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Advanced design approach to the high temperature superconducting magnet
Cryogenics 41 (2001) 27±33
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Advanced design approach to the high temperature superconducting magnet q Young-Sik Jo a,*, Young-Kyoun Kim a, Jung-Pyo Hong a, Ju Lee b, Young-Kil Kwon c, Kang-Sik Ryu c a
Department of Electrical Engineering, Changwon National University, Changwon 641-773, South Korea b Department of Electrical Engineering, Hanyang University, Seoul 133-791, South Korea c Korea Electrotechnology Research Institute, Changwon 641-120, South Korea Received 27 November 2000; accepted 1 March 2001
Abstract The value of critical current (Ic ) in high temperature superconductor (HTS) tape has a great in¯uence on vertical ®eld (B?). Therefore, in the shape design of HTS magnet, a method to reduce the B? should be considered in order to maintain the stability and substantial improvement on the performance. This paper presents the advanced racetrack HTS magnet which is used as ®eld coil in the high temperature superconducting generator (HTSG) and the results of the approach are compared with Ic of the initial design magnet. On the basis of the magnetic ®eld analysis using Biot±Savart's law and 2-dimensional ®nite element analysis (2D FEA), this study is focused on the function of iron plates, which is to obtain smaller B?, and they are applied to both initial and optimized magnets. Moreover, the design approach is veri®ed by comparison with those experimental results. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Racetrack HTS magnet; Iron plates; High temperature superconducting generator
1. Introduction High temperature superconductor (HTS) wire can carry much larger electrical currents, which means remarkably smaller and more powerful system. Manufacturing operations of all kinds could bene®t from systems that take up less space and provide increased working capacity. In addition, substituting HTS wire for conventional wire eliminates energy loss caused by electrical resistance, and generators with HTS increase eciency up to 99.5%. Today's typical generators operate at an eciency rate of 97±98% [1]. In recent years, HTS wire gears up the development of superconducting generator all over the world. High temperature superconducting generators (HTSG) are bene®ted from a liquid nitrogen ambient because the capital costs of the refrigeration plant would be lower and the cryogenic
q
This paper was presented at the ``Korea±Japan Joint Workshop on Applied Superconductivity and Cryogenics'', Cheju-Do, Korea, 2±4 October 2000. * Corresponding author. E-mail address: [email protected] (Y.-S. Jo).
design would be simpler. However, it is well known that Ic in HTS wire is more sensitive to magnetic ®elds directed along the c-axis of the unit cell (B?) than to ®elds in the ab plane Bk [2]. Thus, in shape design of the HTS magnet, a method to reduce the B? should be considered in order to maintain the stability and substantial improvement on the performance. This study deals with two steps to reduce the B? in the process of magnet design, in which stress and strain conditions of Ag-sheathed Bi-2223 37-®lament HTS tapes are considered. At ®rst, the initial magnet is optimized with the help of response surface methodology (RSM) [3]. At this step, characteristics of magnetic ¯ux distribution in the initial magnet are accurately analyzed according to the ®eld coil shape, and the base models are analyzed to predict characteristics of magnetic ¯ux distribution according to the variation of bending radius (half of inner width of ®eld coil). And the maximum B? in the optimized magnet is compared with that of the initial magnet. At the stage of magnet design, stress and strain conditions should be considered because when the HTS tape is handled, it is necessary to know the limiting values of loading, bending and twisting to avoid any
0011-2275/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 1 - 2 2 7 5 ( 0 1 ) 0 0 0 4 4 - 3
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Y.-.S. Jo et al. / Cryogenics 41 (2001) 27±33
damage. The damage is commonly de®ned as degradation of the current carrying capabilities [4]. Thus, before designing the magnet, the tape's stress and strain conditions are determined through the experiment results. The next step involves, the magnetic ¯ux distributions of the initial magnet, optimized magnet, initial magnet with iron plates, and optimized magnet with iron plates are compared with each other. The primary objective of this step is to provide an advanced design approach in the racetrack HTS magnet design, in which the iron plates help reduce the B?. The designed racetrack HTS magnet, which is mainly for 3000 W HTSG, is manufactured and tested by Korea Electrotechnology Research Institute (KERI). Ic vs. strain curve is used in designing and winding condition. I±V test is conducted on the condition the current is supplied at 0.5 A/s rate and at LN2 temperature with no external ®eld. The details of manufacture and its experiment results are described as below.
2. Initial racetrack HTS magnet for the 3000 W HTSG A cross-section of the electromagnetic part is shown in Fig. 1. The excitation system consists of four racetrack HTS magnets. Except for magnetic shield, other parts of HTSG including damper, nitrogen tank, vacuum vessel, ®eld core, and armature core, etc., are nonmagnetic materials. Fig. 2 shows a cross-section view of the initial magnet, dimension of HTS tape where P1, P2, and P3 refer to each position in the doublepancake coil, and x1 ; x2 ; x3 , and x4 refer to design variables. The HTS tape cross-section is 3 mm thick by 0.25 mm wide and the total turn number per magnet is 467 turns.
Fig. 2. Cross-section view of the initial magnet, dimension of HTS tape and design variables.
Table 1 Speci®cations of the 3000 W HTSG Rated power Rated speed Magnet number Superconductor Ic (short sample) Operating temperature Input current
3000 W 1800 rpm 4 Ag-sheathed Bi-2223 37-®lament HTS tape 33 A (at 77 K, 0 T) 30±40 K 46 A
Table 1 shows speci®cations of the 3000 W HTSG: the ®eld coil is wound as a racetrack by using Ag-sheathed Bi-2223 37-®lament HTS tape; its bending radii are 16 and 21.5 mm; and the armature winding method is full pitch. The bending radius is determined through experiment results of the HTS tapes. The speci®cations of armature winding include Y connection, double layer winding, and 2 slots per magnet per phase.
3. Magnetic ®eld distribution of the magnet In order to analyze the magnetic ®eld of one magnet using the Biot±Savart's law, its calculation has been performed by Eq. (1). 0 ! 1 0 ~ R l I d l A; d~ B 0 @ 1 R3 4p
Fig. 1. Cross-section of the electromagnetic part in the 3000 W HTSG.
where d~ B is the magnetic ¯ux density due to a current element Id~ l0 ; ~ R is the distance vector directed from the source point to the ®eld point, and l0 is permeability of free space. Fig. 3 presents an overall view of coil shape of the racetrack magnet. In the case that there is no magnetic material around the magnet, the characteristic of magnetic ®eld distribution of the racetrack HTS magnet is shown in Fig. 4. In the winding of the magnet, the maximum B? occurring at P1 and P3 in x±y plane, determines Ic , and it is a principle factor in the sense of stability and performance. Therefore, the shape design
Y.-.S. Jo et al. / Cryogenics 41 (2001) 27±33
29
Fig. 3. Overall view of coil shape of the racetrack magnet.
Fig. 5. Cross-section view of coil shape of the base model.
Table 2 Classi®cations of the base model to obtain the characteristics of magnetic ®eld distributions according to the variation of bending radius Bending radius (mm) W1 W2 W3 W4 W5
10 12 14 16 18
Fig. 4. Magnetic ®eld distribution of the initial racetrack HTS magnet at x±y plane.
of ®eld coil should be optimized as the following conditions that its maximum B? at HTS tape is decreased at P1 and P3. Fig. 5 shows a cross-section view of coil shape of the base model to predict characteristics of magnetic ®eld distributions according to the variation of bending radius (W), through the 2-dimensional ®nite element analysis (2D FEA). In Fig. 5, P1, P2, and P3 refer to each position in the double-pancake coil, W is half of inner width of ®eld coil, the total turn number per magnet is 477 turn, and input current is 46 A. Table 2 shows classi®cations of the base model according to the variation of W. Analysis results of the base model are shown in Fig. 6 where the maximum B? and radial component of ¯ux density at armature (Br) are normalized on the basis of W1. At W5, maximum B? at HTS tape and Br at armature are increased by 7% and 34%, respectively. When Br ¯ows in the armature coil,
Fig. 6. Analysis results of base model according to the variation of W.
an electromotive force is induced in accordance with Faraday's law. Therefore, Br should be examined when HTS magnet is used as ®eld coil of the HTSG.
30
Y.-.S. Jo et al. / Cryogenics 41 (2001) 27±33
4. Optimization and an advanced design approach to the magnet Generally, a design problem is a natural process to optimize a solution vs. a speci®ed requirement. The problem can be complex because there are many numbers of design variables and these design variables frequently interact with each other. Moreover, when designing certain devices by using a conventional optimization algorithm, their performance cannot be satis®ed as the desired one in certain cases. It is due to the limitations on the manufacturing tolerances and job requirements, which could not be considered, adjustments for improving the eciency of manufacturing, and experiences of a skilled labor, etc., Whenever these cases appear, it is time consuming that we perform the optimization repeatedly or recalculate the system performance. However, using the RSM for design constraints in design optimization provides the designer with an overall perspective of the system response according to the behavior of design variables within a design space, and oers a systematized approach to study the eect of design variables on the system performance by the variance analysis [5]. However, a conventional optimization technique cannot provide these. For the sake of shape optimization of the racetrack HTS magnet, RSM is chosen because the eects of the various HTS tape thicknesses need to be considered. 4.1. RSM procedures Fig. 7 shows ¯ow chart of RSM procedures. RSM procedures seek to ®nd the relationship between design variable and response and determine the optimum system response through statistical ®tting method using observed datum [6,7]. RSM provides an approximate relationship between a true response g and k design variables, which is based on the observed data from the process or system. The response is generally obtained from real experiments or computer simulations, and the true response g is expected the response. Thus, computer
simulations are performed in this paper. We suppose that the true response g can be written as g F f1 ; f2 ; . . . ; fk ;
2
where the variables f1 ; f2 ; . . . ; fk in Eq. (2) are expressed in natural units of a measurement, so-called the natural variables. The experimentally obtained response y diers from the expected value g due to random experimental error. Because the form of the true response function F is unknown and perhaps very complicated, we must approximate it. The relation between y and g may be written as y F x1 ; x2 ; . . . ; xk e;
3
where, e denotes the random error, which includes measurement error on the response and is inherent in the process or system and the variables x1 ; x2 ; . . . ; xk are the coded variables of the natural variables. We treat e as a statistical error, often assuming it to have a normal distribution with mean zero and variance r2 . In many cases, the approximating function F of the true response g is normally chosen to be either a ®rstorder or a second-order polynomial model, which is based on Taylor series expansion. In general, the ®rst-order model is g b0 b1 x1 bk xk
4
and the second-order model is g b0
k X j1
bj xj
k X j1
bij x2j
XX i6j
bij xj xj :
5
From the above approximating function, the estimated response y^ at n data point can be written in matrix form as ^ ^y Xb;
6
where, the caret (^) denotes estimated values. In Eq. (6), X is a matrix of model terms evaluated at the data points and the vector b^ contains the unknown coecients which are usually estimated to minimize the sum of the squares of the error term, which is a process known as regression. 4.2. Optimization RSM was applied to ®nd the optimal point for reducing the maximum B?. With four design variables, 25 experiments were conducted. The surface response y(estimated value of the maximum B?) of the secondorder model was obtained with the least squares method. y^ b0
Fig. 7. Flow chart of RSM procedures.
k X j1
bj xj
k X j1
bij x2j
XX i6j
bij xi xj ;
7
Y.-.S. Jo et al. / Cryogenics 41 (2001) 27±33
31
Table 3 Regression coecients of the predictive model Coecients
Estimated value
Coecients
1
1:6683 10 1:3802 10 2 2:3643 10 3 2:3812 10 34 1:5505 10 3
b0 b1 b2 b3 b4
Estimated value
b11 b22 b33 b44 b12
3:9818 10 9:5552 10 6:7333 10 2:5771 10 1:9281 10
where the regression coecients b0 ; bj ; bjj ; bij are shown in Table 3. Stationary point conditions are obtained by using sequential quadratic programming (SQP) method, which is one of the optimization algorithms, has been commonly used to minimize the objective function that satis®es the constraint at this paper. The objective function is de®ned by Objective function: f x y^ b0
k X j1
bj xj
k X j1
bjj x2j
XX
bij xi xj :
8
4 5 5 5 4
Coecients
Estimated value
b13 b14 b23 b24 b34
1:3059 10 3:3312 10 4:1469 10 7:5000 10 6:0312 10
4 5 5 6 6
4.3. Results of those designs Fig. 8 is cross-sectional view of the optimized magnet with iron plates, where current distribution of the magnet is optimized with the help of RSM. Iron plate is included in the optimized magnet in order to decrease the B? because it can change the magnetic path at nearby P1 and P3. The role of iron plate is to decrease B? as low as possible where Br is taken into account. The shape, position and kinds of iron plates are chosen by way of 2D FEA, which considers magnetic saturation of iron plates. Considering symmetry, the analysis
i6j
From the limitation of the manufacturing magnet, the constraints are de®ned by h x
x1 2x2 2x3 x4 0:25
109:1
467;
9
24:6 6 x1 6 32:6;
10
34:35 6 x2 6 42:35;
11
34:35 6 x3 6 42:35;
12
29:1 6 x4 6 37:1;
13
where, x1 ; x2 ; x3 , and x4 , respectively, refer to the outer width of ®eld coil as shown in Fig. 2.
Fig. 8. Cross-section view of the optimized magnet with iron plates: (a) the optimized magnet; (b) the optimized magnet with iron plates.
Fig. 9. 2D FE ¯ux plot of each magnet in the 3000 W HTSG.
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Y.-.S. Jo et al. / Cryogenics 41 (2001) 27±33
Fig. 10. 2D FEA results of the maximum B ? and Br.
model is set as 1/8 of its full model. The detail of the ®eld plot around the magnet region of ¯ux path changing is shown in Fig. 9. The 2D FEA results of the maximum B? and Br are normalized on the basis of the initial magnet, and this is shown in Fig. 10. The maximum B? occurs at P3 and the maximum B? of the initial magnet, optimized magnet, initial magnet with iron plates and optimized magnet with iron plates is 0.445, 0.412, 0.371 and 0.377 T, respectively, and each of maximum Br is 0.0844, 0.089, 0.089 and 0.091 T. 5. Experimental results and discussions Fig. 11 shows the plot of normalized critical current vs. B. The current is signi®cantly higher when the ®eld is being reduced and is also higher when the applied ®eld is parallel to the wide face of the tape. Fig. 12 shows the percentage drop in Ic vs. bending strain. The tape is bent around dierent ®xed-radii cylinders.
Fig. 11. Critical current vs. B.
Fig. 12. The percentage drop in critical current vs. bending strain.
Ic vs. tensile strain characteristic of the HTS tapes is measured by use of strain gauge and micrometer at 77 K under zero bias ®elds. Fig. 13 shows the percentage drop in Ic against tensile strain. Twisting is of little concern when HTS magnets are made. Through the test, the minimum-bending radius is determined by 16 mm under 10 MPa tensile strain. The magnet is wound with Ag-sheathed Bi-2223 37®lament HTS tape onto a stainless steel bobbin and the length of initial magnet is 241 m and that of optimized magnet is 249 m. The electrical insulation between HTS tape and bobbin is made by 0.06 mm thick of Kapton adhesive tape. Each three double-pancake coil is jointed and its jointing length is one turn length. The magnet is impregnated by epoxy resin at a room temperature 25°C for 24 h. Fig. 14 shows photographs of each wound magnet and the iron plates.
Fig. 13. The percentage drop in critical current vs. tensile strain: (a) the initial target; (b) the optimized target; (c) upper and lower parts of the iron plates.
Y.-.S. Jo et al. / Cryogenics 41 (2001) 27±33
33
Fig. 15. Test results of I±V at P3 of each magnet.
Fig. 14. Photographs of each wound magnet and the iron plates.
I±V test is conducted on the condition that the current is supplied at 0.5 A/s rate and at 77 K with no external ®eld. The test results at P3 of each magnet are shown in Fig. 15. The length of each P3 in the initial magnet and the optimized magnet is 70 and 68 m, respectively. The values of Ic obtained in the initial magnet, optimized magnet, initial magnet with iron plates, and optimized magnet with iron plates are 10, 12, 15 and 15 A, respectively. 6. Conclusions The advanced design approach for racetrack HTS magnet is presented through manufacturing and testing
these magnets. When Ic at P3 of the initial magnet with iron plates is compared with the initial magnet, it is increased to 5 A at 77 K. A racetrack HTS magnet can provide very high ®eld in which a complete air-cored magnet is usually used. However, in this paper iron plates are proposed for the following three reasons: 1. Iron plates help reduce B? while changing the magnetic path at nearby magnet. 2. They contribute to increase Br while reducing the leakage ¯ux. 3. They provide a simple shape of racetrack magnet which is easily to be wound and jointed. Set against these bene®ts: 1. Eddy-current loss at time-varying ®elds occurs in the iron plates, but this magnet can be excluded if excitation winding is avoided.
References [1] Superconductivity and its Electric Power Applications, US DoE Report, July 1998. [2] Picard JF, et al. Technologies for high ®eld HTS magnets. IEEE Trans Appl Superconduc 1999;9(2):535±40. [3] Myers RH. Response surface methodology. New York: Wiley; 1995. [4] Skov-Hansen P, et al. Stresses and strains in multi-®lament HTS tapes. IEEE Trans Appl Superconduc 1999;9(2):2617±20. [5] Venter G, et al. Construction of response surface approximations for design optimization. AIAA J 1998;36:12. [6] Alauddin M, et al. Prediction of tool life in end milling by response surface methodology. J Mater Process Technol 1997;71:456±65. [7] Rong R, et al. Applying response surface methodology in the design and optimization of electromagnetic devices. IEEE Trans Magnet 1999;33(2):1916±9.
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Advanced design approach to the high temperature superconducting magnet q Young-Sik Jo a,*, Young-Kyoun Kim a, Jung-Pyo Hong a, Ju Lee b, Young-Kil Kwon c, Kang-Sik Ryu c a
Department of Electrical Engineering, Changwon National University, Changwon 641-773, South Korea b Department of Electrical Engineering, Hanyang University, Seoul 133-791, South Korea c Korea Electrotechnology Research Institute, Changwon 641-120, South Korea Received 27 November 2000; accepted 1 March 2001
Abstract The value of critical current (Ic ) in high temperature superconductor (HTS) tape has a great in¯uence on vertical ®eld (B?). Therefore, in the shape design of HTS magnet, a method to reduce the B? should be considered in order to maintain the stability and substantial improvement on the performance. This paper presents the advanced racetrack HTS magnet which is used as ®eld coil in the high temperature superconducting generator (HTSG) and the results of the approach are compared with Ic of the initial design magnet. On the basis of the magnetic ®eld analysis using Biot±Savart's law and 2-dimensional ®nite element analysis (2D FEA), this study is focused on the function of iron plates, which is to obtain smaller B?, and they are applied to both initial and optimized magnets. Moreover, the design approach is veri®ed by comparison with those experimental results. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Racetrack HTS magnet; Iron plates; High temperature superconducting generator
1. Introduction High temperature superconductor (HTS) wire can carry much larger electrical currents, which means remarkably smaller and more powerful system. Manufacturing operations of all kinds could bene®t from systems that take up less space and provide increased working capacity. In addition, substituting HTS wire for conventional wire eliminates energy loss caused by electrical resistance, and generators with HTS increase eciency up to 99.5%. Today's typical generators operate at an eciency rate of 97±98% [1]. In recent years, HTS wire gears up the development of superconducting generator all over the world. High temperature superconducting generators (HTSG) are bene®ted from a liquid nitrogen ambient because the capital costs of the refrigeration plant would be lower and the cryogenic
q
This paper was presented at the ``Korea±Japan Joint Workshop on Applied Superconductivity and Cryogenics'', Cheju-Do, Korea, 2±4 October 2000. * Corresponding author. E-mail address: [email protected] (Y.-S. Jo).
design would be simpler. However, it is well known that Ic in HTS wire is more sensitive to magnetic ®elds directed along the c-axis of the unit cell (B?) than to ®elds in the ab plane Bk [2]. Thus, in shape design of the HTS magnet, a method to reduce the B? should be considered in order to maintain the stability and substantial improvement on the performance. This study deals with two steps to reduce the B? in the process of magnet design, in which stress and strain conditions of Ag-sheathed Bi-2223 37-®lament HTS tapes are considered. At ®rst, the initial magnet is optimized with the help of response surface methodology (RSM) [3]. At this step, characteristics of magnetic ¯ux distribution in the initial magnet are accurately analyzed according to the ®eld coil shape, and the base models are analyzed to predict characteristics of magnetic ¯ux distribution according to the variation of bending radius (half of inner width of ®eld coil). And the maximum B? in the optimized magnet is compared with that of the initial magnet. At the stage of magnet design, stress and strain conditions should be considered because when the HTS tape is handled, it is necessary to know the limiting values of loading, bending and twisting to avoid any
0011-2275/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 1 - 2 2 7 5 ( 0 1 ) 0 0 0 4 4 - 3
28
Y.-.S. Jo et al. / Cryogenics 41 (2001) 27±33
damage. The damage is commonly de®ned as degradation of the current carrying capabilities [4]. Thus, before designing the magnet, the tape's stress and strain conditions are determined through the experiment results. The next step involves, the magnetic ¯ux distributions of the initial magnet, optimized magnet, initial magnet with iron plates, and optimized magnet with iron plates are compared with each other. The primary objective of this step is to provide an advanced design approach in the racetrack HTS magnet design, in which the iron plates help reduce the B?. The designed racetrack HTS magnet, which is mainly for 3000 W HTSG, is manufactured and tested by Korea Electrotechnology Research Institute (KERI). Ic vs. strain curve is used in designing and winding condition. I±V test is conducted on the condition the current is supplied at 0.5 A/s rate and at LN2 temperature with no external ®eld. The details of manufacture and its experiment results are described as below.
2. Initial racetrack HTS magnet for the 3000 W HTSG A cross-section of the electromagnetic part is shown in Fig. 1. The excitation system consists of four racetrack HTS magnets. Except for magnetic shield, other parts of HTSG including damper, nitrogen tank, vacuum vessel, ®eld core, and armature core, etc., are nonmagnetic materials. Fig. 2 shows a cross-section view of the initial magnet, dimension of HTS tape where P1, P2, and P3 refer to each position in the doublepancake coil, and x1 ; x2 ; x3 , and x4 refer to design variables. The HTS tape cross-section is 3 mm thick by 0.25 mm wide and the total turn number per magnet is 467 turns.
Fig. 2. Cross-section view of the initial magnet, dimension of HTS tape and design variables.
Table 1 Speci®cations of the 3000 W HTSG Rated power Rated speed Magnet number Superconductor Ic (short sample) Operating temperature Input current
3000 W 1800 rpm 4 Ag-sheathed Bi-2223 37-®lament HTS tape 33 A (at 77 K, 0 T) 30±40 K 46 A
Table 1 shows speci®cations of the 3000 W HTSG: the ®eld coil is wound as a racetrack by using Ag-sheathed Bi-2223 37-®lament HTS tape; its bending radii are 16 and 21.5 mm; and the armature winding method is full pitch. The bending radius is determined through experiment results of the HTS tapes. The speci®cations of armature winding include Y connection, double layer winding, and 2 slots per magnet per phase.
3. Magnetic ®eld distribution of the magnet In order to analyze the magnetic ®eld of one magnet using the Biot±Savart's law, its calculation has been performed by Eq. (1). 0 ! 1 0 ~ R l I d l A; d~ B 0 @ 1 R3 4p
Fig. 1. Cross-section of the electromagnetic part in the 3000 W HTSG.
where d~ B is the magnetic ¯ux density due to a current element Id~ l0 ; ~ R is the distance vector directed from the source point to the ®eld point, and l0 is permeability of free space. Fig. 3 presents an overall view of coil shape of the racetrack magnet. In the case that there is no magnetic material around the magnet, the characteristic of magnetic ®eld distribution of the racetrack HTS magnet is shown in Fig. 4. In the winding of the magnet, the maximum B? occurring at P1 and P3 in x±y plane, determines Ic , and it is a principle factor in the sense of stability and performance. Therefore, the shape design
Y.-.S. Jo et al. / Cryogenics 41 (2001) 27±33
29
Fig. 3. Overall view of coil shape of the racetrack magnet.
Fig. 5. Cross-section view of coil shape of the base model.
Table 2 Classi®cations of the base model to obtain the characteristics of magnetic ®eld distributions according to the variation of bending radius Bending radius (mm) W1 W2 W3 W4 W5
10 12 14 16 18
Fig. 4. Magnetic ®eld distribution of the initial racetrack HTS magnet at x±y plane.
of ®eld coil should be optimized as the following conditions that its maximum B? at HTS tape is decreased at P1 and P3. Fig. 5 shows a cross-section view of coil shape of the base model to predict characteristics of magnetic ®eld distributions according to the variation of bending radius (W), through the 2-dimensional ®nite element analysis (2D FEA). In Fig. 5, P1, P2, and P3 refer to each position in the double-pancake coil, W is half of inner width of ®eld coil, the total turn number per magnet is 477 turn, and input current is 46 A. Table 2 shows classi®cations of the base model according to the variation of W. Analysis results of the base model are shown in Fig. 6 where the maximum B? and radial component of ¯ux density at armature (Br) are normalized on the basis of W1. At W5, maximum B? at HTS tape and Br at armature are increased by 7% and 34%, respectively. When Br ¯ows in the armature coil,
Fig. 6. Analysis results of base model according to the variation of W.
an electromotive force is induced in accordance with Faraday's law. Therefore, Br should be examined when HTS magnet is used as ®eld coil of the HTSG.
30
Y.-.S. Jo et al. / Cryogenics 41 (2001) 27±33
4. Optimization and an advanced design approach to the magnet Generally, a design problem is a natural process to optimize a solution vs. a speci®ed requirement. The problem can be complex because there are many numbers of design variables and these design variables frequently interact with each other. Moreover, when designing certain devices by using a conventional optimization algorithm, their performance cannot be satis®ed as the desired one in certain cases. It is due to the limitations on the manufacturing tolerances and job requirements, which could not be considered, adjustments for improving the eciency of manufacturing, and experiences of a skilled labor, etc., Whenever these cases appear, it is time consuming that we perform the optimization repeatedly or recalculate the system performance. However, using the RSM for design constraints in design optimization provides the designer with an overall perspective of the system response according to the behavior of design variables within a design space, and oers a systematized approach to study the eect of design variables on the system performance by the variance analysis [5]. However, a conventional optimization technique cannot provide these. For the sake of shape optimization of the racetrack HTS magnet, RSM is chosen because the eects of the various HTS tape thicknesses need to be considered. 4.1. RSM procedures Fig. 7 shows ¯ow chart of RSM procedures. RSM procedures seek to ®nd the relationship between design variable and response and determine the optimum system response through statistical ®tting method using observed datum [6,7]. RSM provides an approximate relationship between a true response g and k design variables, which is based on the observed data from the process or system. The response is generally obtained from real experiments or computer simulations, and the true response g is expected the response. Thus, computer
simulations are performed in this paper. We suppose that the true response g can be written as g F f1 ; f2 ; . . . ; fk ;
2
where the variables f1 ; f2 ; . . . ; fk in Eq. (2) are expressed in natural units of a measurement, so-called the natural variables. The experimentally obtained response y diers from the expected value g due to random experimental error. Because the form of the true response function F is unknown and perhaps very complicated, we must approximate it. The relation between y and g may be written as y F x1 ; x2 ; . . . ; xk e;
3
where, e denotes the random error, which includes measurement error on the response and is inherent in the process or system and the variables x1 ; x2 ; . . . ; xk are the coded variables of the natural variables. We treat e as a statistical error, often assuming it to have a normal distribution with mean zero and variance r2 . In many cases, the approximating function F of the true response g is normally chosen to be either a ®rstorder or a second-order polynomial model, which is based on Taylor series expansion. In general, the ®rst-order model is g b0 b1 x1 bk xk
4
and the second-order model is g b0
k X j1
bj xj
k X j1
bij x2j
XX i6j
bij xj xj :
5
From the above approximating function, the estimated response y^ at n data point can be written in matrix form as ^ ^y Xb;
6
where, the caret (^) denotes estimated values. In Eq. (6), X is a matrix of model terms evaluated at the data points and the vector b^ contains the unknown coecients which are usually estimated to minimize the sum of the squares of the error term, which is a process known as regression. 4.2. Optimization RSM was applied to ®nd the optimal point for reducing the maximum B?. With four design variables, 25 experiments were conducted. The surface response y(estimated value of the maximum B?) of the secondorder model was obtained with the least squares method. y^ b0
Fig. 7. Flow chart of RSM procedures.
k X j1
bj xj
k X j1
bij x2j
XX i6j
bij xi xj ;
7
Y.-.S. Jo et al. / Cryogenics 41 (2001) 27±33
31
Table 3 Regression coecients of the predictive model Coecients
Estimated value
Coecients
1
1:6683 10 1:3802 10 2 2:3643 10 3 2:3812 10 34 1:5505 10 3
b0 b1 b2 b3 b4
Estimated value
b11 b22 b33 b44 b12
3:9818 10 9:5552 10 6:7333 10 2:5771 10 1:9281 10
where the regression coecients b0 ; bj ; bjj ; bij are shown in Table 3. Stationary point conditions are obtained by using sequential quadratic programming (SQP) method, which is one of the optimization algorithms, has been commonly used to minimize the objective function that satis®es the constraint at this paper. The objective function is de®ned by Objective function: f x y^ b0
k X j1
bj xj
k X j1
bjj x2j
XX
bij xi xj :
8
4 5 5 5 4
Coecients
Estimated value
b13 b14 b23 b24 b34
1:3059 10 3:3312 10 4:1469 10 7:5000 10 6:0312 10
4 5 5 6 6
4.3. Results of those designs Fig. 8 is cross-sectional view of the optimized magnet with iron plates, where current distribution of the magnet is optimized with the help of RSM. Iron plate is included in the optimized magnet in order to decrease the B? because it can change the magnetic path at nearby P1 and P3. The role of iron plate is to decrease B? as low as possible where Br is taken into account. The shape, position and kinds of iron plates are chosen by way of 2D FEA, which considers magnetic saturation of iron plates. Considering symmetry, the analysis
i6j
From the limitation of the manufacturing magnet, the constraints are de®ned by h x
x1 2x2 2x3 x4 0:25
109:1
467;
9
24:6 6 x1 6 32:6;
10
34:35 6 x2 6 42:35;
11
34:35 6 x3 6 42:35;
12
29:1 6 x4 6 37:1;
13
where, x1 ; x2 ; x3 , and x4 , respectively, refer to the outer width of ®eld coil as shown in Fig. 2.
Fig. 8. Cross-section view of the optimized magnet with iron plates: (a) the optimized magnet; (b) the optimized magnet with iron plates.
Fig. 9. 2D FE ¯ux plot of each magnet in the 3000 W HTSG.
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Y.-.S. Jo et al. / Cryogenics 41 (2001) 27±33
Fig. 10. 2D FEA results of the maximum B ? and Br.
model is set as 1/8 of its full model. The detail of the ®eld plot around the magnet region of ¯ux path changing is shown in Fig. 9. The 2D FEA results of the maximum B? and Br are normalized on the basis of the initial magnet, and this is shown in Fig. 10. The maximum B? occurs at P3 and the maximum B? of the initial magnet, optimized magnet, initial magnet with iron plates and optimized magnet with iron plates is 0.445, 0.412, 0.371 and 0.377 T, respectively, and each of maximum Br is 0.0844, 0.089, 0.089 and 0.091 T. 5. Experimental results and discussions Fig. 11 shows the plot of normalized critical current vs. B. The current is signi®cantly higher when the ®eld is being reduced and is also higher when the applied ®eld is parallel to the wide face of the tape. Fig. 12 shows the percentage drop in Ic vs. bending strain. The tape is bent around dierent ®xed-radii cylinders.
Fig. 11. Critical current vs. B.
Fig. 12. The percentage drop in critical current vs. bending strain.
Ic vs. tensile strain characteristic of the HTS tapes is measured by use of strain gauge and micrometer at 77 K under zero bias ®elds. Fig. 13 shows the percentage drop in Ic against tensile strain. Twisting is of little concern when HTS magnets are made. Through the test, the minimum-bending radius is determined by 16 mm under 10 MPa tensile strain. The magnet is wound with Ag-sheathed Bi-2223 37®lament HTS tape onto a stainless steel bobbin and the length of initial magnet is 241 m and that of optimized magnet is 249 m. The electrical insulation between HTS tape and bobbin is made by 0.06 mm thick of Kapton adhesive tape. Each three double-pancake coil is jointed and its jointing length is one turn length. The magnet is impregnated by epoxy resin at a room temperature 25°C for 24 h. Fig. 14 shows photographs of each wound magnet and the iron plates.
Fig. 13. The percentage drop in critical current vs. tensile strain: (a) the initial target; (b) the optimized target; (c) upper and lower parts of the iron plates.
Y.-.S. Jo et al. / Cryogenics 41 (2001) 27±33
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Fig. 15. Test results of I±V at P3 of each magnet.
Fig. 14. Photographs of each wound magnet and the iron plates.
I±V test is conducted on the condition that the current is supplied at 0.5 A/s rate and at 77 K with no external ®eld. The test results at P3 of each magnet are shown in Fig. 15. The length of each P3 in the initial magnet and the optimized magnet is 70 and 68 m, respectively. The values of Ic obtained in the initial magnet, optimized magnet, initial magnet with iron plates, and optimized magnet with iron plates are 10, 12, 15 and 15 A, respectively. 6. Conclusions The advanced design approach for racetrack HTS magnet is presented through manufacturing and testing
these magnets. When Ic at P3 of the initial magnet with iron plates is compared with the initial magnet, it is increased to 5 A at 77 K. A racetrack HTS magnet can provide very high ®eld in which a complete air-cored magnet is usually used. However, in this paper iron plates are proposed for the following three reasons: 1. Iron plates help reduce B? while changing the magnetic path at nearby magnet. 2. They contribute to increase Br while reducing the leakage ¯ux. 3. They provide a simple shape of racetrack magnet which is easily to be wound and jointed. Set against these bene®ts: 1. Eddy-current loss at time-varying ®elds occurs in the iron plates, but this magnet can be excluded if excitation winding is avoided.
References [1] Superconductivity and its Electric Power Applications, US DoE Report, July 1998. [2] Picard JF, et al. Technologies for high ®eld HTS magnets. IEEE Trans Appl Superconduc 1999;9(2):535±40. [3] Myers RH. Response surface methodology. New York: Wiley; 1995. [4] Skov-Hansen P, et al. Stresses and strains in multi-®lament HTS tapes. IEEE Trans Appl Superconduc 1999;9(2):2617±20. [5] Venter G, et al. Construction of response surface approximations for design optimization. AIAA J 1998;36:12. [6] Alauddin M, et al. Prediction of tool life in end milling by response surface methodology. J Mater Process Technol 1997;71:456±65. [7] Rong R, et al. Applying response surface methodology in the design and optimization of electromagnetic devices. IEEE Trans Magnet 1999;33(2):1916±9.