4 mm; the dashed lines: analytical esults of [3] and [4]; the solid line: expeiment [5]; the cicles, the tiangles and the ectangles ae the points computed with COMSOL fo the same conditions.

4 Validation of the model is pefomed by compaing computed magnetic field and cuent density distibutions ove the stip width with the theoy of Bandt [3]. The compaison shows a fai ageement of the computed and analytical pofiles. 4.2 HTS stip with a tanspot cuent Specifications of the model stips n. 2a and 2b ae listed in Table 3. Futhemoe, in Fig. 7 calculated AC losses of the stip with 50 Hz sinusoidal tanspot cuent ae compaed to the theoy of Bandt fo a thin stip [3] and the ageement is easonable. Obseved deviation between theoy and computation at lowe cuent amplitudes will be addessed elsewhee. 4.3 Aay of HTS stips (x-stack) Fo infinite aays the symmeties [2] whee applicable ae used to speed up the computation pocess, hee below we pesent an example of COMSOL computation fo a finite x-stack of YBCO stips. In this case an aay of ten filaments is made by stiating a 12 mm wide tape into the aay of 0.84 mm-wide filaments sepaated by gaps of 0.4 mm [5]. Specifications of the compising stips n.2a ae listed in Table 3. Using the symmeties, one quate of the actual cosssection shown in Fig. 8 can be modeled. Example of the mesh is also shown in the figue. Figue 6. Computed magnetic field distibutions ove the width of the stip n. 2a, scaled cuent distibution J z /J c is also shown. Validation of the model is pefomed by compaing computed magnetic field and cuent density distibutions ove the stip width with those given by the theoy [3], again a fai ageement is found. Fo the 50 Hz sinusoidal tanspot cuent examples of the computed magnetic field pofiles (plotted at 1, 2, 3, 4 and 5 ms) and the cuent density distibution (at 30 ms) ae shown in Fig. 6 fo the stip 2a. Nomalised AC loss 1E+0 1E-1 1E-2 1E-3 Theoy of E. Bandt Computed fo stip 2 1E Nomalised tanspot cuent amplitude I t0/i c Figue 7. Computed AC losses as a function of the tanspot cuent fo a thin stip 2b (boxes), compaed to the theoy [3] (dashed line). The losses ae nomalized by the facto 2 fi π. c µ 0 Figue 8. Example of the mesh used fo the 12 mmwide stiated YBCO tape. AC losses ae computed fo the case when the aay is in extenal pependicula magnetic field oscillating at 50 Hz. The esults ae compaed to the theoy [4] and to the expeiment [5] as shown in Fig. 6. Hee the boxes epesent the computed points, the solid line epesents the expeiment [5] pefomed at 100 Hz and 200 Hz and the dashed line epesents the theoy [4]. Ou conclusion is that computation esults ae in excellent ageement with both the expeiment and the theoy ove the entie ange of magnetic field amplitudes. Theeby, this model (of stiated tape) is also validated. In addition, Fig. 9 shows an example of the computed cuent distibution J z /J c inside one of the middle filaments of the stiated tape.

5 Figue 10. Example of the computed cuent distibution Jz/Jc inside one of the filaments of the stiated tape (at 50 Hz, 30 ms, and 60 mtms). 4.4 AC HTS cable (single phase) Finally, we use COMSOL to model a HTS cable made of YBCO tapes. Example of a simplified model fo a single laye, single phase HTS cable made in this case with eight coated tapes assembled into a polygon cable conducto [6] is shown in Fig. 10. Natually, depending on the diamete of the cable fome, on the tape width, numbe of layes, etc., a diffeent numbe of tapes can be in the polygon cable conducto, see fo instance [7]. Figue 10. Model geomety fo a single laye cable made of coated conducto tapes, afte [6]. Because YBCO tapes ae thin and in a cable conducto numbe of the tapes can be substantial, the computation pocess of HTS cable can be time consuming. In ode to speed up the computation pocess, one can use the symmety consideations and model a secto that includes just a few tapes o even a (pat of a) singe tape as futhe illustated in Fig. 11fo the case of single laye model cable. In this case a symmety axis is the line connecting the cente of the cable fome with the middle point of the gap between two adjacent tapes, case A (and/o with the middle of the tape, case B). In the case A, Neumann bounday condition is set along the symmety lines, and in the case B it is the condition H y = 0. Using this symmety appoach, one can compute AC losses in a single laye polygon Figue 11. Example of the symmety use in COMSOL geomety fo a single laye polygon cable conducto. conducto compised fo instance of 19 tapes sepaated by 0.2 mm gaps (fo the case shown in the figue) by modeling just one o two tapes (n. 3, Table 3) and then multiplying the AC losses by the numbe of tapes. Example of the AC losses computed this way is shown in Fig. 12. Hee the citical cuent of each tape is 320 A, the citical cuent of the cable is about 6 ka. AC Losses, W/m at 50 Hz 1E+1 1E+0 1E-1 1E-2 1E-3 single polygon laye Tanspot cuent, A ms Figue 12. Computed AC loss of a model single-laye and single-phase YBCO cable made of nineteen 4-mm wide YBCO tapes sepaated by 0.2 mm gaps and placed aound a 25 mm diamete fome. Using the appoach, it is possible to model moe complex stuctues (e.g., a cable with moe tape layes, a thee-phase cable, a tiax cable, etc.), all this togethe with the elevant expeimental data and detailed validations will be epoted elsewhee. 7. Conclusions We developed and validated a set of numeical models fo coated conducto HTS tapes, aays and cables using COMSOL Multiphysics. Since existing COMSOL example fom AC/DC module is used, the changes in COMSOL elevant to modeling of HTS tapes ae kept to the minimum. A use aiming to model HTS tapes and devices with COMSOL, can access the modeling pocess moe diectly, without a need to wite PDE s fist.

6 8. Refeences 1. R. Bambilla, F. Gilli, Simulating Supeconductos in AC Envionment: Two Complementay COMSOL Models, Poceedings of the COMSOL Confeence 2009 Milan, Italy (2009) 2. R. Bambilla e. a.., Development of an edgeelement model fo AC loss computation of hightempeatue supeconductos, Supecond. Sci. Technol (2007) 3. E. H. Bandt and M. Indenbom, Type IIsupeconducto stip with cuent in a pependicula magnetic field, Phys. Rev. B, (1993) 4. Y. Mawatai, Citical state of peiodically aanged supeconducting-stip lines in pependicula fields, Phys. Rev B 54, (1996) 5. M. Machevsky, e.a., AC losses and magnetic coupling in multifilamentay coated HTS conductos and tape aays, IEEE Tans. On Applied Supeconductivity, 19, n.3, p. 3, (2009) 6. Y. Mawatai, K. Kajikawa, Hysteetic ac loss of polygonally aanged supeconducting stips caying ac tanspot cuent, Applied Phys. Lettes, 92, (2008) 7. G. Kona, e.a., Simulation of AC Loss in High Tempeatue Supeconducting Cable using COMSOL Multiphysics, Poceedings of the COMSOL Confeence, Bangaloe, India (2009). 9. Acknowledgements The suppot of the COMSOL application team is geatly acknowledged.