Frozen Image Model Research Articles

Vertical and lateral forces, torque and potential energy as a magnetic dipole with arbitrary angle moves laterally above a flat high-temperature superconductor

In the magnetic levitation system consisted of permanent magnets (PMs) and high-temperature superconductors (HTSs), the magnetization direction of the PM is often inconsistent with the c-axis of the HTS. Therefore, the interaction between the PM with an angle and the HTS should be studied to obtain the effects of the angle on the levitation properties due to the anisotropy property of the critical current density. In this paper, as the PM dipole oriented with the angle with respect to the c-axis of the HTS moves laterally above the semi-infinite HTS, the exact expressions for the lateral and vertical forces, the torque, and the total potential energy are obtained by using the advanced frozen-image model. The dependence of these physical quantities upon the angle and the initial cooling height have been systematically studied. Under the same conditions, the predicted force and torque calculations provided by the advanced frozen-image model qualitatively agree with the previous experimental and theoretical results. The study can provide some theoretical references for designing the HTS levitation system with the angle.

Frozen image analysis of a superconducting magnetic levitation system consisting of multi-surface superconductor and Halbach array permanent magnet configuration

Levitation strength provided by high temperature superconductors are limited for device applications. Although superconducting material properties are continuously improving, there is still strong necessity of efficient design mechanisms for the superconducting magnetic levitation systems. Studies in the last decades have shown that combining multi-surface superconductor and permanent magnet components in optimum configurations has improved the levitation forces. In this respect, Halbach arraying permanent magnets interacting with multi-surface superconductors has become one of the most utilized methods. This paper investigates frozen image modeling of the levitation and guidance forces on a particular levitation system, which consists of a permanent magnet guideway and high temperature superconductor car body. The levitation enhancement is investigated for three configurations according to force interactions between the guideway and car body. These configurations are based on the use of single permanent magnet-single superconductor, Halbach array permanent magnets-single superconductor and Halbach array permanent magnets-multi-surface superconductors. The vertical and guidance forces for the present configurations were calculated in terms of field cooling and zero field cooling conditions by using frozen image model with magnetic dipole approximation. The predicted force calculations are analyzed in terms of vertical and lateral traverses of the car body respect to guideway for particular measurement distances. The force analysis provided by frozen image model qualitatively agree with the previously obtained experimental data.

Angular dependence of vertical force and torque when magnetic dipole moves vertically above flat high-temperature superconductor* *Projects supported by the National Natural Science Foundation of China (Grant No. 11572232) and the China Three Gorges Corporation Research Project (Grant No.202103407).

The interaction between a permanent magnet (PM) assumed as a magnetic dipole and a flat high-temperature superconductor (HTS) is calculated by the advanced frozen-image model. When the dipole vertically moves above the semi-infinite HTS, the general analytical expression of vertical force and that of torque are obtained for an arbitrary angle between the magnetization direction of the PM and the c axis of the HTS. The variations of the force and torque are analyzed for angle and vertical movements in both zero-field cooling (ZFC) condition and field cooling (FC) condition. It is found that the maximum vertical repulsive or attractive force has the positive or negative cosine relation with the angle. However, the maximum torque has the positive or negative sine relation. From the viewpoint of the rotational equilibrium, the orientation of the magnetic dipole with zero angle is deemed to be an unstable equilibrium point in ZFC, but the same orientation is considered as a stable equilibrium point in FC. In addition, both of the variation cycles of the maximum force and torque with the angle are π.

Reduced Embedded Magnetic Field in Type-II Superconductor of Finite Dimension

This article maps the magnetic field within a type-II superconductor of finite dimension that is magnetically flux-pinned. The measured field is lower in magnitude than anticipated from the frozen image model and changes shape dependent on the field-cooled image location. A proposed refined model more accurately reflects the measured field.

Flux-Pinned Dynamics Model Parameterization and Sensitivity Study

Flux-pinned interfaces maintain a passively stable equilibrium between two spacecraft in close proximity. Although flux-pinning physics has been studied from a materials science perspective and at the system level, the sensitivities and implications of system-level designs on the dynamics need to be better understood, especially in interfaces with multiple magnets and superconductors. These interfaces have highly nonlinear coupled dynamics that are influenced by physical parameters, including strength of magnetic-field sources, field-cooled position, and superconductor geometry. Kordyuk’s frozen-image model (“Magnetic Levitation for Hard Superconductors,” Journal of Applied Physics, Vol. 83, No. 1, Jan. 1998, pp. 610–612) successfully approximates the characteristics of flux-pinning dynamics, but could provide a more precise state prediction with the addition of these physical parameter refinements. This paper addresses that gap by offering parametric terms to improve the dynamics model, which may better simulate the behavior of a multiple-magnet-and-multiple-superconductor interface. The sensitivity of the general flux-pinned dynamics model is studied by varying the physical parameters and simulating the system-level dynamics. This work represents a critical step in the development of a model suited to spacecraft performance verification.

Rotor optimization in a superconducting magnetic bearing by using frozen image model and Amperian current approximation

Developments in superconducting magnetic bearings led the key technological progresses for flywheel energy storage systems. However, the present status of the superconducting magnetic bearings still deprived from the sufficient stability. Since the magnetization characteristics of the superconducting materials reaches the limits the necessity of more efficient designs for bearing systems is inevitable. For instance, strongly stable bearing designs would be possible by using various magnetic and superconducting components in the bearing. In this context, we have investigated the effect of particular levitation configuration on the stability of the rotor in the bearing mechanism. Vertical and lateral forces acting on the rotor are determined in terms of the various shape and dimensions of the superconductors and permanent magnets. In this context, the Amperian current approximation was used to model the interaction force between the permanent magnets. In the case of determining the interaction force between the permanent magnet and high temperature superconductor, the frozen image model was used to implement the Amperian current approximation according to the cooling procedure of the superconductor. It was determined that the optimum levitation of the rotor strongly depends on the use of superconductor and permanent magnet configurations.

Effects of magnetization conditions on dynamic characteristics of spacecrafts with superconducting flux pinning docking interfaces

Depending on the self-stable levitation characteristics of superconductors with trapped magnetic flux, superconducting flux pinning docking interfaces (FPDIs) provide a promising approach for space applications. In this paper, we investigate the influence of magnetization conditions on the dynamic response between two satellites equipped with FPDIs. The flux pinning interactions between a high-temperature superconductor and a source magnet are approximated using a frozen image model. An analytic model of the docking process is derived based on the Kane method to simulate the dynamic response. A linearized dynamic model is derived, and linear quadratic regulator (LQR) control is applied to the FPDI to minimize the relative attitude deviations. The results show that improving the frozen magnetic flux density of superconductors is beneficial to the capture ability of the FPDI. Increasing the damping coefficient is also an efficient approach to improving the docking performance, but this approach is limited by the field cooling height. Damping materials with a lower density and a higher damping coefficient are proposed to be employed in the FPDI. The LQR control contributes to suppressing the amplitude of the oscillation and reducing the settling time in the docking process. Although the amplitude of oscillation in a noncontrol scheme converges more slowly than the LQR control scheme, it does not require the high dynamic response speed of a control system like the LQR control scheme, which contributes toward simplifying the structure of the FPDI. The principles presented above provide a guide for the design and implementation of FPDI in space docking missions.

Linearized Dynamics of General Flux-Pinned Interfaces

A flux-pinned interface offers a passively stable equilibrium that otherwise cannot occur between magnets because electromagnetic fields are divergenceless. The contactless, compliant nature of flux pinning offers many benefits for close-proximity robotic maneuvers, such as rendezvous, docking, and actuation. This paper derives the six degree-of-freedom linear dynamics about an equilibrium for any magnet/superconductor configuration. Linearized dynamics are well suited to predicting close-proximity maneuvers, provide insights into the character of the dynamic system, and are essential for linear control synthesis. The equilibria and stability of a flux-pinned interface are found using Villani's equations for magnetic dipoles. Kordyuk's frozen-image model provides the nonlinear flux-pinning response to these magnetic forces and torques, all of which are then linearized. Comparing simulation results of the nonlinear and linear dynamics shows the extent of the linear model's applicability. Nevertheless, these simple models offer computational speed and physical intuition that a nonlinear model does not.

Optimization of the Force and Stiffness in a Superconducting Magnetic Bearing Based on Particular Permanent-Magnet Superconductor Configuration

One of the most challenging issues in superconducting-bearing technology is the rotor's stability, which is closely related to the levitation configuration. Establishing a stable and fixed position of the rotor requires strong stabilizing forces among the bulk superconductor and permanent or electromagnets. The proposed configuration for the superconducting magnetic bearing in this paper consists of a permanent magnet on the rotor and a combination of a cylindrical bulk superconductor and a ring permanent magnet base. The load-carrying capability of the proposed bearing configuration is analyzed in terms of the vertical force on the rotor as a function of the vertical and lateral displacements. In addition, the lateral stability mechanism is evaluated by determining the lateral forces on the rotor as a function of the vertical and lateral displacements. The forces on the rotor are determined with a quick and effective calculation by using MATLAB for various design configurations based on the considerations of the cooling procedure of the superconductors, the Amperian current approximation, and the frozen-image model. According to the results of the analysis, for the same levitation heights the proposed bearing design having ring permanent magnet in the base provides levitation force higher than that of without having the ring permanent magnet. The present levitation configuration provides better clearance between the rotor and stator in the bearing.

The connection characteristics of flux pinned docking interface

This paper presents the mechanism and potential advantages of flux pinned docking interface mainly composed of a high temperature superconductor and an electromagnet. In order to readily assess the connection characteristics of flux pinned docking interface, the force between a high temperature superconductor and an electromagnet needs to be investigated. Based on the magnetic dipole method and the Ampere law method, the force between two current coils can be compared, which shows that the Ampere law method has the higher calculated accuracy. Based on the improved frozen image model and the Ampere law method, the force between high temperature superconductor bulk and permanent magnet can be calculated, which is validated experimentally. Moreover, the force between high temperature superconductor and electromagnet applied to flux pinned docking interface is able to be predicted and analyzed. The connection stiffness between high temperature superconductor and permanent magnet can be calculated based on the improved frozen image model and Hooke's law. The relationship between the connection stiffness and field cooling height is analyzed. Furthermore, the connection stiffness of the flux pinned docking interface is predicted and optimized, and its effective working range is defined and analyzed in case of some different parameters.

Optimization of levitation and guidance forces in a superconducting Maglev system

Optimization of the levitation for superconducting Maglev systems requires effective use of vertical and guidance forces during the operation. In this respect the levitation and guidance forces in terms of various permanent magnet array configurations are analyzed. The arrangements of permanent magnet arrays interacting with the superconductor are configured for the purpose of increasing the magnetic flux density. According to configurations, modeling the interaction forces between the permanent magnet and the superconductor are established in terms of the frozen image model. The model is complemented with the analytical calculations and provides a reasonable agreement with the experiments. The agreement of the analytical calculation associated with the frozen image model indicates a strong case to establish an optimization, in which provides preliminary analysis before constructing more complex Maglev system.

Analysis of the levitation force of pure and starch/polystyrene/MWCNT added bulk MgB2 superconductors using frozen image model under zero field cooling condition

The measurement of superconducting levitation force between permanent magnet and polycrystalline samples of pure and MgB2 added with starch, polystyrene (PS) and multiwall carbon nanotube (MWCNT) have been performed under zero field cooling (ZFC) condition at 20K in both descending and ascending modes. For this, the bulk pellets were synthesized by conventional solid state sintering technique. The XRD data indicate well developed MgB2 phase. However, a decrease in lattice parameter ‘a=b’ have been observed for doped MgB2 samples. Superconducting transition temperature of MgB2 also decreases with starch/PS/MWCNT addition. Unlike MWCNT, the addition of starch/polystyrene is found to enhance the levitation force of MgB2 superconductor. The levitation force between PM and investigated pellets in ZFC condition is explained well in terms of the updated version of modified frozen image model and the magnetic moment originated due to vertical motion of the superconductors have been estimated. It may be noted that except for MWCNT, addition of starch/PS in MgB2 improves the magnetic moment generated by vertical movement of pure MgB2. However, this improvement is more pronounced for 1wt.% of PS added MgB2, which indicates more flux trapping and hence better levitation properties in 1wt.% of PS added MgB2. The vertical stiffness estimated for pure and starch/PS/MWCNT doped MgB2 samples indicate that the levitation force are more sensitive in the region close to the PM.

The design considerations for a superconducting magnetic bearing system

In this paper a high temperature superconducting magnetic bearing is studied with various design considerations. The design of the bearing consists of a rotor with 7.5kg mass. The stable levitation of the rotor is provided with the Evershed type and superconducting components. The dynamic stability of the rotor is strengthened with the electromagnetic and electrodynamic levitation techniques. The force on the rotor is predicted in terms of semi-analytical frozen image model. The designed driving system sustains stable levitation during the rotation of the rotor and achieves higher rotational speed than that of the torque driver. The results indicate that the designed rotor and driving system have potential solutions for the development of the superconducting flywheel energy storage.

An Effective Noncontact Torque Mechanism and Design Considerations for an Evershed-Type Superconducting Magnetic Bearing System

An Evershed type of superconducting magnetic bearing is designed and fabricated. An alternative torque mechanism is proposed to drive the rotor of the designed bearing. The design consideration for the rotor consists of a permanent magnet with 7.5-kg mass and its force interaction with the superconductor defined in terms of a semi-analytical frozen image model. The driving mechanism is based on eddy-current induction on a conducting disk placed on the rotor of the bearing. The driving system is intended not only to sustain stable levitation during the variation of the rotor speed but also to rotate the rotor with a speed higher than that of the torque provider. Levitation, drag, and lift forces are discussed via considering various conditions such as the rotor configuration and conducting material. The thickness of the conductor strongly affects the optimum working conditions of the magnetic bearing. In order to compensate the repulsive force due to lift force on the rotor, the top of the rotor was covered with iron sheet. The results indicate that the designed rotor and driving system have potential solutions for more advanced bearing applications for superconducting flywheel energy storage.

Modeling of hysteretic behavior of the levitation force between superconductor and permanent magnet

The hysteretic behavior of the levitation force between a permanent magnet and a melt-textured-growth YBCO bulk has been investigated under both zero-field cooling (ZFC) and field cooling (FC) processes. It is found that both in ZFC and FC measurements, the hysteresis loop for the first descent/ascent cycle of magnet is relatively larger than that for the second or third cycle, and the hysteresis loops for Cycle 2–4 have the same area. These results can be qualitatively understood in terms of the critical state model. To describe these experimental results, we develop an updated frozen-image model, which is obtained by modifying the change rules of the vertical movement image in the advanced frozen-image model proposed by Yang et al. Comparing with the advanced frozen-image model proposed by Yang et al., our model cannot only give the hysteretic characteristic in the first descent–ascent cycle of magnet, but also show the hysteresis loops with the same area for the second and subsequent cycles.

Optimization of the Force Modeling between High Temperature Superconductor and Permanent Magnet

Abstract The determination of the optimum force between a permanent magnet and a high temperature superconductor is studied in terms of frozen image model. The Amperian surface current assumption is assumed for the magnetization of the permanent magnet in the force calculation. The modeling is incorporated with the magnetic configurations of the magnet-superconductor system, such as vertical force for different cooling procedures. In the optimization study, the Amperian bulk current assumption is introduced to the force calculation. Comparing the theoretical results obtained from both assumptions with the experimental results showed that the force estimation is better with the bulk current assumption than that of the surface current. The results indicate that this improvement in the calculation of the force between the superconductor and permanent magnet is useful for larger geometries such as superconducting energy storage systems.

An improvement of frozen-image model and its application in a HTS levitation system

This paper mainly reports an improvement of frozen-image model which can qualitatively describe the influence of lateral moving speed on vertical force in a HTS levitation system under lateral movement with field-cooling condition. The model is improved by introducing a dipole which represents the influence of lateral moving speed and modifying the rule of diamagnetic dipole based on frozen-image model. The vertical and lateral forces that are obtained by this improved model agree with the previous measurements qualitatively. This model can also describe the effect of finite scale of superconductor sample in a levitation system.

Determination of levitation force and pinning properties of (Sm123)1−x(Yb211)x bulk superconductor depending on Yb211 doping ratio

AbstractThe vertical force density curves of (Sm123)1−x(Yb211)x high temperature superconductor (HTS) samples prepared by a melt‐powder‐melt‐growth technique were measured during vertical and lateral traverses in zero field cooling (ZFC) regime for various doping levels (x). It was determined that the maximum vertical force density of the samples increases when the doping level increases and after certain level of the doping the force saturates. The pinning properties, which create attractive force, were enhanced up to a certain Yb211 doping level by introducing locally suppressed superconducting regions. These results indicate that the optimization fabrication conditions and Yb211 doping level should be further investigated for larger dimensional samples to enhance technological applicability. The frozen image model was used to predict the experimentally measured levitation force between a permanent magnet (PM) and the superconductor samples. Various configurations of the PM/HTS force interaction were tested according to the magnet and superconductor size. The doped HTS samples do not show very strong diamagnetic behavior as the frozen image model predicts. The levitation force behavior regarding the dimensions of the PM and HTS indicates an illustration how the diamagnetic behavior of the superconductor transforms into the flux pinning.

Vertical, radial and drag force analysis of superconducting magnetic bearings

The behavior of the force between a permanent magnet (PM) and a high temperaturesuperconductor (HTS) was tested with the frozen-image model based on flux pinning. Itwas found that the associated dipole moment assumptions of the method of the frozenimage underestimate the force somewhat; thus a quadrupole moment analysis is proposed.The radial and drag forces associated with the rotation of the PM levitated above the HTSwere measured by using a force transducer and by means of a cantilevered beam technique.The radial force was found not to be dependent on the radial direction, and the least radialforce was found to be periodic with an angular displacement during the slow rotation of thePM relative to the HTS. The periodicity behavior of the force is attributed tothe geometric eccentricity from the magnetization distribution of the PM andHTS. The drag force associated with the torsional stiffness of the levitated PMduring the low and high rotational speeds was incorporated with the data from theliterature.

Interaction between a point magnetic dipole and a high-temperature superconducting sphere

Abstract The interaction between a small magnet and a high-temperature superconducting sphere in the field cooled state is investigated using the frozen image model. Analytical expression of the interaction potential and force between the magnet and the superconductor is derived as function of the geometrical parameters of the system. The angular dependence of the force and energy is analyzed. The interaction between a straight magnetic line and superconducting sphere is investigated.