Magnetic Multipoles, Magnet Design

Transcription

1 Magnetic Multipoles, Magnet Design Alex Bogacz, Geoff Krafft and Timofey Zolkin Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

2 Maxwell s Equations for Magnets - Outline Solutions to Maxwell s equations for magnetostatic fields: in two dimensions (multipole fields) in three dimensions (fringe fields, end effects, insertion devices...) How to construct multipole fields in two dimensions, using electric currents and magnetic materials, considering idealized situations. A. Wolski, University of Liverpool and the Cockcroft Institute, CAS Specialised Course on Magnets, 2009, Lecture 5 Magnetic Multipoles USPAS, Hampton, VA, Jan ,

3 Basis Vector calculus in Cartesian and polar coordinate systems; Stokes and Gauss theorems Maxwell s equations and their physical significance Types of magnets commonly used in accelerators. following notation used in: A. Chao and M. Tigner, Handbook of Accelerator Physics and Engineering, World Scientific (1999). Lecture 5 Magnetic Multipoles USPAS, Hampton, VA, Jan ,

4 Maxwell s equations 4

5 Maxwell s equations 5

6 Physical interpretation of 6

7 Physical interpretation of 7

8 Linearity and superposition 8

9 Multipole fields 9

10 Multipole fields 10

11 Multipole fields 11

12 Multipole fields 12

13 Multipole fields 13

14 Multipole fields 14

15 Generating multipole fields from a current distribution 15

16 Multipole fields from a current distribution 16

17 Multipole fields from a current distribution 17

18 Multipole fields from a current distribution 18

19 Multipole fields from a current distribution 19

20 Multipole fields from a current distribution 20

21 Multipole fields from a current distribution 21

22 Multipole fields from a current distribution 22

23 Multipole fields from a current distribution 23

24 Multipole fields from a current distribution 24

25 Multipole fields from a current distribution 25

26 Superconducting quadrupole - collider final focus 26

27 Multipole fields in an iron-core magnet 27

28 Multipole fields in an iron-core magnet 28

29 Multipole fields in an iron-core magnet 29

30 Multipole fields in an iron-core magnet 30

31 Multipole fields in an iron-core magnet 31

32 Multipole fields in an iron-core magnet 32

33 Multipole fields in an iron-core magnet 33

34 Multipole fields in an iron-core magnet 34

35 Multipole fields in an iron-core magnet 35

36 Multipole fields in an iron-core magnet 36

37 Multipole fields in an iron-core magnet 37

38 Generating multipole fields in an iron-core magnet 38

39 Generating multipole fields in an iron-core magnet 39

40 Generating multipole fields in an iron-core magnet 40

41 Generating multipole fields in an iron-core magnet 41

42 Maxwell s Equations for Magnets - Summary 42

43 Multipoles in Magnets - Outline Deduce that the symmetry of a magnet imposes constraints on the possible multipole field components, even if we relax the constraints on the material properties and other geometrical properties; Consider different techniques for deriving the multipole field components from measurements of the fields within a magnet; Discuss the solutions to Maxwell s equations that may be used for describing fields in three dimensions. Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

44 Previous lecture re-cap Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

45 Previous lecture re-cap Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

46 Allowed and forbidden harmonics Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

47 Allowed and forbidden harmonics Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

48 Allowed and forbidden harmonics Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

49 Allowed and forbidden harmonics Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

50 Allowed and forbidden harmonics 50

51 Allowed and forbidden harmonics 51

52 Measuring multipoles Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

53 Measuring multipoles in Cartesian basis Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

54 Measuring multipoles in Cartesian basis 54

55 Measuring multipoles in Polar basis 55

56 Measuring multipoles in Polar basis 56

57 Measuring multipoles in Polar basis 57

58 Advantages of mode decompositions 58

59 Three-dimensional fields 59

60 Three-dimensional fields 60

61 Three-dimensional fields 61

62 Three-dimensional fields 62

63 Three-dimensional fields 63

64 Summary Part II Symmetries in multipole magnets restrict the multipole components that can be present in the field. It is useful to be able to find the multipole components in a given field from numerical field data: but this must be done carefully, if the results are to be accurate. Usually, it is advisable to calculate multipole components using field data on a surface enclosing the region of interest: any errors or residuals will decrease exponentially within that region, away from the boundary. Outside the boundary, residuals will increase exponentially. Techniques for finding multipole components in two dimensional fields can be generalized to three dimensions, allowing analysis of fringe fields and insertion devices. In two or three dimensions, it is possible to use a Cartesian basis for the field modes; but a polar basis is sometimes more convenient. Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

65 Appendix A - Field Error Tolerances Focusing point error perturbs the betatron motion leading to the Courant-Snyder invariant change: Beam envelope and beta-function oscillate at double the betatron frequency Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

66 Appendix A - Field Error Tolerances Single point mismatch as measured by the Courant-Snyder invariant change: ε = βθ ( + δθ) + 2 αθ ( + δθ) x + γx ( x) 2 2 = ε + βθ + α δθ + βδθ 2 2, ε x = εβ sin μ, θ = sin μ( cos μ -αsin μ) β Each source of field error (magnet) contributes the following Courant-Snyder variation δε = εβ μ δθ + βδθ 2 2 cos, grad B dl m δθ = = δφmx, where δφm = δ kmdl B ρ m= 1 here, m =1 quadrupole, m =2 sextupole, m=3 octupole, etc m m ( ) m ( ) m m δε = 2 εβ εβ δφ cosμ sin μ + β εβ δφm sin μ, m= 1 m=

67 Appendix A - Field Error Tolerances multipole expansion coefficients of the azimuthal magnetic field, B θ - Fourier series representation in polar coordinates at a given point along the trajectory): m-1 r B r B m A m (, ) = ( cos + sin ) θ θ m θ m θ m= 2 r0 multipole gradient and integrated geometric gradient: G m 1 m = B m m 1 kgauss cm Gn ( n+ 1) + r kn = cm Bρ 0 n Gdl n φ = n ρ cm B Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

68 Appendix A - Field Error Tolerances Cumulative mismatch along the lattice (N sources): N 2 m 1 m 1 m 2 m εn = ε 1+ 2β ( εβ ) δφm cosμ sin μ + β ( εβ ) δφm sin μ, n= 1 m= 1 m= 1 Standard deviation of the Courant-Snyder invariant is given by: δε N ( ) m m cos sin ( ) m m βi εβi δφm μ μ βi εβi δφm sin μ ε i= 1 m= 1 m= 1 σ δε ε = = + ε Assuming weakly focusing lattice (uniform beta modulation) the following averaging (over the betatron phase) can by applied: 2 π 1... = dμ... 2π 0 Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

69 Appendix A - Field Error Tolerances Some useful integrals. : m cos μ sin μ = 0, m 1 m ( m ) m m 2 sin μ = sin μ = 1!! 0 m odd m!! m even will reduce the coherent contribution to the C-S variance as follows: N ( ) m m cos sin ( ) m m = βi εβi δφm μ μ + βi εβi δφm sin μ ε i= 1 m= 1 m= 1 σ ε 2 σ ε ε i= 1 Including the first five multipoles yields: N { β ( ) ( ) ( ) } i δφ sin μ εβ 2 sin 2 2 sin... 1 i δφ δφ δφ μ εβ i δφ δφ δφ δφ δφ μ = Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

70 Appendix A - Field Error Tolerances Beam radius at a given magnet is : = 1 2 εβ a i i One can define a good fileld radius for a given type of magnet as: a = i Max( a ) Assuming the same multipole content for all magnets in the class one gets: N σ ε ε i= = βi δφ1 + a ( δφ2 + 2δφδφ 1 3) + a ( δφ3 + 2δφδφ δφ2δφ4) The first factor purely depends on the beamline optics (focusing), while the second one describes field tolerance (nonlinearities) of the magnets: ΔΦ = δφ1 + 3 a ( δφ2 + 2δφ1δφ3) + 5 a ( δφ3 + 2δφ1δφ5 + 2 δφ2δφ4) Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21,

71 Appendix B - The vector potential A scalar potential description of the magnetic field has been very useful to derive the shape for the pole face of a multipole magnet. 71

72 Appendix B - The vector potential 72

73 Appendix B - The vector potential 73

74 Appendix B - The vector potential 74

75 Appendix B - The vector potential 75

76 Appendix B - The vector potential 76

77 Appendix B - The vector potential 77

78 Appendix B - The vector potential 78

79 Appendix B - The vector potential 79

80 Appendix B - The vector potential 80

81 Appendix B - The vector potential 81