Magnetic Multipoles, Magnet Design S.A. Bogacz, G.A. Krafft, S. DeSilva and R. Gamage Jefferson Lab and Old Dominion University Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 13-24, 2016 1
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, 2009 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. 17-28, 2011 2
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. Lecture 5 - Magnetic Multipoles USPAS, Hampton, VA, Jan. 17-28, 2011 3
Maxwell s equations 4
Maxwell s equations 5
Physical interpretation of 6
Physical interpretation of 7
Linearity and superposition 8
Multipole fields 9
Multipole fields 10
Multipole fields 11
Multipole fields 12
Multipole fields 13
Multipole fields 14
Generating multipole fields from a current distribution 15
Multipole fields from a current distribution 16
Multipole fields from a current distribution 17
Multipole fields from a current distribution 18
Multipole fields from a current distribution 19
Multipole fields from a current distribution 20
Multipole fields from a current distribution 21
Multipole fields from a current distribution 22
Fields from a cylindrical current distribution 23
Fields from a cylindrical current distribution 24
Multipole fields from a current distribution 25
Multipole fields from a current distribution 26
Superconducting quadrupole - collider final focus 27
Multipole fields in an iron-core magnet 28
Multipole fields in an iron-core magnet 29
Multipole fields in an iron-core magnet 30
Multipole fields in an iron-core magnet 31
Multipole fields in an iron-core magnet 32
Multipole fields in an iron-core magnet 33
Multipole fields in an iron-core magnet 34
Multipole fields in an iron-core magnet 35
Multipole fields in an iron-core magnet 36
Multipole fields in an iron-core magnet 37
Multipole fields in an iron-core magnet 38
Generating multipole fields in an iron-core magnet 39
Generating multipole fields in an iron-core magnet 40
Generating multipole fields in an iron-core magnet 41
Generating multipole fields in an iron-core magnet 42
Maxwell s Equations for Magnets - Summary 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 13-24, 2016 44
Previous lecture re-cap Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 13-24, 2016 45
Previous lecture re-cap Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 13-24, 2016 46
Allowed and forbidden harmonics Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 13-24, 2016 47
Allowed and forbidden harmonics Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 13-24, 2016 48
Allowed and forbidden harmonics Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 13-24, 2016 49
Allowed and forbidden harmonics Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 13-24, 2016 50
Allowed and forbidden harmonics 51
Allowed and forbidden harmonics 52
Measuring multipoles Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 13-24, 2016 53
Measuring multipoles in Cartesian basis Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 13-24, 2016 54
Measuring multipoles in Cartesian basis 55
Measuring multipoles in Polar basis 56
Measuring multipoles in Polar basis 57
Measuring multipoles in Polar basis 58
Advantages of mode decompositions 59
Three-dimensional fields 60
Three-dimensional fields 61
Three-dimensional fields 62
Three-dimensional fields 63
Three-dimensional fields 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 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 Lecture 5 - Magnetic Multipoles 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: εʹ = β( θ + δθ) + 2 α( θ + δθ) x+ γx ( x) 2 2 = ε + βθ + α δθ + βδθ 2 2, ε x = εβ sin µ, θ = sin µ ( cos µ - α sin 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 δφ m sin µ, m= 1 m= 1 2 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 13-24, 2016 68
Appendix A - Field Error Tolerances Cumulative mismatch along the lattice (N sources): N 2 m 1 m 1 m 2 m ε N = ε 1+ 2β ( εβ ) δφ m cos µ sin µ + β ( εβ ) δφ m sin µ, n= 1 m= 1 m= 1 Standard deviation of the Courant-Snyder invariant is given by: σ ε ε 2 2 2 2 δε N 1 1 2 2 ( ) m m cos sin ( ) m m βi εβi δφ m µ µ βi εβi δφ m sin µ ε i= 1 m= 1 m= 1 δε = = + 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 13-24, 2016 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 1 1 2 2 ( ) m m cos sin ( ) m m = βi εβi δφ m µ µ + βi εβi δφ m 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 ( ) ( ) ( ) } 2 2 2 2 4 2 2 6 i δφ sin µ εβ 2 sin 2 2 sin... 1 i δφ δφδφ µ εβ 2 1 3 i δφ δφδφ δφ δφ µ 3 1 5 2 4 = + + + + + + 1 2 1 3 2 4 Lecture 5 - Magnetic Multipoles 1 3 5 2 4 6 USPAS, Fort Collins, CO, June 13-24, 2016 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= 1 1 2 2 3 2 2 5 4 2 = βi δφ1 + a ( δφ2 + 2δφδφ 1 3) + a ( δφ3 + 2δφδφ 1 5+ 2 δφ2δφ4) +... 2 2 2 The first factor purely depends on the beamline optics (focusing), while the second one describes field tolerance (nonlinearities) of the magnets: 2 2 2 4 2 ΔΦ = δφ1 + 3 a ( δφ2 + 2δφ1δφ3) + 5 a ( δφ3 + 2δφ1δφ5+ 2 δφ2δφ4) +... 2 2 Lecture 5 - Magnetic Multipoles 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. 72
Appendix B - The vector potential 73
Appendix B - The vector potential 74
Appendix B - The vector potential 75
Appendix B - The vector potential 76
Appendix B - The vector potential 77
Appendix B - The vector potential 78
Appendix B - The vector potential 79
Appendix B - The vector potential 80
Appendix B - The vector potential 81
Appendix B - The vector potential 82