Design and Modelling of LTS Superconducting Magnets. Magnetic design

Transcription

1 ASC 212 Short Course Design and Modelling of LTS Superconducting Magnets Magnetic design Paolo Ferracin European Organization for Nuclear Research (CERN)

2 Introduction The magnetic design is one of the first steps in the a superconducting magnet development It starts from the requirements (from accelerator physicists, researchers, medical doctors others) A field shape Dipole, quadrupole, etc A field magnitude Usually with low T superconductors from 5 to 2 T A field homogeneity Uniformity inside a solenoid, harmonics in a accelerator magnet A given aperture (and volume) Some cm diameter for accelerator magnets, much more for detectors and fusion magnets Design and modeling of LTS superconducting magnets, October 7, Magnetic design 2

3 Goal of this chapter How much conductor do we need to meet the requirements? And in which configuration? Outline Different field shape and their function solenoid, dipole, quadrupole How do we create a perfect field? How do we express the field and its imperfections? How do we design a coil to minimize field errors? Which is the maximum field we can get? Overview of different designs Introduction Design and modeling of LTS superconducting magnets, October 7, Magnetic design 3

4 References 1. P. Ferracin, E. Todesco, S. Prestemon, H. Felice, Superconducting accelerator magnets, US Particle Accelerator School, Units 2, 5, 8, and 9 by Ezio Todesco 2. A. Jain, asic theory of magnets, CERN 98-5 (1998) Classes given by A. Jain at USPAS, 4. K.-H. Mess, P. Schmuser, S. Wolff, Superconducting accelerator magnets, Singapore: World Scientific, Chapter 4 5. Martin N. Wilson, "Superconducting Magnets", Chapter 1 6. Fred M. Asner, "High Field Superconducting Magnets", Chapter 9 7. Classes given by A. Devred at USPAS, 8. L. Rossi, E. Todesco, Electromagnetic design of superconducting quadrupoles, Phys. Rev. ST Accel. eams 9 (26) S. Russenschuck, Field computation for accelerator magnets, J. Wiley & Sons (21). Design and modeling of LTS superconducting magnets, October 7, Magnetic design 4

5 Field shape: dipoles Main field components is y Perpendicular to the axis of the magnet z Magnetic field steers (bends) the particles in a circular orbit F ev p er Driving particles in the same accelerating structure several times Particle accelerated energy increased magnetic field increased ( synchro ) to keep the particles on the same orbit of curvature r Design and modeling of LTS superconducting magnets, October 7, Magnetic design 5

6 Field shape: quadrupoles The force necessary to stabilize linear motion is provided by the quadrupoles Quadrupoles provide a field which is proportional to the transverse deviation from the orbit, acting like a spring A series of alternating quadrupole focus the beam in both planes y x Gx Gy Design and modeling of LTS superconducting magnets, October 7, Magnetic design 6

7 Field shape: solenoids Main field components is z Parallel to the axis of the magnet z Applications in accelerators A solenoid providing a field parallel to the beam direction (example: LHC CMS) A series of toroidal coils to provide a circular field around the beam (example: LHC ATLAS) The particle is bent with a curvature radius z Other applications Fusion magnets, NMR, MRI Design and modeling of LTS superconducting magnets, October 7, Magnetic design 7

8 Goal of this chapter How much conductor do we need to meet the requirements? And in which configuration? Outline Different field shape and their function solenoid, dipole, quadrupole How do we create a perfect field? How do we express the field and its imperfections? How do we design a coil to minimize field errors? Which is the maximum field we can get? Overview of different designs Introduction Design and modeling of LTS superconducting magnets, October 7, Magnetic design 8

9 Perfect dipole field Intercepting ellipses or circles A uniform current density in the area of two intersecting ellipses produces a pure dipolar field The aperture is not circular Not easy to simulate with a flat cable The same for intercepting circles Within a cylinder carrying uniform current j, the field is perpendicular to the radial direction and proportional to the distance to the centre r: jr 2 Combining the effect of the two cylinders jr x r1 sin1 r2 sin 2 2 jr j y r1 cos 1 r2 cos s Similar proof for intersecting ellipses M. Wilson, [5] pg. 28 Design and modeling of LTS superconducting magnets, October 7, Magnetic design 9

10 Perfect dipole field Thick shell with cos current distribution If we assume J = J cos where J [A/m 2 ] is to the crosssection plane Inner (outer) radius of the coils = a1 (a2) The generated field is a pure dipole y J ( a2 a 2 1 ) Linear dependence on coil width Easier to reproduce with a flat rectangular cable Design and modeling of LTS superconducting magnets, October 7, Magnetic design 1

11 Perfect quadrupole field Intercepting ellipses or circles Thick shell with cos2 current distribution If we assume J = J cos2 where J [A/m 2 ] is to the crosssection plane Inner (outer) radius of the coils = a1 (a2) G r y J a ln 2 a 2 1 And so on Perfect sextupoles: cos3 or three intersecting ellipses Perfect 2n-poles: cosn or n intersecting ellipses Design and modeling of LTS superconducting magnets, October 7, Magnetic design 11

12 Perfect solenoid field Let s consider the ideal case of a infinitely long thin-walled solenoid, with thickness d, radius a, and current density J θ. The field outside the solenoid is zero. The field inside the solenoid is uniform, directed along the solenoid axis and given by J The field inside the winding decreases linearly from at a to zero at a +. The average field on the conductor element is /2 Twice more efficient than thick shell dipole!!! Design and modeling of LTS superconducting magnets, October 7, Magnetic design 12

13 From ideal to practical configuration How can I reproduce thick shell with a cos distribution with a cable? Rectangular cross-section and constant current density First rough approximation Sector dipole etter ones Single layer with wedges to reduce current towards 9 More layers with different angles As a result, the field is not perfect anymore How can I express in improve the imperfect field inside the aperture? Design and modeling of LTS superconducting magnets, October 7, Magnetic design 13

14 Goal of this chapter How much conductor do we need to meet the requirements? And in which configuration? Outline Different field shape and their function solenoid, dipole, quadrupole How do we create a perfect field? How do we express the field and its imperfections? How do we design a coil to minimize field errors? Which is the maximum field we can get? Overview of different designs Introduction Design and modeling of LTS superconducting magnets, October 7, Magnetic design 14

15 Maxwell equations for magnetic field In absence of charge and magnetized material If (constant longitudinal field), then Field representation Maxwell equations Design and modeling of LTS superconducting magnets, October 7, Magnetic design 15 z y x z y x t E J,, x y z x y z y x x z z y z z y x y x x y y x

16 Field representation Analytic functions If Maxwell gives and therefore the function y +i x is analytic where C n are complex coefficients Advantage: we reduce the description of the field to a (simple) series of complex coefficients Design and modeling of LTS superconducting magnets, October 7, Magnetic design 16 z z y x x y x y x y x f y f y f x f y x y x 1 1 ), ( ), ( n n n x y iy x C y x i y x Cauchy-Riemann conditions ), ( ), ( n n n n n n n x y iy x ia iy x C y x i y x

17 Field representation Harmonics What are these coefficients (or harmonics)? y ( x, y) i x ( x, y) n1 C n n1 n 1 x iy ia x iy n1 n n For n=1 dipole y i x 1 ia 1 1 A 1 For n=2 quadrupole 2 A 2 ia x iy x i y ia x A y y ix [K.-H. Mess, et al, [4] pg. 5] Design and modeling of LTS superconducting magnets, October 7, Magnetic design 17

18 Field representation Harmonics So, each coefficient corresponds to a pure multipolar field y ( x, y) i x ( x, y) n1 C n n1 n 1 x iy ia x iy n1 n n [K.-H. Mess, et al, [4] pg. 5] The field harmonics are rewritten as (EU notation) n ( ) x iy y ix bn ian n1 Rref We factorize the main component ( 1 for dipoles, 2 for quadrupoles) We introduce a reference radius R ref to have dimensionless coefficients We factorize 1-4 since the deviations from ideal field are.1% The coefficients b n, a n are called normalized multipoles b n are the normal, a n are the skew (adimensional) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 18

19 Field representation Harmonics y i x n1 ( b n ia n ) x iy R ref n1 The coefficients b n, a n are called normalized multipoles b n are the normal, a n are the skew (adimensional) Reference radius is usually chosen as 2/3 of the aperture radius y R ref r + + x - - Design and modeling of LTS superconducting magnets, October 7, Magnetic design 19

20 Field representation Harmonics Important property: starting by the multipolar expansion of a current line (iot-savart law) Design and modeling of LTS superconducting magnets, October 7, Magnetic design ) ( n n ref n ref n n R iy x z R z I z z z I z ) ( 2 ) ( z z z I z z I z z =x +iy x y z=x+iy = y +i x ) ( ) ( ) ( z i z z x y ) ( 1 n n ref n n x y R iy x ia b i n ref n n z R z I ia b

21 Field representation Harmonics One can demonstrate that with line currents with a dipole or a quadrupole symmetry, most of the multipoles cancelled For n=1 dipole Only b 3, b 5, b 7,.. are present For n=2 quadrupole Only b 6, b 1, b 14,.. are present K.-H. Mess, et al, [4] and so on These multipoles are called allowed multipoles The field quality optimization of a coil lay-out concerns only a few quantities For a dipole, usually b3, b5, b7, and possibly b9, b11 Design and modeling of LTS superconducting magnets, October 7, Magnetic design 21

22 ack to the original issue: From ideal to practical configuration How can I reproduce thick shell with a cos distribution with a cable? Rectangular cross-section and constant current density First rough approximation Sector dipole etter ones Single layer with wedges to reduce current towards 9 More layers with different angles Now, I can use the multipolar expansion to optimize my practical cross-section Design and modeling of LTS superconducting magnets, October 7, Magnetic design 22

23 Goal of this chapter How much conductor do we need to meet the requirements? And in which configuration? Outline Different field shape and their function solenoid, dipole, quadrupole How do we create a perfect field? How do we express the field and its imperfections? How do we design a coil to minimize field errors? Which is the maximum field we can get? Overview of different designs Introduction Design and modeling of LTS superconducting magnets, October 7, Magnetic design 23

24 A good field quality dipole Sector dipole We compute the central field given by a sector dipole with uniform current density j We start from iot-savart law and integrate w And we obtain I jrdrd a rw 1 2 w j 2 a r cos 2 j rdrd r sina n1 j R 2n 2n ref 2sin( an) ( r w) r n n 2 n Multipoles n are proportional to sin (n angle of the sector) They can be made equal to zero! - - r a + + Design and modeling of LTS superconducting magnets, October 7, Magnetic design 24

25 A good field quality dipole Sector dipole First allowed multipole 3 (sextupole) 3 jr 2 ref 1 sin(3a ) 1 3 r r w - w a + for a=/3 (i.e. a 6 sector coil) one has 3 = - r + Second allowed multipole 5 (decapole) 4 jrref sin(5a ) r r w for a=/5 (i.e. a 36 sector coil) or for a=2/5 (i.e. a 72 sector coil) one has 5 = With one sector one cannot set to zero both multipoles let us try with more sectors! Design and modeling of LTS superconducting magnets, October 7, Magnetic design 25

26 A good field quality dipole Sector dipole Coil with two sectors jr jr 2 ref 1 sin 3a 3 sin 3a 2 sin 3a r 4 ref sin 5a 3 sin 5a sin 5a r r w 1 ( r w) a 3 a 2 a 1 Note: we have to work with non-normalized multipoles, which can be added together Equations to set to zero 3 and 5 sin(3a 3) sin(3a 2) sin(3a 1) sin(5a 3) sin(5a 2) sin(5a 1) There is a one-parameter family of solutions, for instance (48,6,72 ) or (36,44,64 ) are solutions Design and modeling of LTS superconducting magnets, October 7, Magnetic design 26

27 A good field quality dipole Sector dipole We have seen that with one wedge one can set to zero three multipoles ( 3, 5 and 7 ) What about two wedges? sin( 3a ) sin(3a 4 ) sin(3a 3) sin(3a 2 ) sin(3a 1) 5 sin( 5a ) sin(5a 4 ) sin(5a 3) sin(5a 2 ) sin(5a 1) 5 sin( 7a ) sin(7a 4 ) sin(7a 3) sin(7a 2 ) sin(7a 1) 5 One wedge, b 3 =b 5 =b 7 = [-43.2, ] sin( 9a ) sin(9a 4 ) sin(9a 3) sin(9a 2 ) sin(9a 1) 5 sin( 11a ) sin(11a 4 ) sin(11a 3) sin(11a 2 ) sin(11a 1) 5 One can set to zero five multipoles ( 3, 5, 7, 9 and 11 ) ~[ -33.3, , ] Two wedges, b 3 =b 5 =b 7 =b 9 =b 11 = [-33.3, , ] Design and modeling of LTS superconducting magnets, October 7, Magnetic design 27

28 A good field quality dipole Sector dipole y (mm) Let us see two coil lay-outs of real magnets The RHIC dipole has four blocks x (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 28

29 A good field quality dipole Sector dipole y (mm) Limits due to the cable geometry Finite thickness one cannot produce sectors of any width Cables cannot be key-stoned beyond a certain angle, some wedges can be used to better follow the arch One does not always aim at having zero multipoles There are other contributions (iron, persistent currents ) x (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 29

30 A good field quality dipole Sector quadrupole Let s look at the quadrupoles First allowed multipole 6 (dodecapole) 6 5 jrref sin(6a ) r r w - - for a=/6 (i.e. a 3 sector coil) one has 6 = Second allowed multipole r w + jr sin(1a ) r 1 r w 8 ref 1 8 for a=/1 (i.e. a 18 sector coil) or for a=/5 (i.e. a 36 sector coil) one has 5 = The conditions look similar to the dipole case Design and modeling of LTS superconducting magnets, October 7, Magnetic design 3

31 y (mm) A good field quality dipole Sector quadrupole For a sector coil with one layer, the same results of the dipole case hold with the following transformation Angles have to be divided by two Multipole orders have to be multiplied by two Examples One wedge coil: ~[ -48, 6-72 ] sets to zero b 3 and b 5 in dipoles One wedge coil: ~[ -24, 3-36 ] sets to zero b 6 and b 1 in quadrupoles The LHC main quadrupole is based on this lay-out: one wedge between 24 and 3, plus one on the outer layer for keystoning the coil x (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 31

32 y (mm) A good field quality dipole Sector quadrupole y (mm) Examples One wedge coil: ~[ -12, 18-3 ] sets to zero b 6 and b 1 in quadrupoles The Tevatron main quadrupole is based on this lay-out: a 3 sector coil with one wedge between 12 and 18 (inner layer) to zero b 6 and b 1 the outer layer can be at 3 without wedges since it does not affect much b 1 One wedge coil: ~[ -18, ] sets to zero b 6 and b 1 The RHIC main quadrupole is based on this lay-out its is the solution with the smallest angular width of the wedge x (mm) x (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 32

33 Goal of this chapter How much conductor do we need to meet the requirements? And in which configuration? Outline Different field shape and their function solenoid, dipole, quadrupole How do we create a perfect field? How do we express the field and its imperfections? How do we design a coil to minimize field errors? Which is the maximum field we can get? Overview of different designs Introduction Design and modeling of LTS superconducting magnets, October 7, Magnetic design 33

34 Maximum field and coil thickness Dipoles jsc(a/mm 2 ) We recall the equations for the critical surface Nb-Ti: linear approximation is good with s~6.1 8 [A/(T m 2 )] and * c2~1 T at 4.2 K or 13 T at 1.9 K This is a typical mature and very good Nb-Ti strand Tevatron had half of it! j sc *, c( ) s( c 2 ), Nb-Ti at 1.9 K Nb-Ti at 4.2 K (T) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 34

35 Maximum field and coil thickness Dipoles The current density in the coil is lower Strand made of superconductor and normal conducting (copper) There are voids and insulation The current density flowing in the insulated cable is reduced by a factor (filling ratio) It ranges from ¼ to 1/3 The critical surface for j (engineering current density) j j c ( ) j, ( ) c sc c * ( ) s( 2 ) c Design and modeling of LTS superconducting magnets, October 7, Magnetic design 35

36 Maximum field and coil thickness Dipoles We characterize the coil by two parameters c j c : how much field in the centre is given per unit of current density p c j Cos dipole 1 j 2 w Sector dipole 2 1 w j sina : ratio between peak field and central field for a cos dipole, =1 For a sector and in general is = Design and modeling of LTS superconducting magnets, October 7, Magnetic design 36

37 Maximum field and coil thickness Dipoles Current density j (A/mm 2 ) We can now compute what is the highest peak field that can be reached in the dipole in the case of a linear critical surface j c * ( ) s( 2 ) * p, ss c jc cs( c2 p, ss p, ss c cs 1 s c * c2 ) j ss ss p = c j j=s( * c2-) [ p,ss,j ss ] Magnetic field (T) We can now compute the maximum current density that can be tolerated by the superconductor (short sample limit) c j and the bore short sample field (in the centre not on the conductor) p c j s 1 c s c * c2 cs 1 s * c2 Design and modeling of LTS superconducting magnets, October 7, Magnetic design 37

38 y (mm) Maximum field and coil thickness Dipoles y (mm) y (mm) We got an equation giving the field reachable for a dipole with a superconductor having a linear critical surface ss cs 1 s The plan: use c from sector coils and try to find an estimate for which characterize the lay-out What is (ratio between peak field and bore field)? To compute the peak field one has to compute the field everywhere in the coil, and take the maximum c * c x (mm) x (mm) x (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 38

39 Maximum field and coil thickness Dipoles Numerical evaluation of for different sector coils (adim) layer 6 1 layer layer layer three blocks Fit width (mm) For interesting widths (1 to 3 mm) is This simply means that peak field 5-15% lager The cos approx having =1 is not so bad for w>2 mm Typical hyperbolic fit ar ( w, r) ~ 1 with a~.45 w Design and modeling of LTS superconducting magnets, October 7, Magnetic design 39

40 Maximum field and coil thickness Dipoles We now can write the short sample field for a sector coil as a function of s Material parameters c, * c2 Cable parameters Aperture r and coil width w a=.45 c = [Tm/A] for Nb-Ti s~6.1 8 [A/(T m 2 )] and * c2~1 T at 4.2 K or 13 T at 1.9 K ss c 1 s c * c2 ( w, r) ~ c c w ~ 1 ar w ss c * ~ c2 1 1 ws ar w c ws Design and modeling of LTS superconducting magnets, October 7, Magnetic design 4

41 Maximum field and coil thickness Dipoles Nb 3 Sn at 4.2 K = ss (T) 1 5 Nb-Ti at 4.2 K =.35 r=3 mm r=6 mm r=12 mm Cos theta equivalent width (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 41

42 Maximum gradient and coil thickness Quadrupoles The same approach can be used for a quadrupole We define c G j the only difference is that now c gives the gradient per unit of current density, and in p we multiply by r for having T and not T/m We compute the quantities at the short sample limit for a material with a linear critical surface (as Nb-Ti) r cs * s c * p, ss j c2 ss c2 ss c2 1 r cs 1 r cs 1 r cs Please note that is not any more proportional to w and not any more independent of r! p rg G c c ln1 w r s * Design and modeling of LTS superconducting magnets, October 7, Magnetic design 42

43 y (mm) y (mm) Maximum gradient and coil thickness Quadrupoles The ratio is defined as ratio between peak field and gradient times aperture (central field is zero ) Numerically, one finds that for large coils Peak field is going outside for large widths x (mm) (adim) [,3] [-24,3-36] [-18,22-32] x (mm) w/r (adim) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 43

44 Maximum gradient and coil thickness Quadrupoles The ratio is defined as ratio between peak field and gradient times aperture (central field is zero ) The ratio depends on w/r A good fit is r ( w, r) a 1 1 a1 w a -1 ~.4 and a 1 ~.11 for the [ -24,3-36 ] coil A reasonable approximation is ~ =1.15 for ¼<w/r<1 w r [adim] ISR MQ TEV MQ HERA MQ SSC MQ LEP I MQC LEP II MQC RHIC MQ RHIC MQY LHC MQ LHC MQM LHC MQY LHC MQX LHC MQXA current grading aspect ratio w eq /r (adim) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 44

45 Maximum gradient and coil thickness Quadrupoles We now can write the short sample gradient for a sector coil as a function of Material parameters s, * c2 (linear case as Nb-Ti) Cable parameters Aperture r and coil width w Relevant feature: for very large coil widths w the short sample gradient tends to zero! Design and modeling of LTS superconducting magnets, October 7, Magnetic design 45 * 2 1 c c c ss s r s G r w a w r a r w ~ ), ( r w r w c c 1 ln ), ( * * 2 1 ln ln 1 c c c c c c ss s r w r r w a w r a s r w s r s G

46 G ss (T/m) Maximum gradient and coil thickness Quadrupoles Evaluation of short sample gradient in several sector lay-outs for a given aperture (r=3 mm) 4 3 G * = * c2 /r -3% 2 1 [,3] [-24,3-36] [-18,22-32] sector width w (mm) No point in making coils larger than 3 mm! Max gradient is 3 T/m and not 13/.3=433 T/m!! We lose 3%!! Design and modeling of LTS superconducting magnets, October 7, Magnetic design 46

47 Gradient [T /m] Maximum gradient and coil thickness Quadrupoles G ss as a function of aperture (maximum achievable) 6 5 Nb 3 Sn at 1.9 K = Nb-Ti at 1.9 K Aperture radius r [mm] Gain of Nb 3 Sn is ~5% in gradient for the same aperture (at 35 mm) Gain of Nb 3 Sn is ~7% in aperture for the same gradient (at 2 T/m) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 47

48 Gradient [T /m] Maximum gradient and coil thickness Quadrupoles G ss as a function of aperture (maximum achievable and with different coil thicknesses) 3 Nb 3 Sn at 1.9 K w/r=.25 w/r=.5 w/r=1 w/r= Aperture [mm] Design and modeling of LTS superconducting magnets, October 7, Magnetic design 48

49 Operational margin Magnets have to work at a given distance from the critical surface, i.e. they are never operated at short sample conditions At short sample, any small perturbation quenches the magnet One usually operates at a fraction of the loadline which ranges from 6% to 9% Loadline with 2% operational margin Operational margin and temperature margin This fraction translates into a temperature margin Design and modeling of LTS superconducting magnets, October 7, Magnetic design 49

50 Operational gradient [T /m] Gradient [T /m] Operational margin 6 5 Nb 3 Sn at 1.9 K = Nb-Ti at 1.9 K Aperture radius r [mm] 6 8% of Nb 3 Sn at 1.9 K Aperture [mm] 8% of Nb-Ti at 1.9 K Design and modeling of LTS superconducting magnets, October 7, Magnetic design 5

51 Iron yoke An iron yoke usually surrounds the collared coil it has several functions Keep the return magnetic flux close to the coils, thus avoiding fringe fields In some cases the iron is partially or totally contributing to the mechanical structure RHIC magnets: no collars, plastic spacers, iron holds the Lorentz forces LHC dipole: very thick collars, iron give little contribution Considerably enhance the field for a given current density The increase is relevant (1-3%), getting higher for thin coils This allows using lower currents, easing the protection Increase the short sample field The increase is small (a few percent) for large coils, but can be considerable for small widths This action is effective when we are far from reaching the asymptotic limit of * c2 Design and modeling of LTS superconducting magnets, October 7, Magnetic design 51

52 Iron yoke A rough estimate of the iron thickness necessary to avoid fields outside the magnet The iron cannot withstand more than 2 T Shielding condition for dipoles: r ~ t iron sat i.e., the iron thickness times 2 T is equal to the central field times the magnet aperture One assumes that all the field lines in the aperture go through the iron (and not for instance through the collars) Example: in the LHC main dipole the iron thickness is 15 mm t iron r ~ Shielding condition for quadrupoles: sat 28*8.3 2 ~ 1 mm r 2 2 G ~ t iron sat Design and modeling of LTS superconducting magnets, October 7, Magnetic design 52

53 Iron yoke The iron yoke contribution can be estimated analytically for simple geometries Circular, non-saturated iron: image currents method Iron effect is equivalent to add to each current line a second one at a distance with current 2 R I r' r 1 I' I 1 Limit of the approximation: iron is not saturated (less than 2 T) R I r r ' Design and modeling of LTS superconducting magnets, October 7, Magnetic design 53

54 y (mm) Gain in ss [%] y (mm) Iron yoke Impact of the iron yoke on short sample field Large effect (25%) on RHIC dipoles (thin coil and collars) etween 4% and 1% for most of the others (both Nb-Ti and Nb 3 Sn) TEV M SSC M LHC M MSUT HERA M RHIC M Fresca D x (mm) D2 and yoke 15 HFDA NED equivalent width w (mm) x (mm) RHIC main dipole and yoke Design and modeling of LTS superconducting magnets, October 7, Magnetic design 54

55 y (mm) y (mm) Iron yoke Similar approach can be used in quadrupoles Large effect on RHIC quadrupoles (thin coil and collars) etween 2% and 5% for most of the others The effect is smaller than in dipoles since the contribution to 2 is smaller than to 1 4 Gss increase due to iron [%] ISR MQ HERA MQ RHIC MQ LHC MQY LHC MQM LHC MQX TEV MQ SSC MQ RHIC MQY LHC MQ LHC MQXA w eq /r (adim) x (mm) x (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 55

56 Goal of this chapter How much conductor do we need to meet the requirements? And in which configuration? Outline Different field shape and their function solenoid, dipole, quadrupole How do we create a perfect field? How do we express the field and its imperfections? How do we design a coil to minimize field errors? Which is the maximum field we can get? Overview of different designs Conclusions Design and modeling of LTS superconducting magnets, October 7, Magnetic design 56

57 Appendix I: A review of dipole lay-outs Design and modeling of LTS superconducting magnets, October 7, Magnetic design 57

58 (T) y (mm) Appendix I: A review of dipole lay-outs RHIC M 1 8 Main dipole of the RHIC 296 magnets built in 4/94 1/96 Nb-Ti, 4.2 K w eq ~9 mm ~.23 1 layer, 4 blocks no grading * c x (mm) sector [-48,6-72] No iron With iron w (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 58

59 (T) y (mm) Appendix I: A review of dipole lay-outs Tevatron M 1 8 Main dipole of the Tevatron 774 magnets built in 198 Nb-Ti, 4.2 K w eq ~14 mm ~.23 2 layer, 2 blocks no grading * c x (mm) sector [-48,6-72] No iron With iron w (mm) Unit Design 11: and Electromagnetic modeling of LTS design superconducting episode III magnets, October 7, Magnetic design

60 (T) y (mm) Appendix I: A review of dipole lay-outs HERA M 1 8 Main dipole of the HERA 416 magnets built in 1985/87 Nb-Ti, 4.2 K w eq ~19 mm ~.26 2 layer, 4 blocks no grading * c x (mm) sector [-48,6-72] No iron With iron w (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 6

61 (T) y (mm) Appendix I: A review of dipole lay-outs SSC M 1 8 Main dipole of the ill-fated SSC 18 prototypes built in Nb-Ti, 4.2 K w eq ~22 mm ~.3 4 layer, 6 blocks 3% grading * c x (mm) sector [-48,6-72] No iron With iron w (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 61

62 (T) y (mm) Appendix I: A review of dipole lay-outs HFDA dipole 1 8 Nb 3 Sn model built at FNAL 6 models built in 2-25 Nb 3 Sn, 4.2 K j c ~2 to 25 A/mm 2 at 12 T, 4.2 K (different strands) w eq ~23 mm 2 layers, 6 blocks no grading ~ * c x (mm) 1 5 sector [-48,6-72] No iron With iron w (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 62

63 (T) y (mm) Appendix I: A review of dipole lay-outs LHC M 1 8 Main dipole of the LHC 1276 magnets built in 21-6 Nb-Ti, 1.9 K w eq ~27 mm ~.29 2 layers, 6 blocks 23% grading * c x (mm) sector [-48,6-72] No iron With iron w (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 63

64 (T) y (mm) Appendix I: A review of dipole lay-outs FRESCA 1 8 Dipole for cable test station at CERN 1 magnet built in 21 Nb-Ti, 1.9 K w eq ~3 mm ~.29 2 layers, 7 blocks 24% grading * c x (mm) sector [-48,6-72] No iron With iron w (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 64

65 (T) y (mm) 6. A REVIEW OF DIPOLE LAY-OUTS MSUT dipole 1 8 Nb 3 Sn model built at Twente U. 1 model built in 1995 Nb 3 Sn, 4.2 K j c ~11 A/mm 2 at 12 T, 4.2 K w eq ~35 mm ~.33 2 layers, 5 blocks 65% grading * c x (mm) 1 5 sector [-48,6-72] No iron With iron w (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 65

66 (T) y (mm) 6. A REVIEW OF DIPOLE LAY-OUTS D2 dipole 1 8 Nb 3 Sn model built at LNL (USA) 1 model built in??? Nb 3 Sn, 4.2 K j c ~11 A/mm 2 at 12 T, 4.2 K w eq ~45 mm ~.48 4 layers, 13 blocks 65% grading * c x (mm) 1 5 sector [-48,6-72] No iron With iron w (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 66

67 (T) y (mm) 6. A REVIEW OF DIPOLE LAY-OUTS HD2 1 8 Nb 3 Sn model being built in LNL 1 model to be built in 28 Nb 3 Sn, 4.2 K j c ~25 A/mm 2 at 12 T, 4.2 K w eq ~46 mm ~.35 2 layers, racetrack, no grading * c x (mm) 1 5 sector [-48,6-72] No iron With iron w (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 67

68 Appendix II: A review of quadrupole lay-outs Design and modeling of LTS superconducting magnets, October 7, Magnetic design 68

69 y (mm) RHIC MQX Appendix II: A review of quadrupole lay-outs Quadrupole in the IR regions of the RHIC 79 magnets built in July 1993/ December 1997 Nb-Ti, 4.2 K w/r~.18 ~.27 1 layer, 3 blocks, no grading * c2/r x (mm) Gss (T/m) 1 5 sector [-24,3-36] No iron With iron w eq /r (adim) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 69

70 y (mm) RHIC MQ Appendix II: A review of quadrupole lay-outs Main quadrupole of the RHIC 38 magnets built in June 1994 October 1995 Nb-Ti, 4.2 K w/r~.25 ~.23 1 layer, 2 blocks, no grading x (mm) Gss (T/m) * c2/r sector [-24,3-36] No iron With iron w eq /r (adim) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 7

71 y (mm) LEP II MQC Appendix II: A review of quadrupole lay-outs Interaction region quadrupole of the LEP II 8 magnets built in Nb-Ti, 4.2 K, no iron w/r~.27 ~.31 1 layers, 2 blocks, no grading Gss (T/m) * c2/r sector [-24,3-36] x (mm) 2 No iron w eq /r (adim) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 71

72 y (mm) LEP I MQC Appendix II: A review of quadrupole lay-outs Interaction region quadrupole of the LEP I 8 magnets built in ~ Nb-Ti, 4.2 K, no iron w/r~.29 ~.33 1 layers, 2 blocks, no grading x (mm) Gss (T/m) * c2/r sector [-24,3-36] No iron w eq /r (adim) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 72

73 y (mm) Tevatron MQ Appendix II: A review of quadrupole lay-outs Main quadrupole of the Tevatron 216 magnets built in ~198 Nb-Ti, 4.2 K w/r~.35 ~.25 2 layers, 3 blocks, no grading 6 25 * c2/r x (mm) Gss (T/m) sector [-24,3-36] No iron With iron w eq /r (adim) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 73

74 y (mm) HERA MQ Appendix II: A review of quadrupole lay-outs Main quadrupole of the HERA Nb-Ti, 1.9 K w/r~.52 ~.27 2 layers, 3 blocks, grading 1% 6 3 * c2/r x (mm) Gss (T/m) 2 1 sector [-24,3-36] No iron With iron w eq /r (adim) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 74

75 y (mm) LHC MQM Appendix II: A review of quadrupole lay-outs Low- gradient quadrupole in the IR regions of the LHC 98 magnets built in Nb-Ti, 1.9 K (and 4.2 K) w/r~.61 ~.26 2 layers, 4 blocks, no grading 6 5 * c2/r x (mm) Gss (T/m) sector [-24,3-36] No iron With iron w eq /r (adim) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 75

76 y (mm) LHC MQY Appendix II: A review of quadrupole lay-outs Large aperture quadrupole in the IR regions of the LHC 3 magnets built in Nb-Ti, 4.2 K w/r~.79 ~.34 4 layers, 5 blocks, special grading 43% 6 3 * c2/r x (mm) Gss (T/m) 2 1 sector [-24,3-36] No iron With iron w eq /r (adim) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 76

77 y (mm) LHC MQX Appendix II: A review of quadrupole lay-outs Large aperture quadrupole in the LHC IR 8 magnets built in Nb-Ti, 1.9 K w/r~.89 ~.33 2 layers, 4 blocks, grading 24% 6 4 * c2/r x (mm) Gss (T/m) 2 1 sector [-24,3-36] No iron With iron w eq /r (adim) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 77

78 y (mm) SSC MQ Appendix II: A review of quadrupole lay-outs Main quadrupole of the ill-fated SSC Nb-Ti, 1.9 K w/r~.92 ~.27 2 layers, 4 blocks, no grading 6 5 * c2/r x (mm) Gss (T/m) sector [-24,3-36] No iron With iron w eq /r (adim) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 78

79 y (mm) LHC MQ Appendix II: A review of quadrupole lay-outs Main quadrupole of the LHC 4 magnets built in Nb-Ti, 1.9 K w/r~1. ~.25 2 layers, 4 blocks, no grading 6 5 * c2/r x (mm) Gss (T/m) sector [-24,3-36] No iron With iron w eq /r (adim) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 79

80 y (mm) LHC MQXA Appendix II: A review of quadrupole lay-outs Large aperture quadrupole in the LHC IR 18 magnets built in Nb-Ti, 1.9 K w/r~1.8 ~.34 4 layers, 6 blocks, special grading 1% 6 4 * c2/r x (mm) Gss (T/m) 2 1 sector [-24,3-36] No iron With iron w eq /r (adim) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 8

81 y (mm) Appendix II: A review of quadrupole lay-outs LHC MQXC Nb-Ti option for the LHC upgrade LHC dipole cable, graded coil 1-m-long model built in to be tested in 212 w/r~.5 ~.33 2 layers, 4 blocks x (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 81

82 y (mm) Appendix II: A review of quadrupole lay-outs LARP HQ 12 mm aperture Nb 3 Sn option for the LHC upgrade (IR triplet) 1-m-long model tested in 211, more to come plus a 3.4-m-long w/r~.5 ~.33 2 layers, 4 blocks x (mm) Design and modeling of LTS superconducting magnets, October 7, Magnetic design 82

83 Appendix III: Numerical tools Design and modeling of LTS superconducting magnets, October 7, Magnetic design 83

84 Appendix III: Numerical tools Main critical aspects for computational tools Electromagnetic: 2D, 3D Coil ends Persistent currents Iron saturation Mechanical: 2D, 3D Coupling with magnetic model to estimate impact of Loretz forces Interfaces between different components Quench analysis, stability and protection: Thermal properties of materials Hypothesis on cooling Multiphysics codes: Can couple magnetic-mechanic-thermal Design and modeling of LTS superconducting magnets, October 7, Magnetic design 84

85 Appendix III: Numerical tools Computational tools are needed for the design: Electromagnetic: Roxie, Poisson, Opera, (Ansys), Cast3m Mechanical: Ansys, Cast3m, Abaqus, Quench analysis, stability and protection: Roxie, Multiphysics codes: Comsol, Stress due to electromagnetic forces in Tevatron dipole, through ANSYS Design and modeling of LTS superconducting magnets, October 7, Magnetic design 85

86 Appendix III: Numerical tools The ROXIE code Project led by S. Russenschuck, developed at CERN, specific on superconducting accelerator magnet design Easy input file with coil geometry, iron geometry, and coil ends Several routines for optimization Evaluation of field quality, including iron saturation (EM-FEM methods) and persistent currents Evaluation of quench Design and modeling of LTS superconducting magnets, October 7, Magnetic design 86

87 The ROXIE code Appendix III: Numerical tools Quench propagation in an LHC main dipole Splices in the LHC main dipole Interconnection between LHC main dipoles Design and modeling of LTS superconducting magnets, October 7, Magnetic design 87