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Simulation of Dynamic Levitation Force Measurements Taking the Flux Creep into Account
R. B. Kasal, ELETROBRÁS, R. de Andrade Jr., G. G. Sotelo, and A. C. Ferreira, PEE/COPPE - UFRJ
Abstract—This work compares dynamic levitation force measurements, at different approaching speeds, with levitation force simulations that take the flux creep into account. This was done using a power-law electric field current-voltage relationship that can be derived from the Anderson-Kim model. The algorithm for the simulation starts from the analytical expressions of the classical electromagnetic theory to write the integral equation of the time derivative of the current density inside the superconductor, depending on the geometry of the system and the configurations of the applied field. Then, using the Method of Moments, the analytical integral equation was written in its matrix formulation. The current density in each time step is obtained by a simple integral rule (Method of Euler). From the current density profile in the superconductor, the levitation force between a permanent magnet and a superconductor, with finite cylindrical geometry, is calculated. The current density profile depends on the approaching speed. The results of the simulations were successfully compared with the experimental measurements.
Index Terms— Critical State Model, Dynamic Levitation Force, Flux Creep.
I.INTRODUCTION
T
HE force between a permanent magnet and a superconductor could decrease with time due the flux creep into the superconductor. As the relaxation of the flux lines obey an exponential decay with a large time constant, in applications where the transient response of superconductors are not relevant the flux creep can be neglected. However, when superconductors are used in devices that are submitted to dynamical forces, such as superconducting magnetic bearings for flywheels, the flux creep becomes an important phenomenon to consider. Thus, the appropriate behavior of these systems can not be predicted by models that employ simple and direct rules for the relation between the critical current (Jc) and the applied magnetic field (B). Whether Bean’s [1] and Kim’s [2] models were used to simulate the superconductor, it is impossible to take the flux creep in account.
In this paper, a logarithmic dependence between the critical current and the thermodynamic activation energy, together with the Arrhenius Law, is used to simulate the flux creep into the superconductor, as suggested in [3]. The resulting exponential law between the local electrical field and the current density inside the superconductor, can be written as:
(1)
where Ec is the critical electric field of the superconductor and n=Uc/(kB T), with Uc corresponding to the critical activation energy, is the parameter that establishes the type of the superconductor. The limits of n are the ohmic conductor (n=1) and the Bean’s model (n → ∞).
II.MODELING AND IMPLEMENTATION
A.System description and analytical modeling
The geometry of the arrangement considered for the simulations, with the cylindrical coordinates indicated, is presented in Fig. 1.
Fig. 1. Finite cylindrical superconductor in the presence of a cylindrical permanent magnet.
The magnetic field of the permanent magnet (PM) is modeled by a superficial current density limited to the external surface of the PM, with a remanant magnetic flux density Bres. One can use the Biot-Savart’s Law together with the definition of the vector potential to obtain the expression for the applied field:
(2)
with r=(ρ,z). As equation (2) can not be analytically solved, a numerical approximation of this integral was used, as suggested in [3].
The magnetic field produced by the superconducting currents can be also derived by the application of the Biot-Savart’s Law and the definition of the vector potential:
(3)
where the integral kernel is expressed by:
(4)
The total magnetic field of the system is obtained adding (2) and (3), resulting:
(5)
A linear rule for the movement between the permanent magnet and the superconductor, was used, defined by the expression:
(6)
where z00+z0 is the maximum distance between the PM and the superconductor, z00 is the minimum distance, and vs is the velocity of the superconductor. The stationary reference system adopted for the simulations was located at the upper surface of the PM.
Taking the time derivative of (5) and substituting the vector potential using Faraday`s Induction Law (∂A/∂t = - E), it can be written:
(7)
The flux creep is implemented in the model by substitution of E in (7) with the exponential rule described in (1). This leads to the integral equation:
(8)
B.Numerical Implementation
The section of the superconductor was mapped using a grid with equidistant points, as shown in Fig. 2.
Fig. 2. Mapping of the superconducting cylindrical section with an equidistant grid.
Fig. 2 shows, as an example, Nρ=6 and Nz=6, resulting in a grid with N=36 points. Equation (8) must be solved for each point of this grid. As the integral equation in (8) does not have an analytical solution, the Method of Moments, described in [4], had to be employed to proceed with the numerical approximation. This operation leads to the following equation in matricial form:
(9)
The equidistant spacing of the grid is defined by dρ=a/Nρ and dz=(2b)/Nz.The kernel matrix Qcilpm(NxN) is obtained by the application of (4) for each point of the grid. In order to assure stability and velocity to the computation algorithm, each element of the kernel matrix was multiplied by a weight equal to:
(10)
The dynamic solution for the current density can be calculated by a simple integral rule, such as the Method of Euler:
(11)
The calculations start with the initial current density equal to zero and an initial maximum distance between the superconductor and the PM. The time step that brought the most fast and stable results was:
(12)
where Min(ρ(t)(Nx1)) is the minimum value, in each time step, of the resistivity of the point of the grid, with the resistivity calculated by:
(13)
As suggested in [5], the vertical levitation force is calculated using Lorentz’s Force:
(14)
with the radial component of the permanent magnet flux density given by:
(15)
with (16)
In (15), K and E are the elliptical integrals of first and second order, respectively. For the calculations, we used reduced units defined below:
(17)
III.Measurements Procedure
In order to implement the tests that would be compared with simulation results obtained with the model described above, a measurement system was constructed. The tests were made using a Nd-Fe-B permanent magnet disk (diameter=22mm and height=10mm) and a melt textured YBCO superconductor cylinder (diameter=28mm and height=10mm). The magnetic flux density at the center face of the magnetic disk was 415 mT. To control the vertical position of the Nd-Fe-B disk above the superconductor, a linear guide was used. Vertical positioning was done by a stepper motor driven from an electronic converter. An acquisition board was used with a microcomputer to generate the signals responsible to move the stepper motor and to process the voltage provided from a load cell that was connected to the Nd-Fe-B disk. Fig. 3 shows the measurement system with its components.
Fig. 3. Measurement system used during the experiments.
The YBCO disk was cooled in LN2 without the presence of magnetic field (zero field cooling – ZFC) and then the permanent magnet was vertically approximated to it at a constant speed. After a minimal possible gap of 1 mm was reached, the moving direction was inverted and the Nd-Fe-B disk started to ascend. These tests were made for two different speeds: 0.09375 mm/s and 0.75 mm/s. The comparison of these measurements with the simulations will be presented in the next session.
IV.Results
Fig. 4 shows a comparison between the dynamic levitation force measured in the laboratory and the respective simulation using the model described in the previous sessions. The velocity of the relative movement between the PM and the superconductor is vs=0.75 mm/s. For the YBCO disk, it was used n=21, Jc=107 A/cm2 and Ec=1 μV/cm.
Fig. 4. Comparison between the dynamic levitation force measured in laboratory (circles) and the simulation (bold line), considering the velocity between the PM and the superconductor equal to 0.75 mm/s.
It can be noted that there is a slight difference between the curve profiles, mainly in the region where the hysteretic effect is more stressed. Whether this discrepancy is due to the simulation parameters or it has experimental reasons, is yet to be more carefully investigated. However the maximum value of the levitation force, obtained from the simulation, seems to match with great accordance to the measurements.
Fig. 5 presents two curves for the levitation force: the first one, with the PM moving with vs=0.093875 mm/s (points), and the second one, with vs=0.75 mm/s (circles).
Fig. 5. Vertical levitation force measured for two different speeds: vs=0.093875 mm/s (points) and vs=0.75 mm/s (circles).
Fig. 6 shows the calculated curves using the same velocities mentioned above.
Fig. 6. Vertical levitation force calculated using two different speeds: vs=0.093875 mm/s (dashed line) and vs=0.75 mm/s (bold line).
The comparison between Fig. 5 and Fig. 6 indicates that the model can correctly predict the increase of the maximum value of the levitation force and the reduction of the hysteretic effect, when the velocity of the PM increases. This result is the main goal of this model: the introduction of the exponential relationship between the local electric field and the superconducting current density, defined by (1), enables to write the integral equation (8) that brings the dynamic characteristic of the flux creep into the calculations. When the speed increases, the vortices in the superconductor have less time to relax throughout the material and the hysteretic effect becomes smaller. The intensity of the reduction of the hysteretic effect is determined by the material of the superconductor, which in this model, is represented by the parameter n=Uc/(kB T).
V.Conclusion
The model presented in this work was able to reproduce the dynamic levitation force curves between a superconductor disk and a permanent magnet with cylindrical geometry. The introduction of the flux creep in the model proved to be useful when time dependency is relevant in the calculations. The comparison with experimental results showed a slight discrepancy in the region where the hysteretic effect is more significant. The reasons for this difference are yet to be understood. Some parameters used in the simulation should be further investigated, such as the formation rule of the kernel of the dynamic integral equation for the current density inside the superconductor Qcilpm.
Acknowledgment
The authors thank M. J. Qin, from the Institute for Superconducting and Electronic Materials, University of Wollongong, Wollongong NSW 2522, Australia, and E. H. Brandt, from the Max-Planck-Institut für Metallforschung, D-70506 Stuttgart, Germany, for the helpful hints during the computational implementation of the model.
References
[1] C. P. Bean, “Magnetization of High-Field Superconductors”, Reviews fo Modern Physics, pp. 31–39, January 1964.
[2] Y. B. Kim, C. F. Hempstead, A. R. Strnad, “Magnetization and Critical Supercurrents”, Physical Review, vol. 129, n. 2, pp. 528–535, January 1963.
[3]E. H. Brandt, “Superconductor disks and cylinders in an axial magnetic field. I. Flux penetration and magnetization curves”, Physical Review B, vol. 58, n. 10, pp. 6506–6522, September 1998.
[4]M. N. O. Sadiku. (2001). Numerical Techniques in Electromagnetics. (2nd ed.). [Press]. Boca Ratton, CRC Press.
[5]M. J. Qin, G. Li, H. K. Liu, S. X. Dou, E. H. Brandt, “Calculation of the hysteretic force between a superconductor and a magnet”, Physical Review B, vol. 66, pp. 024516-1–024516-11, July 2002.
Manuscript received August 24, 2006. This work as supported by the Brazilian agencies CNPq and FAPERJ.
R. B. Kasal is with the Industrial and Technological Development Department, ELETROBRÁS SA, Rio de Janeiro, Brazil (phone: 55 21 2514-5234; fax: 55 21 2514-5409; e-mail: ).
R. de Andrade Jr., G. G. Sotelo and A. C. Ferreira are with PEE/COPPE/UFRJ, Federal University of Rio de Janeiro, RJ 21945-972, Brazil (e-mail: , , ).