Field Simulation for Accelerator Magnets

VOLUME 1: FOUNDATIONS OF FIELD COMPUTATION AND MAGNETIC MEASUREMENTS

1 Algebraic Structures and Vector Fields
1.1 Groups, Rings, and Fields
1.2 Mappings
1.3 Vector Spaces
1.4 Affine Space
1.5 Inner Product Space
1.6 Orientation
1.7 Topological Spaces
1.8 Exterior Products
1.9 Identities of Vector Algebra
1.10 Vector Fields
1.11 Phase Portraits
1.12 Matrix Algebra
1.13 The Physical Dimension System

2 Classical Vector Analysis
2.1 Space Curves
2.2 The Directional Derivative
2.3 Gradient, Divergence, and Curl
2.4 Identities of Vector Analysis
2.5 Surfaces in E3
2.6 The Differential
2.7 Differential Operators on Scalar and Vector Fields in r and r'
2.8 Path Integrals
2.9 Coordinate-Free Definitions of the Differential Operators
2.10 A Descriptive Representation of the Curl
2.11 Integral Theorems
2.12 Curvilinear Coordinates
2.13 Integration on Space Elements
2.14 Orthogonal Coordinate Systems
2.15 Transformation of Vector Components
2.16 The Lemmata of Poincaré
2.17 De Rham Cohomology
2.18 Fourier Series and the Discrete Fourier Transform

3 Maxwell's Equations and Boundary-Value Problems in Magnetostatics
3.1 Maxwell's Equations
3.2 Kirchhoff's Laws
3.3 Constitutive Equations
3.4 Energy in Electromagnetic Fields
3.5 Boundary and Interface Conditions
3.6 Magnetic Materials
3.7 Classification Diagrams for Electromagnetism
3.8 Field Quantities Expressed as Differential Forms
3.9 Field Lines
3.10 Boundary-Value Problems 1: Magnetostatics
3.11 Boundary-Value Problems 2: Magnetic Diffusion

4 Fields and Potentials of Line-Currents
4.1 The Dirac Delta
4.2 Green Functions
4.3 Helmholtz Decomposition
4.4 Potentials on Bounded Domains
4.5 The Boundary-Integral Equation for the Magnetic Flux Density
4.6 Properties of Harmonic Fields
4.7 The Biot?Savart Law
4.8 A One-Dimensional Inversion of the Biot-Savart Law
4.9 The Magnetic Dipole Moment
4.10 Magnetic Double Layers
4.11 The Image-Current Method
4.12 Stored Energy in a Magnetostatic Field
4.13 Magnetic Energy in Nonlinear Circuits
4.14 Magnetic Forces and the Maxwell Stress Tensor
4.15 Magnetic Force Calculation by Virtual Displacement
4.16 Fields and Potentials of Magnetization Currents
4.17 The Torque on a Magnetic Dipole Moment
4.18 Magnetic Levitation

5 Harmonic Fields
5.1 Circular Harmonics
5.2 Field Harmonics in Rectangular Domains
5.3 Plane Elliptic Harmonics
5.4 Bipolar Harmonics
5.5 Integrated Multipoles in Accelerator Magnets
5.6 Generalized Gradients
5.7 Zonal Harmonics
5.8 Toroidal Harmonics

6 Complex Analysis Methods for Magnet Design
6.1 The Field of Complex Numbers
6.2 Holomorphic Functions and the Cauchy-Riemann Equations
6.3 Power Series
6.4 Complex Representation of the Field Homogeneity
6.5 Complex Potentials
6.6 Harmonic Polynomials
6.7 Conformal Mapping
6.8 Complex Integration
6.9 The Field and Potential of Line Currents
6.10 Multipoles Generated by a Magnetic Dipole Moment
6.11 The Current-Sheet Theorem
6.12 Electromagnetic Forces on the Dipole Coil
6.13 The Field of a Polygonal Conductor
6.14 Magnetic Flux Density Inside Elliptical Conductors

7 Faraday's Law of Induction
7.1 The Electromotive Force
7.2 Definitions of the EMF
7.3 EMF Formulas
7.4 Faraday Paradoxes

8 Field Diffusion
8.1 Time-Constants and Penetration Depths
8.2 The Laplace Transform
8.3 Conductive Slab in a Time-Transient Applied Field
8.4 Conductive Tube in a Transient Dipole Field
8.5 The LHC Cold Bore
8.6 The LHC Beam Screen

9 Synchrotron Radiation
9.1 The InhomogeneousWave Equations in Free Space
9.2 The Liénard-Wichert Potentials
9.3 The Fields of Moving Charges
9.4 Power Radiated by an Accelerated Charge
9.5 Nonrelativistic Motion
9.6 Bremsstrahlung
9.7 Synchrotron Radiation
9.8 Wigglers and Undulators

10 The Theory of the Coil Magnetometer
10.1 Terminology for Magnetic Field Transducers
10.2 Production Techniques for Coil-Magnetometers
10.3 Coil-Sensitivity Factors
10.4 Reparametrization to the Arc-Length
10.5 The Sensitivity Factor as a Convolution Kernel
10.6 Calibration of Rotating-Coil Magnetometers
10.7 An Alternative Radius-Calibration Procedure
10.8 The Metrological Characterization of the Measurement System
10.9 The Transversal Multipole Mapper
10.10 The Harmonic Field Scanner
10.11 The Translating-Coil Magnetometer
10.12 The Solenoidal-Field Transducer
10.13 The Quench Antenna

11 Stretched-Wire Field Measurements
11.1 The System Architecture of Wire Systems
11.2 Caternaries and Sag Parameters of a Taut String
11.3 The Single Stretched-Wire Method
11.4 Quadrupole Axis Alignement
11.5 Alternating-Current Powering of the Magnet
11.6 The Vibrating-Wire Method
11.7 The Oscillating-Wire Method
11.8 Sampling Techniques
11.9 Applications

Appendix A: Differential Forms
Appendix B: The Vibration of the Taut String
Appendix C: Uncertainty in Measurement and Approximation
Appendix D: Orthogonal Array Testing

Glossary

VOLUME 2: FIELD COMPUTATION FOR MAGNET DESIGN AND OPTIMIZATION

1 Magnets for Accelerators
1.1 High-Energy Circular Accelerators
1.2 A Magnet Metamorphosis
1.3 Superconductor Technology
1.4 Critical Surface Modeling
1.5 The LHC Dipole Coldmass
1.6 Superfluid Helium Physics and Cryogenic Engineering
1.7 Cryostat Design and Cryogenic Temperature Levels
1.8 Vacuum Technology
1.9 Powering and Electrical Quality Assurance
1.10 Electromagnetic Design Challenges


2 Elementary Beam Optics and Field Requirements
2.1 The Equations of Charged Particle Motion in a Magnetic Field
2.2 Magnetic Rigidity and the Bending Magnets
2.3 The Linear Equations of Motion
2.4 Weak Focusing
2.5 Thin-Lens Approximations
2.6 Transfer Matrices
2.7 Strong Focusing and the FODO Cell
2.8 The Beta Function, Tune, and Transverse Resonances
2.9 Off-Momentum Particles
2.10 Field Error Specifications for the LHC

3 Reference Frames and Magnet Polarities
3.1 Magnet Polarity Conventions
3.2 Reference Frames
3.3 Multipole Expansions
3.4 Orbit Correctors
3.5 Position of the Connection Terminals
3.6 Turned Magnets and Magnet Assemblies
3.7 Electrical Circuits in the LHC Machine


4 Iron-Dominated Magnets
4.1 C-Shaped Dipoles
4.2 Magnetic Dipole Circuit with Varying Yoke Width
4.3 Quadrupoles
4.4 Ohmic Losses in Dipole and Quadrupole Coils
4.5 Ideal Pole Shapes of Iron-Dominated Magnets
4.6 The Mass of the Dipole Yoke
4.7 Rogowski Profiles
4.8 Combined-Function Magnets
4.9 Permanent Magnet Excitation
4.10 Wigglers and Undulators
4.11 Cooling of Normal-Conducting Magnets

5 Coil-Dominated Magnets
5.1 Generation of Pure Multipole Fields
5.2 Electromagnetic Force Distribution
5.3 Margins on the Load Line
5.4 Combined-Function Magnets and the Unipolar Current Dipole
5.5 Rectangular Block-Coil Structures
5.6 Field Enhancement in Coil Ends of Accelerator Magnets
5.7 Magnetic Force Distribution in the LHC Dipole Coil Ends
5.8 Canted-Cosine-Theta Magnets
5.9 The On-Axis Field of the Double-Helix Wiggler
5.10 Helmholtz and Maxwell Coils
5.11 Solenoids

6 Finite-Element Formulations
6.1 One-Dimensional Finite-Element Analysis
6.2 FEM with the Vector-Potential (Curl-Curl) Formulation
6.3 Complementary Formulations

7 Discretization
7.1 Quadrilateral Mesh Generation
7.2 Finite-Element Shape Functions

8 Coupling of Boundary and Finite Elements
8.1 The Boundary-Element Method
8.2.The Finite-Element Domain
9.3 BEM-FEM Coupling
9.4 BEM-FEM Coupling using the Total Scalar Potential
9.5 BEM-FEM Coupling for Time-Transient Field Problems
9.6 Eddy Currents in TransformerWindings
9.7 Applications
9.8 BEM Postprocessing of Field Measurements

10 Superconductor Magnetization
10.1 Superconductor Magnetization
10.2 Critical Surface Modeling
10.3 The Critical State Model
10.4 The Ellipse on a Cylinder Model
10.5 Nested Intersecting Circles and Ellipses
10.6 Hysteresis Modeling
10.7 Magnet Field Errors caused by Filament Magnetization
10.8 The M(B) Iteration
10.9 Software Implementation
10.10 Applications to Magnet Design
10.11 The Vector-Hysteresis Model
11.12 Combined Function Dipole
11.13 Experimental Validation of the Vector-Hysteresis Model

11 Interstrand Coupling Currents
11.1 Analysis of Linear Networks
11.2 A Network Model for the Interstrand Coupling Currents
11.3 Steady-State Calculations
11.4 Time-Transient Analysis
11.5 The M(B) Iteration of Interstrand Coupling Currents
11.6 Approximation for the Interstrand Coupling Currents
11.7 Interfilament Coupling Currents
11.8 Applications to Magnet Design

12 Quench Simulation
12.1 The Heat Balance Equation
12.2 Electrical Network Models of Superconductors
12.3 Current Sharing
12.4 Winding Schemes and Equivalent Electrical Circuit Diagrams
12.5 Quench Detection
12.6 Magnet Protection
12.7 Current-Pulse Protection
12.8 Numerical Quench Simulation
12.9 Coupling-Loss Induced Quench Protection
12.10 The Time-Stepping Algorithm
12.9 Applications

13 Differential Geometry Applied to Coil-End Design
13.1 Constant-Perimeter Coil Ends
13.2 Differential Geometry of the Strip Surfaces
13.3 Discrete Theory of the Strip Surface
13.4 Optimization of the Strip Surface
13.5 Ribbon-Cables
13.6 Curved Magnet Heads
13.7 Roll, Swing, and Tilt Transformations
13.8 Spacer Manufacturing
13.9 Splice Connections

14 Mathematical Optimization Techniques
14.1 Mathematical Formulation of the Optimization Problem
14.2 Optimality Criteria for Unconstrained Problems
14.3 Karush-Kuhn-Tucker Conditions
14.4 Pareto Optimality
14.5 Methods for Decision Making
14.6 Box Constraints
14.7 Treatment of Nonlinear Constraints
14.8 Deterministic Optimization Algorithms
14.9 Genetic Optimization Algorithms
14.10 Applications

15 Model-Based Systems Engineering
15.1 Domain-Specific Challenges in Accelerator Magnet Development
15.2 Models and Systems
15.3 Coherence Between Beam Physics, Magnet Technology, and Magnetic Measurements
15.4 Uncertainty and Ignorance
15.5 Twins
15.6 Applications
15.7 The MBSE Software Environment
15.8 Systems Generation Engineering

Appendix A: Material-Property Data for Field Simulation
Appendix B: The LHC Magnet Catalog
Appendix C: Ramping the LHC Dipoles

Glossary
Index