Numerical Modeling of Iron Yoke Levitation Using the ... - IEEE Xplore

IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 5, MAY 2007

2001

Numerical Modeling of Iron Yoke Levitation Using the Pinning Effect of High-Temperature Superconductors Mojtaba Ghodsi1 , Toshiyuki Ueno1 , Hidekazu Teshima2 , Hosei Hirano2 , and Toshiro Higuchi2 The University of Tokyo, Tokyo 113-8656, Japan Nippon Steel Corporation, Chiba 293-8511, Japan

A ferromagnetic material can be levitated by the pinning effect of a field-cooled superconductor. This paper presents two methods for modeling this effect: 1) an approximate calculation to determine the relationship between attractive force and air gap at both room temperature and superconductive temperature (77 K) and 2) a novel way of modeling the pinning effect by a finite-element method (FEM). A comparison of analytical and FEM results with experimental results verifies the validity of the methods. The methods can be used to estimate the system’s behavior when the cylindrical yoke is replaced by a ring yoke. The stiffness of the system will increase by 70% (to 5.3 N/mm) when a ring yoke with the same surface area is used instead of a cylindrical yoke. Index Terms—Analytical model, field-cooled high temperature superconductor, finite-element method (FEM), Maxwell theory, numerical modeling, pinning effect, shape effect.

I. INTRODUCTION IGH-TEMPERATURE superconductors (HTSs) can be used in levitation technologies. On account of the diamagnetic property of superconductive material, a permanent magnet (PM) can be levitated over HTS [1]. Passive levitation and high levitation force are two considerable advantages of this conventional suspension system. This suspension system can be used for public transportation systems, but the cost would be high as the rail has to be made of permanent magnet. Tsutusi and Hull proposed a levitation system of a ferromagnetic yoke using a combination of a permanent magnet and a bulk HTS sample, operated in liquid nitrogen [2], [3]. The maximum levitation force in Tsutsi and Hull systems was about 10 N, whereas, this value of force is too small for practical applications. Tsutsui believed that the positive stiffness to realize the levitation is attributed to the pinning effect, but this theory has not been supported by experimental results [2]. Hull postulated that the diamagnetic property of HTS, when the magnetization of steel induces shielding currents in the HTS, results in a repulsive force between the HTS and the yoke [3]. So far, we highlighted the role of the thickness of the HTS sample in this levitation system and a levitation of heavy iron mass, 8.6 kg, was demonstrated [4]. Furthermore, we confirmed the principle of this levitation system proposed by Tsutsui and supported this idea by novel experimental method [5]. In this paper, both a rough analytical method to calculate the attractive force and a new method for modeling the pinning effect by commercially available FEM software are presented. Moreover, since the robustness against the external disturbance plays an important role in passive lev-

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itation systems, the effect of the yoke shape on the stiffness of this system is investigated. To confirm the validity of proposed models, the relationship between attractive force and gap are calculated and measured with different yoke diameters (10, 15, 17, and 20 mm) and different yoke shapes (ring and cylindrical). II. PRINCIPLE OF LEVITATION The basic geometry of the levitation system is shown in Fig. 1. According to the Maxwell theory, there exists a magnetic pressure acting normally on the surface boundary between a region 1 (with the permeability ) and a region 2 (with the permeability ) in a magnetic field [6]. This magnetic pressure creates the force. The magnetic pressure on the boundary between regions 1 and 2 can be calculated by [6]

(1) where is the normal component of the magnetic induction and is the tangential component of the magnetic field strength. The direction of magnetic pressure is from region 2 to region 1. Based on the boundary condition, and are constant at the surface boundary 1-2. When the region 1 is air and region 2 is iron, the field is normal to the boundary or [6]. Since , (1) becomes

(2) Therefore, the vertical force that is acting on the yoke surface , is determined by the integration of the pressure over the surface as

Digital Object Identifier 10.1109/TMAG.2006.890218 Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. 0018-9464/$25.00 © 2007 IEEE

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Fig. 2. Attractive force versus air gap at RT (300 K) and 77 K; HTS thickness is 10 mm, ring yoke. Fig. 1. Principle of levitation system; rough analytical model.

If the magnetic field is determined in the air gap, the attractive force can be calculated. The magnetic flux can be trapped in type-II of high temperature superconductor (HTS) by a so-called “pinning effect.” Since the fluxes are pinned by impurities inside of the HTS sample, the freedom of fluxes, expelling from the surface of the HTS, is limited. Realizing levitation consists of two processes: cooling process and levitation condition. In the cooling process, the HTS sample is placed between a PM and a ferromagnetic yoke and cooled by liquid nitrogen for 30 min Fig. 1(a). In the next process, when a cylindrical yoke approaches to HTS from Fig. 1(b) to (c), the trapped magnetic fluxes emanated from the surface of HTS gather toward the yoke and the attractive force increases. However, as the yoke approaches closer to the surface of HTS as is shown in Fig. 1(d), the magnetic fluxes which pass through the face surface of yoke will decrease and some of the fluxes will enter from the side surface of the yoke. Thereby, the reduction of the normal flux density causes a decrease of the attractive force in (3). As a result, a positive stiffness in the curve of attractive force versus gap allows passive stable levitation of the yoke. By using this idea, a transportation system can be constructed with low cost, e.g., the rail can be constructed with iron. For instance, the relationships between the attractive force and the gap at 77 K and room temperature (RT) in approach/retreat cycle are shown in Fig. 2. It is obvious that in case of RT, the attractive force increases when the gap decreases. This system at RT is intrinsically unstable, because the stiffness over the complete range of the gap is negative. In contrast, the general shape of this relationship will change at 77 K. The positive stiffness in the small air gap ( 1.6 mm) allows a stable passive levitation. As defined in [4], three items are presented as “fundamental parameters” in this levitation system. In the curve of attractive force versus gap at 77 K, the finite variation of force over finite variation of displacement in each point is called stiffness (Fig. 2). The air gap range in which the system’s stiffness is positive, is called “positive stiffness gap” (PSG). Moreover, the average of stiffness values throughout the PSG is presented as “stiffness value” and the maximum value of attractive force is called “max. attractive force.” In this paper, the fundamental

parameters are compared to each other for different presented models to investigate the behavior of the system. III. ESTIMATION OF ATTRACTIVE FORCE In this section, an approximate analytical model for calculating the relationship between attractive force and gap will be explained. On account of this purpose firstly, the fluxes trapped inside HTS during cooling process will be determined. Secondly, by realizing the trapped flux density and using the Maxwell theory the attractive force at 77 K will be calculated. On the way of proposing the analytical models, a correction coefficient, , is defined based on the experimental results [4] to reduce the errors percentage arisen from taken assumptions. Cooling Process: In this process, the fluxes will be trapped in the bulk HTS samples during cooling condition. As shown in Fig. 1, the cooling is done when a ferromagnetic yoke is placed in an initial cooling air gap . Using an equivalent circuit model presented in Appendix A, the rough value of trapped flux density trapped inside the HTS can be calculated. This are trapped uniassumption is taken that the gap fluxes . formly in a part of the HTS area covered by the yoke area Therefore, the uniform flux density can be derived from (4) Analysis of Force at 77 K by Rough Model: It will be described how by using the pinning effect and Maxwell equations, the positive stiffness is achievable. After the cooling process, in which fluxes are trapped inside HTS samples, the movable yoke is approached to the surface of the HTS samples. In [5], 10 mm) it was highlighted that for thick HTS samples ( the variation of magnetic flux at 77 K can be ignored. In other words, the trapped magnetic flux density inside the HTS will not change by approaching and retreating of the iron yoke to the surface of the HTS samples. By ignoring the leakage fluxes, we propose a rough model for pinning and investigate the effect of the yoke diameter and yoke shape on the system. In the rough model Fig. 1(a), it is assumed that only part of trapped fluxes in the HTS will gather at the yoke surface. These fluxes are emanating from circle 1 and rim 2 of the HTS’s surface. As shown

GHODSI et al.: NUMERICAL MODELING OF IRON YOKE LEVITATION

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in Fig. 1(a), area 1 is equal to the yoke area and area 2 is a rim area assumed to be proportional to the air gap. In Fig. 1(b)– (d), the cross section of flux related to area 1 and 2 are denoted by and , respectively. Therefore, the sum of areas 1 and 2 of HTS is given by (5) in which is the diameter of yoke, is the air gap, and based on the experimental results in [4], a correction coefficient is defined 0.1 when the yoke diameter is between 10 and 25 mm. Having determined the trapped flux density in (4), flux denpassing through the yoke surface is given by sity

(6) Since is the normal flux density, substituting (4) and (6) in (3) gives the attractive force. Therefore, the approximate value for attractive force can be derived from

(7) Regarding (7), the attractive force will decrease by approaching the yoke from Fig. 1(b) to (d). Therefore, the PSG estimated by this method is . Analysis of Force at 77 K by Modify Model: Although the rough model helps to describe the concept of positive stiffness, it represents the positive stiffness for all air gap range which is not correct because in the real system there is a transition between negative and positive stiffness. Now a modified model will be presented in which the leakage fluxes are considered. In the modified model, the air gap is divided into two volumes [Fig. 3(a)]. The circle and rim areas are denoted by 1 and 2, respectively. It is assumed that the central flux emanating from the HTS, passing through the central part of yoke (diameter of central area of yoke is ) is constant during approach and retreat. In Fig. 3(b)–(d), the cross section of flux related to circle 1 and rim 2 are denoted by and , respectively. As shown in Fig. 3(b), when the air gap is big the leakage fluxes are not passing through the face surface of the yoke. By approaching the yoke to the HTS surface Fig. 3(c), part of the leakage fluxes are passing through the yoke and as a result the attractive force will increase. Although in the small air gap Fig. 3(d) most of the leakages intend to pass through the face surface, the gap is small. Therefore, they cannot enter through the face surface of the yoke and the attractive force will decrease. In other words, the parameter in the rough model can be replaced by a multiplication of and cosine function of . The function is selected as cosine because the leakage flux passing through the face surface of yoke tends to increase from its minimum value in gap to its maximum value in zero gap length. By dividing the area of yoke to a central area and a rim area , it can be concluded that (8)

Fig. 3. Modified analytical model.

in which is the total flux passing through the face of the and are the fluxes entering through the central yoke, (area 1) and the rim (area 2) areas of the yoke, respectively. As mentioned before, the flux density at the center of the yoke is constant. However, the rim flux density of yoke is given by

(9) where

(10) , the cenBy the assumption that tral area of yoke is and yoke area can be calculated. Consequently, the rim area of yoke is . By substituting (4) and (10) in (9), can be calculated. It is necessary to mention that . By substituting and in (3), the total attractive force exerted on the central area and rim area of yoke can be calculated by

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IV. MODELING OF PINNING EFFECT BY FEM To prove the principle of the levitation system, the analytical model was presented in the previous section. Because of taken assumptions, the correction coefficient was defined. In this section, a numerical model for pinning effect in a field-cooled HTS sample will be presented. This model can prove the principle, whereas, it is not required to define any correction factor. The design and optimization of levitation system structure using HTS is a costly and time-consuming process. It can be

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TABLE I SPECIFICATIONS OF COMPONENTS

Fig. 4. Modeling and meshing by ANSYS. (a) Cylindrical yoke. (b) Ring yoke.

reduced by software tools able to describe the superconductivity phenomenon. However, the superconductor element and hysteresis effect are not implemented in the present commercially available field analysis tools. In [7], a new method for modeling a zero-field-cooled HTS was presented in which HTS was simulated as a diamagnetic material whose relative perme. Here, we present a modeling method for ability a field-cooled HTS sample. In [5], it was shown that for thick HTS samples ( 10 mm), the pinned fluxes remain approximately constant when the yoke approaches/retreats to the HTS surface. This means that (12) and are the trapped flux and surface area of where the bulk HTS, respectively. Using the magnetic vector potential , the governing equation for a 3-D analysis could be found as (13) Replacing (13) in (12) causes (14) To avoid long-term program development, the ANSYS software is used to simulate the pinning effect in the HTS. Equation (14) shows that the vector potential of each node inside the HTS , is constant during the approach and retreat cycles. sample, Therefore (15)

Fig. 5. Schematic configuration of experimental setup. (a) Cylindrical yoke. (b) Ring yoke.

When the vector potential is known, it can be considered as a load condition. The magnetic fluxes are trapped inside the HTS samples during cooling process. As it was explained in cooling process, the cooling is done when the distance between the HTS and the yoke is . Thereby, the fluxes correspond to the air gap are trapped in the HTS samples. To simulate the pinning effect, the following guideline should be followed. • Since the model is symmetric, create only half of the 3-D model and mesh it (Fig. 4). • Apply the flux parallel condition as a boundary condition to the cross section of model. • Analyze the generated magnetic model when the air gap is and save the vector potentials of all HTS’s nodes. • Apply the obtained vector potential results as a load condition to all corresponding nodes of the HTS and repeat the analysis for different air gaps. Fig. 4 shows a schematic view of modeling and meshing of levitation system. To investigate both the validity of FEM modeling


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Fig. 6. Relationship between attractive force and air gap at RT; analytical, numerical, and experimental results for different diameters: (a) D = 10 mm; (b) D = 15 mm; (c) D = 17 mm; (d) D = 20 mm.

and effect of yoke shape, two shape patterns of a ring and cylindrical-shaped yoke are modeled. The specifications of the exploited bulk HTS samples and the PM are mentioned in Table I. V. EXPERIMENTAL DETAILS The methods for determining relationship between attractive force and air gap were presented. To confirm their validity and to investigate the effect of yoke shape, the experimental setup shown in Fig. 5 is used to measure the relationship between attractive force and air gap at both RT and 77 K. The specifications of the setup are mentioned in Appendix B. As explained in [5], the magnetic circuit comprises two main parts, stationary and moveable parts. The stationary part consists of two bulk HTS disks attached by two permanent magnets (PMs) to a back iron yoke. In the first experiment, the movable yoke is made of iron with cylindrical yoke. The relationship between the attractive force and air gap is measured when the diameter of yoke is 10, 15, 17, and 20 mm. In [8], it was shown that among different shapes of yoke, the stiffness of the system will increase if the cylindrical yoke is replaced by the ring yoke. To investigate the effect of yoke shape, the cylindrical yoke ( mm) is replaced with a ring yoke ( mm and mm) with the same surface area. Therefore, the stationary and the movable parts make a close magnetic flux loop. The relationship between the attractive force and the air gap is measured by a load cell (A&D LC1205-K200) and a displacement laser sensor

(Keyence LC-2440), respectively, in approaching/retreating cycles. The measurement is repeated in superconductivity state (77 K). On account of this purpose, the cryostat is filled with air liquid nitrogen when the movable yoke is placed in the gap. Thereby, the behavior of attractive force versus air gap when movable yoke equipped with ring and cylindrical yokes is investigated at both RT and superconductivity state (77 K). VI. RESULTS AND DISCUSSIONS To investigate the accuracy of proposed methods, the relationships between attractive force and air gap are measured at RT with different diameters of cylindrical yoke (Fig. 6). The general shape of curves is similar in which decreasing of the gap causes an increase of force. It is obvious that by increasing the diameter of the yoke, the attractive force will increase. The FEM method simulates the force behavior more accurate than the rough analytical model. However, the rough model can be used to calculate the force and amount of flux in the air gap with a good estimation. This system at RT is intrinsically unstable, because the stiffness over the complete range of the gap is negative. The measurement, simulation, and calculation of attractive force versus gap are repeated at 77 K (Fig. 7). The rough model predicts the relationship between the force and gap as a line with a positive slope. It means that by approaching the yoke towards the HTS from the initial cooling gap ), the attractive force will decrease. This model can estimate roughly the

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Fig. 7. Relationship between attractive force and air gap at 77 K; analytical, numerical, and experimental results for different diameters: (a) D = 10 mm; (b) D = 15 mm; (c) D = 17 mm; (d) D = 20 mm.

stiffness and attractive force in the small air gap, while PSG can not be predicted correctly. The general shapes of the modified model, FEM, and experimental results are similar. By using the FEM model, the maximum attractive force can be estimated with a maximum 20% error. Even though the PSG cannot be predicted by rough model correctly, the modified method can predict the PSG with good accuracy. For example, the modified model predicts the PSG by maximum 15% error. The FEM model shows that by enlarging the yoke diameter, the PSG will increase. Despite good prediction of the PSG by FEM model for 10–17 mm yoke, the PSG is poorly predicted for the 20 mm yoke (38% error). Attractive force as a function of air gap at RT and 77 K for magnetic circuits equipped with both ring and cylindrical yoke patterns is shown in Fig. 8. The analytical model shows the same results for ring and cylindrical yokes at RT. Fig. 8(a) shows that FEM, analytical model, and experimental results have a good agreement with each other at RT. The measurement is repeated at 77 K for ring and cylindrical yokes [Fig. 8(b)]. The reduction of the attractive force in the small air gap causes the positive stiffness. The most important point is the enhancement of stiffness by exploiting the ring yoke, which is assessed by all methods. If the ring yoke is used instead of the cylindrical yoke, the modified and the FEM results predict the enhancement of 133% and 180% for stiffness, respectively. This enhancement is verified by the experimental results. The stiffness of the ring yoke reaches up to 5.3 N/mm, which is approximately 70%

higher than system’s stiffness with cylindrical yoke. The difference between estimated values of enhancement can be accounted for assumptions taken in the models; however, the enhancement of stiffness by using the ring yoke is demonstrated by all models. It can be found from analytical model that the flux passing trough the central part of yoke is constant, while the flux in the rim area of yoke decreases in approaching. However, in the case of ring yoke, there are two rim areas (internal and external) and the reduction of flux is bigger than cylindrical yoke. It is required to mention two points. The first is that, in simulation of PM at 77 K the value of residual flux density must be considered 1.47 T, whereas, it is 1.24 T at RT. The second point is the limitations of ANSYS software and analytical model that they cannot model hysteresis. The presented modeling methods provide good results which can be used to prove the principle of this levitation system and predict the dependency of force behavior on the shape of yoke. VII. CONCLUSION We have studied the behavior of a field-cooled HTS sample in the levitation system of a ferromagnetic material by the HTS. Two methods for modeling of pinning effect are presented. The relationship between the attractive force and the air gap is analytically calculated, numerically analyzed, and experimentally measured at both room temperature (RT) and superconductive temperature (77 K). The results obtained from proposed models,


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is the flux, is the PM magnetomotive force in which is the permeance (MMF), is the air-gap reluctance, and of the air gap. To determine , the magnetic flux lines are approximated by arcs and straight lines and the flux tube reluctances are then calculated [9]. The equivalent magnetic circuit is drawn and then solved using Kirchoff’s laws. According to Fig. 9, the fluxes expelled from the surface of PM are passing through three routes. The main path of flux is air gap and other routes are for leakage between PMs and leakage between opposite poles of each magnet. Therefore (17) Moreover, by applying Ampere’s law, it can be written (18) (19) and are the permeances between PMs, permeances between the opposite poles of each PM, and air gap, respectively. Furthermore, for the PM, the – characteristic is a straight line in the second quadrant [6], and is given by (20) and are the operating point and the remanence where fluxes of PM. By substituting (18), (19), and (20) in (17), a closed-form equation for the flux in the air gap can be presented by Fig. 8. Comparison of attractive force versus air gap with ring and cylindrical yokes at 77 K and RT; HTS thickness is 10 mm. (a) RT. (b) 77 K.

(21) analytical and numerical, show a poor agreement with experimental results and also they cannot model the hysteresis phenomenon. However, the fact that pinning effect causes positive stiffness is showed by the proposed models. Furthermore, the enhancement of the stiffness by using ring yoke instead of cylindrical yoke is demonstrated by the presented FEM and modified model. It is found that the stiffness of the system reaches to 5.3 N/mm with the ring yoke, while it is about 3.1 N/mm for the cylindrical yoke. Comparison of system’s stiffness between ring and cylindrical yoke highlights the effect of rim area in the yoke. In cylindrical yoke the flux passing through the central part of yoke is constant, whereas, the flux in the rim area decreases in approaching. Since there are two rim areas in the ring yoke, the reduction of flux is bigger than the cylindrical yoke. Therefore, the stiffness of ring yoke is higher than cylindrical yoke. In summary, the presented methods are able to prove the principle of the levitation system. Furthermore, the stiffness of the system will increase if the ring yoke is used instead of the cylindrical yoke. APPENDIX A

APPENDIX B Vector potential. Flux density. Flux density of yoke. Flux density of HTS. Flux density of rim area of yoke. Flux density of central area of yoke. Remanence flux density of PM (1.24 T at RT and 1.47 T at 77 K). Diameter of central area of yoke. Diameter of yoke. Diameter of magnet (30 mm). Attractive force. Magnetic field strength.

To determine the magnetic field, we use the so-called Ohm’s magnetic law (16)

Distance between the wall of the yokes . Distance between center of the yokes (50 mm).

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Rim area of yoke. Surface area of yoke. Surface area of magnet. Width of back iron perpendicular to the page (15 mm). Correction coefficient (0.1). Magnetic flux. Vacuum permeability

.

Relative permeability of ferromagnetic material .

REFERENCES

Fig. 9. Schematic configuration of the magnetic circuit. (a) Magnetic flux in magnetic circuit. (b) Equivalent circuit.

Air gap. Air gap during cooling (3 mm). Magnet thickness (10 mm). Superconductor thickness (10 mm). Permeance between magnets. Permeance of air gap. Permeance between the poles of each magnet. Permeance of magnet. Magnetic pressure.

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Surface area. Central area of yoke.

Manuscript received November 7, 2005; revised December 20, 2006. Corresponding author: M. Ghodsi (e-mail: [email protected]).