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Magnetic Multipoles, Magnet Design

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1 Magnetic Multipoles, Magnet Design
S.A. Bogacz, G.A. Krafft, S. DeSilva and R. Gamage Jefferson Lab and Old Dominion University USPAS, Fort Collins, CO, June 13-24, 2016

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, /Lectures/PDFs/Wolski-1.pdf USPAS, Hampton, VA, Jan , 2011

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). USPAS, Hampton, VA, Jan , 2011

4 Maxwell’s equations USPAS, Fort Collins, CO, June 13-24, 2016

5 Maxwell’s equations USPAS, Fort Collins, CO, June 13-24, 2016

6 Physical interpretation of
USPAS, Fort Collins, CO, June 13-24, 2016

7 Physical interpretation of
USPAS, Fort Collins, CO, June 13-24, 2016

8 Linearity and superposition
USPAS, Fort Collins, CO, June 13-24, 2016

9 Multipole fields USPAS, Fort Collins, CO, June 13-24, 2016

10 Multipole fields USPAS, Fort Collins, CO, June 13-24, 2016

11 Multipole fields USPAS, Fort Collins, CO, June 13-24, 2016

12 Multipole fields USPAS, Fort Collins, CO, June 13-24, 2016

13 Multipole fields USPAS, Fort Collins, CO, June 13-24, 2016

14 Multipole fields USPAS, Fort Collins, CO, June 13-24, 2016

15 Generating multipole fields from a current distribution
USPAS, Fort Collins, CO, June 13-24, 2016

16 Multipole fields from a current distribution
USPAS, Fort Collins, CO, June 13-24, 2016

17 Multipole fields from a current distribution
USPAS, Fort Collins, CO, June 13-24, 2016

18 Multipole fields from a current distribution
USPAS, Fort Collins, CO, June 13-24, 2016

19 Multipole fields from a current distribution
USPAS, Fort Collins, CO, June 13-24, 2016

20 Multipole fields from a current distribution
USPAS, Fort Collins, CO, June 13-24, 2016

21 Multipole fields from a current distribution
USPAS, Fort Collins, CO, June 13-24, 2016

22 Multipole fields from a current distribution
USPAS, Fort Collins, CO, June 13-24, 2016

23 Multipole fields from a current distribution
USPAS, Fort Collins, CO, June 13-24, 2016

24 Multipole fields from a current distribution
USPAS, Fort Collins, CO, June 13-24, 2016

25 Multipole fields from a current distribution
USPAS, Fort Collins, CO, June 13-24, 2016

26 Superconducting quadrupole - collider final focus
USPAS, Fort Collins, CO, June 13-24, 2016

27 Multipole fields in an iron-core magnet
USPAS, Fort Collins, CO, June 13-24, 2016

28 Multipole fields in an iron-core magnet
USPAS, Fort Collins, CO, June 13-24, 2016

29 Multipole fields in an iron-core magnet
USPAS, Fort Collins, CO, June 13-24, 2016

30 Multipole fields in an iron-core magnet
USPAS, Fort Collins, CO, June 13-24, 2016

31 Multipole fields in an iron-core magnet
USPAS, Fort Collins, CO, June 13-24, 2016

32 Multipole fields in an iron-core magnet
USPAS, Fort Collins, CO, June 13-24, 2016

33 Multipole fields in an iron-core magnet
USPAS, Fort Collins, CO, June 13-24, 2016

34 Multipole fields in an iron-core magnet
USPAS, Fort Collins, CO, June 13-24, 2016

35 Multipole fields in an iron-core magnet
USPAS, Fort Collins, CO, June 13-24, 2016

36 Multipole fields in an iron-core magnet
USPAS, Fort Collins, CO, June 13-24, 2016

37 Multipole fields in an iron-core magnet
USPAS, Fort Collins, CO, June 13-24, 2016

38 Generating multipole fields in an iron-core magnet
USPAS, Fort Collins, CO, June 13-24, 2016

39 Generating multipole fields in an iron-core magnet
USPAS, Fort Collins, CO, June 13-24, 2016

40 Generating multipole fields in an iron-core magnet
USPAS, Fort Collins, CO, June 13-24, 2016

41 Generating multipole fields in an iron-core magnet
USPAS, Fort Collins, CO, June 13-24, 2016

42 Maxwell’s Equations for Magnets - Summary
USPAS, Fort Collins, CO, June 13-24, 2016

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. USPAS, Fort Collins, CO, June 13-24, 2016

44 Previous lecture re-cap
USPAS, Fort Collins, CO, June 13-24, 2016

45 Previous lecture re-cap
USPAS, Fort Collins, CO, June 13-24, 2016

46 Allowed and forbidden harmonics
USPAS, Fort Collins, CO, June 13-24, 2016

47 Allowed and forbidden harmonics
USPAS, Fort Collins, CO, June 13-24, 2016

48 Allowed and forbidden harmonics
USPAS, Fort Collins, CO, June 13-24, 2016

49 Allowed and forbidden harmonics
USPAS, Fort Collins, CO, June 13-24, 2016

50 Allowed and forbidden harmonics
USPAS, Fort Collins, CO, June 13-24, 2016

51 Allowed and forbidden harmonics
USPAS, Fort Collins, CO, June 13-24, 2016

52 Measuring multipoles USPAS, Fort Collins, CO, June 13-24, 2016

53 Measuring multipoles in Cartesian basis
USPAS, Fort Collins, CO, June 13-24, 2016

54 Measuring multipoles in Cartesian basis
USPAS, Fort Collins, CO, June 13-24, 2016

55 Measuring multipoles in Polar basis
USPAS, Fort Collins, CO, June 13-24, 2016

56 Measuring multipoles in Polar basis
USPAS, Fort Collins, CO, June 13-24, 2016

57 Measuring multipoles in Polar basis
USPAS, Fort Collins, CO, June 13-24, 2016

58 Advantages of mode decompositions
USPAS, Fort Collins, CO, June 13-24, 2016

59 Three-dimensional fields
USPAS, Fort Collins, CO, June 13-24, 2016

60 Three-dimensional fields
USPAS, Fort Collins, CO, June 13-24, 2016

61 Three-dimensional fields
USPAS, Fort Collins, CO, June 13-24, 2016

62 Three-dimensional fields
USPAS, Fort Collins, CO, June 13-24, 2016

63 Three-dimensional fields
USPAS, Fort Collins, CO, June 13-24, 2016

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. USPAS, Fort Collins, CO, June 13-24, 2016

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 USPAS, Fort Collins, CO, June 13-24, 2016

66 Appendix A - Field Error Tolerances
Single point mismatch as measured by the Courant-Snyder invariant change: Each source of field error (magnet) contributes the following Courant-Snyder variation here, m =1 quadrupole, m =2 sextupole, m=3 octupole, etc USPAS, Fort Collins, CO, June 13-24, 2016

67 Appendix A - Field Error Tolerances
multipole expansion coefficients of the azimuthal magnetic field, Bq - Fourier series representation in polar coordinates at a given point along the trajectory): multipole gradient and integrated geometric gradient: USPAS, Fort Collins, CO, June 13-24, 2016

68 Appendix A - Field Error Tolerances
Cumulative mismatch along the lattice (N sources): Standard deviation of the Courant-Snyder invariant is given by: Assuming weakly focusing lattice (uniform beta modulation) the following averaging (over the betatron phase) can by applied: USPAS, Fort Collins, CO, June 13-24, 2016

69 Appendix A - Field Error Tolerances
Some useful integrals …. : will reduce the coherent contribution to the C-S variance as follows: Including the first five multipoles yields: USPAS, Fort Collins, CO, June 13-24, 2016

70 Appendix A - Field Error Tolerances
Beam radius at a given magnet is : One can define a ‘good fileld radius’ for a given type of magnet as: Assuming the same multipole content for all magnets in the class one gets: The first factor purely depends on the beamline optics (focusing), while the second one describes field tolerance (nonlinearities) of the magnets: USPAS, Fort Collins, CO, June 13-24, 2016

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. USPAS, Fort Collins, CO, June 13-24, 2016

72 Appendix B - The vector potential
USPAS, Fort Collins, CO, June 13-24, 2016

73 Appendix B - The vector potential
USPAS, Fort Collins, CO, June 13-24, 2016

74 Appendix B - The vector potential
USPAS, Fort Collins, CO, June 13-24, 2016

75 Appendix B - The vector potential
USPAS, Fort Collins, CO, June 13-24, 2016

76 Appendix B - The vector potential
USPAS, Fort Collins, CO, June 13-24, 2016

77 Appendix B - The vector potential
USPAS, Fort Collins, CO, June 13-24, 2016

78 Appendix B - The vector potential
USPAS, Fort Collins, CO, June 13-24, 2016

79 Appendix B - The vector potential
USPAS, Fort Collins, CO, June 13-24, 2016

80 Appendix B - The vector potential
USPAS, Fort Collins, CO, June 13-24, 2016

81 Appendix B - The vector potential
USPAS, Fort Collins, CO, June 13-24, 2016

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