non Sinusoidal Conditions

Transcription

1 AC Losses in High Temperature Superondutors under non Sinusoidal Conditions By Spetor Marat B.S., Ben-Gurion University, 7 maratspetor@gmail.om Masters Thesis, Otober, Submitted in partial fulfillment of the requirements for the degree of Master in Physis in the Faulty of Natural Sienes at Ben-Gurion University of the Negev, Beer-Sheva, Israel Author signature Date Supervisor approval Supervisor approval Date Date Department Counil approval Date

2 Dediated to my beloved family

3 Abstrat 3 AC losses in superondutors haraterize their physial properties (the mirosopi motion of the Abrikosov vorties, phase state of the vortex lattie, et) as well as determining ranges of the rated urrents and magneti fields for superonduting devies that are required for the optimal operation power of ryogeni equipment and eonomial gain. The methods of AC loss alulation are usually based on onsideration of the non-linear Maxwell equations in whih a superondutor is simulated by a media with non-linear voltage urrent harateristi, in the general ase whih depends on a loal magneti field and a loal temperature. Many investigations are devoted to a onsideration of AC losses in superondutors of various forms suh as wires, tapes, oated ondutors, et., under different onditions. However, most of the works onsider sinusoidal magneti fields or/and urrents. In reality, urrents in physial experiments and in eletri power systems exhibit a distribution of harmoni frequenies. The main goal of this thesis is the treatment of AC losses in superondutors and oated ondutors under non-sinusoidal onditions. Analytial expressions were obtained for various onfigurations of superondutors suh as: bulk materials, thin strips, oated ondutors in the framework of ritial state model as well as in high magneti field approximation for a power law superondutor. An analysis of losses in power law superondutors was arried out using the MatLab software to perform numerial solution of the integral equation. The analysis shows that in devies with bulk superondutors, higher harmonis an substantially hange losses. Thus, the 5% seond harmoni an ause a loss inrease of % in superondutors in omparison to % for a normal metal. Moreover, the ontribution of the higher harmonis depends on their phases. In a ertain range of phases, the odd harmonis an even redue AC losses. These peuliarities distinguish the behavior of superonduting devies from that of onventional ones.

4 4 Influene of higher harmonis losses in oated ondutors is markedly stronger. Thus, the 5% third urrent harmoni auses a loss inrease of 45% in the superonduting part of a oated ondutor and 8% in the normal-metal parts at a urrent lose to the ritial value. Numerial alulations of the total losses (sum of losses in superonduting and normal metal parts) in a oated ondutor within the power-law approximation showed that the relative ontribution of higher harmonis inreases with their amplitude. The relative ontribution of the third % harmoni to the total losses an ahieve up to % of the losses aused by the main harmoni and is about ten times larger than losses aused by the same higher harmoni in the normal-metal ondutor of the same form. Even at a low power index (n = 4) the predited losses are substantially higher than AC losses in normal metals: relative ontribution an be 4 times higher and ahieves it 44% of the main harmoni losses replae of %.

5 Aknowledgements 5 The author wishes to thank Dr. Vladimir Sokolovsky for his extraordinary patiene lose guidane, support and teahing in the theory of superondutivity and the solid state physis of superonduting materials. The world of superondutivity would not be aessible to me without him. To Prof. Shuker Reuven for the unique teahing and guidane, using his knowledge and experiene in many areas of physis. His unique knowledge alleviated my understanding. To Prof. Leonid Prigozhin for his important teahing in theory, and his important help in the numerial analysis and simulation of power law superondutors. To Dr Vitor Meerovih for his assistane and help.

6 Content 6. Introdution AC losses in superondutors under sinusoidal onditions.... Origin of AC Losses in Superondutors.... Critial State Model Analyti alulation of AC losses in a superonduting slab AC losses in thin superonduting strip Strip with transport urrent Strip in a perpendiular magneti field Further development of Bean s Model Losses in HTSC Peuliarities of HTSC Analytial alulation of AC losses in superonduting slab... 3 under non-sinusoidal onditions Summary Coated Condutors AC power losses in oated ondutor Coated ondutor in a perpendiular magneti field Coated ondutor with a transport urrent Results and disussion Analytial approah for AC losses in striped oated ondutors Power law superonduting strip Summary Numerial analysis of AC losses in oated ondutors Introdution Mathematial Model Numerial algorithm Results and disussion Summary Conlusion Bibliography... 65

7 . Introdution 7 A superondutor has zero resistane only under DC onditions, under AC onditions a hanging magneti field (either self generated or externally applied) ating on the material interats with transport and leads to energy dissipation. The subjet of AC losses in superondutors may be onsidered under various aspets. Firstly, the physial properties of high temperature superondutors (HTSC) suh as: the mirosopi motion of Abrikosov vorties, phase state of the vortex lattie, urrent density, ritial temperature and also with developing the theory where the mirosopi proesses are desribed by a nonlinear urrent voltage harateristi E E J, B ( E is the loal eletri field aused by the urrent density J and depending also on a loal magneti indution B ) and by reversible magnetization urve H H B [-3]. Seondly, from the tehnial point of view AC loss values determine ranges of the rated urrents and magneti fields for superonduting devies, required power of ryogeni equipment and eonomial gain. The methods of AC loss alulation are usually based on onsideration of the non-linear Maxwell equations in whih a superondutor is simulated by a metal with non-linear voltage urrent harateristi, in the general ase, depending on a loal magneti field and a loal temperature. A lot of investigations are devoted to a onsideration of AC losses in superondutors of various forms, wires, tapes, oated ondutors and. under different onditions (see for example [4-6] and works referened in them). However, most of the works onsider sinusoidal magneti fields or/and urrents. In reality, urrents It in physial experiments and eletri power systems ontain a wide variety of harmonis and an be represented as

8 8 I os kt (.) I t k k k where f, f is the frequeny of the first harmoni, I k and k are the amplitude and phase of the k-th harmoni, (k =,,..). The frequeny f for power systems is 5 or 6 Hz, while for speial eletri systems (airplanes, ships, et.) frequently is 4-8 Hz. For the normal metal parts of a devie the Joule losses P j aused by a urrent It and the eddy losses P ed aused by a magneti field Ht are presented in the form: R f I Pj r i kk k K f H P ed p k h k k where R f is the ondutor resistane at the frequeny f, r k (.) gives the frequeny dependene of the resistane, i k I I k, h k H H k, k H is the k-th harmoni amplitude of the magneti field, K f H / is the eddy losses aused by the main harmoni, p k haraterize the dependene of the losses on the frequeny. The expliit forms of R, K, r k, p k depend on the ondutor shape and magneti field diretion. For example, losses per unit length in a thin normal metal strip in an external magneti field direted perpendiular to the wide surfae are ad K p k 3 and k (.3) where is the magneti permeability of the vauum, is the resistivity, d is the strip thikness, a is a half of the strip width. Aording to the requirements of power quality supported by power systems, the ontribution of higher harmonis is a few perent and urrents are very lose to sinusoidal. Usually higher harmonis derease proportionally to k at least and it is suffiient to take into aount

9 9 several first harmonis. I beause a diret urrent annot be transferred by the transformer. So, with the auray of about % the losses in normal onduting parts of the power devies are determined by the main harmoni. Currents beome strongly non-sinusoidal at the operation of the onverters, non-linear reators and during transient and overload onditions. In this ase all harmonis have to be taken into aount at the loss estimation. Sine superondutors posses a strongly non-linear urrent voltage harateristi, one an expet substantial ontribution of higher harmonis to AC losses in superonduting elements. The main goal of this thesis is the onsideration of AC losses in superondutors and oated ondutors under non-sinusoidal onditions. The objetives are the following - analysis of penetration of a non-sinusoidal magneti field into superonduting slab, thin film (strip), and oated ondutor; - development of analytial methods for AC loss alulation based on the ritial state model (CSM); - numerial alulation of AC losses in power law superondutors and oated ondutors. The thesis struture is the following: Chapter - ontains a brief introdution to the subjet of AC losses in superondutors inluding desription of the mehanism of AC losses, analytial methods for alulation of AC losses in magneti fields or with transport urrents; Chapter 3- reveals an analytial analysis of AC losses under non-sinusoidal onditions for superonduting slab and thin film (strip); Chapter 4- represents analytial approah for alulation of AC losses in oated ondutors under non-sinusoidal onditions; Chapter 5- represents results of numerial simulation of AC losses based on the power law approximation for oating ondutors.

10 The main obtained results have been presented in the following papers:. V.Sokolovsky, V.Meerovih, M.Spetor, G.Levin, and I, Vajda Losses in superondutors under non-sinusoidal urrents and magneti fields, IEEE Trans. Appl. Superon., v.9 pp: (9). G.Furman, M.Spetor, V.Meerovih, and V.Sokolovsky Losses in oated ondutors under non-sinusoidal urrents and magneti fields, in print in Journal of Superondutivity and Novel Magnetism and at the onferenes. V. Sokolovsky, V. Meerovih, M. Spetor, G. Levin, and I, Vajda, Losses in superondutors under non-sinusoidal urrents and magneti fields, Applied Superondutivity Conferene 8, August 7-, Hyatt Regeny Chiago, Illinois, Chiago USA, Conferene Program p 8, (MPB9). Poster Presentation. G. Furman, M. Spetor, V. Meerovih, and V. Sokolovsky Losses in oated ondutors under non-sinusoidal urrents and magneti fields, International Conferene on Superondutivity and Magnetism 5-3 April, Antalya, Turkey, Abstrat Book: p776, (LSA-P-). Poster Presentation

11 . AC losses in superondutors under sinusoidal onditions. Origin of AC Losses in Superondutors The vorties in a type II superondutor reate a lattie of parallel flux lines through the speimen, eah arrying a quantum of magneti flux h/ e.7 5 T m (where h is the Plank onstant and e is the eletron harge). The magneti flux penetrates into the speimen through the vortex ore, whih is in the normal state and loated in the entre of the vortex. The magneti field is maximum near the enter and exponentially deays with the distane from the entre over the London penetration depth of the order of radius is of the order of oherene length of 7 m. The ore 8 m. The density of the vorties in the lattie is determined by a loal magneti field B as B n. This mixed state is observed in the superondutors up to the upper ritial magneti field B. The upper ritial magneti field is of the order of T for low temperature superondutors and - T for high temperature superondutors at 4 K. The vorties an move freely in a homogenous material, their density is proportional to the flux density and they an be pinned by inhomogeneities in the material. Any motion of vorties results in energy dissipation. Let us assume that the urrent is applied in the presene of a magneti field. In homogenous superondutors the vorties start to move under the influene of the Lorenz fore: fl j ( j is the urrent density), or per unit of volume of the lattie FL j B. As a result the moving magneti flux generates an eletri field E v B (Fig.). The motion of vorties results in the energy loss per unit of volume of the lattie P = J E. This motion an be presented as motion opposite the visous drag fore density fd v ( is the visosity oeffiient of the flux lines, v is the veloity of the vorties). In reality superondutors do

12 ontain defets and interation of vorties with the defets (pinning fore) prevents the movement of the vorties until the Lorenz fore exeeds the pinning fore. There is a urrent density j, the ritial urrent density at whih the Lorentz fore is equal to the pinning one f p f L j. At higher urrents we observe the vortex motion whih is known as flux flow. The vortex veloity is determined from the equality of the Lorentz fore and a sum of the pinning and visous drag fores. The power loss per unit of volume is proportional to the vortex density n and determined as: p flvn J n( J J) FF J( J J) (.) where FF ( B / ) is the flux flow resistivity depending on a loal magneti field and temperature. Magneti field B J B F L J Current Density Fig. A superondutor with a urrent in an external magneti field It was shown by Bardeen and Stephen [] that the expression for FF an be given approximately as: ( B, T ) B ( T ) / B ( T) (.) FF n where n is the resisitivity in the normal state at temperature T. The visosity oeffiient an be approximated by:

13 3 (.3) B / n As a result the urrent density an be presented in superondutors in the form: J J E / (.4) FF where the urrent density J flows without dissipation, and an additional part J E/ FF is related to the motion of the eletrons of the normal ondutivity. Another mehanism of AC losses predited by Anderson is the eletri field generation in superondutors onneted with the thermally ativated flux jumps out of pinning entres over the pinning barrier ( flux reep [7] ). A usual method of the flux reep study is based on the measurement of a magneti field penetration in a hollow superonduting ylinder (deay of persistent urrent in the ylinder). The deay of the ylinder urrent in the flux reep regime is usually desribed by the logarithmi law while the urrent deay in the flux flow regime obeys to an exponential law. This differene allows one to separate these regimes experimentally. There are two mehanisms of energy dissipation as a result of the motion of vorties [8,9]: - moving vorties ross the urrent lines (due to ontinuity of the later ). Hene a urrent flows through a vortex parts (vortex ores) whih are in the normal state. This leads to energy dissipation. - when a vortex passes through some point of the superondutor a phase transition ours in this point from the superonduting to normal state and vie versa. From thermodynamis it is known that suh a proess an our without energy dissipation only if a yli phase transition is infinitely slow. A veloity of vortex filaments is finite and the phase transition aused by the vortex motion leads to dissipation of energy. Estimations [,] showed that ontributions of these loss mehanisms are about the same. Let us disuss briefly models for analytial alulation of AC losses in superondutors and their range of appliation.

14 . Critial State Model 4 The marosopi desription of magneti properties type-ii low temperature superondutors (LTS) was developed in sixties by C. Bean [] and is known as the ritial state model (CSM). Bean has proposed to use a ontinuum desription of a vortex lattie: a loal magneti field is proportional to vortex density. The onditions when the quantum desription ould be replaed by the ontinuum one disussed in many publiations and books [for example 3-4]. Brandt noted that a ontinuum desription of a vortex lattie is valid at B B [7]. The basi postulate of CSM is that there is a limiting marosopi superonduting urrent density J that every superondutor an arry without resistane and any eletromotive fore will indue this urrent density. This assumption was based on two fats known for LTS. First, the flux reep regime is observed in a very narrow region near J. Seond the addition of J to the ritial urrent density is very small. The following estimations onfirm the last point. The LTS are usually used in the form of the multifilament wires where the diameter d of a multifilament is about -5 m. From the Maxwell equations, the indued eletri field in a filament plaed in AC magneti field with the frequeny f and amplitude B an be estimated as E fdb. Taking into aount Eqs. (.) and (.4), the additional urrent density is J n fdb T. The upper ritial field B and normal resistivity T n of LTS are the order of the magnitude of T and 8 Ω m, orrespondingly, and J A/m. Therefore, J / J ~ 3 at the frequeny of 5 Hz. Thus, in real onditions the additional urrent density is muh less than the ritial urrent density. The basi points of Bean s model an be summarized as follows: - urrent density in a superondutor is equal to J or ; - ritial urrent density appears whereas there is nonzero eletri field; - urrent density remains onstant with time when an eletri field equals to zero; - urrent density is field independent;

15 5 - magneti field penetrates a superondutor from its surfae. Note that Bean s model desribes magneti properties of a hard superondutor in a steadystate limit. Sine any variation of the magneti flux in the sample results in an indued eletri field, superondutors in the dynami regime are always in the resistive state (a urrent density is higher than the ritial value). This fat is usually negleted for the analysis of the magneti properties of LTS s in dynami regime. The tehnique of the AC loss alulation is based on lassial eletrodynamis (Maxwell s equations): B E ;_ H J; _ D ; _ B ; _ B H t (.5) omplemented by Bean s assumptions. Here D is the eletri field indution whih is proportional to E and the linear relation between B and H is valid only for an isotropi superondutor at B B [5]. There are various methods for alulation of losses in the superondutor per a period of the urrent: - integrating the Poynting vetor S E He (where H e is magneti field at the superondutor surfae), on the sample surfae over the period; - integrating the power Joulle losses E J through the sample volume - alulation of the work of mehanial fores put out opposite the pinning fores at the motion of the vortex lattie. Loss power per unit volume is equal to p J Bv, where the veloity v of the vorties an be determined by the onservation law of vortex density B div B v. t - using the surfae impedane onept. The surfae impedane is defined as the relation of the first harmoni of eletri and magneti fields on the surfae:

16 6 Z E x,texp it d t (.6) H where i is the unit imaginary number. The loss power P dissipated in a sample is determined by the real part of the surfae impedane: P Re z H This onept is widely used for haraterization of superondutors in high magneti fields. - alulation using AC suseptibility. For sinusoidal external magneti field one may define the omplex suseptibility i ; m =,,3..;,,, m m m i m H, M texp imt d t (.7) H where M t is the omponent of the (dipolar) magneti moment of M along the external magneti field. The magneti moment is M t r J r,td 3 r (.8) The energy onverted into heat in the whole of the speimen during one yle of the AC field is proportional to the imaginary part of the first harmoni P H (.9),, The real part of the omplex AC suseptibility determines the time average of the magneti energy stored in the volume oupied by the sample. The approah is widely used to desribe the linear and nonlinear response of HTSC speimens in various geometries (slab, infinite and finite ylinders, strip, dis, and et. [6,7]). The AC suseptibility allows one not only to determine value of AC losses but also to onlude the physial properties of the material.

17 7.. Analyti alulation of AC losses in a superonduting slab Let us onsider a superonduting slab with thikness whose inward surfae normal points in the positive x-diretion. The external sinusoidal magneti field: H H sint e is applied in the z-diretion, whih is parallel to the surfae of the slab while the eletri field E and urrent J are indued along the y-axis. Fig. shows two different ases of a distribution of the magneti field and urrent in a superonduting slab; the first one (top) relates to the slab plaed in a magneti field while the seond ase (bottom) orresponds the slab with a transport urrent. J J B J B B y z Slab in a magneti field x J J B J B B Slab with transport urrent Fig. The profiles of the magneti field and urrent in a slab in an external magneti field (top) and with a transport urrent (bottom). The problem is symmetri and hereafter it is suffiient to onsider only one half of the slab thikness from x = till x = and one half of the period of the applied magneti field e.g. between the minimum He H to the maximum He H. Aording to the Bean model for the initial state at t / the urrent density is J J for all x reahed by the

18 8 penetrating magneti field. For the slab, in the area in whih magneti field penetrates, Eqs. (.5) are redued to H x E x J H t (.) (.) with boundary onditions: H H at x and x. e This initial profile of the magneti field is marked as in Fig. 3. In the ase of an inomplete penetration (the amplitude of the applied magneti field H is less than the omplete penetration field Hp J ) there is a region when magneti field and urrent are absent. The region with J Jextends to the Bean penetration depth p x H / J. In the ase of the omplete penetration, H Hp, magneti field penetrates the whole of the slab and the region with J x J extends to p (Fig. 3 b). As the magneti field inreases, the profile of the urrent density hanges and starts to ontain a region with J=J. Let us denote by x the point at whih the urrent reverses its sign. We have J = J for x x and J = -J for x>x. The solution of Eq. (.) is H He Jx, x x H J x, x x (.) The point x is determined from a ontinuum for the magneti field in this point x ( H H ) / J (.3) e In this point the eletri field equals zero: Ex ( ). This is the boundary ondition for Eq. (.). The eletri field in the superondutor is e H x x, x x E t, x x (.4)

19 9 H e H J x p x J H e H J J J J H e x H e J J J H e H H p J J H e H J J J J J a) b) Fig. 3 The profiles of the magneti field and urrent in a slab at the inrease of external magneti field from -H to H for the ases of an inomplete penetration (left olumn) and the omplete penetration (right olumn). Note that in the ase of the omplete penetration the inrease of an applied magneti field above H p H does not lead to inrease of the urrent density in the superondutor. At the external magneti field above H p H, x, and the boundary ondition for the Eq. (.) is zero eletri field in the entre of the slab. There are several different approahes for

20 alulation of AC losses per a period of time in a superondutor. Here we will use the formula for the power Joule losses, i.e. by integrating E J through the sample volume and period. The alulation proedure is in details presented in many papers (see for example [,]), here we represent the results for the ase when a superonduting slab and an infinitely long ylinder are subjeted to a sinusoidal external field or a transport urrent. For the ase of the inomplete penetration H Hp, the loss density per a period and per a surfae unit is determined as per one side of a slab: for a ylinder: p 3 H ; (.5) 3J H p H ( ) ; (.6) 3J J R 3 for the ase of omplete penetration: for one side of a slab: H p J 3 p ( H H ); (.7) p for a ylinder: 3 H p H H p ( H H p ) J 3 3H H p p ; (.8) where for the ylinder H p JR, R is the radius of the ylinder. Total power losses are determined by multiplying Eqs. (.6)-(.8) by the sample surfae and frequeny. Losses in the slab with the alternating transport urrent an be determined using equation (.) where the magneti field amplitude H is replaed by the I / (where I is the amplitude of the urrent per a height unit of the slab). AC losses per a period per unit of length of the superonduting ylinder with a transport urrent along its axis are: 4 i p JR {( i)[ln( i) i] } (.9)

21 where i I I a ; Ia is the amplitude of transport urrent; I J R is the ritial urrent of the ylinder..3 AC losses in thin superonduting strip In the previous setion we onsidered AC losses in bulk superondutors using the model of an infinitely long slab (ylinder) with a transport urrent or in an external magneti field parallel to the superondutor surfae. In many ases of real appliations superondutors, suh as thin films, tapes, oated ondutors, an be simulated by infinitely long thin strips arrying a transport urrent or in an external magneti field [4, 7, 8]. Below we will briefly disuss the main points of the mathematial model developed in [7,8] for analysis of magnetization and AC losses in the strips. Let us onsider a superonduting strip filling the spae x d /, y a, z with d << a (Fig. 4) and assume that the ritial urrent density j is independent of a loal magneti field. Taking into aount that d << a onsideration of a real urrent distribution in the strip ross-setion is redued to onsideration of a sheet urrent determined as [4,7]: d /, J y j x y dx. d / In the frame of this approximation the y-omponent of a magneti field on the superondutor surfae is H d /, y J y (upper surfae, y > ) and H d /, y J y y surfae, y < ) and the x-omponent H d /, y H d /, y H y [4,7]: H y a a J u du H y u x x y (lower and is given by e. (.)

22 The magneti field penetrates through the superonduting strip only in the area, where d /. Where J < J the x-omponent of the magneti field is zero at. d / J J j dx The urrent distribution is sought in the form: J =± J near the strip ends and as a solution of Eq. (.) with H(y) = in a entral part of the strip. The negative magneti moment M and the total magneti flux per length unit of the strip are defined by a (.) a M yj y dy a a a a u H y dy ahe J uln du. (.) a a u I e d H e a x z y Fig. 4 Strip arrying a transport urrent in an external magneti field H e..3. Strip with transport urrent In this setion we onsider a ase when a superonduting strip arries a transport urrent I t in zero external magneti field. A sheet urrent density distribution, as a solution of Eq. (.), is given by the following expression [7]:

23 3 / J a b I artan, y b I J y bi y. (.3) J, bi y a Integrating this one gets the total urrent (transport urrent) / I J a b (.4) t I / and the penetration width b a I / I (.5) I t where I aj is the maximum ritial urrent ourring at full penetration b I =. We may obtain the magneti field profiles by using Ampere s law and Eq. (.3) [4]., y bi / Hy y b I H y artanh, bi y a y a bi / Hy a b I artanh, y a y y bi (.6) where H J /. The results (.3)-(.6) apply to the initial virgin state where I t is inreased from zero. The orresponding expressions for an alternating applied urrent with amplitude I are obtained by looking e.g., at the situation when I t is redued monotonially from I to - I. We an see that the resulting urrent J and field H profiles are linear superpositions of the form [7] e e J y,i,j J y,i,j J y,i I, J H y,i,j H y,i,j H y,i I, J e e (.7) with J y,i,j and e H y,i,j given by Eqs. (.3) and Eqs. (.6). e

24 4 When Ie I is reahed, the original state is reestablished, with J and H having hanged sign, J y, I,J J y,i,j and H y, I,J H y,i,j. In the half period with inreasing I e one has J y,i,j J y, I,J and H y,i,j H y, I,J e e. e e When I e inreases from I to I the eletri field in the strip is E y t y with the magneti flux y is then y H u du. The energy dissipated in the strip during this half yle o / a a a bi b y U dt E y J y dy J dy H u du. (.8) HC Inserting (.6) in (.8) and performing integration one obtains the dissipated power P=fU HC per length unit of the strip [7]: P f I q I / I (.9) q x x ln x x ln x x. where For small and large amplitudes this gives P ln f / 6 I I, I I 4 f / I, I I (.3).3. Strip in a perpendiular magneti field The urrent and field profiles in a thin strip in a perpendiular field H e and with zero transport urrent an be alulated in a similar way as for the urrent arrying strip in Se.3.. Here we present the results for urrent and magneti field profiles for a thin strip in a perpendiular field [7]:

25 5 J H y artan, y b / H J y bh y Jy / y, bh y a, y bh / y bh H y H artanh, bh y a H y H y H artanh, y a / y bh (.3) (.3) where b a / H / H, H / H osh tanh. The results (.3)-(.3) apply to the H e H e initial virgin state where H e is inreased from zero. At the end one an obtain in a similar way as desribed in Se.3. the expression for the dissipated power when the strip is in an external sinusoidal magneti field [7] H P 4 f a J Hg H where g x ln osh x tanh x. x (.33) For small and large amplitudes H this gives P a / 3 H H, H H 4 P 4 f a J H H H H. (.34).4 Further development of Bean s Model Bean s ritial model has been suessfully applied for the alulation of AC losses and analysis of magneti behaviour of low temperature type-ii superondutors and gives a good agreement with experimental data. For high magneti fields deviations between the experimental data and the predition given by this model have been found.

26 6 One of the main reasons for these deviations is the dependene of the ritial urrent density on a loal magneti field. This dependene is partiularly strong for HTSCs and manifests itself even for low magneti fields. The development of the Bean model was direted towards onsideration of the magneti field dependene of the ritial urrent density. One of the widely used models is the Kim- Anderson model [9]: j j H H / on (.35) where j and H on are the onstants of the material. Other widely used models are J J ( / ) n H H, Yeshurun [] J J exp( H / H ), on Fietz [] J J /( H / H ), Watson and Shi [] on J / K / H, Matshushita [3]. The fitting parameters of the models jo, H on, H, K an be determined experimentally from the ritial urrent measurements in DC magneti field, from experimental investigation of the dependene of the AC losses and suseptibility on the magneti field [6], and of the magneti shielding properties of superondutors [4]. In the framework of the models listed above, the Bean dependenies of the losses on the magneti field amplitude are disturbed. Note, however, that the losses per a period are frequeny independent as it is given by the original Bean model..5 Losses in HTSC After the disovery of high temperature superondutivity the Bean model and the other models based on the oneption of the ritial state are used for the desription of the magneti field

27 7 response of new lass superondutors [5]. It is noted in many papers the CSM gives the good agreement with experimental data. However, pronouned deviations from CSM preditions are observed in many experiments. In partiular, the penetration threshold of magneti field depends on the rate of magneti field inrease. A pronouned frequeny dependene of AC losses per yle in HTSC materials was found [6]. It was noted that an HTSC sample demonstrated pronouned deviations from the CSM under varying external onditions..5. Peuliarities of HTSC HTSC have learly defined the granular and the layered struture and hene, these superondutors are highly anisotropi. The granularity is aused by weak links at grain boundaries. These links have a very low J as ompared with granules and magneti field penetrates to the entre of the sample leaving the grains unpenetrated. Other peuliarities of HTSC are assoiated with partiular properties of the vortex lattie desribed in reviews [5,5]. It is noted that the oxidant vaations in HTSC an be the pinning materials. These pinning entres posses a lower potential barrier than defets of the material struture. It redues the ritial urrent density and inreases the effet of flux reep. In HTSC the thermal depinning is observed in wide temperature and urrent intervals below the ritial values. A strong and rapid flux reep, i.e. the thermally ativated motion of vorties, is the harateristi feature of HTSC. This "giant flux reep" [] an be observed till very high eletri fields 3 V/m in omparison with LTSC where the reep is observed only at eletri fields of the order of 6 V/m and in a very narrow range of urrent near the ritial value. The riterion of µv/m for the ritial urrent of a LTS learly determining the demaration line between the flux reep and flux flow regimes loses

28 8 its meaning in HTSC. For HTSC, the transition point from one regime to the other depends on the superondutor type, preparation tehnology and external magneti field. The thermally ativated flux reep leads to a marked deay of persistent urrents in the HTSC sample and dereases the magnetization and the gradient of the magneti field in the superondutor. In LTSC the derease of the magnetization is well desribed by logarithmi law [6]. For HTSC, the law of the derease is dependent on the state of the vortex lattie [7]. In many ases the law is also lose to the logarithmi one [,7]. Formally, this flux reep is equivalent to a nonlinear urrent dependent resistivity ~ exp J / J. A novel feature in HTSC is the (TAFF) thermally assisted flux flow with a linear (ohmi) resistivity that is observed pratially from zero urrent [8]. The both, flux reep and TAFF are limiting ases of Anderson s general expressions for the eletri field E H,T,J whih may be written as [9]: E J exp U / k T sinh JU / J k T, J J (.36) B B is the resistivity at J J where H,T Boltzman onstant. U H,T is the pinning potential; k B is the ; The eletri field in HTSC appears from motion of the vorties. At a urrent density lower than the ritial value, the average vortex veloity is determined by the probability of the diretional jump of the vorties through a potential barrier. This veloity is determined as B v v j / j exp U J / k T (.37) where v is onstant ; U J is the ativation energy. The E J harateristi of a superondutor is determined by the substitution of (.37) in the expression E Hv. Generally the ativation energy depends on the temperature, magneti field, vortex lattie state, and pinning type. For LTSC, the usual expression for the ativation energy follows from the Kim-Anderson model [7,9]. For HTSC, there are a number of different models onfirmed experimentally for different samples. One of the widely used models proposes

29 U J J U J 9 (.38) Where the exponent haraterizes the vortex pinning model: single vortex, vortex glass or olletive reep models. The parameter depends on the properties of HTSC, the external magneti field and also on the urrent density and temperature. In the framework of the Kim- Anderson theory and the ativation energy is: J J U U J. (.39) Theoretial treatment of the olletive pinning showed that there are three different regimes depending on the urrent density J [3]: 7 / 9, j j ; 3 /, j j ; (.4) ~, j ~ j. Other frequently used forms for the dependene of the ativation energy on the urrent U J density are the power law J U J and the logarithmi law U J J U ln J. The last dependene leads to a power law E J harateristi in the flux reep region: E J E J n (.4) U where n. kt B Expression (.4) is frequently used for approximation of a measured E-J dependene in the flux reep regime n and E are determined from fitting experimental data and for an analysis of magneti properties and alulation of AC losses in HTSC [6]. Calulation of AC losses is based on numerial methods or approximately approahes [37].

30 3 All the models listed above have been applied for alulation of AC losses in superondutors under sinusoidal onditions (sinusoidal magneti fields or/and urrents). In reality, urrents It in eletri power systems ontain a wide variety of harmonis and an be represented by Eq (.). Sine superondutors posses a strongly non-linear urrent voltage harateristi, one an expet substantial ontribution of higher harmonis to AC losses in superonduting elements. In the next setions we will onsider AC losses in superondutors under non-sinusoidal onditions. We are going to take into aount the first two harmonis k = or k = 3. The seond harmoni is the main one appearing in two-phase iruits and the third for three-phase iruits. Usually higher harmonis rapidly deay.

31 H(t) 3 3. Analytial alulation of AC losses in superonduting slab under non-sinusoidal onditions The waveforms of non-sinusoidal urrents (magneti fields) an be divided into three types: I-symmetrial ase a urrent monotonially inreases (dereases) from the minimum (maximum) to the maximum (minimum) and the maximum equals the module of the minimum (Fig. 5, line ); II asymmetrial ase a urrent monotonially inreases (dereases) from the minimum (maximum) to the maximum (minimum) and the maximum does not equals the module of the minimum (Fig. 5, line ); III a urrent hange is no-monotoni from the minimum to the maximum (Fig. 5, line 3). The type of the waveform depends on the number of harmonis, their frequenies, amplitudes and phases. If e H k the monotoni behavior is observed at k H k H t H os t H os k t..5 H max A C B H min t Fig. 5 Non-sinusoidal urrent waveform: symmetrial ase; - asymmetrial ase; 3 - non-monotoni ase. In previous hapter we presented the expressions for alulation of AC losses in the frame of CSM in the ases of sinusoidal external magneti field or transport urrent. Using the same approah and mathematial models developed in [8,9,3] we obtained the following

32 3 expressions for the AC power losses for non- sinusoidal monotoni urrent waveforms (symmetrial and asymmetrial ases): (a) losses per surfae unit of a slab in a parallel magneti field are H H 3 max min f, H H H 6 j P 4 H p Hmax Hmin H p 3 f, H H H j max min p max min p (3.) (b) losses per length unit of a thin strip in a perpendiular magneti field are P 4 a fjhhg h, (3.) where h H H H max min, g x is given by Eq. (.33) () losses per length unit of a thin strip with a transport urrent are P fi I I max min q I. (3.3) where q x is given by Eq. (.9) Note, that AC losses aused by a urrent in a slab an be estimated using the first equation of (3.) where values of the magneti field are replaed by the urrent per height unit of the slab. As seen from Eqs. (3.)-(3.3) the losses in superondutors under non-sinusoidal monotoni urrent waveforms are independent of frequeny of higher harmonis, while in the ase of normal metal parts of the devie power of the Joule losses and eddy urrent losses are given by Eqs. (.) and both have a very strong dependene on the frequeny of the higher harmonis. Losses per a urrent period in a superondutor are determined only by the differene Hmax Hmin and are independent of frequeny of the main harmoni and higher harmonis. The AC loss alulation aounting the main harmoni only leads to an error determined by H H H H. the differene max min To a first approximation, the AC losses an be estimated as

33 33 P P KH / H (3.4) where P is the AC losses aused by the main harmoni, K is the oeffiient depending on the shape of a superondutor and on the value of H. So, K = 3 for a slab at H H H ; max min p K = 4 for a strip at Imax Imin I or at H max H min H ; K= for both onfigurations at high magneti fields. A relatively low differene H / H leads to a notieable inrease of AC losses. For example, losses inrease by up to % at H / H. 5. At the same time, in the normal metal, the five perent seond harmoni auses the loss inrease by %. The differene 6). H is determined not only by the harmoni amplitudes but also their phases (Fig. And what is more this differene an be negative (Fig. 6b), hene the appearane of higher harmonis an redue AC losses. At design of superonduting power devies, one should alulate AC losses for the worst ase when the losses are maximal. For k = the maximum of H is ahieved when and inreases as 4. H, for k = 3 the maximum equals H 3 8. at 3. Now, we onsider a non-monotoni urrent hange when additional maximums and minimums appear in the waveforms of magneti field and urrent (Fig. 5, line 3). The field ylially hanges from point A to point B through point C. The AC losses indued during this yle an be alulated using Eqs. (3.)-(3.3) where H max and H min are replaed by H A and H C, respetively. The number of wells and their depths, A C H H, depend on the number of harmonis, their frequenies, amplitudes and phases. If H t H os t H os k t, the maximum well number is k-. The total AC e k k losses per period are a sum of the losses ontributed by all yles. Fig. 7 presents the harateristi values of a two-harmoni waveform for / at k =.

34 (H max - H min )/H max (h 3 =) - (H max -H min )/H max (h =) h = (a) h 3 = (b) Fig. 6 Dependenes of H on phase a) seond harmonis b) third harmonis. At H /H <.5, and H max -H min = H, the losses are determined only by the main harmoni amplitude. If H is higher, the losses inrease due to the growth of H max -H min and the ontribution of an additional, "inner", yle. If H A H C < H p, the last ontribution ahieves 5% at H /H = ; for H A H C >> H p the 5% level is observed at H /H =.4. So, one has to aount the loss part appearing due to wells at relatively high amplitudes of higher harmonis.

35 Normalized magneti field H /H Fig. 7 Normalized harateristi values for H( t ) Hsin( t ) H sin( t / ) as funtions of H / H : - (H max -H min )/H ; - (H A H C )/H ; 3 - (H A H C )/(H max -H min ). 3. Summary The alulation results show that in devies with superonduting elements, higher harmonis an substantially hange the loss values. While, in the normal metal, the 5% seond harmoni auses the loss inrease by %, in superondutors this inrease an ahieve %. Moreover, the ontribution of the harmonis depends on their phases: in a ertain range of phases, the odd harmonis an even redue AC losses. These peuliarities distinguish the behavior of superonduting devies from that of onventional ones. Non-monotoni harater of the waveforms of a urrent or a magneti field influenes AC losses only at very high amplitudes of additional harmonis.

36 4. Coated Condutors 36 In previous hapters we have onsidered an AC loss problem for the model ases (slab, strip) and shown that existene of higher harmonis an ause pronouned hanges of AC losses. For example, the 5% seond harmoni an ause the loss inrease by % in superondutors. Moreover, the ontribution of the harmonis depends on their phases: in a ertain range of phases, the odd harmonis an even redue AC losses. These peuliarities distinguish the behavior of superonduting materials from that of onventional ones. In this setion we will onsider AC losses in the superonduting wire under non-sinusoidal onditions. At present the main andidate for broader ommerialization of HTSC wires is the seond generation (G) YBa Cu 3 O 6+x (YBCO)-oated ondutor [3,3]. These ondutors are produed by deposition of a -3 m film of YBa Cu 3 O 6+x on - m thik metalli substrate, and superonduting film is subsequently overed by protetive silver and stabilizing opper layers (Fig. ). Buffer layers, typially oxides (CeO, MgO, GdZO and et.), are sandwihed between the YBCO and substrate, the buffer layers are funtioned as a texture base, a reation barrier between the YBCO and the metalli substrate layer and a layer prevents diffusion of metal atoms into the superondutor. The nature of buffer layer is seleted depending on the prodution proess of the oated ondutor. The typial harateristis of oated ondutors are presented in Tabl.. The G superondutors are haraterized by very high ritial urrent density, up to 4 MA/m at 77 K, lose to that in the epitaxial thin films. The sheet ritial urrent density is usually of the order of A/m, for the best samples this density an ahieve A/m [3]. The implementation of superondutors in AC power appliations suh as transformers, motors and generators depends ruially on the suess in designing ondutors with low losses in time-varying magneti environment. AC loss values determine ranges of the rated urrents and magneti fields for superonduting devies, required power of ryogeni equipment, and eonomial gain. A lot of investigations are devoted to a onsideration of AC

37 37 losses in oated ondutors and oils winded using these ondutors under different onditions (see for example [33, 34] and referenes in them). However, most of the works onsider sinusoidal magneti fields or/and urrents. In this setion we will analyze losses in oated ondutors under non-sinusoidal onditions when a urrent monotonially inreases (dereases) from the minimum (maximum) to the maximum (minimum) (see Fig. 5, lines and ). Fig. Sketh of a oated ondutor. The typial harateristis of oated ondutors Table Thikness of layer Resistivity at 77 K ( n m) Metal substrate (Ni- - m 4.6 alloy) Buffer layer - nm dieletri SC layer (YBCO) Protetive layer ( Ag) Stabilizer (Cu) 5m - 5 3m.65 3 m.

38 4. AC power losses in oated ondutor 38 The total losses in a oated ondutor are determined as a sum of losses in a superondutor (hysteresis losses) and in a normal-metal parts (eddy urrent losses). For loss alulation the oated ondutors an be presented as a metal strip and a type II superonduting strip plaed one on top of the other as shown in Fig. Losses in oated ondutors will be alulated for ases: (i) a non-sinusoidal magneti field is applied perpendiular to the ondutor wide surfae (in the z-diretion) and (ii) non-sinusoidal urrent flows through the ondutor in the y-diretion without an external magneti field. In this setion losses are estimated in the framework of the ritial state model negleting the response of the normal-metal substrate and stabilization layers. For a sinusoidal magneti field perpendiular to a normal-metal strip the last is valid if ad m, (4.) where is the resistivity of the metal, d m and a are the thikness and half-width of the metal strip. In the mathematial model of a oated ondutor the substrate, protetive silver and stabilizing opper layers an be represented by an effetive normal-metal strip where d m / is replaed by d sub / sub + d sil / sil + d st / st (here sub, sil, st, and d sub, d sil, d st are the resistivities and thiknesses of the substrate, protetive and stabilizing layers, respetively). For a nonsinusoidal magneti field the ondition (4.) has to be valid for eah harmoni taken into aount. If the ondition (4.) is valid, the field produed by a urrent in the normal-metal strip may be negleted and this strip an be onsidered in a total field whih is a sum of the applied field and field produed by a urrent in the superondutor. We assume that the normal-metal and superonduting strips are of half-width a and length l and {d s, d m }<< a << l. The model of infinitely long thin strips an be applied. In the framework of the thin strip approximation the magneti field indued by a urrent in the ondutor is perpendiular to the surfae and is independent of the z-oordinate inside a strip.

39 39 The real urrent distribution through the strip thikness an be replaed by the sheet urrent determined as an integral of the urrent density through the strip thikness [4, 7]. Expressions for AC losses in a superonduting strip under non-sinusoidal onditions were obtained in the previous hapter, see Eqs. (3.) and (3.3). In this hapter we onsider eddy urrent losses in a normal-metal strip and ompare them with hysteresis losses. The peuliarity of this onsideration is that this strip is subjeted to an external non-sinusoidal non-uniform magneti field. l I B d s a d m z y x Fig. Sketh of a oated ondutor (thin strip approximation), where d d a l, s m d s is the superonduting layer thikness 4.. Coated ondutor in a perpendiular magneti field A non-sinusoidal magneti field H e (t) is applied perpendiular to the surfae of a oated ondutor. If an external magneti field monotonially inreases, in the framework of CSM AC losses per a period in a superondutor are determined only by the extreme values and do not depend on the waveform and frequeny. If the ritial urrent density is independent of a magneti field, value of these losses is given by Eqs. (3.). When an external field inreases (dereases) the magneti field is desribed by the following equations, respetively:

40 4 Hz Hz( x, Hmax, J) Hz x, Hmax He,J (4.) Hz Hz( x, Hmin, J) Hz x, Hmin He,J here, (4.3) H z is the z-omponent of the magneti field whih is the sum of the external field and the z-omponent of the magneti field produed by the urrents in the superonduting layer and it is given by, similarly to the ase of a sinusoidal external field [33], the following: H e x b H artanh, b x a H z x,h e,j x, x b (4.4) where b a, tanh He/ H H / H osh e, H =J /. In opposite to a superonduting strip, losses in the normal strip depend on the field extremums as well as on the field waveform. In the onsidered task there are the y- omponents of the eletri field E y and urrent density j y,ed, only. The eddy urrent loss power p ed per unit of volume in an infinity long thin strip is a funtion of x and time t: x, te x t ped jy, ed y,. (4.5) The eletri field and urrent density are related aording to Ohm s law: E y j y, ed. (4.6) The eletri field E y is indued by the time varying magneti field x H z u, t Ey x, t du t. (4.7) The eddy loss power per length unit is: x,t d / a H m z Ped dt dx a t. (4.8) Integration of the last expression through x gives: P ed 3 dma H Fh (4.9) 3 where

41 4 t 3 Fh he dt 3 t osh Y osh Y / t t 3 he dt 3 osh X osh X (4.) where h e e H H t t H He H Y, min H X e H H, max, t, is t are the points at whih the external field reahes its maximum and minimum, respetively. We obtain the following expression for the total loss power (the sum of the loss powers in a superondutor and normal-metal) per length unit in a oated ondutor: P Htot p 3 H Fh hqh dma / 3 H, (4.) where Q 6 g h h h, h h ad, H h. H m Here losses are normalized to the eddy losses aused by the uniform magneti field with the amplitude and frequeny equaled to those of the main harmoni when a superondutor is absent Coated ondutor with a transport urrent Let us onsider losses aused by a non-sinusoidal transport urrent I(t) flowing in the y- diretion without an external magneti field. We study the ase when the urrent maximum I max and module of the urrent minimum I min are less than the ritial value I = aj. It is assumed that the magneti field produed by a urrent indued in the metal is negligible in omparison with the field from the urrent in the superondutor. Losses in the superondutor are given by Eq. (3.3). To alulate the eddy urrent loss power P ied per

42 4 length unit, Eq. (4.8) is used where, at a derease (inrease) of the urrent from its maximum (minimum) till the minimum (maximum), the magneti field is determined as [7]: ~ ~ H x,t H x,i,i H x,i I t, I (4.) z z max z max ~ ~ H x,t H x,i,i H x,i I t, I (4.3) z z min z min, x bi ~ / H z x,i,i x x b, i H artanh, b i x a x a bi where and bi a I I. Using the approah developed in setion 4.. the total loss power per length unit of a oated ondutor arrying a non-sinusoidal transport urrent is: P p F Q i Itot 3 3 I i h i dma / I, (4.4) I where max I i min, Q i (x) = q(x)/i, i = I /I, I / t t F i i t G b i dt i t G b i dt, t t x x G x x x Log Log x x Imax I t, Imin I t b i I b i I, di t it I dt., 4. Results and disussion In the ase of a sinusoidal waveform ( H t H t e sin or I(t) = I sin(t)), the dependene of the funtions F h, F i, Q h and Q i on the magneti field and urrent has been analyzed in

43 43 [7,33]. Under non-sinusoidal onditions when I(t)=I sin(t)+i k sin(kt+ k ), the monotoni behavior of the waveform is observed at I k /I </k. Losses in the normal metal also depend on a phase of the higher harmoni. Our alulation showed that the maximum of the losses is ahieved at the same phases at whih losses in a superondutor are maximal: = for the seond harmoni and 3 = for the third one. At design of superonduting power devies, one should alulate AC losses for the worst ase when the losses are maximal, therefore here we will analyze the ases of the seond harmoni with = and of the third harmoni with 3 =. The numerial integration and graphial presentation were arried out using Mathematia. For monotoni waveforms funtions F h, Q h and Q i, F i are shown in Figs. and 3, respetively. Note, that funtions F h and F i haraterizing the eddy losses as well as funtions Q h and Q i orresponding to losses in superondutor whih are independent of frequeny of the main harmoni. Despite this fat the total losses in a oated ondutor are frequeny dependent: eddy urrent losses are proportional to and losses in a superondutor ~. Using our normalization it is learly shown that the relative ontribution of losses in a superondutor losses depend on the parameter h whih is frequeny dependent, ~. Thus the relative ontribution of AC losses in a superondutor is given h by the inverse relation of the main harmoni frequeny. The influene of higher harmonis on losses in a superonduting strip in perpendiular magneti field has been analyzed in hapter 3, where it was shown that the hysteresis losses per a urrent period are independent of frequeny of the harmonis. These losses depend only on the maximum and minimum of the urrent or magneti field. Figs.4 and 5 present the relative ontribution P P P / P of higher harmonis to losses (here P is losses aused by the main harmoni). The ontribution is maximal at a low magneti field and dereases when the field inreases. For example at H H. 5 for the third harmoni, the ontribution to AC losses ahieves % in a superondutor and 35% in the normal metal parts (Fig 5, green line). At a high field when sreen properties of a superondutor an be negleted, eddy

44 44 losses are determined as for a normal-metal strip in the uniform external non-sinusoidal field and the relative ontribution of the k-th harmoni is given by k P k H H and for the 5% third harmoni losses in the metal inrease by % and in the superondutor by 5%. F h,q h F h, Q h h h a) b) Fig. Dependenes of the funtions F h and Q h on amplitudes of two first harmonis: a) the ase of the main and seond harmonis at = (blue, Q h, and red, F h, lines at H =; blak, Q h, and green, F h, lines H =.5 H ) and b) main and third harmonis at 3 = (blue, Q h, and red, F h, lines at H 3 =; blak, Q h, and green, F h, lines at H 3 =. H ). F i,q i F i,q i i i a) b) Fig. 3 Dependenes of the funtions F i and Q i on amplitudes of two first harmonis: a) the main and seond harmonis at = (blue, Q i, and red, F i, lines at I =, blak, Q i, and green, F i, lines at I =.5 I ) and b) main and third harmonis at 3 = ( blue, Q i, and red, F i, lines at I 3 =; blak, Q i, and green, F i, lines at I 3 =. I )

45 P 45 P h h a) b) Fig. 4 Relative ontribution of the seond harmoni of the magneti field to losses in a superondutor (a) and normal-metal parts (b): blue line H =.5 H ; green H =.5 H, red line H 3 =.5 H. P P h h a) b) Fig. 5 Relative ontribution of the third harmoni of the magneti field to losses in a superondutor (a) and normal-metal parts (b): blue line - H 3 =. H ; green H 3 =.5 H, red H 3 =. H Fig. 8 shows the dependene of losses in a superondutor as funtion of third a) magneti field and b) transport urrent harmonis, qualitatively similar behavior is expeted for the seond harmonis. Another dependene is obtained for a oated ondutor with a transport urrent (Figs. 6, 7, 8b). At I max -I min <<I losses in a superondutor are determined by the expression similar to (3.4) with K = 4, where the magneti field values are replaed by the orresponding urrents. At I/I =.5 the losses inrease also by about %. However, in ontrast to the ase of a magneti field, the relative ontribution inreases with the urrent and ahieves about 35% at I I (Fig. 7a, blue line). Qualitatively similar behavior is observed

46 46 for losses in the normal-metal parts: the relative ontribution inreases with the urrent (Figs. 6b and 7b) P P i i a) b) Fig. 6 Relative ontribution of the seond urrent harmoni to losses in a superondutor (a) and normal-metal parts (b): red line I =.5I, blue line I =.5I, green I =.5I. P P i i a) b) Fig. 7 Relative ontribution of the third urrent harmoni to losses in a superondutor (a) and normal-metal parts (b): red line - I 3 =.I, blue line - I 3 =.5I, green - I 3 =.I. a) b) Fig. 8 Relative ontribution of higher harmonis to losses in the superonduting part of a oated ondutor: a) ontribution of the third magneti field harmoni; b) ontribution of the third urrent harmoni.

47 47 However, the relative ontribution an be more substantial. For example, the 5%-third harmoni auses inrease of eddy losses by about 35% at low urrents and ahieve about 8% at I (Fig. 7b, blue line). At the same time losses in a superondutor inrease by from I % up to 45% (Fig. 7a, blue line). The total losses in a oated ondutor are determined by the sum of the losses in the superonduting and normal-metal strips. The relative ontributions of these parts are determined by the ratio F h and Q h in ase of the magneti field and by the ratio F i and Q i for the urrent as well as by the onstant h. At h << both funtions, F h and Q h, are well fitted by power laws with exponents 4 and, respetively. At h >> Q h dereases proportionally to /h while F h tends to the limiting value +(kh k /H ). Thus, the relative ontribution of eddy urrent losses to the total losses inreases with the applied field (Fig. ). In spite of the fat that both funtions F i and Q i inrease (Fig. 3) with the urrent, the relative ontribution of losses in a normal-metal strip to the total losses also inrease. For example, for the 5% third harmoni the ratio F i /Q i =. at i =. and this ratio inreases up to about.6 at i =.9. Let us estimate the parameter h whih is the same in the expressions for the total losses in both ases and depends on a frequeny of the main harmoni and harateristis of the normal-metal strip (resistivity, thikness, and width) only. For a oated ondutor without protetive silver layer and stabilizer [33] the parameter h is evaluated as ( 6 s - )/f, for a ondutor with the silver layer with thikness of ~ k [35] h (8 4 s - )/f, for a well stabilized oated ondutor with ooper stabilizer thikness of k [36] the parameter h is evaluated as (.6 3 s - )/f. The estimations were done for the oated ondutor with the width of m, the resistivity of a hastelloy-c substrate was taken.4-6 m and resistivity of silver and ooper -9 m at 77 K. One an see that the losses in a superondutor dominate in the two first ondutor types till high frequenies of the order of MHz. The losses in the normal-metal parts of a well stabilized oated ondutor an be omparable or even dominate at eletrotehnial

48 48 frequenies ~ khz (the range of frequeny is used in speial eletri power systems suh as airplanes, ships, et.). At 5-6 Hz losses in the normal-metal parts will dominate in magneti field h > 3, e.i. when H >. T at the typial value of J = 4 A/m ( H =.4 T). This value is muh less than the working magneti fields in many power devies, for example, a working field in a generator is of the order of T, and losses in the normal metal parts of a oated ondutor dominate. In superondutor power ables the working fields are of the order of.3 T and the total losses are mainly determined by losses in a superondutor. Basing on the fat that AC loss values in a normal-metal parts an not exeed losses in a superondutor. Thus, AC loss values in a nominal regime of power devies imposes a restrition on the possible thikness of the stabilization layers. For a oated ondutor without a protetive layer AC loss value in normal metal parts prevails losses in superondutor at the typial value of T whih is signifiantly higher than all the working magneti fields in modern power devies. 4.3 Analytial approah for AC losses in striped oated ondutors A suffiiently AC-tolerant YBCO oated ondutor an enable the fully superonduting version for motors and generators in whih both the field and armature windings are superonduting [3]. From expressions (3.) and (3.3) one an see that the value of hysteresis losses is proportional to the strip width in square. The width of the developed oated ondutor is m with plans to redue it to 4 mm. When suh a wide superonduting tape is exposed to time varying external field a large amount of heat is generated [5,37]. Thus, a signifiant redution of hysteresis losses in oated ondutors is a prerequisite for their use in AC power appliations. It is also important for suh a modifiation to be ompatible with urrent tehniques of manufaturing the oated ondutors. To redue hysteresis losses in tapes several millimeters wide, subdividing of the

49 49 tape into narrow parallel strips has been proposed [37]. The resulting ondutor is a multifilamentary tape with parallel thin strips (filaments) separated by narrow gaps (Fig. 8). Let us onsider losses aused by a perpendiular non-sinusoidal magneti field in a striped oated ondutor. a) b) Fig.9 a) One m wide uniform oated ondutor and 33-filament striated sample shown side by side the top visible layer is silver ; b) mirophotograph of the ross-setion of the striated sample in the groove area. The Hastelloy substrate, YBCO and silver layer are indiated. It was shown [6] that the losses an be presented as a sum of the loss in a normal onduting substrate, the transverse oupling urrent loss and the loss in the filamentary strips. For a suffiiently high magneti field these losses an be estimated separately assuming that the magneti field equals the external one. To find the area of appliability of this approah we will use the solution obtained in the framework of the model of a thin strip [7]: PxB P g h (4.5) where P f a J H. 4 At h, g h. 386 / h. The last term,. 386 / h desribes the derease of losses during the lower field part of a period due to partial field penetration and also due to the differene among the loal magneti field to the external one. The term an be negleted at h 5 with an auray of %.

50 The value of 5 J is typially of the order of 4 A/m for modern oated ondutors [3]. Therefore H ~ 4 T and the approximation that the magneti field equals the external one an be used for external magneti fields with amplitudes higher than.6 T, typial for transformers, eletrial motors and generators. Under this assumption, the oupling loss and loss in a substrate per length unit are given by the following expression P ed K f H p k h k k (4.6) where K f H / is the eddy losses aused by the main harmoni, p haraterize the k dependene of the losses on frequeny. Generalizing Eq. (44) from [5], we obtained for both losses p k k, for a strip K = ad m 3, while for the oupling losses per the length unite this oeffiient is a L 3 N Rg, where L is the ondutor length or the twisting step length, N is the number of filamentary strips, R g is the resistane between two neighboring filaments. Losses in superonduting strips an be estimated using the results obtained above Power law superonduting strip The results presented above have been obtained in the framework of the ritial state model with the ritial urrent density independent of a loal magneti field. A more realisti piture is given by the models based on a power law fitting of the voltage-urrent harateristi of a superondutor or/and taking into aount the dependene of the ritial urrent on a magneti field. It was shown [6] that at high magneti fields the loss values obtained for power-law dependene even with a high index of power (-3) are markedly different from those given by the ritial state models. Let us analyze the influene of higher harmonis on losses in a thin superonduting strip with the power-law relation between the sheet urrent

51 5 density J and eletri field E and dependene of the ritial urrent on a magneti field aording the Kim-Anderson model: n E E J J, J H J b, (4.7) H Hb where E = V/m is the eletri field aused by the ritial urrent density j and j is its value at H =, H b is a onstant determined by the properties of the superondutor. At high magneti fields losses per unit of the strip length are /n n wh P P F, 4n E (4.8) /n h k os k kt T k F k dt, h b h k sinkt k k where hb H / Hb, P f J a H is the loss given by CSM in asymptotially high 4 magneti fields. The dependene of losses on the properties of a superondutor, n, E, H b, has been investigated in [6] for a sinusoidal magneti field. The influene of higher harmonis on losses appears in the value F of the integral in (4.8). For a sinusoidal field (h k = for k >) and at n, h b =, (the ritial state model), this integral equals 4 and P = P. Let us analyze the value of F for two ases of a two-harmoni waveform: (a) k =, ; (b) k =, 3. For pratial appliations it is important to determinate the maximum attainable values of AC losses. F has the maximum at = in the ase (a) and 3 = in the ase (b). The results of numerial integration are presented in Fig. at h b = and ( =, 3 = π). The value of F dereases with an inrease of h b approximately as /(+.85 h b ).65 for both ases and inreases with the amplitudes of higher harmonis proportional to +a h.8 for the first ase and +a 3 h 3 for the seond ase. The oeffiients a and a 3 depend on the power index n and parameter h b. For h b < and n >, within the auray of %, a and a 3 an be

52 5 taken.3 and.9, respetively. The inrease of n above does not lead to a marked growth (the differene of the values of F at n = and n = 4 is less than %). a) b) Fig. Dependene of F on n and a) h for H t H sin t H sin t sin sin 3 H t H t H3 t., b) h 3 for To determine the appliability of the approah developed in this setion let us ompare the asymptoti result for n>> and h b = (Eqs. (4.8)), and analytial exat results obtained for CSM (see setion 4..). Fig. shows that for the first harmoni amplitude higher than 5H one an use the approximation of asymptoti high magneti field. P.5 P h 5 5 h a) b) Fig. Relative ontribution of higher harmonis to AC losses vs. amplitudes the main harmonis: a) the seond harmonis; yan line CSM, blue - asymptoti high magneti field approximation, b) the third harmonis: yan line CSM, blue - asymptoti high magneti field approximation.