CERN Accelerator School Superconductivity for Accelerators Case study 3 Paolo Ferracin ( ) European Organization for Nuclear Research (CERN)

Case study 3 High field - large aperture magnet for a cable test facility Introduction High field (B bore >10 T) magnets are needed to upgrade existing accelerators in Europe and to prepare for new projects on a longer timescale. Nb 3 Sn is today the right candidate to meet those objectives, because of its superconducting properties and its industrial availability. On the very long term, further upgrades could require dipole magnets with a field of around 20 Tesla (T): a possible solution is to combine an outer Nb 3 Sn coil with an inner coil of High Critical Temperature (HTS) conductor, both contributing to the field. In addition, an high-field dipole magnet with a large aperture could be used to upgrade the Fresca test facility at CERN, in the aim of meeting the strong need to qualify conductor at higher fields. Goal Design a superconducting dipole with an 100 mm aperture and capable of reaching 15 T at 1.9 K (~90% of I ss ). Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 2

High field - large aperture magnet for a cable test facility Questions 1. Determine maximum gradient and coil size (using sector coil scaling laws) 2. Define strands and cable parameters 1. Strand diameter and number of strands 2. Cu to SC ratio and pitch angle 3. Cable width, cable mid-thickness and insulation thickness 4. Filling factor κ 3. Determine load-line (no iron) and “short sample” conditions 1. Compute j sc_ss, j o_ss, I ss, B ss, B peak_ss 4. Determine “operational” conditions (80% of I ss ) and margins 1. Compute j sc_op, j o_op, I op, B op, B peak_op 2. Compute T, j sc, B peak margins 5. Compare “short sample”, “operational” conditions and margins if the same design uses Nb-Ti superconducting technology 6. Define a possible coil lay-out to minimize field errors 7. Determine e.m forces F x and F y and the accumulated stress on the coil mid- plane in the operational conditions (80% of I ss ) 8. Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of I ss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 3

Additional questions Evaluate, compare, discuss, take a stand (… and justify it …) regarding the following issues High temperature superconductor: YBCO vs. Bi2212 Superconducting coil design: block vs. cos Support structures: collar-based vs. shell-based Assembly procedure: high pre-stress vs. low pre-stress Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 4

High field - large aperture magnet for a cable test facility Questions 1. Determine maximum gradient and coil size (using sector coil scaling laws) 2. Define strands and cable parameters 1. Strand diameter and number of strands 2. Cu to SC ratio and pitch angle 3. Cable width, cable mid-thickness and insulation thickness 4. Filling factor κ 3. Determine load-line (no iron) and “short sample” conditions 1. Compute j sc_ss, j o_ss, I ss, B ss, B peak_ss 4. Determine “operational” conditions (80% of I ss ) and margins 1. Compute j sc_op, j o_op, I op, B op, B peak_op 2. Compute T, j sc, B peak margins 5. Compare “short sample”, “operational” conditions and margins if the same design uses Nb-Ti superconducting technology 6. Define a possible coil lay-out to minimize field errors 7. Determine e.m forces F x and F y and the accumulated stress on the coil mid- plane in the operational conditions (80% of I ss ) 8. Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of I ss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 5

Case study 3 solution Maximum field and coil size The max. field that one could reach with 60 mm wide coil is about 16.5 T Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 6

Case study 3 solution Maximum gradient and coil size With a w/r of 60/50 = 1.2  λ of 1.04 Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 7

Case study 3 solution Maximum gradient and coil size Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 8

High field - large aperture magnet for a cable test facility Questions 1. Determine maximum gradient and coil size (using sector coil scaling laws) 2. Define strands and cable parameters 1. Strand diameter and number of strands 2. Cu to SC ratio and pitch angle 3. Cable width, cable mid-thickness and insulation thickness 4. Filling factor κ 3. Determine load-line (no iron) and “short sample” conditions 1. Compute j sc_ss, j o_ss, I ss, B ss, B peak_ss 4. Determine “operational” conditions (80% of I ss ) and margins 1. Compute j sc_op, j o_op, I op, B op, B peak_op 2. Compute T, j sc, B peak margins 5. Compare “short sample”, “operational” conditions and margins if the same design uses Nb-Ti superconducting technology 6. Define a possible coil lay-out to minimize field errors 7. Determine e.m forces F x and F y and the accumulated stress on the coil mid- plane in the operational conditions (80% of I ss ) 8. Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of I ss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 9

Case study 3 solution Cable and strand size We assume a strand diameter of 0.80 mm We assume a pitch angle  of 17  Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 10

Case study 3 solution Cable and strand size Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 11 We assume Thick. Comp. = -12 % Width. Comp. = -1.5 % 35 strands Ins. Thick. = 150 μm We obtain Cable width: 15 mm Cable mid-thick.: 1.4 mm

Case study 3 solution Cable and strand size Summary Strand diameter = 0.80 mm Cu to SC ratio = 1.1 Pitch angle  = 17  N strands = 35 Cable width: 15 mm Cable mid-thickness: 1.4 mm Insulation thickness = 150 μm Area insulated conductor = 26.0 mm 2 We obtain a filling factor k = area superconductor/area insulated cable = 0.31 Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 12

Case study 3 High field - large aperture magnet for a cable test facility Questions 1. Determine maximum gradient and coil size (using sector coil scaling laws) 2. Define strands and cable parameters 1. Strand diameter and number of strands 2. Cu to SC ratio and pitch angle 3. Cable width, cable mid-thickness and insulation thickness 4. Filling factor κ 3. Determine load-line (no iron) and “short sample” conditions 1. Compute j sc_ss, j o_ss, I ss, B ss, B peak_ss 4. Determine “operational” conditions (80% of I ss ) and margins 1. Compute j sc_op, j o_op, I op, B op, B peak_op 2. Compute T, j sc, B peak margins 5. Compare “short sample”, “operational” conditions and margins if the same design uses Nb-Ti superconducting technology 6. Define a possible coil lay-out to minimize field errors 7. Determine e.m forces F x and F y and the accumulated stress on the coil mid- plane in the operational conditions (80% of I ss ) 8. Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of I ss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 13

Case study 3 solution Margins Let’s work now on the load-line The bore field is given by So, for a J sc = 1000 A/mm 2 j o = j sc * k = 465 A/mm 2 B bore = 12.8 T B peak = B bore * λ = 12.8 * 1.04 = 13.3 T Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 14

Case study 3 solution Margins Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 15 Nb 3 Sn parameterization Temperature, field, and strain dependence of Jc is given by Summers’ formula where  Nb3Sn is 900 for  = , T Cmo is 18 K, B Cmo is 24 T, and C Nb3Sn,0 is a fitting parameter equal to AT 1/2 mm -2 for a Jc =3000 A/mm 2 at 4.2 K and 12 T.

Case study 3 solution Margins Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 16 Nb-Ti parameterization Temperature and field dependence of B C2 and T C are provided by Lubell’s formulae: where B C20 is the upper critical flux density at zero temperature (~14.5 T), and T C0 is critical temperature at zero field (~9.2 K) Temperature and field dependence of Jc is given by Bottura’s formula where J C,Ref is critical current density at 4.2 K and 5 T (~3000 A/mm2) and C Nb-Ti (27 T),  Nb-Ti (0.63),  Nb-Ti (1.0), and  Nb-Ti (2.3) are fitting parameters.

Case study 3 solution Margins Nb 3 Sn Let’s assume  = The load-line intercept the critical (“short-sample” conditions) curve at j sc_ss = 1230 mm 2 j o_ss = j sc_ss * k = 381 mm 2 I ss = j o_ss * A ins_cable = 9900 A B bore_ss = 15.8 T B peak_ss = 16.4 T Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 17

Case study 3 solution Margins Nb 3 Sn The operational conditions (80% of I ss ) j sc_op = 984 mm 2 j o_op = j sc_op * k = 305 mm 2 I op = j o_op * A ins_cable = 7930 A B bore_op = 12.7 T B peak_op = 13.2 T Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 18

Case study 3 solution Margins Nb 3 Sn In the operational conditions (80% of I ss ) 4.6 K of T margin ( ) A/mm 2 of j sc margin ( ) T of field margin Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 19

Case study 3 solution Margins Nb-Ti “Short-sample” conditions j sc_ss = 850 mm 2 j o_ss = j sc_ss * k = 264 mm 2 I ss = j o_ss * A ins_cable = 6900 A B bore_ss = 11.0 T B peak_ss = 11.4 T The operational conditions (80% of I ss ) j sc_op = 680 mm 2 j o_op = j sc_op * k = 244 mm 2 I op = j o_op * A ins_cable = 6350 A B bore_op = 8.8 T B peak_op = 9.1 T 2.1 K of T margin ( ) A/mm 2 of j sc margin ( ) T of field margin Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 20

Case study 3 High field - large aperture magnet for a cable test facility Questions 1. Determine maximum gradient and coil size (using sector coil scaling laws) 2. Define strands and cable parameters 1. Strand diameter and number of strands 2. Cu to SC ratio and pitch angle 3. Cable width, cable mid-thickness and insulation thickness 4. Filling factor κ 3. Determine load-line (no iron) and “short sample” conditions 1. Compute j sc_ss, j o_ss, I ss, B ss, B peak_ss 4. Determine “operational” conditions (80% of I ss ) and margins 1. Compute j sc_op, j o_op, I op, B op, B peak_op 2. Compute T, j sc, B peak margins 5. Compare “short sample”, “operational” conditions and margins if the same design uses Nb-Ti superconducting technology 6. Define a possible coil lay-out to minimize field errors 7. Determine e.m forces F x and F y and the accumulated stress on the coil mid- plane in the operational conditions (80% of I ss ) 8. Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of I ss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 21

Case study 3 solution Coil layout One wedge coil sets to zero b 3 and b 5 in quadrupoles ~[0°-48°, 60°-72°] ~[0°-36°, 44°-64°] Some examples Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 22

Case study 3 High field - large aperture magnet for a cable test facility Questions 1. Determine maximum gradient and coil size (using sector coil scaling laws) 2. Define strands and cable parameters 1. Strand diameter and number of strands 2. Cu to SC ratio and pitch angle 3. Cable width, cable mid-thickness and insulation thickness 4. Filling factor κ 3. Determine load-line (no iron) and “short sample” conditions 1. Compute j sc_ss, j o_ss, I ss, B ss, B peak_ss 4. Determine “operational” conditions (80% of I ss ) and margins 1. Compute j sc_op, j o_op, I op, B op, B peak_op 2. Compute T, j sc, B peak margins 5. Compare “short sample”, “operational” conditions and margins if the same design uses Nb-Ti superconducting technology 6. Define a possible coil lay-out to minimize field errors 7. Determine e.m forces F x and F y and the accumulated stress on the coil mid- plane in the operational conditions (80% of I ss ) 8. Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of I ss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 23

Case study 3 solution E.m. forces and stresses For a dipole sector coil, with an inner radius a 1, an outer radius a 2 and an overall current density j o, each block (quadrant) see Horizontal force outwards Vertical force towards the mid-plan In case of frictionless and “free-motion” conditions, no shear, and infinitely rigid radial support, the forces accumulated on the mid- plane produce a stress of Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 24

Case study 3 solution E.m. forces and stresses In the operational conditions ( B bore_op = 12.7 T) F x (quadrant) = MN/m F y (quadrant) = MN/m The accumulates stress on the coil mid-plane is Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 25

Case study 3 High field - large aperture magnet for a cable test facility Questions 1. Determine maximum gradient and coil size (using sector coil scaling laws) 2. Define strands and cable parameters 1. Strand diameter and number of strands 2. Cu to SC ratio and pitch angle 3. Cable width, cable mid-thickness and insulation thickness 4. Filling factor κ 3. Determine load-line (no iron) and “short sample” conditions 1. Compute j sc_ss, j o_ss, I ss, B ss, B peak_ss 4. Determine “operational” conditions (80% of I ss ) and margins 1. Compute j sc_op, j o_op, I op, B op, B peak_op 2. Compute T, j sc, B peak margins 5. Compare “short sample”, “operational” conditions and margins if the same design uses Nb-Ti superconducting technology 6. Define a possible coil lay-out to minimize field errors 7. Determine e.m forces F x and F y and the accumulated stress on the coil mid- plane in the operational conditions (80% of I ss ) 8. Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of I ss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 26

Case study 3 solution Dimension of the yoke The iron yoke thickness can be estimated with Therefore, being B bore = 14.2 T (at 90% of I ss ) r = 50 mm and B sat = 2 T we obtain t iron = ~360 mm Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 27

Case study 3 solution Dimension of the support structure We assume a 25 mm thick collar Images not in scale Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 28

Case study 3 solution Dimension of the support structure Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 29 We assume that the shell will close the yoke halves with the same force as the total horizontal e.m. force at 90% of I ss F x_total = F x_octant * 2 = MN/m Assuming an azimuthal shell stress after cool-down of  shell = 200 MPa The thickness of the shell is t shell = F x_total /2/1000/  shell ~ 36 mm

Case study 3 solution Magnet cross-section Coil inner radius: 50 mm Coil outer radius: 110 mm The operational conditions (80% of I ss ) j sc_op = 984 mm 2 j o_op = j sc_op * k = 305 mm 2 I op = j o_op * A ins_cable = 7930 A B bore_op = 12.7 T B peak_op = 13.2 T Collar thickness: 25 mm Yoke thickness: 330 mm Shell thickness: 36 mm OD: 1 m Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 30

Comparison Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 3 31