Abstract:
A superconductor coating inclusive, tape-like electrical conductor and windings using such conductor for magnets and electrical machines, etc. The described windings are suited for inclusion of successor superconductor materials such as yttrium barium copper oxide wherein magnetic flux related losses can potentially be excessive and preclude successful machine operation. Winding orientation and configuration of the conductor in an alternating current machine for lower losses are disclosed along with methods and apparatus for achieving the desired windings. Windings intended for differing locations within a machine of this type are made possible by the invention. Equations relating to magnetic losses incurred in such windings are also disclosed.

Description:
RIGHTS OF THE GOVERNMENT 
     The invention described herein may be manufactured and used by or for the Government of the United States for all governmental purposes without the payment of any royalty. 
    
    
     BACKGROUND OF THE INVENTION 
     High temperature superconducting (HTS) electrical motors, generators, and transformers can be significantly lighter in weight and smaller than their conventional counterpart machines. Continuing development of such machines is needed for use in advanced military and civilian equipment especially in the development of space and airborne systems where both weight and size considerations are of prime importance. The use of winding materials based on a flat uniform or filamented tape of high current density second generation superconductor materials such as yttrium barium copper oxide (YBCO, YBa 2 Cu 3 O 7x ) in the machine windings appears to offer a promising avenue toward machines of the needed types. 
     Two major shortcomings of superconductors, such as yttrium barium copper oxide coated superconductors, need to be overcome, however, in order to permit their widespread implementation into alternating current electrical machinery applications such as armature and field winding in motors and generators and transformer windings [1]. (Bracketed numbers such as this [1] refer to the list of reference documents appearing at the end of this specification; these documents and each other document identified in this text are hereby incorporated by reference herein.) 
     One issue associated with yttrium barium copper oxide coated conductors, manufactured in the form of thin and relatively wide tapes, for example, is the high hysteresis loss occurring when such a conductor is disposed in a time-varying magnetic field. Another issue concerns attendant mechanical properties of the conductor that are very different from the properties of traditional material such as copper Litz wire. Bending strain limitations restrict the types of winding configurations that are possible when such conductors are compared to copper. A route to hysteresis (and overall) loss reduction explored in recent years is replacement of the uniform wide yttrium barium copper oxide film with a set of parallel narrow filaments or stripes or striations [2-6]. Early work has suggested that in time the hysteresis loss in experimental multifilamentary samples can be reduced by at least an order of magnitude. 
     Notwithstanding such hysteresis loss improvement however, another type of loss specific to multifilamentary coated conductors—i.e., coupling loss—can become comparable in size to the hysteresis loss at a sweep rate Bf of a few Tesla per second when the conductor twist pitch is for example equal to 20 centimeters (here B is the amplitude of the magnetic field and f is the field change frequency) [1]. In order to achieve a substantial—i.e., one or two orders of magnitude—reduction in total losses (hysteresis and coupling) at an operating sweep rate of at least 10 Tesla per second, measures need to be taken to reduce both hysteresis and coupling losses. 
     Another shortcoming of coated superconductors is their low tolerance to bending and twisting strain. This conductor characteristic requires an almost complete reexamination of the winding techniques used with such conductors. The problems of alternating current losses and mechanical properties of the conductor become intertwined because twisting of the multifilamentary conductor is necessary in order to limit coupling losses. The present invention presents novel approaches to arranging magnets and coils with second generation superconductors such as yttrium barium copper oxide coated conductors. 
     SUMMARY OF THE INVENTION 
     The present invention provides conductor geometry and winding arrangements improving on the performance of superconductor based winding materials; the invention is particularly concerned with high temperature superconductor coated conductor winding material, for example, the yttrium barium copper oxide-coated superconductor. 
     It is an object of the present invention therefore to provide second generation superconductor materials in alternating and direct current electrical machines. 
     It is an object of the present invention to provide second generation superconductor winding configurations for alternating and direct current electrical machines. 
     It is an object of the present invention to provide magnet and coil windings usable in electromagnetic applications in general. 
     It is an object of the present invention to provide second generation superconductor winding arrangements usable in both rotor and stator portions of an electrical machine. 
     It is another object of the invention to provide exemplary processes for forming superconductor windings while observing restrictive properties of the materials used. 
     It is another object of the invention to provide a convenient method for fabricating high temperature superconductor coated conductor winding materials. 
     It is another object of the invention to provide a method for achieving electrical winding arrangements usable in a plurality of electrical machines. 
     It is another object of the invention to provide tools for achieving desirable superconductor inclusive electrical winding arrangements. 
     These and other objects of the invention will become apparent as the description of the representative embodiments proceeds. 
     These and other objects of the invention are achieved by a superconductor film inclusive alternating current electrical machine winding comprising the combination of: 
     a superconductor film layer included tape-like electrical conductor having a tape width, W, greater than a tape thickness, T, said conductor being disposed into a magnetic pole-generating plurality of turns of said machine winding; 
     said machine winding electrical conductor tape including a plurality of lengthwise extending segregated parallel filament striations disposed across said tape width, W into said superconductor film layer; 
     said electrical conductor tape plurality of turns each including a filament striation direction-altering winding turn curvature portion wherein each generally coplanar and parallel filament active segment striation curves into an inactive segment filament striation interconnection region of cusp like profile and interconnecting conductors having parallel disposition and orthogonal orientation with respect to said filament active segment striations. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWING 
       The accompanying drawings, incorporated in and forming a part of the specification, illustrate several aspects of the present invention and together with the description serve to explain the principles of the invention. In the drawings: 
         FIG. 1  includes the views of  FIG. 1   a  and  FIG. 1   b  and shows a cross sectional view of a yttrium barium copper oxide inclusive superconductor element and comparison views of non striated and a striated superconductor elements respectively. 
         FIG. 2  includes the views of  FIG. 2   a  and  FIG. 2   b  and shows views of two pancake coils according to the present invention. 
         FIG. 3  shows a conductor material disposed in a difficult superconductor bend pattern. 
         FIG. 4  includes the views of  FIG. 4   a  and  FIG. 4   b  and shows two views of a slit superconductor winding element. 
         FIG. 5  shows a winding mandrel or bobbin for a conductor as shown in  FIG. 4 . 
         FIG. 6  includes the views of  FIG. 6   a  and  FIG. 6   b  and shows a comparison view of tape like superconductor material having an axial twist and a bending twist respectively. 
         FIG. 7  shows striated superconductor details. 
         FIG. 8  shows details of a superconductor coil winding. 
         FIG. 9  shows a coil winding formation method. 
         FIG. 10  includes the views of  FIG. 10   a  and  FIG. 10   b  and shows two alternate arrangements for joining superconductor inclusive materials into a single winding conductor. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Second generation high temperature superconductor structures may be formed into wires and tape-like conductors in which a thin superconducting film is deposited on a metallic substrate. On top of the superconducting film a layer of silver and copper, a stabilizer, is often attached. An example of such a coated conductor is shown in the  FIG. 1(   a ) drawing herein wherein a drawn representation of a profile microphotograph of an alternating current superconductor material is shown. In this  FIG. 1  drawing a substrate made of for example Hastelloy material appears at  100 , a thin layer of yttrium barium copper oxide superconductor material appears at  102  and a silver protective layer is shown at  104 . The groove at  106  in  FIG. 1  represents a laser ablation achieved electrical segregation between adjacent striation conductors and is accomplished for electrical insulation purposes; more on this topic later herein. 
     Coated conductors with a non-segregated superconducting layer can be used in direct current winding applications, such as in field coils for motors and generators. In alternating current winding applications, such as in transformers, alternating current transmission lines and armatures of motors and generators, the winding conductors are exposed to time-varying magnetic field. This exposure can lead to large energy losses through hysteresis effects. In order to decrease magnetic hysteresis losses in alternating magnetic fields a superconducting film can be subdivided into thin filaments or striations. The resulting multifilamentary structure of the conductor may be described as a tape including parallel thin strips of high temperature superconductor material separated by non-superconducting, resistive barriers. Such material as achieved by laser ablation is represented in  FIG. 1   b , right hand conductor; in  FIG. 7  and indeed in  FIG. 1   a  herein. The left hand conductor in  FIG. 1   b  is of the non striated superconductor type, the  FIG. 1  background represents a standard U.S. penny coin included for size comparison with a typical winding conductor. The hysteresis loss in a superconducting tape is directly proportional to the width of the tape when the tape is fully penetrated by a magnetic field. When the superconductor tape is subdivided into striations or filaments as in  FIG. 1   b , the hysteresis loss is directly proportional to the width of an individual superconducting filament. 
     An example of a low loss high temperature superconductor tape patterned into a multifilamentary structure and subdivided by electrically resistive barriers as shown at  106  in  FIG. 1  is described in reference [1] herein. Thus, in order to decrease the loss and/or accommodate the mechanical limitations of second generation superconductor materials such as yttrium barium copper oxide coated conductors, the use of multifilament coated conductor in combination with the new approaches to winding coils is desirable. Such windings for rotor and stator portions of a rotating machine and for static “machines” such as transformers or solenoids may take on several different physical forms as are disclosed in the following paragraphs. 
     Double Pancake Coil 
     The double pancake coil is a preferred form of making the field coil windings in rotating machinery because each of the coil ends are located on the coil exterior as is opposed to being located in outside and inside locations as occurs in a simple single pancake coil [2].  FIG. 2   a  of the present drawings illustrates how the winding start turn  202  remains within the interior of a coil  200  and the winding finish turn  204  is located on the coil exterior in a single pancake coil winding. In contrast  FIG. 2   b  in the drawings illustrates how a first winding terminal  208  and a second winding terminal  210  each remain on the exterior of a double pancake coil formed around the winding mandrel  212  if the winding connection arrangement described subsequently herein is used. 
       FIG. 3  in the drawings illustrates a winding interconnection arrangement often used in forming the turns of a high current winding portion  300  when the winding in question is made of flexible material such as copper.  FIG. 3  type winding shapes often appear in automotive starter motors for example. The interconnection region  306  of the winding portion  300  is however of special concern when superconductor materials are used. There are indications in reference [3] for example that the  FIG. 3  illustrated lateral or sideways bending, in the region  306  of  FIG. 3  where the innermost turns of a double pancake coil join the outermost turns, degrades the current-carrying capacity of first generation superconductor wires, conductor wires based on the bismuth strontium calcium copper oxide superconductor material. Making a double pancake coil from the second generation wire preferred for present invention use appears even more problematic with the  FIG. 3  bending situation because of the much higher rigidity of present invention flat metal tape with respect to the sideways bending shown in  FIG. 3 . This leads to one aspect of the present invention. 
       FIG. 4  in the drawings shows one way to overcome the  FIG. 3  problem of lateral deformation in flat wide superconductor inclusive tapes such as are preferred for the present invention. In the  FIG. 4   a  portion of  FIG. 4  a long tape conductor  400  is shown to be cut into two branches  402  and  404  with the remaining (uncut) part of the conductor tape at  406  allowing the current to flow between the tape branches (as is shown by the arrow  408 ). If W and L are respectively the width and length of the initial conductor, as shown at  410  and  412  in  FIG. 4 , the resultant cut conductor will have approximately a width of W/2 and will be twice as long as the initial conductor segment. 
     The uncut area  406  in the  FIG. 4  conductor need only be (W×W/2) or greater in dimensions in order to maintain the same current-carrying capacity (i.e., the critical current characteristics) in the conductor as that in both conductor branches. Coated conductors are currently produced as wider tapes that may be later mechanically sliced into conductors of a desired width. For example, the manufacturer SuperPower Inc. of 450 Duane Avenue, Schenectady, N.Y., [4] makes 12 millimeter wide tape which may be sliced into three 4 millimeter wide tapes of the same length in the manner of  FIG. 4   a . According to the invention therefore, a 12 millimeter wide tape of length L can be sliced as shown in  FIG. 4   a  into a conductor of approximately 6 millimeters width and length  2 L.  FIG. 4   b  shows a view of a partially expanded conductor  400  cut as shown in  FIG. 4   a  and ready for use in winding fabrication. The two different conductor surfaces are also apparent in the  FIG. 4   b  drawing. 
     In  FIG. 5  a useful tool in working with conductor of the  FIG. 4  type is shown. In the  FIG. 5  instance, the end face of the coil former  500  includes a radial slot  502  and two quarter circle (of radius r) curvilinear slots  504  and  506  leading to the outer rim or periphery  508  of the coil former  500 . In the coil former or mandrel or bobbin  500  the quarter circle slots  504  and  506  may be permanent, as achieved by cuts in the mandrel  500 , or formed by fixed or removable inserts. Similarly the caps  510  and  511  may be embodied as permanent or removable members. In a later drawing herein the use of a cap structure that is split in the thickness dimension is found to be advantageous. The coil former or mandrel  500  may be fabricated from metallic or nonmetallic materials including the reinforced phenol based easily machined plastic materials depending on the tool life needed. A wood based coil former or mandrel  500  has proven useful for early development of the invention. 
     In using the  FIG. 5  coil former  500 , the joined or uncut part of the  FIG. 4   a  and  FIG. 4   b  conductor  400  is inserted into the slot  502  and each branch is then threaded though the respective quarter circle curvilinear slots  504  and  506  to be wound in opposite directions around the outer rim or periphery  508  of the coil former  500 . In the resulting winding, the conductor experiences only the bending strain determined by the radius of curvature of the slots at  504  and  506 , but no lateral strain. A small section of the conductor inside the quarter circle slots  504  and  506  (of r dependent length) is doubtless subjected to the largest strain as is determined by the radius of the slots  504  and  506 . It is notable that in double pancake winding coils of the types implied in either  FIG. 4  or  FIG. 5  the conductor portions leading into and leading out of the illustrated winding interconnections are indeed additive in nature with respect to their magnetic flux generating characteristics. This follows from the constant flow direction of the winding currents and may be further verified by way of application of the usual right hand rule concerning current flow and generated magnetic flux. 
     When a superconductor is exposed to a time-varying magnetic field it suffers high losses. In order to reduce these losses in coated conductors the uniform superconducting layer should be replaced by a number of parallel superconducting filaments (stripes). In such multifilament coated conductor the total magnetization loss is the sum of losses in the superconducting layer Q s  and in the normal metal Q n  of the superconductor substrate (predominantly the coupling loss). In the limit of full field penetration this magnetization loss is given by the relationship [1] 
     
       
         
           
             
               
                 
                   
                     Q 
                     = 
                     
                       
                         
                           q 
                           s 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         Bf 
                       
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                           q 
                           n 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ( 
                             Bf 
                             ) 
                           
                           2 
                         
                       
                     
                   
                   ; 
                   
                     
                       q 
                       s 
                     
                     ≈ 
                     
                       
                         W 
                         n 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         I 
                         c 
                       
                     
                   
                   ; 
                   
                     
                       q 
                       n 
                     
                     = 
                     
                       
                         
                           π 
                           2 
                         
                         6 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           L 
                           2 
                         
                         
                           R 
                           eff 
                         
                       
                       ⁢ 
                       
                           
                       
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                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
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     Here I c  is the critical current, W n  is the width of an individual stripe, L is half of the twist pitch, R eff  is the phenomenological effective coupling resistance that characterizes the coupling loss, and W is the width of the conductor. In a uniform magnetic field a twisted conductor exposes half of its length to the magnetic field face-up and another half face-down. As the result the current loops induced in the superconducting filaments by the changing field have length equal to half of the twist pitch. The length of the current loops L determines the coupling losses according to Eq. (1). 
     Reduction of the coupling loss can be achieved by increasing conductor effective resistance R eff  and by twisting the conductor. In the present invention we concentrate on the latter part of this two-prong effort. 
     Usually in the literature [5] one can find a description of the axial twist shown in the conductor of  FIG. 6   a  herein. In this case the tape is twisted about its longitudinal or long axis. In  FIG. 6   b  we show again another deformation option—a conductor “bending twist” arrangement. The bending twist is obtained in the same way as for the conductor shown in  FIG. 4   a  and  FIG. 4   b . In an applied magnetic field the conductor shown in  FIG. 6   b  will expose each conductor face (indicated by denser and thinner conductor shadings in the drawing) to the magnetic field. The effect of the bending twist on coupling loss is thus the same as that of the regular axial twist shown in  FIG. 6   a . Both types of twist reverse the effective direction of the magnetic flux through the superconductor tape, thereby reducing the coupling losses. Each type of twist has its advantages and disadvantages. There are situations where one twist may be more suitable than the other. In certain situations both types of twist may be employed in order to achieve the maximum benefit. 
     A type of striation that can be used in conjunction with the bending twist is shown in the drawing of  FIG. 7 . Here a 12 millimeter wide conductor  700  is divided at  706  into 0.5 millimeter wide athletic track lane-like parallel stripes e.g.  702  and  704  by for example laser ablation. The resulting conductor is similar to that shown in the  FIG. 1  drawing herein. The  FIG. 7  conductor is then cut along the centerline so that the two branches can be bent in opposite directions to form a bending twist as shown in the  FIG. 4   b  and  FIG. 6   b  drawings. The  FIG. 4   b  and  FIG. 6   b  bending twist conductor arrangement may also be described as a cusp like profile or a winged seagull profile as a result of the overall appearance of the conductor. The name “cusp” appears in the mathematics field and is defined, for example, in a classic  1950 &#39;s Thorndike-Barnhart dictionary as “A pointed end.” The  FIG. 7  conductor includes magnetically active regions  708  and  710  that contribute to the magnetic pole being generated and magnetically inactive region  712  that is oriented orthogonally to the active regions  708  and  710 . The magnetically active regions  708  and  710  may extend beyond the lengths shown in  FIG. 7  in the manner suggested in the  FIG. 9  discussion and in order to connect with additional oppositely oriented right-facing magnetically inactive regions  712 , Such a length of alternately right facing and left facing active regions  712  joined by active regions of the  708  and  710  types may be appreciated to define a meander pattern. 
       FIG. 8  in the drawings shows a double pancake coil made from a conductor of the type shown in  FIG. 4   a ,  FIG. 4   b  and  FIG. 6   b  using a mandrel of the type shown in the  FIG. 5  drawing. In  FIG. 8  the two branches of the coil are shown by the conductor length surface shading used to be transposed (twisted) with respect to each other as is described in the  FIG. 6  discussion above. The  FIG. 8  double pancake coil also uses the  FIG. 4  described conductor interconnection arrangement to advantage in achieving the side by side or axially displaced windings of a double pancake coil. 
     Coil Construction 
     A double pancake coil of the type shown in  FIG. 8  can be fabricated from a coated conductor as shown in  FIG. 4   a ,  FIG. 4   b  and  FIG. 7  as follows. A fabrication mandrel  900  inclusive of one or two removable caps  902  and  904  is shown in the  FIG. 9  drawing. Initially a conductor length  906  is wound on a reel  908  as shown in  FIG. 9 ; the uncut area is wound on to the periphery  910  of the reel. The uncut area of the reel-contained conductor is then placed in the radial slot  912  of the mandrel  900 , then the conductor is covered by the removable caps  902  and  904  and both conductor branches are wound on to the mandrel  900 . When the conductor  906  is almost completely wound on the mandrel  900 , one of the branches is disconnected from the reel  908  and attached temporarily to the periphery of the mandrel coil. Then the direction of rotation of the mandrel and the reel are reversed and the branch yet attached to the reel is wound back on to the reel. The appropriate one of the caps  902  and  904  is then temporarily removed allowing the mandrel  900  to continue to continue to rotate and is then replaced while rotation continues in the same direction. 
     Pancake Coil with Resistive Joint. 
     In some situations it may be advantageous to make a pancake coil using conventional second or first generation tape-like conductors. In order to avoid a hard bend of such conductor, two conductors can be spliced using a wider segment of coated conductor as is shown in the drawing of  FIG. 10 . For example, two long 4 millimeter wide coated conductors  1002  and  1004  can be soldered to a 12 millimeter wide coated conductor  1006  as is shown in the  FIG. 10  drawing. Such conductors need to be soldered “face to face”, so that the yttrium barium copper oxide layers of the soldered conductors are separated by the minimum amount of normal metal including the silver cap layer and copper stabilizer as shown in  FIG. 1  herein. The length of the soldered conductor part  1006  in  FIG. 10  can be comparable to the internal circumference of the coil and therefore can be of several centimeters extent. Unlike the previously described  FIG. 3  arrangement wherein the superconductor current path is entirely superconducting, in the  FIG. 10  arrangement the transitional area between the two conductor branches has finite electrical resistance. The advantage of this arrangement which makes use of the wider coated conductors manufactured today is that the resistance of the joint can be made very small. The current between the two  FIG. 10  branches flows through the superconductor. Therefore, the resistance of the joint does not depend on the distance between the two branches. The resistance of the  FIG. 10  joint is determined by the resistance of the interface between the yttrium barium copper oxide and the silver cap layer and by the normal resistance of the solder and stabilizer in the second generation superconductor of the present invention: 
     
       
         
           
             
               
                 
                   R 
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           R 
                           0 
                         
                       
                       Wl 
                     
                     + 
                     
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         d 
                       
                       Wl 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Here R 0  is the interface resistivity. As shown in Reference [6] the value of R 0 ≈5×10 −8  Ωcm 2  is appropriate. Here p and d respectively are predominately the resistivity and thickness of the copper stabilizer. At a temperature of T=77 K the resistivity of copper ρ≈0.2×10 −6  Ωcm. The thickness of the stabilizer d≈80 μm. Thus, ρd≈1.6×10 −9  Ωcm 2 , which is much smaller than the interface resistivity and, therefore, the main contribution to the resistance of the joint is the interface resistance. If the width of the conductors is 4 mm and the length l along which they are soldered to the connecting coated conductor l=1 cm, the resultant resistance is R≈2.5×10 −7 Ω. If, for example, the current I flowing through these spliced conductors is 100 A, the total power dissipation: Q=RI 2 ≈2.5×10 −3  W, which is an acceptable level of power loss. 
     If the superconductor containing conductors are spliced as shown in  FIG. 10   a  the resultant conductor can be wound as a conventional double pancake coil. The conductors spliced as shown in  FIG. 10   b  are similar to the all-superconducting tape as shown in  FIG. 4   a  and  FIG. 4   b  and can be wound as described above. 
     SUMMARY 
     Herein is presented a novel approach to accomplishing a bending twist of tape-like conductors similar to the 2 nd  generation YBCO coated conductors. The construction of both superconductor DC field coils and AC transformer coils, as well as superconducting stator windings may benefit from the described approach. The approach is based on an unusual manner of cutting wide sheets of coated conductors into narrow tapes of filaments or striations as has been illustrated in  FIG. 1  through  FIG. 3 . Although the illustrations given here are simple, they indicate the potential of new winding configurations based on coated conductor technology. 
     The foregoing description of the preferred embodiment has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Obvious modifications or variations are possible in light of the above teachings. The embodiment was chosen and described to provide the best illustration of the principles of the invention and its practical application to thereby enable one of ordinary skill in the art to utilize the inventions in various embodiments and with various modifications as are suited to the particular scope of the invention as determined by the appended claims when interpreted in accordance with the breadth to which they are fairly, legally and equitably entitled. 
     REFERENCES 
     Each Hereby Incorporated by Reference Herein 
     
         
         1. G. A. Levin, P. N. Barnes, J. W. Kell, N. Amemiya, Z. Jiang, K. Yoda, and F. Kimura, Appl. Phys. Lett. 89, 012506 (2006) (Including references identified therein) 
         2. M. Polak, E. Demencik, L. Jansak, P. Mozola, D. Aized, C. L. H. Thieme, G. A. Levin, and P. N. Barnes, Appl. Phys. Lett. 88, 232501 (2006) 
         3. S. W. Kim et al. IEEE Trans. Appl. Supercond. 13, 1784 (2003). 
         4. Y.-Y. Xie, et al. Physica C 426-431, 849-857 (2005). 
         5. C. E. Oberly et al. Cryogenics 41, 117 (2001). 
         6. Polak, G. A. Levin, and P. N. Barnes, Superconductor Science and Technology 19, 817 (2006)