Current Distribution Analysis for High Temperature Superconducting Cable Considering Hysteresis Characteristics

Transcription

1 Curren Dsrbuon Analyss for Hgh Temperaure uperconucng Cable Conserng Hyseress Characerscs huhong WANG a, Je QIU a, Xnyng LIU a, Zhxuan WANG a, Janguo ZHU b, Youguang GUO b, Zh We LIN b, Y WANG b an We XU b a Faculy of Elecrcal Engneerng, X'an Jaoong Unversy, X'an 7149, Chna Tel.: ; E-mal: shwang@mal.xju.eu.cn b chool of Elecrcal, Mechancal an Mecharonc ysems, Unversy of Technology, yney, NW 7, Ausrala Absrac. Ths paper presens a hyseress moel for Type-II hgh emperaure superconucor (HT) by usng smplfe Presach Moel, n whch he Presach srbuon funcon μ k (α, β) s eermne only base on he B- H lmng loop. The nonlnear ynamc crcu moel of superconucor s esablshe. In he crcu moel, he hyseress nucance an hyseress loss escrbe by smplfe Presach Moel are euce. Applyng he hyseress crcu moel, he currens flowng n fferen superconucor layers of hgh emperaure superconucng cable are smulae, as well as he hyseress loss of he superconucng cable. The smulaon resuls are verfe by comparson wh aa recore n leraures. Fnally, he nfluences of hyseress on superconucng cable are analyze an scusse. Keywors. uperconucng cable, Hyseress, Presach Moel, Curren srbuon 1 Inroucon HT cables for large curren ransmsson n general have a mul-layer srucure conssng of parallel connece apes, wse n each layer. A warm-elecrc HT cable s shown n Fg. 1. The conrol of curren srbuon among hese layers s an mporan ssue for esgn an opmzaon of an HT cable because hs sgnfcanly affecs he curren ransmsson capacy an power losses. The pnne magnec flux consues a memory ha gves rse o a hyseress wh corresponng losses an prouces an aonal volage n he HT cable. Fg. 1. A HT cable. A basc propery of he magnec maeral s he vecor B-H relaonshp, known as he vecor magnec hyseress, whch escrbes he phenomenon ha he varaon of B lags ha

2 of H n eher magnue or recon when he rae of varaon s very slow. Owng o he very complex mechansms of magnec hyseress, whch s no ye fully unersoo so far, he evelopmen of mahemacal moels of magnesaon processes has no been successful especally for engneerng applcaons espe he fac ha many aemps have been mae by varous researchers. By now, varous moels have been evelope wh a beer unersanng of magnec hyseress phenomenon. Among hem, he mos noceable works are he oner an Wohlfarh (-W) moel [19], he Presach moel [] an he combne moel [1]. In 1948, oner an Wohlfarh posulae a heory base on he roaon of magnec momens of sngle oman parcles wh respec o her easy axes. In 199, Aheron an Beae [] suggese a mofcaon o ncorporae parcle neracons no he -W moel by he aon of a mean fel erm, bu he numercal mplemenaon an expermenal verfcaon of he heory was no repore. The Presach moel has been grealy mprove an generalse o moel ynamc hyseress [3], whch accouns for he nfluence of he rae of varaon of magnec fel srengh on branchng, an magnesaon epenen hyseress [4,5], whch assumes ha he magnesaon for a gven fel srengh epens on boh he prevous magnec hsory an he curren value of magnesaon. The Presach moel has also been generalse o represen vecor hyseress [6]. The vecor hyseress s smulae by a se of scalar Presach moels, each havng a fferen magnesaon recon an responng o he componen of he apple fel n ha recon. The Presach moel has been use o escrbe he hyseress n superconucng cable [1]. J. G. Zhu, e al. propose he crcu moel for magnec core by usng smplfe hyseress Presach moel []. In hs paper, he smplfe Presach moel propose n [] s apple o escrbe he magnec performance of ype-ii superconucors n econ II. In econ III, he hyseress nucance s euce accorng o he smplfe moel. The nonlnear ynamc crcu moel conserng hyseress nucance s presene. The crcu equaons are solve by usng Back-Eular meho an Newon-Raphson eraon. In econ IV, comparson beween he smulae resuls n [1] s conuce an scusse. mplfe Presach Moel To moel he nonlnear magnezaon propery an he hyseress loss, he Presach moel s aope for s parameer enfcaon an conserable accuracy. However, srbuon funcon μ(α, β) s ffcul o be eermne. asc meho s apple o calculae he srbuon funcon accorng o a mass of repeaable expermens. In [], a smplfe meho s propose o euce he srbuon funon only base on he measure lmng hyseress loop. Fg. shows he hyseress loop of B-3 an he general Presach agram. 1 Expermenal resul mulaon resul ß a=ß - M/ 1Gs o - H sa a H/ 1 4 Oe Fg.. (a) Hyseress loop of B-3 an (b) General Presach agram. -H sa L() u()

3 In Fg. (b), n he rangle, -H sa α β H sa, H sa s he sauraon magnec fel srengh. In smplfe Presach moel, when a magnec maeral s magneze from he nal unmagneze sae, he flux ensy B ue o magnec fel H s expresse as ˆ B =Γ H = µαβγ, ( H ) αβ ( ) ( ) ( ) ( ) α β αβ (, ) (, ) T( H, H) = µαβ α β µαβ α β= (1) γ αβ where, n he rangle, γ s a sgn funcon, αβ (H) s equal o 1 on enclose area + an s -1 on -. The nerface beween + an -, L(), s euce by he hsory of magnezaon. T(α, β), he area negraon of hyseress srbuon funcon, s scrbe as ( αβ, ) = α α µαβ (, ) = α x µαβ (, ) T xy xy () β y β β On a ownwar rajecory of lmng B-H loop, (1) s rewren as B H B Hsa T Hsa H ( ) = µαβ (, ) α β µαβ (, ) α β = ( ) (, ) On a upwar rajecory, (1) s can be erve as B H B H T H H ( ) = µαβ (, ) α β µαβ (, ) α β = ( ) + (, ) u sa sa T(α, β) can be rewren as ( β) B ( α) B u T( αβ, ) = + F( α) F( β) (3) where, B an B u, are he flux enses on he ownwar an upwar rajecores of he lmng B-H loop, an ( ) ( ) ( ) B ( H) Bu H B H F H = ( ) = ( ) (when < ) F H B H H (when H ), I shoul be noe ha F(a) s expresse only n erms of flux ensy values of he B-H lmng loop wh ou conserng he srbuon funcon. Thus, only he lmng B-H loop s neee for he mplemenaon of he hyseress moel. Conserng he amagnesm of superconucor, Eq. (1) may be rewren as ˆ M =Γ H = µαβγ, ( H ) αβ (5) ( ) ( ) ( ) ( ) α β In Fg. 1(a), he smulae hyseress loop by usng smplfe presach moel s smlar o expermenal loop of B-3, a kn of ype-ii superconucng maeral.. The hyseress loss can be calculae as αβ (, )( ) Q = µαβ α β α β (6) Ω (4)

4 3 Hyseress Inucance Moel The nucance of he superconucng apes s erve base on smplfe Presach moel [1]. In a HT cable, he axal magnec flux lnkage of he h Layer, φ z,, can be calculae as ( r ) ( ) ( r ) ( ) ϕ πρ I β ρ ρ r z, = Bz, nz, = f an r1 Γ π r r + r L r ρ I ρ = an β an r1 f r β ρ π r r r an he crcumferenal axal magnec flux lnkage φ θ, s ϕ I ρ r B n f ρ r, =,, = Γ r 1 πρ ( r + ) r θ θ θ where, B z, an B θ, are he axal an crcumferenal axal magnec flux enses of he h layer, respecvely. f(x) s he smplfe Presach moel. I s he curren flowng hrough he h superconucng layer. r, β L an are he raus, ape pch angle, lengh an hckness of he h superconucng layer, respecvely. The negraon can be calculae by usng Gauss-Legenre meho. The oal magnec flux lnkage φ s ϕ = ϕ + ϕ θ z,, Thus, he hyseress nucance of he h superconucng layer L s, s 1 L s, ϕ = I 4 Nonlnear Dynamc crcu Moel Conserng Hyseress The mofe equvalen crcu of mul-layer hgh superconucng cable s shown n Fg. 3 [1][3]. The eermnaon of self-, muual- nucances an nonlnear ressances superconucng layers may refer o [3]. 1 R 1 L s,1 L 1 R L s, M 1, L R 3 L 3 L 3 s,3 M,4 M 1,4 4 R 4 L s,4 L 4

5 Fg. 3. A equvalen crcu of HT cable conserng hyseress. The crcu equaons can be euce as 1() 1() () n () R 1 1() + L1 + Ls,1 + M1 + + M1 n = U() 1() () () n () M1 + R () + L + Ls, + + Mn = U() 1() () n() n() Mn1 + Mn + + R nn() + Ln + Lsn, = U() 1() + () + 3 () + + n () = () These nonlnear fferenal equaons may be solve by usng Newon-Raphson meho. 5 Example In orer o verfy he correcon of smplfe Presach moel, he superconucng cable prooypes, whch s smulae n [1], s employe. Table I lss he srucural parameers of hese prooypes. B-3 s aope as he superconucor. Table I rucural parameers of superconucng cable prooypes Peak of Curren No. of Layer r /mm Prooype a I p =38A θ / o Orenaon r /mm Prooype b I p =38A θ / o Orenaon r /mm Prooype c I p =48A θ / o Orenaon r /mm Prooype I p =48A θ / o Orenaon The curren resuls smulae n [1] an by he presene meho n hs paper are pcure (a) n Fg [15In Fg. 4, he sol, ashoe, ashe an oe lnes represen he currens o,15 o,15 o,15 o ],Ip=38A flowng hrough Layer 1,, 3 an 4, respecvely. I can be seen ha he resuls calculae by he smplfe Presach moel s smlar o hose presene n [1]. Curren Dsrbuon/A [35 o,35 o,-5 o,-5 o ],Ip=38A [5 o,1 o,-13 o,-36 o ],Ip=48A [3 o,14 o,18 o,34 o ],Ip=48A Tme/s Curren Dsrbuon/A (b) - [15 o,15 o,15 o,15 o ],Ip=38A (c) [35 - o,35 o,-5 o,-5 o ],Ip=38A [5 - o,1 o,-13 () o,-36 o ],Ip=48A - [3 o,14 o,18 o,34 o ],Ip=48A Tme/s (a) (b) (c) ()

6 (a) Currens smulae n [1] Fg. 4. The smulae currens. (b) Currens smulae n hs paper Fg. 5. llusraes he curren srbuon of he 4-layer HT cable, he magnec hyserss loop of he 1s layer an he hyseress loss wh oal curren 48A, ape pch angles of layers +5, +1, -14 an -37, respecvely. In hs case, he curren srbuon becomes unform. A sac operaon, he hyseress loss per cycle approaches o a consan. Curren / A 1-1 Layer 1 Layer Layer 3 Layer Tme/s M/mT H/(A/m) (a) Curren srbuon (b) Hyseress loop (c) Hyseress loss Fg. 5. The smulae resuls of a 4-layer HT cable. P h / mj Tme / s 6 Concluson The hyseress of superconucor causes he non-unform curren srbuon n HT cable. In hs paper, a smplfe Presach moel, whch jus consers he lmng loop of hyseress loop, s apple o escrbe he magnec hyseress performance of superconucng cable. The hyseress loss s presene base on he smplfe moel. The crcu moel of HT cable conserng hyseress s euce. Comparson wh he resuls n leraures, he correcon of hs smplfe Presach moel an crcu moel s verfe. Choosng proper srucure parameers, he curren srbuon of HT layer can be unform. References [19] E.C. oner an E.P. Wohlfarh, A mechansm of magnec hyseress n heerogeneous alloys, Phl. Trans. Roy. oc., vol. 4A, May 1948, pp [] V.F. Presach, Uber e magnesche Nachwrkung, Z. fur Physk, B. 94, 1935, pp [1] J. O an E.D. Torre, A vecor movng moel of boh reversble an rreversble magnesng processes, J. Appl. Phys., vol. 67, no. 9, May 199, pp [] D.L. Aheron, an J.R. Beae, "A mean fel oner-wohlfarh hyseress moel", IEEE Trans. Magn., vol. 6, no. 6, Nov. 199, pp [3] I.D. Mayergoyz, "Dynamc Presach moels of hyseress", IEEE Trans. Magn., vol. 4, no. 6, Nov. 1988, pp [4] J. O, F. Vaja, an E.D. Torre, Ienfcaon of parameers n a movng moel, J. Appl. Phys., vol. 69, no. 8, Apr. 1991, pp [5] E.D. Torre, "Exsence of magnesaon-epenen Presach moels", IEEE Trans. Magn., vol. 7, no. 4, July 1991, pp [6] A.A. Aly an I.D. Mayergoyz, A new vecor Presach-ype moel of hyseress, J. Appl. Phys., vol. 73, no. 1, May 1993, pp [1] F. Grll an M. josrom, Precon of ressve an hyserec losses n a mul-layer hgh- Tc superconucng cable, uperconucor cence an Technology, 17 (4), []. Y. R. Hu, Jan Guo Zhu, an V.. Ramsen. A generalze ynamc crcu moel of magnec cores for low- an hgh-frequency applcaons. II. Crcu moel formulaon an mplemenaon. IEEE Transacons on POWER ELECTRONIC, 11 (1996), [3] huhong Wang, Je Qu, Zhen Zhao, Jan Guo Zhu an Youguang Guo. rucural Parameer Desgn on Mullayer Conucors of HT AC Transmsson Cable by Means of Parcle warm Opmzaon. Journal of he Japan ocey of Apple Elecromagnecs an Mechancs, Vol.I5, No.3 (7), 9-93