Superconductor and magnet levitation devices

REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 74, NUMBER 12 DECEMBER 2003 REVIEW ARTICLE Superconductor and magnet levitation devices K. B. Ma,a) Y. V. Postrekhin, and W. K. Chu Texas Center for Superconductivity and Advanced Materials, University of Houston, Houston, Texas 77204 共Received 13 November 2001; accepted 29 June 2003兲 This article reviews levitation devices using superconductors and magnets. Device concepts and their applications such as noncontact bearings, flywheels, and momentum wheels are discussed, following an exposition of the principles behind these devices. The basic magneto–mechanical phenomenon responsible for levitation in these devices is a result of flux pinning inherent in the interaction between a magnet and a type II superconductor, described and explained in this article by comparison with behavior expected of a perfect conductor or a nearly perfect conductor. The perfect conductor model is used to illustrate why there is a difference between the forces observed when the superconductor is cooled after or before the magnet is brought into position. The same model also establishes the principle that a resisting force or torque arises only in response to those motions of the magnet that changes the magnet field at the superconductor. A corollary of the converse, that no drag torque appears when an axisymmetric magnet levitated above a superconductor rotates, is the guiding concept in the design of superconductor magnet levitation bearings, which is the common component in a majority of levitation devices. The perfect conductor model is extended to a nearly perfect conductor to provide a qualitative understanding of the dissipative aspects such as creep and hysteresis in the interaction between magnets and superconductors. What all these entail in terms of forces, torques, and power loss is expounded further in the context of generic cases of a cylindrical permanent magnet levitated above a superconductor and a superconductor rotating in a transverse magnetic field. Then we proceed to compare the pros and cons of levitation bearings based on the first arrangement with conventional mechanical bearings and active magnetic bearings, and discuss how the weak points of the levitation bearing may be partially overcome. In the latter half, we examine designs of devices using superconductor magnet levitation, focusing more on issues specific to the application. We note that applications of superconductor magnet levitation devices tend to be most attractive in situations where energy conservation is critical. The most advanced in development are flywheel kinetic energy storage systems incorporating superconductor magnet bearings. Variations in the designs to enhance the performance in some specific regards are examined case by case. Next we present a reaction wheel for attitude control on small satellites, similar in overall design to the flywheel kinetic energy storage systems, but with subtle differences in details of emphasis, due to the difference in purpose and environment. Finally, we take a brief look at the case of vibration isolation devices as an example of a rectilinear modification of the more familiar rotational bearing applications. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1622973兴 I. INTRODUCTION ries or support. Having such equipment levitated would also eliminate other ills of contact, such as wear and tear, slip and stick motion, as well as friction, energy loss, and heat generation at the interface. Thus, the general purpose is to create a noncontact interface between moving parts for refined control of force and torque transmission between levitated equipment and supporting framework. Levitation is a phenomenon in which an object occupies a fixed position in a gravitational field without the benefit of support by direct physical contact with other objects, and does not necessarily involve magnetism. Brandt1 gave a comprehensive review of levitation in physics by magnetic as well as other means as of 1989. The primary motive in the application of levitation is to eliminate otherwise needed support structure for a piece of equipment designed to move or rotate relative to its accesso- II. APPROACHES TO LEVITATION Levitation by electromagnetic forces works even in vacuum. Of these, the strongest is the magnetic forces between magnets and magnetic materials, including superconductors. The magnitudes of the levitation forces that one can a兲 Author to whom correspondence should be addressed; electronic mail: kma@uh.edu 0034-6748/2003/74(12)/4989/29/$20.00 4989 © 2003 American Institute of Physics 4990 Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 obtain depend upon the sizes and shapes of the interacting magnets and superconductors and their relative positions and orientations, as well as the intrinsic properties of the magnetic materials and superconductors, such as the magnetization of the magnets and the critical current densities of the superconductors. It even depends on the processing of the superconductor and the history of motion and cooling of magnet and the superconductor in relation to each other. Occasionally, when the noncontact forces supporting the object do not come from a device below the levitated object, the levitation device is also referred to as a suspension device. The topic of this review will include both levitation and suspension devices using superconductors and magnets. One general approach to achieve levitation is to place the body to be levitated in a position of stable equilibrium under the combined action of gravity and other long range noncontact forces, notably electromagnetic forces. However, when we examine the known long range noncontact forces of nature including gravity, electricity, and magnetism, all the states of equilibrium turn out to be unstable or neutral, when taken individually or in combination if the distribution of mass, charge, current, or magnetic moment involved remains constant. For magnetic forces, this is achieved if we hold the magnetizing currents in the coils constant for electromagnets, or use hard permanent magnets with a large rectangular hysteresis loop, under fields weaker than their coercive forces. A. From active to passive levitation If, instead of holding the currents constant, we vary the currents in the electromagnetic coils in a manner conditioned by feedback from the actual state of motion of the body to be levitated, stability can be achieved. This concept leads to the development of active magnetic bearings and active magnetic suspension systems. On the other hand, if we choose to replace permanent magnets with soft magnets so that its magnetic moment can be changed to control the motion of the levitated body via feedback, we end up with active levitation using an electromagnetic coil. Left to themselves, soft magnetic materials together with permanent magnets are always unstable. The paramagnetic response of soft magnetic materials to permanent magnets is akin to positive feedback in control systems. This result for the magnetic case is known as Earnshaw’s theorem.2 From this point of view, the stabilization by negative feedback in an active system could be emulated by a passive diamagnetic response. One well known case is Lenz’s law in electromagnetic induction. Ordinarily, the induced current would begin to decay even as it is being built up, due to normal electrical resistance, turning what could have been an elastic restoring force into a damping one. However, most of the loss can be recovered by increasing the rate of change of magnetic flux in the induction process. For instance, a magnet moving at constant height above a horizontal sheet of metal with high conductivity will generate both a drag force and a lift force. At low speeds, ␯, the drag force is proportional to ␯, but the lift force is proportional to ␯ 2 and so the drag force dominates. As speed is increased, due to the ac skin effect, the drag force eventually peaks and comes back Ma, Postrekhin, and Chu down as ␯ ⫺1/2 as demonstrated by Rossing and Hull,3 while the lift force increases monotonically to the value expected from an image force as in perfect diamagnetism. Eventually, the lift force is the only force remaining. This is one of the principles employed in magnetic levitation trains utilizing the repulsive mode. Somewhat unexpectedly, as soon as the speed is high enough so that the lift force becomes greater than the drag force, a negative damping coefficient for vertical oscillations could develop.4,5 Later, a very similar approach has been applied to construct a passive magnetic bearing that springs into action after reaching a high enough rotating speed. Here, the magnets on the train become a multipole array of magnets and the ground plane is replaced by a circular array of coil circuits. The end result is that the lift forces are now directed radially and serve as an extra source of stiffness against lateral instability while the axial stability and support force are taken care of with permanent magnets in the conventional manner. The drag force turns into the direction of the circumference and becomes a drag torque. With appropriate coil circuitry, the drag torque was claimed to be negligible. B. Levitation by diamagnetism Moving magnets are not essential, nor do we need to simulate them with multiphase coils. A diamagnetic response can be elicited by coupling parametrically to the inductance of an L – R – C resonance circuit with a high quality factor, carrying ac current at a frequency slightly above resonance. The cost is having to maintain a high frequency ac current through the circuit all the time. This technique was developed in passive magnetic suspension systems which were predecessors to the active systems.6 The simplest case would be if we could have materials that have a response that is diamagnetic instead of paramagnetic. In fact, we can expect a diamagnetic response from all materials since Lenz’s law should be universally valid, including organic matter and live organisms. The diamagnetic response is usually weak. Even so, live objects have been levitated, using this meager response.7 In many other cases, though, this diamagnetic response is overwhelmed by other mechanisms that give rise to paramagnetism or even ferromagnetism. The only class of materials known today that display a sizable diamagnetic response are the superconductors. Indeed, perfect diamagnetism is one of the hallmarks of superconductivity, the other being zero electrical resistance. Perfect diamagnetism means that the inside of the bulk of a superconductor is always shielded from magnetic fields by appropriate surface currents. The interaction between a perfect diamagnet and a permanent magnet gives rise to a conservative force field that can and does exhibit positions of stable equilibrium with an appropriate geometry of the diamagnet and permanent magnet combination. Stable levitation can be achieved with a permanent magnet and a superconductor in its perfectly diamagnetic state. Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 III. MAGNETIC PROPERTIES OF SUPERCONDUCTORS However, the perfectly diamagnetic state of a superconductor can only be maintained so long as the total magnetic field is everywhere lower than the critical field which depends on temperature and the superconducting material used. This critical field is one component that determines the critical current density that can flow inside the superconductor without destroying its superconductivity. The critical current density is one of the main factors governing the levitation pressure that can be sustained by the superconductor. In general, the temperatures for which this levitation pressure could be high enough to be of practical value would be in the liquid helium range or lower, making it difficult in practice from the point of view of thermal management. A. Type I versus type II superconductors When the magnetic field on the surface of the superconductor exceeds the critical value, it starts to penetrate the superconductor, which then phase separates. The superconducting phase now coexists with the normal phase which contains the magnetic field in the bulk of the material. Now, a surface separating these two phases could be expected to exist. As a matter of fact, it does in some materials, but in others, this surface becomes unstable, and the normal phase disperses itself finely into the superconducting phase in the form of vortices containing discrete quanta of the magnetic flux. The materials in which the phase interface between the superconducting and the normal state is stable are designated as type I, while the others are type II. All the high temperature ceramic superconductors 共HTSs兲 that were discovered since 1986 are type II materials. Furthermore, it is behavior of the HTS materials with high critical current densities prepared by the melt-texture growth methods8,9 that will be the focus of our attention from here on, since it is this type of material that yields significant levitation forces and made the most contribution to the development of levitation devices. Type II materials enter into a mixed state when the magnetic field begins to penetrate the bulk of the superconductor. Magnetic vortex lines start to nucleate at the surface and drift into the bulk of the material when the magnetic field on the surface becomes stronger than the lower critical field. Magnetic vortex lines continue to accumulate inside the material until they become so dense that they destroy superconductivity in the material altogether. This happens at the upper critical field. The lower critical field is lower, and the upper critical field is higher than the thermodynamic critical field, which is the critical field expected on the basis of the difference in free energy between the superconducting and normal states of the material. B. Reversible versus irreversible regimes of type II superconductors To complicate matters further, the mixed state itself is subdivided in a magnetic induction–temperature (B – T) diagram into at least two regimes, a reversible regime and an irreversible regime, by a line known as the irreversible line. This irreversible line is an important characteristic of high Superconductor and magnet levitation 4991 temperature superconductors.10,11 The reversible regime occurs at higher temperatures or higher fields above the irreversibility line. In the reversible regime, the magnetic fluxoids inside the superconductor can flow around easily and the material settles into its thermodynamic equilibrium state fairly quickly after an external disturbance. However the flow of fluxoids is a dissipative process, and it does not seem that it would offer as good a prospect for levitation applications as the irreversible regime. In the irreversible regime, the fluxoids are pinned, allowing persistent supercurrent flows to remain in existence. The superconductor takes an extremely long time to reach thermodynamic equilibrium. Thus, the material behaves as a hysteretic diamagnet. It has been demonstrated that extrinsic impurities and defects of all kinds can enhance the pinning effect, but it has never been possible to eliminate the pinning effect altogether. It has been hypothesized that there is an intrinsic component to the pinning effect. The pinning effect is quantitatively measured by the critical current density. It is the least amount of current density required to exert a sufficiently strong Lorentz force to dislodge the pinned fluxoids, thereby destroying the zero resistance nature of the superconductor. The same pinning force is eventually responsible for the strength of levitation forces and stiffnesses between permanent magnets and these superconductors. Hence, from the point of view of levitation applications, superconductor samples with higher critical current densities give better overall performance. Almost all of the levitation studies have been carried out at fields and temperatures such that most of the superconductor is in the irreversible mixed state. Henceforth, we will restrict our discussion to this state. IV. LEVITATION WITH SUPERCONDUCTORS Briefly, the levitation force between a magnet and a superconductor arises from the Lorenz force acting on the supercurrents with density J, flowing in the superconductor by the magnetic flux density, B, from the magnet. This magnetic flux density, B, can be taken to be that which would have been present in the volume occupied by the superconductor in the absence of the superconductor itself. The force can be written as F⫽ 兰 J⫻B dV, which can be shown to be equivalent to F⫽ 兰 M"ⵜB dV under most common circumstances. Here M is the magnetization of the superconductor induced in response to the magnetic field from the magnet. With proper geometric arrangements, stability of the levitation object under this levitation force is made possible by the diamagnetic nature of this magnetic response of the superconductor, acting as a negative feedback to the object to return to equilibrium. Very often, the supercurrent current density responsible for M is at the critical value. Consequently, we can expect that superconductors with higher critical current densities would produce high levitation forces and stiffnesses. It is perhaps best to start with the limiting case of infinite critical current density for a qualitative understanding. 4992 Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 A. Perfect conductor versus perfect diamagnet This limiting case with infinite critical current density yields a perfect conductor. The equation of state of a perfect conductor may be written as ⳵ B/ ⳵ t⫽0. Compared to the equation of state of a perfect diamagnet, B⫽0, the only difference is the initial flux density distribution, B(r,t⫽0). Physically, this can be identified with the field that is present when the superconductor last made the transition from the normal state. It also implies that, subsequently, the electromagnetic behavior of the perfect conductor can be expected to behave exactly like that of the perfect diamagnet, but with the initial frozen-in field superimposed. For example, the levitation force developed on a perfect diamagnet in a magnetic field is the same whether it is cooled before or after the magnetic field is approached, but a perfect conductor would develop the same levitation force only when cooled before the magnetic field is approached, since there is no magnetic field inside the material in both cases. The perfect conductor cooled after the magnetic field is approached would retain this field inside and not distort the magnetic field. As a result, no levitation force is developed with a perfect conductor under such circumstances. Thus, the ambient history of a superconductor is important for type II materials in the irreversible mixed state, but not important for type I materials. B. Zero field cooled versus field cooled conditions Crudely speaking, one distinguishes between zero field cooled 共ZFC兲 conditions when the superconductor is cooled to its superconducting state before the magnetic field is approached, and field cooled 共FC兲 conditions when the superconductor in cooled in the presence of the field that it was intended to work under. With a perfect conductor, no levitation force develops under FC conditions, but an appreciable force appears under ZFC conditions. In stark contrast, the levitation stiffnesses developed under ZFC and FC conditions is comparable. Assuming that we have the superconductor in a definite location, P, in an external magnetic field, the FC superconductor 共in the perfect conductor model兲 would not contain any induced currents in its volume, while the ZFC superconductor would exhibit surface currents that shield its volume entirely from the external flux. These surface currents are then responsible for the difference in the stiffnesses developed in ZFC over that of the FC case. Otherwise, the ZFC and FC superconductors yield the same stiffnesses, identical to the case of a perfect diamagnet, coming from the same infinitesimal force response to the same infinitesimal displacement which causes the same infinitesimal variations of the magnetization in all three cases. Furthermore, this response is the same as due to the surface currents in the ZFC case, by reciprocity, so that in total, the ZFC stiffness is exactly twice that of the FC stiffness. These results are shown in the context of the frozen image model, and experimentally verified in the work of Hull and Cansiz.12 It may appear that we would have to give up FC conditions and live with the inconveniences that come with having to set up ZFC conditions. In practice, neither is used, for there is actually a continuous spectrum of conditions in which the superconductor is cooled in one field and then Ma, Postrekhin, and Chu displaced to another. The FC and ZFC conditions are merely the extremes. Let us take the simple case of levitating a permanent magnet over a superconductor. The superconductor is cooled under the magnet, which is supported by other means. When the superconductor is cold enough, the support for the magnet is removed. Assuming that the weight of the magnet is not excessive, it will be levitated after a drop in height, which can vary from imperceptible to substantial. If the magnet was very far above, then the superconductor was cooled practically in the absence of a field and thus approaches ZFC conditions, and it will drop a substantial amount to a point where its weight is exactly balanced by the levitation force developed as the displacement of the magnet downward brings the superconductor under the influence of a higher magnetic field. On the other hand, if the magnet was near, it will be levitated at wherever it was and appear not to drop any distance at all. This case clearly qualifies as FC, since the superconductor was cooled in the field of the magnet and it remains in the same magnetic field throughout. However, there are intermediate cases in which the magnet is cooled at some distance, and then sags a little when released. There is no natural dividing line between FC and ZFC behavior. From the device utility point of view, FC or nearly FC conditions are more convenient to implement. C. Levitation versus suspension If we set up a permanent magnet levitated under FC conditions and turn it upside down so that the magnet is underneath instead of above the superconductor, the magnet will remain suspended below the superconductor. The same result occurs if we started cooling the superconductor in the field of a permanent magnet placed underneath it, held the superconductor in place, and removed all mechanical support for the magnet. If the weight of the magnet is moderate, it can be seen to drop down a little. In this case, the superconductor was cooled in the field of the magnet, but the subsequent motion of the magnet partially removes the field acting on the superconductor, and an attractive force is developed between the magnet and the superconductor, counterbalancing the downward pull of the weight of the magnet. This phenomenon shows clearly that flux pinning forces are involved, and strongly suggests that the same flux pinning forces are responsible for the main action even in the case of levitation. Suspension cannot be achieved under ZFC conditions. This can be understood as the superconductor, being a perfect conductor in the limit of infinite flux pinning forces, behaves exactly the same way as a perfect diamagnet under ZFC conditions. The force developed between the superconductor and a permanent magnet is always repulsive under ZFC conditions, but under FC conditions, the force is repulsive if the magnet and the superconductor are pushed together from their initial positions on cooling, and attractive if pulled apart. D. Lateral forces and flux pinning Another difference between the perfect conductor and the perfect diamagnet reveals itself when we examine the case of FC superconductor and magnet displaced laterally Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 FIG. 1. Top and side views of four different magnets on HTS configurations to illustrate the utilization of the field gradient effects in different devices: 共a兲 and 共b兲 shows rotational devices made up of a magnet in the shape of a circular disk levitated above a disk of HTS of the same shape. 共a兲 the magnet has the same polarity all around the circumference but opposite to the polarity at the center. Angular displacements does not change this magnetic field and hence the magnet rotates freely, making it a superior bearing. In 共b兲 the cylindrical magnet is divided into four sectors of alternating polarities. Changes in the magnetic field as this magnet is rotated produce a response from the superconductor to follow the motion of the magnet in its attempt to keep the magnetic flux passing through its own bulk constant. This can be used as a magnetic clutch. 共c兲 and 共d兲 Linear analogs of parts 共b兲 and 共a兲, respectively, made up of a magnet in the shape of a square plate, magnetized in the direction of its thickness in stripes of alternating polarity, levitated on a square plate of HTS of the same shape. In 共c兲, the magnet is vibrated in a direction perpendicular to that of the stripes. Changes of the magnetic field around the superconductor brought about by this vibrational motion excite a drag force from the superconductor and energy is dissipated in the form of magnetic hysteresis in the superconductor. This device is a vibration damper. If the direction of the vibration is parallel to the directions of the stripes as in part 共d兲, no changes of the magnetic field occurs except at two of the edges, and little drag force results. When the stripes are extended indefinitely in both directions, the HTS square plate becomes a linear slider along the track of magnetic stripes. with respect to each other. Assume that the superconductor is in the form of a large plane sheet. In the perfect conductor model, a memory of the original position is created in the conductor in the form of an image of the magnet with opposite polarity which recreates the original magnetic field within the conductor. This represents perfect flux pinning, and as a result, the magnet is attracted back to that original position. With the perfect diamagnet model, no additional force develops. No magnetic field was ever present in the interior of the perfect diamagnet to make any position special, so the same repulsive force is obtained anywhere that is at the same distance from the plane. With a superconductor, experiment shows that an attractive lateral force is indeed developed, as expected from the perfect conductor model. Again, the flux pinning forces from inside the superconductor are responsible for the existence of such lateral forces. E. Field gradient effects: elimination versus enhancement of drag While Lenz’s law guarantees that any displacement of the magnet will not be reinforced leading to instability, it is possible for a displacement to fail to excite an opposing force from the superconductor. This happens with displacements of the magnet that do not change the magnetic field entering the bulk of the superconductor. For instance, if the levitated magnet is a cylinder of uniform magnetization along the axis, the magnetic field would not be changed if the magnet is rotated about this axis, as illustrated in Fig. 1共a兲. Indeed, the magnet is able to rotate freely about its axis. Superconductor and magnet levitation 4993 FIG. 2. A hybrid superconductor magnet bearing of simple design. In practice, slight asymmetry in the magnetic field around the circumference will induce a correspondingly small resisting torque. Nevertheless, nearly frictionless bearings can be formed with this combination. Note that nonuniformity in the superconductor does not really matter in this case, so long as each point of the superconductor still experiences no change in the external field. At the other extreme, if the levitated magnet is made up of sections of alternating polarity, as in Fig. 1共b兲, the rotation of the magnet will drag the superconductor along with it. This would make a very efficient noncontact angular momentum coupler, or a magnetic clutch. Figure 1共c兲 shows how the same principle can be adopted to linear motion. Here, we have a rectangular magnet that is really a composite of strips of alternating polarity. Motion of the magnet perpendicular to the direction of the strips will be met with a restoring force, just like the attractive force with lateral displacement that we discussed above. However, motion along the direction of the stripes will be practically frictionless, so long as we are nowhere near the ends of the length of the magnet. F. Levitation with permanent magnets, stabilized with superconductors So far, we have only been dealing with the interaction of one superconductor with one magnet. Within the validity of the perfect conductor model, the many body problem can be handled with the principle of pairwise linear superposition. One application of this has given rise to the hybrid approach to levitation.13 In this approach, levitation of one magnet is achieved by the combined action of a superconductor and another magnet. Both the levitation force and the stiffness are obtained by simply adding the independent contributions from the magnet and the superconductor. This approach enables us to take advantage of an additional magnet to increase the levitation force over that which can be obtained with the superconductor alone, while the stabilizing influence from the superconductor is still available to counteract any inherent instabilities in the force between the magnets. This concept is illustrated in the design of a hybrid superconductor magnet bearing shown in Fig. 2. The value of this approach lies in applications where large levitation pressures 4994 Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 are required, such as large flywheels for energy storage. A bearing based on the straightforward concept of levitation by HTS only as illustrated in Fig. 1共a兲 will need large quantities of HTS material, while a bearing design based on the hybrid concept of levitation by magnets and stabilization by HTS as illustrated in Fig. 2 can reduce the amount of HTS material required. Thus the advantage of a higher levitation pressure attainable with repulsive forces between rare-earth 共RE兲 magnets is realized, while the disadvantage of inherent instability of forces between magnets as expressed in Earnshaw’s principle is bypassed by including HTS in the system to convert it from being unstable to being stable. The stabilizing role of HTS works on the repulsive forces between the like poles of two magnets as in levitation, as well as the attractive forces between the unlike poles of two magnets as in suspension. V. NONIDEAL BEHAVIOR: FORCE HYSTERESIS AND CREEP While the perfect diamagnetic state with type I materials can be reached, the perfect conductor state with type II materials is unattainable, even in principle, since the physical flux pinning forces must be finite. Energy dissipation shows up in two qualitatively new phenomena: force hysteresis in displacement sweeps and force creep. Not surprisingly, both of these phenomena find parallels in the purely magnetic dimension: hysteresis loops in M – H plots and flux creep. Indeed force hysteresis and creep could be expected on the basis of the magnetic observations. However, studies of the magnetic phenomena are usually carried out with uniform fields, whereas the very occurrence of forces depends on field gradients. It is plausible, but not yet determined, that the role of the field gradient does not go beyond merely reflecting the changes in the magnetization of the superconductor in force hysteresis and force creep phenomena. A. Consequences of force creep Qualitatively, the creep phenomenon is characterized by variables of interest such as the magnetization of a superconductor or the levitation pressure between a magnet and a superconductor showing a long drawnout drifting behavior following a short period of rapid change initiated by an external perturbation. Quantitatively, it seems that the drifting portion of this change is logarithmic in time, meaning that the same change is observed between 1 s and 10 s as that between 10 s and 100 s, or 100 s and 1000 s after a somewhat ill-defined instant of initiation. It is obvious that this behavior cannot be valid for all times, but experimentally, it has been observed on time scales from 10⫺4 s up to weeks and months and found to describe the relaxation process fairly closely.14,15 The amount of creep also depends on the magnitude of the perturbation, diminishing as the perturbation is decreased, but if there is no disturbance at all, the creep naturally continues on at the slow rate that it inherited from the last perturbation. This force creep becomes much more notable as a relaxation or drift of the position of a superconductor levitated in a static magnetic field, which is then further subjected to the influence of an ac magnetic field. Early work by Terentiev16 Ma, Postrekhin, and Chu using free-sintered material 关YBaCuO 共YBCO兲兴 relates this to the magnetization hysteresis loop of the material, as follows. The hysteretic magnetization is combined with the field gradients to yield an effective complex stiffness, with the elastic part governed by the average of the magnetization with increasing or decreasing fields and the loss given by the width of the loop in the static field. A natural vibration frequency can be derived from the elastic part and a relaxation time, from the loss portion. This was then used to explain the dependence of the relaxation time on the frequency of the imposed ac field. This relaxation time becomes longer and longer as frequency is decreased, and reproduces the slow creep in the static limit. On the other hand, the system relaxes quickly to a state corresponding to the mean magnetization without hysteresis, under the influence of an ac field with a frequency higher than the natural frequency. The system then behaves as if the magnetic friction of hysteretic origin has disappeared. No dependence of the amplitude of the ac field is contained in the results of this approach. Further work by Terentiev and Kuznetsov17 shows that the drift in the levitation height occurs only when the ac field amplitude exceeds a certain threshold value assumed to be related to microscopic material parameters 共product of critical current density and London penetration depth兲 governing the depinning of flux lines away from pinning centers. Later studies also target the drifts or oscillations of the levitation position rather than the force, but the exposition of the results is cast in terms of dynamical models. The work of Postrekhin et al.18 describes the vibrations in terms of the driven Duffing nonlinear oscillator. The hysteretic properties of the vibration amplitude exhibited in a frequency sweep characteristic of nonlinear oscillators is nicely demonstrated here, in agreement with a brief report from Lamb et al.19 earlier. However, the occurrence of many complicating factors, such as the off axis equilibrium positions and the anisotropy of the materials, prevents a simple interpretation of the results and their relationship to basic material properties. Such complications do not occur in the experiments of Moon and Hikihara20 as both components, magnet and superconductor, are attached to other equipments in their studies. Another difference of their approach from those adapted by workers mentioned above is that vibration of the superconductor is used in place of an ac field generated by electric coils. The main point they made is that the levitation drift is determined only by the operating position on the force hysteresis loop, independent of the material, processing, or the cooling method 共ZFC versus FC兲. Their results were interpreted in terms of a dynamical model, in which a velocity dependent force used to simulate the force hysteresis is also capable of reproducing the levitation drift observed in their experiments. Indeed, they emphasized that vibrations, as well as any other source of ac fields, will induce levitation drift, and this is a general characteristic of high temperature and magnet levitation systems. Based on the relationship between the drift and hysteresis, they suggested that materials designed with narrow hysteresis loops should be used to limit the levitation drift. In practice, vibrations of a levitated system are equivalent in their effects to ac fields explicitly introduced into the surroundings. Coombs and Campbell21 Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 studied gap decay in superconducting magnetic bearings under the influence of vibration, and found that vibrations induce gap creep above that under steady load, which is expected from the above discussion. They also found that the gap can either decrease and may even collapse, or it can increase instead depending on circumstances. They concluded that the main factor determining the behavior of the gap creep is whether the minor force hysteresis loop executed by the system under vibrations exceeds the major force hysteresis loop. A simple spring-damper model representing the minor hysteresis loop is sufficient for this purpose. For levitation applications, the significance of the creep phenomenon lies in the difficulty in maintaining tight tolerances due to gap creep, which happens as a result of force creep. One approach to avoid this problem is to use gaps large enough so that the slow creep rates could never threaten to close the gap within a reasonable lifetime of the equipment. The main reason for the occurrence of creep in the simple leviation or suspension setups is that the magnet always suffers some displacement when it is released after the field cooling of the superconductor, because the weight of the magnet is not exactly balanced by opposing forces from the superconductor. However, this displacement then triggers the creep processes which continues beyond the initial reaction. In the hybrid levitation devices, where the weight of the magnet is balanced by forces from other magnets both before and after the superconductor is cooled to its superconducting state, this initial adjustment of the magnet position is greatly reduced or eliminated. The subsequent changes due to creep would be correspondingly suppressed. Thus, in addition to enhancing lifting capacity, the hybrid approach such as the example given in Fig. 2 could be used advantageously as an alternative hedge against creep, especially where tight tolerances cannot be avoided for other reasons. B. Critical state model Although creep and hysteresis are but different facets of the onset of irreversibility and dissipative processes in a superconductor, there is as yet no unified theory that encumbers both of these phenomena. The current description of hysteresis in type II superconductors is cast in terms of the critical state models, of which the most often cited is Bean’s model.22 Creep is entirely ignored in these treatments, justified by the pragmatic consideration that under most experimental conditions, the hysteretic changes following the field or displacement sweeps are much larger and more rapid, against which the changes due to creep would appear as an insignificant drift in the background at worst. Nevertheless, the critical state models have shown great utility and enjoyed success aplenty within their framework of negligible creep. The critical state models describe the ultimate state of a type II superconductor in its mixed state as critical everywhere. That is, the current density everywhere inside the superconductor is either zero, or at the critical current density corresponding to the magnetic field found at that locale. In the Bean’s model, the extra assumption is Superconductor and magnet levitation 4995 made that this critical current density is a constant independent of the magnetic field. This expedient approximation is based on the observation that when the critical current density is, in fact, a function of the magnetic flux density, this function is only a slow varying function. The resulting simplification allows several useful observed results to be illustrated without undue complications, given that the functional dependence of the critical current density on the magnetic induction is, in general, not known with any high degree of accuracy. The key feature in this approach that leads to irreversibility is that any change always starts at the surface of the superconductor, where it follows the dictates of the external applied field. Then the change is propagated one way inwards to the interior of the superconductor. The same events happen with normal conductors, where they are manifested as diffusion of magnetic flux, but here, the magnetic flux continues to diffuse even after the external field stops changing. On the contrary, the changes in type II superconductors as described in the critical state models stop as soon as the external field goes constant. Subsequent creep is ignored, and the system does not evolve to a unique final state. The result is hysteresis. It can affect the reproducibility of the ‘‘final equilibrium position’’ of the levitated body. C. Consequences of force hysteresis The hysteresis of the levitation force for a system involving a high temperature superconductor and a permanent magnet was first studied and measured by Moon et al.23 They have also observed hysteretic levitation forces on superconducting thin films.24 The chief consequence of hysteresis in levitation applications is a pronounced damping of vibrations about the stable equilibrium configuration, which is only to be expected as hysteresis represent an energy loss mechanism. Thus incorporation of magnet/superconductor pairs in supporting structures could be effective in suppressing their vibrations. For lateral relative motion between a magnet and a superconductor, the resistive force resembles the familiar Coulomb friction existing between solid surfaces with one difference: on reversing the direction of motion, the resistive force takes an appreciable distance before it follows the direction reversal and regains its former magnitude. The gradient of the force at any point becomes double valued, one applicable for continuing the motion in its current direction, and another for turning back. Stick and slip phenomena have not been observed yet, and so the sliding motion might be controllable to a highly refined precision. Perhaps the angular version of the same phenomena as a bearing occurs much more frequently. With an ideally axisymmetric magnet, such a bearing would be frictionless in principle; but in practice, slight asymmetry of the magnet is always present to a greater or lesser extent, resulting in a small residual bearing loss, and hysteresis is to blame. However, the possibility of very accurate pointing and tracking of extremely slowly moving targets such as stars remains an attractive prospect, and in this case, the intentional introduction of a certain amount of friction without stiction could even be desirable. In the end, force hysteresis is a direct consequence of magnetization hysteresis of the superconductor. An extreme 4996 Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 FIG. 3. Experimental setup to study levitation force between a permanent magnet and a superconductor disk 共see Ref. 26兲. manifestation of magnetization hysteresis is the acquisition of trapped flux by subjecting the superconductor to a pulsed field. This phenomenon was studied by Itoh et al.25 They found that the resulting trapped field magnets were substantially the same as trapped field magnets made by field cooling, and therefore serve as an alternative to the field cool method in applications where the latter is not convenient. The maximum magnetic field attained in the pulse field is higher than the uniform field used in the field cool approach to produce a trapped field magnet of the same strength. Apart from hysteresis and creep, we can expect the behavior of the slightly imperfect conductor to deviate quantitatively, though not qualitatively, from the perfect conductor limit results discussed above, but also these deviations would go to zero in a continuous manner as the perfect conductor limit is approached. Deviations can be expected to be correspondingly smaller under conditions which induce very little current in the superconductor. Thus, stiffnesses which relate only to infinitesimal displacements would be expected to be in closer agreement to estimates based on perfect conductor behavior than forces. VI. A GENERIC SUPERCONDUCTOR MAGNET SYSTEM Up to this point, we have put our emphasis on the general features that we can expect of the magneto–mechanical properties of a system of magnets and superconductors. Next, we are going to examine how these general features are manifested in a simple generic case. Ma, Postrekhin, and Chu FIG. 4. Dependence of levitation force on the distance between superconductor and magnet at different speeds 共see Ref. 26兲. versus t, respectively, for a magnet approaching a superconductor at different speeds, and then stopped at the same point. Note that we cannot represent the force F as a joint function of the position coordinate, z, and the velocity, dz/dt, either, since the force continues to change at a fixed position z and dz/dt⫽0. Time must be explicitly involved. All this is not unexpected, and can be understood as force creep resulting from magnetization creep in the superconductor. Furthermore, the difference between the observed forces for different approach speeds shows that force creep is going on all the time, and does not have to be triggered by the termination of the motion of the magnet. As mentioned above, force creep follows a ‘‘logarithmic’’ decay law, following the behavior of remnant magnetization trapped in a superconductor. For all practical purposes, in any given time interval, the force or magnetization appears to take an abrupt drop in the beginning, followed by an almost imperceptible drop lingering along for the remainder of the interval. This behavior is shown in Fig. 5 for the force between a magnet after initial approach to a ZFC superconductor. The system is said to be in quasistatic condition in this latter portion of the time interval. The value of the force obtained at the end of that time interval is sometimes termed the quasistatic value for that experiment. Of course, the most important parameter for a levitation device is the weight that it can levitate. For a system consisting of a permanent magnet on a superconductor, this is the A. Levitation force: zero field cooled case Let us consider a ZFC superconductor and a permanent magnet approaching each other, as in a typical experimental setup illustrated in Fig. 3. A repulsive force develops between them. A natural first question would be: ‘‘What is the relationship between this force, F, and the distance, z, between the magnet and the superconductor?’’ However, we would observe that whereas the force increases monotonically as the magnet approaches, which is not surprising, it decreases as soon as the magnet stops and stays in one position. Thus, we cannot express F as a function of z. F depends not only on z, but on the entire history of the approach z(t), where t denotes time before the magnet stops. This is illustrated in Figs. 4 and 5 which display F versus z and F FIG. 5. Time evolution of the levitation force for different rates of approach of magnet 共see Ref. 26兲. Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 repulsive force between them when they are almost in direct contact. From our discussion above, this value still varies a bit with the manner of approach of the magnet. Two different practices are commonly adopted: a slow approach followed by a short wait at the position of closest approach, corresponding to quasistatic conditions; or a quick approach followed by a pull back, sometime referred to as the impulsive method. For both of these approaches, it is usual to monitor the force throughout the motion of the magnet. The maximum force recorded at closest approach is further normalized to unit area of cross section of the magnet presented across the direction of approach, assuming that this is less than that of the superconductor disk it is approaching. This force per unit magnet cross-section area is quoted as the levitation pressure of the system and is one measure of the quality of performance that we can expect of this magnet and superconductor pair. Understandably, the levitation pressures obtained with the impulsive method are usually larger then those obtained under quasistatic conditions for the same pair of magnet and superconductor. The difference is not large enough to be significant for orders of magnitude estimates. For comparison purposes, however, one method of choice should be used consistently for the results to be meaningful. With NdFeB magnets and seeded growth melt-textured YBCO disks, a typical levitation pressure is 1 – 2⫻105 Pa, or 1–2 atm. More specifically, the data shown in Figs. 4 and 5 were obtained26 with a YBCO superconducting disk of 40 mm diameter and 20 mm thickness. The critical current density of the sample was on the order of 104 A/cm2 at 77 K. The permanent magnet used is of NdFeB, having a diameter of 6.3 mm and a height of 6.3 mm. The magnetic flux density at the center of the pole faces of this magnet is 2.81 kG. B. Levitation force: field cooled case Now, let us consider reversing the motion of the magnet. We could continue the ZFC experiment described above and pull back the magnet after it was stopped at the position of closest approach. Instead, we consider a FC experiment as follows, and return to the ZFC experiment later. The FC experiment starts with the superconductor warm, and the magnet at the position of closest approach. Then the superconductor is cooled to below its critical temperature for condensing into its superconducting state. Thus, the superconductor is cooled in the field of the magnet. The force on the magnet is monitored as it is pulled away from the superconductor. Its value is displayed against the position of the magnet in Fig. 6. Immediately we notice that the force is negative 共i.e., attractive兲 throughout and the force vanishes both at the position of closest approach in the beginning and in the limit of great distances away in the end. This mandates the appearance of at least one maximum 共in magnitude兲 of the force somewhere in the middle. That the force vanishes at large distance is well known as the force between magnetic dipoles falls off as a high power of the distance between them (r ⫺4 for fixed dipoles and faster if one or both of them are induced兲. That the force should be attractive can be understood as Lenz’s law in action under the almost perfect conductor model discussed above. That no force appeared in the Superconductor and magnet levitation 4997 FIG. 6. Force vs distance observed FC with magnet pulled away from superconductor. beginning when the magnet was closest to the superconductor is consistent with the almost perfect conductor model in the perfect conductor limit, which applies when there has been no displacement of the magnet since the superconductor was cooled. According to this model, there is actually no induced currents or magnetization in the cold superconductor before the magnet is pulled back. As the magnet is pulled away to greater distances, persistent currents are induced in the cold superconductor which then acquires a magnetization, manifested as a distinct trapped field when the magnet is pulled far enough away. If we bring the magnet back, these induced currents will be partially reversed and the magnetization in the superconductor develops opposing components. From outside the superconductor, the trapped field appears to be partially wiped out. The above FC experiment starts out with a warm superconductor in the field of a permanent magnet, which is nonuniform. If we cool a superconductor in a uniform magnetic field instead, we end up with a conventional trapped field magnet. The same end result can be obtained by the application of a short intense pulse of magnetic field. Maximum trapped fields of approximately 2 T at 77 K have been obtained.27 Maximum trapped fields, together with levitation force, are often used as a convenient assessment of the quality of superconductor samples in studies of effects of materials processing and inclusion of minor components.28,29 Needless to say, these trapped field magnets must be maintained under cryogenic temperatures. Following the argument in the last paragraph, we can expect that the total magnetic moments in these trapped field magnets change as they interact with other magnets, contrary to ideal permanent magnets. Nevertheless, it is possible to treat their behavior as a superposition of permanent magnets and ZFC superconductors. The application of trapped field magnets as components in levitation devices will be discussed in Sec. VII A. C. Minor force hysteresis loop and quasistatic stiffness Let us return to the ZFC experiment earlier. Figure 7 shows how the force changes with position as the magnet approaches and then recedes from the superconductor. The solid curve in Fig. 8 shows what is expected if the flux den- 4998 Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 Ma, Postrekhin, and Chu FIG. 9. Expanded plot of a minor loop in the force–distance dependence shown in Fig. 6. FIG. 7. Levitation force vs distance dependence with minors loops at ZFC condition. sities are frozen both on approach of magnet and on its way out, so that flux creep or flow occurs only during the turnaround time interval when the magnet was not moving, but then proceeds to full penetration during that interval. With these assumptions, the qualitative features of this curve can be obtained by applying the perfect conductor model during the entire motion of the magnet, except during the turnaround interval. Of course, this could only be regarded as an idealized scenario at best. Actually, flux penetration occurs continually throughout the entire duration of the motion of the magnet, but it may not attain full penetration. It is plausible that there will be less loss and hence we would observe a smaller loop inside the loop for the idealized case, qualitatively similar to the dotted line in Fig. 8. We also see that this dotted line is similar to the results of measurements as shown in Fig. 7 as well, except that the force did not change sign when the magnet is on its way up, which is a quantitative detail. This constitutes an explicit demonstration of hysteresis in the force between a superconductor and a magnet, illustrating once more that the force cannot be represented as a position and velocity dependent force, but its time evolution involves the entire history of the motion of the magnet from the moment the superconductor turns superconducting. FIG. 8. Levitation force as a function of distance at ZFC condition for ideal 共solid line兲 and experimental 共dashed line兲 cases. In the previous example of a force hysteresis loop, referred to as a major hysteresis loop, one of the end points of the loop is at the position of closest approach to the magnet, the other end point is infinitely far away. In general, the end points can be located anywhere in between, and any manner of traversing back and forth along a single path joining the end points would yield a force hysteresis loop, which closes after a few cycles. If we let the end points approach each other and choose the path of shortest distance between them, we would get a small hysteresis loop that is also narrow. We can define a stiffness associated with this minor hysteresis loop as ⌬F/⌬s, where ⌬F is the difference of the force between the end points and ⌬s is the displacement between the end points. This is analogous to the definition of the stiffness as the gradient of the force for a conservative force field, but in this case, the stiffness is different from the gradient of the force observed on first approach to the vicinity of the minor hysteresis loop by way of the major hysteresis loop, reflecting the double-valued nature of the gradient of the force in the presence of hysteretic effects mentioned earlier. In fact, it is often closer to the gradient of the force observed on reversing the direction of motion. If the minor hysteresis loop is traced out in a quasistatic manner, then the stiffness is referred to as a quasistatic stiffness. Figure 9 shows an example of such a minor hysteresis loop. In addition, just like any force versus position relationship with magnet superconductor systems, it is not surprising to find that the details of the hysteresis loops, such as its shape, may be affected by how the loop was first approached, whether it was ZFC or FC for instance. However, it is plausible that, in this case, the difference between the forces as a function of position on the separate legs of the loop are independent of the initial setup. One consequence of this is that the energy loss per cycle around the loop is invariant with respect to the initial setup and is the same for ZFC or FC. Furthermore, in the limit of small hysteresis loops, the perfect conductor behavior is approached even closer. We can expect the loop to collapse, in the sense that ⌬w/⌬s⬃⌬s, where ⌬w is some measure of the width of the loop and ⌬s may be regarded as a scale length for its size. Then the loss per cycle of the loop, given by the enclosed ‘‘area’’ ⌬w⌬s would scale as (⌬s) 3 . So far, these assertions have not been probed in sufficient detail with experiments to permit any definitive conclusion, Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 Superconductor and magnet levitation 4999 FIG. 11. A sample record of torque vs time 共or angle兲 at FC and ZFC. first guideline with respect to the overall dynamical properties of the system, such as its stability. For a NdFeB magnet with a seeded growth melt-textured YBCO superconductor disk, values of the stiffness anywhere from 1 to 100 N/mm can be obtained, with a quality factor of 2–30 or so.30,31 E. Drag torque and power loss in rotational motion FIG. 10. Distribution of current inside superconductor, rotating in 共b兲 and 共d兲: 共a兲 is at ZFC for initial position; 共b兲 is at ZFC for steady state; 共c兲 is at FC for initial position; and 共d兲 is at FC for steady state. Gray represents current flowing up, dark gray flowing down. one way or the other. We will encounter these same aspects in another context below. D. Small amplitude vibrations and dynamic stiffness More often than not, the stiffness is measured by observation of small amplitude vibrations of the magnet superconductor system. This stiffness is called the ‘‘dynamic’’ stiffness where ‘‘dynamic’’ describes the method of measurement, which renders it automatically applicable to the dynamics of small vibrations to a first approximation. The natural vibrations of a freely levitated magnet contain contributions from all six rigid body modes. In practice, the magnet is attached to a well characterized vibration system such as a cantilever beam, so that it moves in the desired z direction. When the cantilever beam is excited in its lowest frequency mode, the dynamic stiffness of the magnet in the z direction at different positions can be obtained from the shift of the natural frequency of the fundamental vibration mode of the cantilever beam with the magnet located at the desired positions against the known fundamental frequency of the cantilever beam without the magnet attached. In principle, higher vibration modes or a mechanical shaker as an external driver of tunable frequency may be used and some information on the dependence of the dynamic stiffness on the speed of traverse of the cycle path or driver frequency can be gleamed. In practice, the frequency of vibration is only weakly dependent on the stiffness (⬃k 1/2) and additional uncertainty due to appreciable losses results in a relatively large margin of error and casts severe doubt on comparative results, whether an impulse excitation method or a steady state harmonic excitation method is used. The detailed interpretation of these results is further complicated by the fact that these vibrations are obviously nonlinear. Despite all these defects, the spread of the values of the stiffnesses and losses obtained is not too great. These values are still a useful In the above, we have discussed two of the important mechanical characteristics of a magnet superconductor levitation system: the levitation force and the stiffness. We have briefly mentioned a third, the power loss, all as manifested by the back and forth linear vibrational motion of the magnet along a straight line. In what follows, we will examine the power loss as manifested by the rotational motion of the magnet about an axis, labeled the z axis. The rotational motion of a magnet shares with the linear vibrational motion the property of being periodic, but in the rotational motion, the return of the magnet to some earlier position does not involve the reversal of any portion of the motion, and so cannot be represented by a hysteresis loop of any sort. Nevertheless, the power loss can be obtained from the average torque acting on the magnet as it excecutes one revolution around the z axis, just as the power loss can be obtained from the average force acting against the magnet as it goes through one cycle of a linear vibration with due account being taken of the opposite sign of the force for opposite direction of motion. We have pointed out earlier that if the magnetic has a magnetic field that is axisymmetric about the z axis, then the drag torque vanishes indeed, and we have an ideal bearing system in which a rotating magnet with an axisymmetric magnetic field levitated over a superconductor without encountering any drag torque. This is exemplified in the pioneering work of Moon and Chang.32 On the other hand, if the magnetic field of the magnet is not axisymmetric, then a drag torque is observed and a clutch system can be fabricated using this scheme as shown earlier in Fig. 2共b兲. For the same reason, imperfect axisymmetry of a magnet used in a bearing system gives rise to a small amount of residual drag torque in the bearing. Thus, it is of interest to study the relationship between the azimuthal variation of the magnetic field, the drag torque that it creates against the rotation of the magnet, and the energy loss of the system. We begin with an ideal model system that is invariant in the z direction. The magnetic field from the magnet is taken to be transverse to the z axis and hence has no component in the z direction, and the superconductor in the shape of a 5000 Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 circular cylinder rotates instead of the magnet. The superconductor can be ZFC or FC. Then we can imagine each angular segment to be subjected to the action of the tangential components of the field, B t , at the surface. We assume a result with the same behavior as that which follows from the Bean model for the case of a uniform axial magnet field acting on an infinitely long superconductor cylinder. More specifically, flux fronts of constant current density, J c ⫽critical current density, flowing in the z direction, will penetrate inward at a rate governed by the rate of change of B t so long as the direction of change is maintained, but a flux front is created and starts to propagate inward from the surface once the direction of change of B t is reversed. Another region of constant current density, J c , also flowing along the z axis but in the opposite direction, follows the flux inwards. For a uniform external field transverse to the z axis, consistent application of this procedure to every angular segment of the rotating superconductor produces the spiral distribution of current densities shown in Fig. 10 as a first approximation to the steady state after more than one revolution of the superconductor.33,34 This current distribution rotates backwards in the rotating frame of the superconductor at the same rate as the superconductor is rotating forwards. Thus this current distribution is stationary in the frame of the nonrotating magnet. This spiral current distribution is common to ZFC or FC cases. The differences between these two cases lie in an extra ‘‘eye’’ shaped distribution at the core that supports the trapped field in the FC case, but is absent in the ZFC case. This trapped field rotates with the rotating superconductor. If we examine the current distribution in the superconductor immediately before the first revolution of the superconductor, we would discover the same ‘‘eye’’ shaped distribution for the ZFC case, but not in the FC case. This is summarized in Fig. 10. As a result of the current distribution in the superconductor as described above, a torque is exerted on the superconductor. The spiral distribution exerts a constant torque because it is actually stationary in the laboratory magnet frame, and not rotating. The magnitude of this constant torque is the same for both the FC and ZFC cases. In the limit of a weak external magnetic field, this constant torque 3 varies as B ext , implying a power loss per cycle that has the same dependence on the external field intensity. The additional eye shaped current distribution that exists in the FC case represents a constant magnetic moment rotating rigidly with the superconductor. Thus it gives rise to a torque that varies as cos ␾ 共␾ is the azimuth angle around the z axis兲, just like a permanent magnet rotating in a uniform field. This sinusoidal torque is superposed on the constant background torque from the spiral distribution. The amplitude of this 2 instead, in component is expected to be proportional to B ext the weak field limit. If we perform a sweeping field while maintaining constant rotation of the superconductor in the experiment, this torque varies as the B ext at the moment of superconductor FC. As far as power loss is concerned, this sinusoidal component of the torque has no contribution, because its average over one complete cycle vanishes. A large portion of these results is borne out by experiments. The rest awaits further experimentation. Figure 11共a兲 Ma, Postrekhin, and Chu FIG. 12. Average drag torque as function of current of the magnetic coils. shows the drag torque exerted on a ZFC superconductor circular disk steadily rotating in a transverse uniform magnetic field, plotted against time or azimuthal angle, while the same for the FC case is displayed in Fig. 11共b兲. The constant offset can be seen to be equal within experimental uncertainty. In Fig. 12, the magnitude of this constant is plotted against the current in the coils that produce the magnetic field. It shows 3 reasonable agreement with a B ext dependence.33 The sinusoidal torque is clearly present in the FC case, but the ZFC result also has a small oscillating component. This is believed to be due to inhomogeneities in the superconductor samples, as it is more irregular and its most dominant period is not identical to the rotation rate, but a multiple of it. VII. APPLICATION OF SUPERCONDUCTING LEVITATION The universal principle behind all levitation devices with superconductor and magnet is to utilize the capacity of this system to effect or prevent transfer of momentum in preferred directions in the absence of a material media or direct contact, with inherent stability. Application may then be broadly classified into two categories. One category of applications consists of those with the objective of inducing motion in one specific direction, such as in a clutch system, or to stop motion in one or many directions, such as in vibration damping. The other category consists of those application with the objective of preserving the motion of a component part in one specific direction, to create effectively frictionless interfaces as in vibration isolation, or in linear or rotational bearings of all sorts. However, research and development efforts have not been distributed evenly among these categories. In fact, most attention is directed towards the superconductor magnet bearing in the second category of applications. Of these, again, most work goes to the development of one application: flywheel kinetic energy storage systems. In view of this situation, we will first review the rotational superconductor magnet bearing, followed by its application in flywheel kinetic energy storage systems. Next, we will examine a slightly variant application: the momentum or reac- Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 tion wheel for small satellites. After this, we will take a look at vibration free platforms, which can involve both vibration isolation, an application in the second category, and vibration damping, an application in the first category. Another example of an application in the first category is the hysteretic superconductor motors using the principles of the above discussion of the drag torque experienced by a superconductor rotating in a uniform magnetic field about a transverse axis. There has not been enough development work on this device to merit further discussion here. Maglev with high temperature superconductors is another example of a second category application not involving rotational superconductor magnet bearings, but linear ones. Although Maglev is probably the most popular embodiment of levitation in the public fascination with levitation phenomena, we will skip over this particular application as it would stretch the scope of this review a bit too far. A. Superconductor magnet levitation bearings Bearings are essential components in many mechanical devices, designed as an interface that minimizes dissipative effects between moving and stationary parts. For instance, the action of ball bearings relies on a smaller rolling friction substituting for a larger sliding friction. This friction could be further reduced by incorporation of solid or liquid lubricants. However, direct physical contact between moving and stationary solids still exists within such bearings, giving rise to mechanical wear and tear. This ultimately limits the lifetime of the bearing. With fluid film bearings and compressed air bearings, contact between moving and stationary parts is mediated by a moving fluid in fast motion under high pressure. Power loss is reduced to that associated with the fluid flow, and solid grinding on solid is replaced by erosion at the fluid–solid interface. The only bearings that eliminate all contacts between moving and stationary components, be it solid or fluid, are levitation devices. Currently, magnetic bearings represent the most developed version of noncontact levitation bearings. Naturally, magnetic levitation bearings can be classified according to the manner of achieving stable levitation. Two major means that are more or less developed are active electronic control or use of high temperature superconductors. As active magnetic bearings are outside the scope of the present review, we will not discuss them further except for purposes of comparison. Before we launch into a detailed description of superconductor magnet levitation bearings, we like to digress a bit to mention the various aspects of a bearing that would be of interest to a potential user trying to make an informed choice among various possible candidates for utilization. First and foremost, of course, is the capability of the bearing to fulfill its purpose of allowing the free motion of one component of a piece of machinery relative to another. Assuming that we are dealing with a rotary bearing, a quantitative indicator of how good the bearing is is the drag torque exerted by the stator against the rotation of the rotor, this being the smaller the better. Another indicator is the mechanical power loss at the bearing, easily measured by a spindown experiment. This yields directly the heat generated at the bearing, and also gives the minimal power required to maintain the rotational Superconductor and magnet levitation 5001 speed of the rotor. The smaller these quantities are, the better the performance of the bearing. More implicated in the essence for a bearing is that it must also provide the necessary forces to keep the moving component in place relative to the stationary component, while allowing the moving component to move freely in the intended direction. For a rotary bearing, this means that we must look into the weight that the bearing can support, the available bending stiffness that keeps the rotor properly aligned, and the lateral stiffness that keeps the rotor centered. Finally, all but one of these properties, namely, the drag torque, the sustenance power, the dynamic bending, and lateral stiffnesses but not the support force, are all dependent on the rotating speed of the rotor, in principle, with the last two giving rise to large amplitude vibrations and rotodynamic resonant instabilities when not adequately designed for. Ultimately, there is the issue of long term reliability and service lifetime. From the perspective of the practitioner, mechanical bearings are always considered first. Mechanical bearings have a long history of service. They are readily available and easy to apply. They offer more support force and stiffnesses than other bearings possibly could. Only when the shortcomings of a mechanical bearing become severe enough to defeat the purpose of an application would one consider other more ‘‘advanced’’ bearings. Most of the misgivings of mechanical bearings can be traced to the fact that there are moving parts in contact with each other in these devices. Such contact points are sites of inevitable wear and tear, which eventually leads to the failure, sometimes catastrophic, of the bearing. Due to this occurrence of wear and tear, mechanical bearings need to be replaced or rebuilt after a certain length of service life, even in the absence of accident or abuse. Often the degrading effects of wear and tear are aggravated by the gradual accumulation of dust particles from the environment, or the very products of the wear and tear process itself. In the short term, wear and tear contributes to the small but appreciable friction in mechanical bearings. In its turn, friction, due to wear and tear as well as other causes, gives rise to loss of mechanical energy and generates heat, which may lead to other problems if excessive, and is difficult to remove in a vacuum environment. In general, these ills may be mitigated by the use of lubricants, which have to be chosen to fit the working environment, such as vacuum, cryogenic temperatures, or a combination of both. Solid lubricants such as graphite or molybdenum disulphide, which may otherwise be appropriate for a cold vacuum, could introduce torque jitters that interfere with the even operation of the bearing. Since contact points are at the roots of all these trouble, it seems natural to turn to noncontact bearings for improved performance. The most studied among these are probably the magnetic bearings, active or passive using superconductors. These devices are of relatively recent stock, and so there is no track record to back up any claim of long term reliability, although we can almost be certain that if there is a long term failure mode, it would not be coming from any wear and tear mechanism. The realization of almost nonexistent friction, however, is immediately obvious at low to moderate rotating speeds, but the full extent of the benefit for high speeds must 5002 Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 still be confined to operation under vacuum. It is also true that the extremely low friction enables the bearing to function with a corresponding low loss of mechanical energy and heat generation, but again, ingenuity must be exercised to prevent energy saved at the bearing to be squandered at the accessories such as motor/generators that it couples to, the control electronics circuitry for an active setup, or the cryogenic system for an implementation with superconductors and magnets. Yet another consideration with these noncontact bearings is the supporting force and stiffnesses that they can provide. Compared to the contact force and stiffnesses available in mechanical bearings, the magnetic force and stiffnesses that can be garnered from active magnetic bearings is more than 3 orders of magnitude weaker, and that from the passive superconductor magnet bearings, an additional 1 or 2 orders of magnitude weaker. We will be going into some of the methods and designs to alleviate some of these problems later. Certainly, there are applications for which the ultralow levels of energy dissipation offered by the levitation bearings is the enabling technology. We will discuss further details for each of these applications further on. Here, we assume that we have such an application on hand, and address the choice between the active and passive levitation bearings. First, a little thought would convince us that whereas the drag torque can be measured by a spindown experiment with passive levitation bearings, this approach is not available to the active magnetic bearing. It is difficult to arrange so that the energy going into levitation would be strictly excluded from driving the rotor itself. What we are really after here is not so much the drag torque as the power required to sustain the rotation of the rotor at a specified speed. This sustenance power is a useful indicator to compare the active and passive magnetic bearings on an equal footing. Without having to work against contact friction, the sustenance power goes into making up for energy loss due to air resistance and eddy current losses in electrically conducting components subject to varying magnetic fields due to nonideality of parts. Furthermore, power is needed for the electronic circuitry in indispensable accessories such as motor/generators. All of these are common with either the active or the passive magnetic bearings. Where they differ is in the power needed for control to overcome instability of equilibrium under magnetic forces in the case of the active magnetic bearings, and the power needed to run the cryogenic system to maintain low temperatures in the case of the passive magnetic bearings with superconductors. The amount of power needed depends on sensor resolution, the presence of other sources of systematic noise that the sensor is sensitive to such as vibrations coming from the nonideal uniformity of rotor magnets or mechanical rotor imbalance and the amount of background vibrations in the former case, and the ambient temperature in the latter, given the best possible thermal insulation available. Thus, we cannot justify a universal preference of one over the other on the basis of sustenance power—it depends on the particular application we have in mind. We have already assumed that no power is spent to levitate with the active magnetic bearings, this function being substantially taken over by using supplementary permanent mag- Ma, Postrekhin, and Chu nets. Otherwise, the power needed to levitate a rotor is frequently more than that would have been lost with a mechanical bearing. To make matters worse, the concomitant Joule and core losses in the electromagnets concerned pose a problem of heat removal from the vacuum environment necessary to reduce windage losses. Thus, the design of the active magnetic bearing would have to be at a more sophisticated level than the conventional in order to be competitive with the passive magnetic bearing using superconductors. Using additional permanent magnets in a supplementary role is also beneficial with passive magnetic bearings, where they can improve the thrust that the bearing can provide.30 If done skillfully, the stability of the bearing does not suffer either. Overall, supporting force and stiffnesses that can be obtained from passive magnetic bearings using superconductors is still less than that from active magnetic bearings. The fundamental reason for this is that the depth of magnetic charging due to motion of permanent magnets in the vicinity of bulk superconductors is only a tiny fraction of what is potentially possible with the superconducting material. Put in another way, the active components of an active magnetic bearing can direct an external power source to pass currents in the control coils that surpasses that expected of a perfectly diamagnetic response. In principle, this extra power need not be dissipated in the process either, but can be stored as magnetic potential energy in the coils, to be relinquished for use elsewhere when the need changes. In practice, this approach has not been attempted because of the immediate problems of ohmic losses with copper coils, or ac losses with superconducting coils. Active magnetic bearings also offer a wider range of support forces and stiffnesses, as well as the capability to adjust these quantities after assembly, which confers upon them some degree of portability between different pieces of machinery. All things aside, there is still a lot of room for improvement with regard to providing high support forces and stiffnesses with levitation bearings, active or passive, to catch up with what we are accustomed to with mechanical bearings. In Fig. 13, we show the schematics of the very simplest prototypes of a superconducting magnetic bearing 共SMB兲, an active magnetic bearing, and a mechanical ball bearing. As can be seen, the SMB is just a generic superconductor magnet system with the restriction that the permanent magnet should have an axis of rotational symmetry which serves as the rotation axis of the bearing about which the magnet can spin freely. We have discussed the magneto–mechanical properties of the generic superconductor magnet system in some detail above. The active magnetic bearing in Fig. 13共b兲 is converted from this SMB by replacing the superconductor with coils and sensors driven by electronic circuitry so as to mimic the diamagnetic response of the superconductor in Fig. 13共a兲. Figure 13共c兲 is a simple ball bearing, having about the same overall size and shape of the magnetic bearings in Figs. 13共a兲 and 13共b兲. We summarize the pros and cons of these bearings in Table I. As pointed out before, the simple version of the active magnetic bearing shown in Fig. 13共b兲 is not competitive from the viewpoint of sustenance power because the rotor is supported by an electromagnet that is part of the control Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 Superconductor and magnet levitation 5003 FIG. 13. Schematic diagram of 共a兲 superconducting magnetic bearing 共SMB兲; 共b兲 active magnetic bearing 共AMB兲; 共c兲 mechanical ball bearing 共MBB兲; and 共d兲 SMB, 共e兲 AMB, and 共f兲 MBB with supplement permanent magnet. The dotted–dashed lines in these diagrams represent the rotational axis. circuitry. One of the methods to reduce this disadvantage is to replace the electromagnet by a permanent magnet to provide the major portion of the necessary support force, combined with a smaller coil as fine tuning for control purposes. This modified version is shown in Fig. 13共e兲. The same idea of using a permanent magnet to provide most of the support force is also applicable to the SMB. This was also discussed and illustrated in Fig. 2 and presented in Fig. 13共d兲 for comparison purposes. Even for a mechanical bearing, a supplementary permanent magnet may be inserted as shown in Fig. 13共f兲 to reduce the weight on the bearing, thereby reducing its friction. The effect is not as dramatic as with the magnetic bearings. Use of supplementary magnets is one way of improving the performance of superconductor magnet bearings. One example is the patent of Chu et al.13 in which an additional cylindrical permanent magnet is placed on the rotation axis behind the superconductor in the stator to strengthen support of the rotor along the vertical axis. Another example is the patent of Iannello et al.35 in which additional ring magnets concentric with the rotor are used to provide support of rotor spinning on a horizontal axis. These are illustrated schematically in Figs. 14 and 15, respectively. While enhancing the support force available from the superconductor magnet bearing, such use of additional magnets may degrade the stiffness in some directions, unless care is taken in the design of these magnets, taking advantage of configurations in which the stator magnets together provide a position of neutral equilibrium for the rotor magnet. At the very least, there is an increase in the complexity of the design. Further, using additional magnets to provide more force does not work very well if the load is not constant, such as when the bearing is to be tilted at various angles during operation, for instance. One extreme form of using a combination of magnets TABLE I. Comparison of superconducting magnetic bearings, active magnetic bearings, and mechanical ball bearings. Coefficient of friction Wear Control system Auxiliary parts Unlimited angular speed Support pressure Stiffness Superconducting magnetic bearings Active magnetic bearings Mechanical ball bearings 10⫺7 10⫺3 yes 10⫺4 共including the electronics losses兲 no yes yes 共sensor and control electronics兲 yes low low low low high high no no yes 共cryocooler兲 yes no no no 5004 Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 FIG. 14. Hybrid superconducting magnetic bearing with regulating coils 共1,7兲 magnets of thrust bearing, 共3,4兲 magnets of journal bearings, 共5,8兲 superconductor for stabilization, 共2,9兲 regulating coils, and 共6兲 shaft 共see Ref. 13兲. and superconductor for the levitating stator is in the form of a superconductor magnet composite, made by mixing millimeter size granules of permanent magnet material with type II superconductor material, as in the patent of Rigney.36 It is claimed in this patent that the composite combines the properties of the superconductor and magnets with the flexibility and toughness of a polymeric material, yet can be machined easily into a bearing structure. Bearings made from this composite have the load capacity and stiffness of permanent magnet bearings with added stability from a type II superconducting material. However, this combination of desirable attributes has not yet been explicitly demonstrated. It was noted that a weak point of the magnetic bearings is low values for the forces and stiffnesses available. As of this moment, there is no solution that can overcome all of these shortcomings unconditionally. This situation allows the development of all sorts of variations around the basic theme, all with their special niches and peculiar misgivings. The additional of supplementary magnets was examined above. This approach is not restricted to using permanent magnets and electromagnets can be introduced instead. Certainly, this will yield higher support forces and superior stiffnesses with higher magnetic fields and field gradients. It has the same disadvantages as suffered by the use of permanent magnets, but in this case, we have the option of varying the current through the coils for control. However, this improve- FIG. 15. Combination of journal and thrust superconducting magnetic bearing 共see Ref. 35兲. Ma, Postrekhin, and Chu ment would be moot if excessive ohmic losses with copper coils, or ac losses with superconducting coils, appears. It seems best to use superconducting magnets to provide a steady intense magnetic field. The optimal mix of control to implement should be determined on a case by case basis. An interesting variation given in a patent by Moon37 is to mount the superconducting magnet on the rotor instead, as no electrical connection to the stationary environment is necessary when the superconducting magnet is operated in the persistent current mode. The threat of Earnshaw type magnetic instability is avoided by using only superconductors and not magnets in the stator. Here, we face another technical challenge. It is not easy to keep the superconducting magnet on the rotor cold enough for its superconductors to work. The same superconductor that we use in levitation bearings also adds to our list of magnets that we can use. Trapped field magnets, made most conveniently by cooling a superconductor in an intense magnetic field and then switching the field off as mentioned earlier, offers higher flux densities with a softer profile due to the finite depth of its magnetization current distribution. It combines properties of a permanent magnet with that of a type II superconductor. Thus, the trapped field magnet can be used to either substitute for permanent magnets or as additional magnets. For instance, we can construct levitation bearings with a trapped field magnet levitating a permanent magnet or another superconductor cooled in its field. However, it cannot be used as a superconductor in the FC mode levitating a permanent magnet, as the FC process has already been utilized in its fabrication as a trapped field magnet. Consequently, and partly due to the softer profile as well, superconductor magnet bearings incorporating trapped field magnets tend to, but not always, have lower stiffnesses38 and be less stable compared to FC SMB, but similar to the simple SMB assembled under ZFC conditions. Of course, they can produce a higher levitating force,39 just as the use of superconducting magnets do. When assembled into a device, they may present a problem of accessibility for recharging by the FC method. However, this can be overcome by using pulse field magnetization25,39 if provided with adequate coils such as with superconducting magnets or ordinary electromagnets. Trapped field magnets also share with superconducting magnets the problem of cooling if attached to the rotor. Trapped field magnets are simpler, structure wise, than superconducting magnets, but their magnetization is not a controllable variable, resembling permanent magnets in this regard. With both permanent magnets and trapped field magnets, the most one can do is to have them magnetized at less than their respective saturation values during fabrication and be used as such, but they cannot be altered at will during active service. As a type II superconductor, the field from trapped field magnets does relax over time, but this trouble is only slightly worse than the case of superconducting magnet bearings without explicit utilization of trapped field magnets, as trapped fields, but weaker, are already involved there. Apart from remnant magnetization in ferromagnetic materials or type II superconductors, or currents injected into coils, magnetization induced in soft magnetic material by an external field is also effective for levitation by a supercon- Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 FIG. 16. Schematic of basic mixed-mu system, with ferromagnet 共Fe兲 stably levitated between permanent magnet 共PM兲 and HTS pairs 共see Ref. 40兲. ductor. Hull et al.40 demonstrated this approach with a cylindrical rod of soft iron levitated between two YBCO disks and two rare-earth 共RE兲 magnets, all aligned on the same vertical axis, as illustrated in Fig. 16. Both the YBCO disks and the RE magnets are fixed. Only the iron rod can rotate freely. These were called ‘‘mixed-mu’’ systems in view of the fact that both paramagnetic and diamagnetic materials have an equivalent permeability, ␮ or ‘‘mu,’’ larger than and smaller than unity, respectively. Mixed-mu systems with levitation gaps not in the axial direction but in the lateral direction were studied by Fucasawa and Ohsaki.41 The configuration they investigated is appropriate for linear bearings which can easily be adapted for a rotating bearing as follows. A short iron cylinder is levitated by a concentric YBCO ring while being magnetized by a larger ring of rare-earth magnet, also concentric with the other two components. Again, only the center iron cylinder is allowed to rotate freely. Both YBCO and RE magnet rings are fixed. This is shown schematically in Fig. 17. Hull pointed out that one advantage of this system lies in the superior mechanical strength of the soft iron rods compared to rare-earth magnets, allowing iron rods to reach higher rotational speeds than rare-earth magnets mounted onto the rotors. A more significant advantage of this system, however, is that all the economically prized YBCO and RE FIG. 17. Schematic diagram of a mixed-mu system with ring-shape HTS and magnet and a levitated iron disk 共Fe disk兲. HTS and magnet rigid are attached through aluminum ring 共Al兲 共see Ref. 41兲. Superconductor and magnet levitation 5005 magnet pieces are either all on the rotor or all on the stator. They can be fixed on the stator as in these investigations, or they can be levitated together, as one might visualize in a magnetically levitated train, leaving the iron as the rail fixed to the ground. This is attractive from the point of view of mass production. Unfortunately, the forces and stiffnesses that can be delivered by the mixed-mu systems studied so far has not been scaled up to levels practical for this application. Forces and stiffnesses that are already demonstrated do not excel beyond those obtained with the simple superconductor magnet systems. All of the above modifications can improve the force a lot, but less so the stiffnesses. One way to improve the stiffnesses moderately lies in the design of the magnet. The concept behind this is to introduce multiple regions of opposing magnetization into the magnets, consistent with the overall requirement of axisymmetry about the rotational axis, and then take advantage of the field gradient effect discussed earlier. This effect falls faster as a function of the gap between the magnet and the superconductor than the levitating force. Thus the improvement levels off as we subdivide the magnet into finer and finer regions and eventually disappears together with the force unless we follow this with a decreasing gap as well. The optimal point is when we operate at a gap that is the same size as a characteristic length of the regions of alternating magnetization. All these assume that the critical current density of the superconductor is high enough so that the penetration depth of the magnetic field involved is smaller than either of the other length scales. When the penetration depth reaches either of these other length scales, no further significant improvement can be expected. Examples of design that make use of this concept are shown in Fig. 1共a兲. These designs are contained in the patents of Ito et al.,42 Takahata et al.,43 and Weinberger and Lynds.44 Another way of looking at these designs illustrates another simple method of increasing the force and stiffnesses coming out of a superconductor magnet bearing, that of using many simple units integrated as one unit. The designs of Weinberger and Lynds44 and some of Takahata et al.43 may be regarded as multiple units stacked along the direction of the rotation axis, while that of Ito et al.42 and the other designs of Takahata et al.43 have multiple units arranged as a series of concentric rings on a plane. An even simpler adaptation of this method, and one that is very common, is the use of levitation and suspension units on the same rotor— two units, twice the force and stiffnesses, roughly speaking. Examples are found in Weinberger et al.45 and Chen et al.46 The one big disadvantage of multiple units is the increase in the bulk of the bearing itself. Integration, so that some parts may be shared by adjacent units, improves the situation a bit and blurs the distinction between separate individual units and one complex design with multiple subunits. Another example of integration is the use of tapered units as in the patent of Fukuyama et al.47 This design provides both lateral and axial stiffnesses with the same piece of superconducting materials, reducing the amount of material required while yielding individual stiffnesses of approximately the same values. 5006 Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 The above method of scaling up is not viable in some applications where space is a premium. Then we have to resort to options that come with a higher force or stiffness densities, and this can be achieved with superconductors with a higher critical current density. It is known very early on that the same RE–BCO superconductors that are commonly used in bulk form in superconductor magnet bearings have a high critical current density when fabricated as thin films. Indeed, the levitation force of three epitaxial YBCO films and a stack made of these films show a very high force density of about 100 N cm⫺3 and a large hysteresis of the force versus distance relation, with the stack yielding a levitation force that is nearly as high as the sum of the forces of the single films and a magnetic stiffness comparable to that of melt-textured bulk samples.48 Force densities as high as 300 N/cm3 have been observed with thin films.49 In practice, these advantages are compromised by the necessity of putting the thin films on a substrate. Attempts to moderate this dilution effect by using thick films met with the dilemma that the critical current density drops towards the bulk material value as the film thickness is increased. Thin film superconductor magnet bearings have been patented.50 They cannot be used to provide a large support or stiffness, but they are useful in micromachine devices for the dual reason that bulky components are obviously not welcome here and the weight of the components that needs to be supported in this case is proportionately smaller, by definition. B. Flywheel kinetic energy storage with levitation bearings We have argued in the previous section that promising applications of magnetic levitation bearings are likely to be those in which it is essential to keep energy dissipation as low as possible. One of those is long term energy storage. If we decide to store energy in the form of the kinetic energy of a flywheel for a long period of time, then magnetic bearings are very attractive because they exhibit low losses. However, there are alternative forms of energy storage, and whether a flywheel energy storage system 共FESS兲 is the optimal choice depends very much on the specific application. Historically, it was not the needs of a FESS that spawned the development of the levitation bearing with superconductors, but rather that the study of pioneering rotating devices with superconducting bearings51,52 that stimulated the concrete embodiment of the levitation bearing in FESS. Here, we will briefly discuss the pros and cons of FESS compared with other forms of energy storage. Of course, it would be best to store energy in the same form as that to be used. Should that not be possible, the next best situation is to store the form of energy most conveniently or efficiently converted into the form that we need, almost exclusively through electrical energy as an intermediate. Forms of energy that can be stored and subsequently utilized in this fashion include: chemical energy in rechargeable batteries, kinetic energy in flywheels coupled to a motor/generator, magnetic energy in superconducting coils 共SMES兲, electrostatic energy in capacitors holding a charge, gravitational potential energy in a reservoir placed at high Ma, Postrekhin, and Chu FIG. 18. Power and energy density comparison 共see Ref. 53兲. altitude, potential energy in the form of compressed air, or thermal energy in the form of hot water heated by the sun. Although not usually thought of as energy storage, electrical generators powered by gas turbines can be regarded as an alternative vehicle to deliver stored chemical energy as electrical energy. In Fig. 18, the ranges of specific energy and specific power available to electrochemical batteries, FESS, SMES, and capacitor banks are mapped out.53 FESS is second only to electrochemical batteries in terms of specific energy, but is capable of delivering a higher power, though not as much as SMES or capacitor banks. A similar trend occurs with respect to energy and power densities, as the average mass density of all devices is in the same order of magnitude. FIG. 19. Photo of prototype flywheel energy storage system 共TCSAM兲. Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 Superconductor and magnet levitation 5007 First, in order to hold a large amount of energy, the flywheel has to attain a high rotational speed. The construction of the flywheel must be able to contain the mounting centrifugal forces as it spins up. A simple dimensional argument will show us that the energy stored per unit volume of the material of the rotor at burst speed varies directly as the ultimate strength of the material, ␴ u , as the only material parameter that enters for isotropic materials and homogeneous flywheels. Thus we can write 共 E/V 兲 burst FIG. 20. Cut out section diagram of flywheel energy storage system 共Boeing兲. Compared to batteries, FESS has not yet achieved the same holding time for a single charge, but can be expected to be far superior in terms of cycling lifetime, and therefore does not have to be changed out that often. FESS is also more benign to the environment, as there are no consumed waste parts to be disposed of. Although we have justified the use of magnetic levitation bearings in FESS in terms of its low rate of energy dissipation, it is not absolutely necessary. We can live with a moderate rate of energy loss if we plan to use it only for a short time interval. Such is the case with some uninterruptible power supplies 共UPSs兲 intended to maintain the operation of a high power facilities over glitches in the main power line for fractions of a second. The UPSs can be charged from the main line at a very sustainable cost, and the facility is insured against short disruptions. From the same viewpoint, the relatively short holding time of days for the FESS, compared to batteries, is not a disadvantage when the FESS is used to smooth out diurnal variations, but the much larger number of charge–discharge cycles that the FESS can tolerate is a critical asset. This is an example of how the end use of the stored energy determines the most appropriate storage device. The magnetic levitation bearing is what makes the holding time of days possible with a FESS but not achievable with mechanical bearings. The principle behind which a FESS works is very simple. A spinning flywheel has kinetic energy, given by E⫽I ␻ 2 /2 共1兲 that can be increased by any agent which spins it up. Here I is the moment of inertia of the flywheel and ␻ is its rotational speed. If we use an electric motor to spin it up, we are storing electrical energy into the flywheel in the form of its kinetic energy. If we gear the flywheel to the axle of a windmill, we are directly converting and storing wind energy as flywheel kinetic energy. This rotational energy of the flywheel can easily be converted back into electrical energy by directing it to drive an electric generator. However, the devil is in the details. speed⫽K ␴ u . 共2兲 Here K is a dimensionless shape factor that depends on the geometrical configuration of the flywheel and the failure criterion used.54 With anisotropic materials such as composites, or inhomogeneous flywheels containing sections of different materials, this relationship would have to be reformulated in terms of a reference value for the ultimate material strength as some appropriately defined average of the mix with which the failure criterion is to be defined. If the weight of the flywheel is a factor to be minimized, we may consider the specific energy 共 E/M 兲 burst speed⫽K ␴ u / ␳ 共3兲 instead. Here, M is the total mass of the flywheel and ␳ is the average density of the materials contained in the flywheel. The main message here is that flywheels ought to be made of materials with a high ultimate strength and low density. Fortunately, these material selection criteria do not necessitate a gross compromise. Carbon fibers can be made to have a high tensile strength and a low density. Modern flywheels in the forefront of technology intended for performance at high speeds are frequently made out of carbon fiber containing composites as the main bulk of inertia.55 For example, the 1 kW h FESS constructed by Minami et al.56,57 utilizes a flywheel made out of carbon fiber reinforced plastic ring, 60 cm outer diameter, 45 cm inner diameter, and 16 cm thick rotating at 20 000 rpm. The same material is used in a 40 cm outer diameter, 37 kg flywheel by Miyagawa et al.58 to store 0.5 kW h of energy by rotating the flywheel up to 30 000 rpm. Other fibers of comparable strength are available. Flywheels of more sophisticated designs exist. Flywheel design with attention of optimal stress distribution is itself an involving subject. Further discussion would take us to far beyond the scope of this review and the reader is referred to the book by Genta54 for further details. Ideally speaking, with a flywheel supported on a levitation bearing, it should keep on spinning until we choose to extract its energy for use. In practice, it slows down, albeit slowly enough for it to be useful as energy storage for a medium term. The simplest way to characterize this residual rate of energy loss is to observe the rate at which it slows down. Mulcahy et al. suggest the use of an effective coefficient of friction, ␮, as a figure of merit for the bearing59 ␮ ⫽2 ␲ R ␥2 共 ⫺d f /dt 兲 / 共 gR D 兲 . 共4兲 Here, R ␥ is the radius of gyration of the flywheel, d f /dt is the negative rate of change of the rotation speed of the flywheel measured in a spin down experiment, R D is the mean radius at which the drag force acts, and g is the acceleration 5008 Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 of gravity. This quantity can be extracted easily from a spin down measurement with a reasonable assumed value for R D . It gives us a general feel as to the rate of energy dissipation at the bearing, and may be used to compare bearings of different designs in a semiquantitative manner if used with due caution. Typical values for superconductor magnet bearings are in the range from 10⫺6 to 10⫺4 . 60 However, it has not been established that this quantity is a meaningful characterization of the bearing interface for this type of bearing, or in other words, that the drag torque is indeed proportional to the lift for a single bearing. One obvious source of drag, which should not be included in the coefficient of friction definition given above, is air drag. This goes up roughly as ␻ 3 ( ␻ ⫽spin rate of flywheel兲, and is unacceptably high at speeds commonly aimed for in FESS. It causes perceptible slowing down even in laboratory demonstration at speeds as low as 100 rpm. It has been demonstrated in a study by Xia et al.61 that in order for the promised low rate of energy loss to be realized in flywheels with superconductor magnet bearings, the flywheel has to be contained in a vacuum chamber under pressures lower than 10⫺6 Torr. Together with the necessity to keep the superconductors cold by either using a cryocooler or immersion in liquid nitrogen under standard atmospheric pressure, fabrication of the delivery system for cryogens does require some careful planning. Now, consider a flywheel supported by superconductor magnet bearing spinning in a vacuum good enough for the air drag to be ignored. A residual loss still exists, all of which can be traced to the nonidealities of the superconductor or the magnetic component of the bearing, and mostly the latter. First and foremost, the magnet on the rotor should be axisymmetric about the rotation axis. Otherwise, loss will occur via the magnetic hysteresis of the superconductor. We have discussed this mechanism above. In the context of a FESS, the size of the largest monolithic rare-earth magnet available tends to set the scale of the size of the flywheel. Larger magnets would have to be made by joining smaller segments together, and circumferential variation of the magnetic field becomes more serious, particularly across the joints. This deficiency could be somewhat moderated by appropriate insertion of shimmings that reduce the field variations, as demonstrated by Higasa et al.62 Another idea along the same line of thought is to use the shielding effect of thin films of YBCO to smooth out the variations of the rotor magnet from the superconductor. This approach has been shown not to work because the shielding is only effective against the component of the magnetic field that is perpendicular to the surface of the superconductor and not those that are parallel to it.32 The best results still come from monolithic magnets, and they can be obtained with a typical value of 3% in the circumferential variation of the field intensity. Further losses are induced by azimuthal variation of the magnetic field of the rotor magnet in the form of eddy currents in electrically conducting parts in its vicinity. Eddy currents losses increases as ␻ 2 , whereas the hysteretic losses in the superconductor increase only as ␻. This means that hysteretic losses dominate at low speeds, but eddy current losses would eventually take over at high speeds if we do not Ma, Postrekhin, and Chu take steps to keep it out. This poses a further challenge to the fabrication of the cryogenic delivery system into the vacuum chamber to avoid using metallic parts near the superconductors to be cooled. The improvement of the spin down result of Xia et al.61 over that of Chen et al. earlier46 is due, in part, to the replacement of the cryogenic chamber containing the superconductors constructed out of stainless steel to one made out of G10, a fiber glass epoxy composite. This observation also has implication in the use of cold fingers in combination with cryocoolers to maintain the temperature of the superconductors in vacuum down into the liquid nitrogen range. Metallic cold fingers cannot extend all the way to the space near the rotor magnets without inducing some additional eddy current losses at high speed in the presence of slightly nonaxisymmetric rotor magnets. To avoid such situations, the superconductors doing the levitating can always be interposed in between. Eddy currents can also be induced in the rotor magnets if they sense a varying magnetic flux density as the rotor spins around. This condition is created, for instance, when the superconductor magnetic bearing and flywheel is assembled in any way that requires the rotor to adjust its position for mechanical equilibrium after the superconductors are cooled below their critical temperature. If this shift in the rotor position does not magnetize the superconductor symmetrically about the rotation axis, subsequent rotation of the rotor would generate eddy currents in any metallic parts of the rotor including the rotor magnet, under the influence of the magnetic field from the magnetization of the superconductors that is not axisymmetric. It would be ideal to use a permanent magnet that is also an electrical insulator, but none exists that retains a high enough magnetic field. It is more feasible to try to assemble the rotor so that the weight of the rotor is exactly compensated by other noncontact means such as additional magnets before the superconductors are cold, so that the rotor remains in equilibrium in the same position after the superconductors are cold. When the superconductors were still warm, the rotor is stabilized in that position by light contact with mechanical holders, which are released when the superconductors take over the function of stabilization. This philosophy is followed in the hybrid superconductor magnet bearing approach advanced by Chu et al. in the patent,63 and demonstrated in the work of Chen et al. and Xia et al.46,61 If we use a strategy of mixed ZFC and FC so that the rotor settles into a new equilibrium position along the rotation axis and the superconductor that the rotor settles on is axisymmetric, the superconductor will be magnetized symmetrically about the rotation axis and we do not have to be concerned about eddy currents in the rotor either. More frequently, the requirements of any sizable FESS is such that high quality superconductors with a large enough diameter are not available and the superconductors have to be made up of smaller pieces. For example, Chen et al.46 used hightemperature superconducting disks, 19 mm in diameter and 19 mm high. This excess energy lost due to using multiplepiece superconductor components under mixed ZFC and FC assembly has been studied indirectly in terms of the number of superconductors used by Hull et al.64 and Kawashima et al.65 Their results show that a smaller number of pieces Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 implying larger pieces making up the same total size is more detrimental. In a sense, the variation of the magnetic field from conglomerates of larger pieces is rougher. On a separate note, energy dissipation in the rotor can be rotordynamically destabilizing under certain conditions,54 and hence is less benign than energy dissipation in the stator. Even if the rotor magnet is perfectly axisymmetric about its geometric axis, it may still present a time varying flux density to the superconductor and hence induce loss by magnetic hysteresis if the rotor is not spinning about the geometric axis of the magnet, or if the rotational axis keeps changing. This can be due to static imbalance or dynamic imbalance of the flywheel. On the other hand, even if the flywheel is perfectly balanced, but if the magnet is not perfectly symmetric, or if the superconductor is not symmetrically disposed, the levitation force acting on the rotor would not be evenly balanced. Unless constrained, the levitated flywheel could assume an equilibrium that is displaced from the central geometrical axis of design, tilted away from the vertical, and azimuthally biased.61 When the flywheel is spun up, gyroscopic effects from these imbalances will cause the flywheel to precess and vibrate and the spin axis to wander about. This is most serious at the critical speeds when the frequency of rotation coincides with a natural vibration mode of the levitation force and stiffness in the superconductor magnet bearing. Due to the small stiffness of the levitation bearing, the critical speed is usually low so that the useful operational regime of the flywheel is supercritical. Then, the flywheel is rotordynamically self centering,54 limiting the extent of the excursions of the rotational axis, and hence the energy loss. As the flywheel spins down, the vibration grows as it approaches the critical speed from above. The rate of slowing down is enhanced perceptibly. It is also evident in the spin down result of Borneman et al.66 and Xia et al.61 as a more precipitous drop near the end. This more rapid speed drop quickly brings the wheel out of resonance, thus limiting the growth of the vibrations. Away from the critical, vibration amplitudes are typically less than 1 mm and decrease with increasing speed.67 Later, research by Hikihara et al.68 show that when the hysteretic nature of the force and torque between magnets and superconductor is incorporated into the rotordynamics of a flywheel system using HTS bearings, the whirling motion acquires the almost discontinuous rate changes characteristic of hysteresis. The angular deviations are estimated to be on the order of 10⫺4 rad or about 20 s of arc with an angular momentum of 0.2 J s stored on the wheel. This is important in cases where precision pointing using the rotor as a gyroscope is contemplated, in which case the whirling motion can always be decreased either by spinning the wheel faster or incorporating a larger wheel. To preserve the gains of using levitation bearings in a FESS, attention must be paid to keep other avenues of energy losses down to a comparable level. One example occurs in the motor/generator as ohmic loss in the copper windings or core losses in the yoke. In the work of Chen et al.,46 the axial gap dc brushless permanent magnet motor/generator is yokeless, so there are no core losses, but ohmic losses in the copper windings are appreciable, not only when the motor/ generator is active, but even when it is on standby due to Superconductor and magnet levitation 5009 eddy currents induced in the armature coil by the rotating field magnets. When a spin down experiment was performed with the armature coils lifted away from the rotating field magnets after the flywheel is spun to speed, the resulting reduction in the rate of slowing down is detectable. One more step down the line, power is consumed in the commutator or control electronics required to run the motor/ generator. A more subtle form of power consumed is that spent to keep the superconductors cold. This power is either that required to run a small cryocooler, or prepackaged in the form of liquid nitrogen eventually evaporating as cold vapor and discharged into the air after serving its function of countering any heat leaks from the warm environment to the cold superconductors. These two last items are usually not thought of as a loss, but in the overall energy balance sheet, they belong naturally to the same loss column and must be kept to a minimum for the operation of the FESS to be justified. While superconductor magnet levitation bearings makes it possible for a FESS to store energy without losing too much too soon, a unit with a reasonable energy capacity requires flywheels of considerable weight which the levitation bearing must support. Ever since the discovery of the high temperature superconductors, there has been a flurry of research activities in superconductor magnetic bearings and its application in FESS. Most of the major research prototypes and some of their physical properties are listed in Table II. Those that have a capacity of less than 0.2 kW h are intended to demonstrate feasibility of concepts in the levitation bearing design during the earlier years. Excluding these, a typical energy capacity is about 1 kW h, a typical weight is about 100 kg, a typical diameter is about 0.5 m, and a typical angular speed is about 20 000 rpm, with the heavier ones having to rotate slower generally. At a typical levitation pressure of 0.1 MPa for melt-textured YBCO under ZFC conditions, this would require superconductor tiles covering the area of a circle of 0.1 m in diameter. This also implies that we have to use magnetic rings of a similar diameter. While all these are possible, the required components are not easy to come by. Even when the components are secured, the ZFC conditions presumed carry with them the uncertainties of creep and inconveniences in assembly. One way out of this dilemma is to transfer all or most of the load to a set of auxiliary permanent magnets in the stator, as proposed in the patent of Chu et al.63 and demonstrated in the work of Chen et al.46 These works also provide examples of how such auxiliary magnets can be brought to bear without compromising the stability coming from the interaction between the superconductor and the rotor magnet. A similar spirit is embodied in the work of Hull et al.75 with Evershed magnet configurations. Besides a low load supporting capability, the superconductor magnet levitation bearing also suffers from low stiffness. We have mentioned the benefit of being able to work in the self-centering regime of supercritical speeds almost all the time because of this low stiffness, but the inconveniences that come with it mostly outweigh this lone benefit. This low stiffness means that coupling with accessory machinery such as the drive shaft of the motor/generator must be on-axis and 5010 Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 Ma, Postrekhin, and Chu TABLE II. Flywheel energy storage systems. Flywheel Company g ANL TcSUHa,h FZi BNFIESj ISTECk CEl NEDOm TFPn APo Boeing Boeingb Bearings c SMB HSMBd SMBc SMBc AMBe⫹SMBc AMBe⫹SMBc AMBe⫹SMBc BBf⫹AMBe BBf⫹AMBe SMBc/HSMBd SMBc/HSMBd Stored energy 共kW h兲 Diameter 共mm兲 Weight 共kg兲 Angular speed 共rpm兲 Ref. 0.08 0.008 0.004 0.004 0.475 1.4 10 0.25 1.25 2.00 10.00 394 300 190 400 400 600 1000 300 775 — — 9.32 19 2.3 40 37 76 334 68 300 140 — 20 400 6000 15 000 15 000 30 000 20 000 17 200 40 000 7000 20 000 10 000 199959 199446 199566 199969 199958 199756 200070 200171 200172 200073 200374 a See Fig. 19. See Fig. 20. c Superconducting magnetic bearing. d Hybrid superconducting magnetic bearing. e Active magnetic bearing. f Ball bearing. g Argonne National Lab, Argonne, IL. h Texas Center for Superconductivity and Advanced Materials at the University of Houston, TX. i Kernforschungzentrum Kerlsruhe GmbH at the Institute fur Nukleare Festkorperphysik, Germany. j British Nuclear Fuels, International Energy Systems, England. k International Superconductivity Technology Center, Japan. l Chubu Electric Co., Mitsubishi Heavy Industries, Japan. m New Energy and Industrial Technology Development Organization, Japan. n Trinity Flywheel Power Co. o Active Power Co., MGE UPS Systems. b accurately aligned. Off-axis coupling from a parallel shaft, which may be convenient otherwise, will result in wild gyrations. This low stiffness also means that the bearing, by itself, cannot be used with widely time varying loads. Response to vibrations in the environment could become a problem. Thus applications on a mobile platform, such as an automobile, would be much more demanding in its design, just to overcome this weakness. A tempting choice is to use a combination of levitation bearings and active magnetic bearings, with the role of the active magnetic bearing being to boost up the stiffness of the bearing system wherever and whenever needed, and control the vibrations should that get excessive.57,58 Active magnetic bearings have been used all by themselves in FESS.76 Their biggest advantage is that they can operate at room temperature, besides having a larger load capacity and stiffness. However, one has to exercise considerable ingenuity to keep its power consumption down to a satisfactory level, otherwise it could cause a heat dissipation problem when operated under vacuum conditions. One must also provide backup power to an active magnetic bearing to prevent a hard landing in case that the mains fails even for an instant. With the superconductor magnet levitation bearing, the superconductor would take some time to warm up and turn normal after the cooling system stops working, due to thermal inertia. This gives more time for emergency measures to kick in so that the flywheel can have a soft and harmless landing instead. On the other hand, active magnetic bearings are ideal for holding, positioning, and centering the flywheel during start up, and the passive superconductor magnet bearing can be allowed to take over for power sav- ings in steady state operation. We find examples of such in the works of Minami et al.57 and that of Miyagawa et al.58 Passive mechanical bearings can also be used to carry out these initializing functions. Pin bearings are used in the work by Ito.42 A magnetic engagement system is used in the patent of Takahata43 for these purposes. There have been some systematic studies by the Japanese to determine the optimal REBaCuO superconductor to use—NdBaCuO, and the optimal REFeB permanent magnet to use—PrFeB共Cu兲, where RE is rare earth element. The last is recommended in the patent of Ito.42 Certain relation between geometry of a flywheel must be required for stable rotation of the wheel as illustrated in the work of Scarborough.77 Borneman et al.78 claim that a double flywheel with superconducting bearings rotating in the same direction will be stable during operations at conditions when the distance between mass center points of the two flywheels is equal or more than the radius of the flywheel multiplied by 冑 3. One may also find more details about the stability of gyro-systems in the work of Bulgakov.79 Closely related to the issue of stability is the issue of safety in the operation of the FESS in practice. Safety should be of highest priority with FESS, as fast rotating equipment is involved. The main reasons leading to failure of the FESS include rotor unbalance, especially at critical rotation speeds,80 flywheel fatigue, and bearing failure. Modern flywheels, accurately balanced, can be manufactured from graphite fiber epoxy, which disintegrates into very fine filaments instead of irregular chunks with monolithic metals. This minimizes the hazards that follow, should the flywheel Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 fail from fatigue. Furthermore, the flywheel is enclosed in a housing covered by special protection layers.81 Use of superconducting magnet bearings can be expected to eliminate bearing failure due to wear and tear, but long term reliability against gap creep has not been demonstrated. Also, failure of the cryogenic system will lead to failure of the superconductor magnet bearing. Therefore, a set of safety backup mechanical bearings should be equipped with every FESS. C. MomentumÕreaction wheel with levitation bearings While there is much ongoing research on FESS, such as those listed in Table I, research on momentum/reaction wheel 共MW/RW兲 is much less. The hardware is almost the same with different emphasis appropriate to the specific application and hence so is the design. MW/RWs are used on artificial satellites for purposes of attitude control. It is referred as a momentum wheel when it is used as a sensor for change of orientation, and a reaction wheel when it provides the necessary torque to change the orientation of the satellite also. It works on the principle of angular momentum storage instead of energy storage. While flywheel for energy storage has competing technology such as batteries, capacitors, SMESs, flywheel for angular momentum storage is unique, without competition. A large moment of inertia is desirable, but that runs against the current trend of miniaturization for diversification. Use of satellites for military, science, and wireless communications purposes has been rising year after year for the last 10 yr. The number of satellite launches increased from 67 in 1990 to 107 in 1999. As a rough estimate, the number of launches is expected to peak at several hundred some time in the period from 2003 to 2005.82 Thus, it befits us to have satellites that are smaller, both in size and in weight, that would make it cost effective to carry out different missions on different orbits at the same time. Naturally, to make smaller wheels carry the same amount of angular momentum, we would have to spin them faster. Mechanical bearings are usually rated for below a certain speed, and turn lossy at higher speeds rapidly, eventually failing due to wear and tear. Superconductor magnet levitation bearings, on the other hand, are adept at high speeds with only a mild increase in loss as we increase the speed. For example, Moon and Chang32 have demonstrated that a 5 g levitated rotor with journal high-temperature superconductor magnet levitation bearing could rotate up to 120 000 rpm. Rigney and Trivedi83 have conducted highspeed rotor-dynamic testing of the SMB on a rotor at speeds up to 520 000 rpm with extremely stable synchronous vibration of only 0.76 ␮m for a 10 g and 0.36 in diameter rotor. Several natural factors favor the superconductor magnet levitation bearing in the space environment. First, there is no need for a pump to maintain vacuum—it is ‘‘just outside.’’ Then, the bearing does not have to carry a high static load because the wheel itself is small and the satellite is under microgravity or ‘‘weightless’’ ‘‘free-fall’’ conditions for most of its orbit in the mission. The stiffness requirement is an open question at the moment, but again, a smaller wheel means a smaller inertia promising quicker response to changing demands. The ambient temperature in space is certainly Superconductor and magnet levitation 5011 lower than on Earth, making the needed cryogenic conditions easier to maintain. Even taking into consideration that additional power is still required to run small cryocoolers to keep superconductor bearings cold enough for them to work, they still draw less power to sustain than systems using ball bearings or even active magnetic bearings. In this regard, superconductor bearings are estimated to be ten times better than ball bearing systems and more than three times better than active magnetic bearing systems. This is crucial in holding down the power budget of a small satellite. Since the hardware of a MW/RW is similar to that of a FESS, a satellite momentum wheel with superconducting bearings can double up as a FESS to replace, partially at least, the number of rechargeable batteries on board. This reduces the weight of the satellite, with the added bonus that the FESS has a longer service life than the electrochemical batteries in terms of cycling. On the systems level, a welldesigned array of momentum wheels with superconducting bearings can integrate both attitude control and energy storage functions. Thus, instead of separate subsystems to provide energy storage and attitude control, we can have a single system serving dual functions, with attendant savings in the total volume and weight of on board hardware.84 Figure 21 is a schematic drawing and a photo of a prototype momentum wheel built by Ma et al.84 Comparison with a drawing of one of the FESS of other groups shows that there is no significant difference in hardware. It utilizes a flywheel 1.9 kg in mass, 3.25 in diameter, and 3 in height, and stores 3.5 J s of angular momentum at a top speed of 15 000 rpm. One emphasis on the design of this prototype momentum wheel here is to be compact, so that it does not occupy too much space on a small satellite. Care would have to be exercised so that different components do not interfere with each other. One notable example is the field magnets of the driver which is not axisymmetric about the rotation axis, and should not be too close to the superconductors of the bearing, or else the levitation bearing would become lossy due to magnetic hysteresis of the superconductor in the field magnet of the drive. This prototype only represents a first step toward the conception of integrated attitude control and energy storage systems in satellites. Further development would involve putting several similar prototypes into a system and an integrated package to manage both the pointing and power supply functions simultaneously without getting into each other’s way. The extremely low loss of superconducting bearings will show itself to be a key factor that allows the use of even higher angular speeds to boost the energy storage capacities of the system. D. Passive vibration free platforms via superconductor magnet levitation It is natural for humans to associate a vibration free state for a levitated body as there is no visible contact between the levitated body and other solid bodies in its vicinity, and one common everyday experience is the transmission of vibrations through direct contact with sources of vibration. It is true that vibration transmission by contact is eliminated by levitation, but the forces responsible for levitation can medi- 5012 Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 Ma, Postrekhin, and Chu FIG. 22. Schematic diagram of the experimental setup to measure vibration isolation properties of a superconducting levitation system 共see Ref. 86兲. very simply as a mechanical filter for vibrations with frequencies higher than the highest natural frequency of the system. This has been advocated in patents by Chu et al.85 as a simple single stage device and as a multistage construct, one for each of three possible translational degrees of freedom. The transmission of vibration across the levitation gap has been measured by Yu et al.86 and studied by Teshima et al.,87,88 for both vertical and horizontal translational degrees of freedom, and some other selected swing modes that combine translation and rotation of the levitated object. Their levitation arrangement and transmitted vibration mode shape are depicted in Figs. 22 and 23. Their results confirm that vibrational transmission is effectively cut off at frequencies higher than the natural frequencies of the system, despite the nonlinear nature of the vibrations of superconductor magnet levitation systems. The superconductor magnet levitation bearing itself can be viewed as a very effective FIG. 21. 共a兲 Schematic diagram of reaction wheel prototype 共TCSAM兲 and 共b兲 Photo of reaction wheel prototype 共TCSAM兲. ate vibration transmission because their magnitude, direction, and location are subject to modulation by mechanical vibrations. With superconductor magnet levitation, the magnetic flux passing through the superconductor will be changed in general, if the position or orientation of either the magnet or the superconductor is changed by vibrations. This would excite a restoring force trying to return the superconductor magnet levitation system to its earlier equilibrium state. Vibration of the levitation system follows. Effectively, vibration from the outside world has been transmitted through the magnet flux linking the superconductor and the magnet, even though magnetic flux is not a solid physical entity. Nevertheless, due to the small stiffness of the invisible magnetic spring linking the superconductor and the magnet, the vibrations have low natural frequencies. This makes it possible to use the superconductor magnet levitation systems FIG. 23. Transmissibility as a function of frequency of external disturbance 共see Ref. 86兲. Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 vibration isolation device for rotational vibrations, or twisting, about the axis of the bearing. Taking one of these systems studied as prototype, a typical levitated magnet or superconductor has a mass on the order of a few tenths of a kilogram, with natural frequencies in the range of 0.5–5 Hz. Dimensions are typically a few centimeters across. Thus, passive vibration isolation is achieved, down to 10 Hz, with a device that is lightweight and energy saving. It is conceivable that small electronic packages such as infrared detectors may be mounted onto the levitated body, which would then constitute the beginning of a viable scanner for thermal imaging. It is definitely desirable to extend the frequency range over which it is effective. This can be done as a tradeoff with the advantage of being lightweight—we can levitate a more massive platform, using the same techniques that were used with flywheels. If we consider a 1000 kg slab, common with moderate sized optical tables, we can isolate down to 0.1 Hz, in principle. If a higher mass is not tolerable, then we have to design for even lower stiffness. This can be done by combining the stable equilibrium of the superconductor magnet system with the inherently unstable equilibrium from the interaction between magnets only. This possibility is suggested by the observation that there are neutral equilibria in all the six rigid body modes of vibration, namely, the translational, rotational, and cross-coupled modes. With this approach, we may even do away with the superconductor and the associated cryogenic needs altogether, if we stabilize the magnet levitated on the magnet system with weak springs, taking advantage of the fact that vibration isolation platforms do not have to spin as flywheels do. In this case, superior performance depends critically on the precision of magnet fabrication. However, the superconductor magnet levitation system automatically incorporates a certain amount of damping, which is beneficial in the overall characteristics of the system if the vibration to be blocked has frequencies at, or not far above, the highest natural frequency. The damping enhances the transmission of vibrations of higher frequencies relative to what we expect without damping, but the absolute transmission level still drops rapidly towards higher frequencies. This is clarified in Fig. 23. From the viewpoint of operation in space, the superconductor magnet levitation system as a vibration isolation platform is attractive with respect to passive operation and small mass. The natural vibration frequencies are likely to be lower under microgravity conditions, but this has to be confirmed with further experimentation. Cooling may come naturally with some missions, such as infrared surveillance, in which the detectors have to be cooled with liquid helium. In space, in orbit, there is no weight to support, and with our approach of vanishing stiffness, it can even be argued that the best vibration isolation is simply to set the equipment to be isolated temporarily free from the main body of the spacecraft itself. This is not too far from the truth, except that very often, there is a last umbilical that carries the life blood of the equipment that cannot be severed. One example is the line carrying liquid helium to the infrared sensors in a thermal imaging observation. The helium line is also carrying unwanted vibrations which might disturb the steady aim of Superconductor and magnet levitation 5013 FIG. 24. Example of a vibration isolation device. the sensors towards the target of study. Here, blocking of vibration configurations consisting of magnets only, where their geometrical dimensions tend to be highly skewed in order to create conditions that give rise to low natural frequencies for transmission is clearly not permissible, but we can go for vibration suppression instead. The superconductor magnet levitation device can be reconfigured to offer strong vibration damping properties that come with magnetic hysteresis in the superconductor when subjected to the varying magnetic field of nearby magnets under the action of vibrations of the helium line. This action is similar to eddy current dampers made with copper instead of the superconductor, but the superconductor gives a stronger response. Lamb et al.19 illustrated this concept with a prototype noncontact vibration absorber using the coupling between permanent magnets and high temperature superconductors. The results confirmed that high temperature superconducting materials provide significantly better damping than copper, the conventional material used in vibration isolation systems. The device is no longer, strictly speaking, a levitation device as the helium line indirectly connects the superconductor to the magnet. A schematic representation of the concept is shown in Fig. 24. A more realistic design embodying these features is that of a microgravity payload platform with an umbilical connection to the external structure, described by Moon in his work.5 Passive vibration isolation has one glaring disadvantage. It is limited to frequencies higher than the natural frequency. The transmissivity for lower frequencies cannot be made to be much less than unity. For this reason, it might be desirable to consider active control. However, the vibrations of a superconductor magnet levitation system are nonlinear and hysteretic, and sometimes chaotic,89 which means that the motion of the system is not precisely predictable in the long run. Linear control methods are most likely not satisfactory for such systems. Nagaya90 has found that a simple method of direct disturbance cancellation is quite effective in suppressing transmissivity at all frequencies, but tends to become unstable if pushed too far. To remedy this defect, he combines direct disturbance cancellation with velocity feedback, which provides some additional damping at the expense of raising the transmissivity somewhat. Nagaya et al.91 devised an optimal control scheme utilizing these approaches with frequency weighting, with an integral of the square of the displacement of the levitated body as a cost function. The dynamic model92 used for the levitation system is that of a nonlinear oscillator, where the restoring force is a polyno- 5014 Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 mial in the displacement with frequency dependent coefficients. This force is obtained by numerical simulation of the currents induced in the superconductor under the influence of base vibrations at various frequencies and amplitudes, using Maxwell equations with the constitutive relation between electric field and current coming from thermally activated flux creep and the normal resistivity for flux flow. An approximate boundary condition has been used to decouple the electromagnetic fields inside the superconductor from that outside so that it is not necessary to deal with an infinite domain in the numerical analysis. Experiments91 show significant suppression of vibrations over a wide frequency range including the resonance frequency and its subharmonics, verifying the concepts advanced above. VIII. SUMMARY Levitation is a fascinating phenomenon that is key to many applications. Useful strengths of levitation forces can be attained via magnetic forces. However, equilibrium under magnetostatic forces with permanent magnets, ferromagnetic, or paramagnetic materials are unstable and so we must resort to active feedback control or include diamagnetic or superconducting materials. Superconductors 共type II兲 such as the recently discovered high temperature superconducting ceramics can be described as nearly perfect conductors or hysteretic diamagnets. The presence of these materials can stabilize a magnetic levitation system because their intrinsic responses to changing magnetic fields are similar to the transfer functions of an electronic feedback control system. The result is stable passive magnetic levitation. To zeroth order, the mechanical response of a superconductor to external magnetic fields can be understood by treating the superconductor as a perfect conductor. Alternatively, we can say that it acts as a superposition of a perfect diamagnet and a permanent magnet due to the trapped flux. Thus, a permanent magnet brought into the vicinity of a zero field cooled superconductor from afar is repulsed. However, if a superconductor is cooled together with a permanent magnet then no force exists between them. Then when we try to pull the superconductor and magnet away from each other we experience an attractive force instead, due to trapped flux. The most unique feature coming out of the perfect conductor model is the occurrence of lateral forces dictated by the symmetry of the magnetic field pattern. If a magnet can be moved in a way without changing the field at the superconductor, then that motion is unimpeded. Otherwise, a restoring force can be expected. This central principle is utilized in the design of numerous noncontact devices such as the bearing or the clutch, the linear damper, or the linear slider. However, the perfect conductor model does not include energy dissipation which does exist in practice. These losses can be traced to flux creep or magnetic hysteresis. Flux creep disappears soon after initial assembly of a device. Magnetic hysteresis appears as small residual losses in otherwise lossless devices. All of the above general features are manifest in a generic superconductor magnet systems such as a cylindrical permanent magnet 共NdFeB兲 levitating above a supercon- Ma, Postrekhin, and Chu ductor 共YBCO兲 disk. The force and the stiffness that the superconductor magnet system can provide are two significant parameters for magneto–mechanical applications. The force is characterized by a levitation pressure of 1–2 atm, normalized to the cross-sectional area of the magnet assumed to be smaller than that of the superconductor. Stiffnesses under similar conditions are typically 1–100 N/mm. A third significant parameter is the energy loss. With a typical 1%–3% fractional deviation of magnetic field uniformity available from permanent NdFeB magnet manufacturers, the drag torque is not easy to detect directly. More conveniently, the loss is inferred from spin down measurements. Quantitatively, the loss is given by an equivalent coefficient of friction typically on the order of 10⫺6 . So far, application of superconductor magnet levitation is dominated through implementation as ultralow loss bearings in rotary machinery, notably the flywheel. A magnet cylinder levitated on a superconductor disk makes a simple version of a superconductor magnet levitation bearing. The magnet rotates freely about its central axis. Thus the bearing has no loss and hence no heat generated during operation. There is no wear and tear because there is no direct contact between moving parts. However, it is best used in vacuum to avoid air friction, and it must be cooled to cryogenic temperatures to work. Force and stiffness may not measure up with heavy machinery. Considering the pros and cons of the superconductor magnet levitation bearing, it is natural to consider its use only when energy losses are of paramount concern. Flywheel kinetic energy storage comes to mind at once as a prime candidate for its application. Indeed, state of the art units having 10 kW h capacities are being built. This application has a time scale of 12–24 h. Another arena in which energy supply is tight is on board small mini satellites. Space and weight constraints in these mini satellites demand smaller reaction wheels operated at a much higher speed. An almost frictionless bearing such as the superconductor magnet levitation bearing allows such high speeds to be sustained without excessive power expenditure. Even with cryogenic requirements, the overall power balance remains favorable. Furthermore, such reaction wheels can double up as energy storage devices, just like flywheels on Earth. Using the same principles, but with a rectilinear geometry instead of a circular geometry, we can build vibration isolation or suppression devices. With stripes of magnets levitated on a superconductor, vibrations along the stripe direction will not be transmitted while vibrations perpendicular to this direction will be suppressed. Prototype devices show excellent performance for vibration frequencies above the natural frequencies of the design. These devices are simple and robust. They are small, lightweight, and passive. Again, mandatory use of cryogens is the main drawback. IX. CONCLUSION Through the perfect conductor model, we have seen how the lossless feature of electrical transport in type II superconductors is imparted onto the friction free rotation of a magnet levitated over a superconductor. In a sense, this continuous and continuing magnet rotation is reminiscent of the persis- Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 tent flow of electrical current in a type II superconductor in the state of a trapped field magnet. Both are not perfect, but only slightly not so. Residual loss in the form of hysteresis or creep is small, if not outright negligible. Therefore, it is not surprising to find that the most promising application of the levitation phenomena with magnets and superconductors is in areas where it is crucial to keep losses to a minimum. Flywheel kinetic energy storage with superconductor magnet levitation bearings is the most prominent of these. 10–100 kW h units are in the forefront development stage. Actual industrial implementation of such units is in the foreseeable future. Another impetus for the utilization of superconductor magnet levitation devices comes from cryogenic considerations. If cryogenic conditions are already required of a piece of instrumentation, for instance, in infrared imaging, or if such conditions are naturally present, such as in deep space, or cryogen handling equipment, then the extra effort to keep superconductors cold enough to function is usually considered a small price to pay for the potential benefits. Instances can be found in the fields of deep space exploration, infrared astronomy, and observational cosmology. The example of reaction wheels for attitude control of small satellites in near space for remote sensing is a borderline case. Emerging in this group of applications is a tendency to take advantage of other benefits of superconductor magnet levitation such as passive vibration isolation or high compliance 共low stiffness兲 on top of its energy conserving properties. The construction of high quality scientific instruments exploiting these and other special properties of superconductor magnet levitation not found elsewhere is still a wide open field of endeavor. Finally, we would like to conclude with some observations on the development of research in the field of magnet superconductor levitation devices. The initial phase was concerned with the phenomenology of stable levitation with magnets and the high temperature superconductors. A main hurdle was to get high quality superconductor samples and only free-sintered material was available anywhere. As a result, experiments were performed with small pieces of superconductors and the levitation forces achieved were small. Nevertheless, important results that show the fundamental distinctive features of the phenomenon emerged— dominance of flux pinning over Meissner repulsion, continuum of equilibrium points, drag free rotation with axisymmetric magnetic fields, and many others. The next phase was concerned with the magnitude of the levitation force that can be achieved with stability. This, among other incentives, spurred the development of the melt-textured, and then the seeded growth materials of moderate sizes and much higher critical current density. At the same time, prototype devices were constructed to explore concepts that expand the potential of the available materials with regard to the load capacity attainable, such as the use of supplementary magnets to create larger levitation forces while maintaining stability. At this juncture, seeded growth YBCO material has become the superconductor of choice. The prototype devices studied were visualized as passive magnetic bearings, and levitation forces developed are typically in the 1–1000 N range. The third phase gets into the development of commercially viable products, involving a litany of engineering and economic Superconductor and magnet levitation 5015 issues that are more or less just as related to levitation as to any other commercial product. In the field of magnet superconductor levitation devices, the most promising candidate for further development appears to be flywheel energy storage devices for the utilities. The smallest device for an industrial prototype would have a storage capacity of several kilowatt hours, using a flywheel of a few hundred kilograms, calling for levitation forces that are just beyond the range encountered in the second phase. The final goal would be for one that is more than 1000 times as big. The first two phases of research took place mostly within universities such as Cornell University, research centers such as the Texas Center for Superconductivity at the University of Houston 共TCSUH, now Texas Center for Superconductivity and Advanced Materials, or TCSAM兲, or national laboratories such as Argonne National Laboratory. However, the research work cannot remain within any one of these research institutions by the end of the second phase. The sheer size of the research object mainly, and the evolving nature of the research to some extent, mandated that industries with manufacturing capabilities such as Boeing takeover. The satellite momentum wheel project was the result of TCSAM trying to continue on the research and development of the superconductor magnet bearing without the encumbrance of having to handle a massive object against tremendous magnetic forces 共when the superconductor is still warm兲. Thus we stand today 共May, 2003兲. The flywheel energy storage system looks to be the superconductor magnet levitation device most promising to be developed into a viable commercial product in the next decade or so. A few basic issues with the interaction between magnets and superconductors remain not fully resolved, but these are not perceived to have an impact on the development of the devices of most current interest. As awareness of the distinctive properties of superconductor magnet levitation is more widely dispersed, and as material advances improve upon these properties or relax the requirements such as cryogenic temperatures, the field of application of superconductor magnet levitation devices can be expected to expand beyond its present perimeters. ACKNOWLEDGMENTS This work was supported in part by the State of Texas through the Texas Center for Superconductivity and Advanced Materials at the University of Houston and the United States Air Force under AFOSR Grant No. F49620-970101. One of the authors 共W.K.C.兲 would also like to acknowledge support by his endowment from the Robert A. Welch Foundation. The authors would also like to express their gratitude to Boeing for providing them with information and material from their February, 2003, Boeing Flywheel Program Status Report. Personnel at Boeing would like to take this opportunity to express their gratitude for the support of the Superconductivity and Energy Storage Programs of the Department of Energy. E. H. Brandt, Science 243, 349 共1989兲. S. Earnshaw, Trans. Cambridge Philos. Soc. 7, 98 共1842兲. 3 T. D. Rossing and J. R. Hull, Phys. Teach. 29, 552 共1991兲. 4 L. C. Davis and D. F. Wilke, J. Appl. Phys. 42, 4779 共1971兲. 5 F. C. Moon, Superconducting Levitation 共Wiley, New York, 1994兲. 1 2 5016 6 Rev. Sci. Instrum., Vol. 74, No. 12, December 2003 R. H. Frazier, P. J. Gilinson, Jr., and G. A. Oberbeck, Magnetic and Electric Suspensions 共Press, Cambridge, MA, 1974兲. 7 M. V. Berry and A. K. Geim, Eur. J. 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