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 Real functions 1.4 Vector Spaces 1.5 Linear Transformations 1.6 Affine Space 1.7 Inner Product Space 1.8 Orientation 1.9 A Glimpse on Topological Concepts 1.10 Exterior Products 1.11 Identities of Vector Algebra 1.12 Vector Fields 1.13 Phase Portraits 1.14 Matrix Algebra 1.15 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 0 2.8 The Path Integral of a Vector Field 2.9 Coordinate-Free Definitions of the Differential Operators 2.10 Integral Theorems 2.11 Curvilinear Coordinates 2.12 Integration on Space Elements 2.13 Orthogonal Coordinate Systems 2.14 The Lemmata of Poincaré 2.15 De Rham Cohomology 2.16 Fourier Series 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 Lines 3.9 Boundary-Value Problems 1: Magnetostatics 3.10 Boundary-Value Problems 2: Magnetic Diffusion 4 Fields and Potentials of Line-Currents 4.1 Green Functions 4.2 Potentials on Bounded Domains 4.3 The Direct Boundary-Element Formulation 4.4 Properties of Harmonic Fields 4.5 The Biot-Savart Law 4.6 Field Contribution of a Straight Line-Current Segments 4.7 Field of a Ring Current 4.8 The Magnetic Dipole Moment 4.9 Magnetic Double Layers 4.10 The Image-Current Method 4.11 Stored Energy in a Magnetostatic Field 4.12 Magnetic Energy in Nonlinear Circuits 4.13 Magnetic Forces and the Maxwell Stress Tensor 4.14 Fields and Potentials of Magnetization Currents 4.15 The Torque on a Magnetic Dipole Moment 4.16 Magnetic Levitation 5 Field Harmonics 5.1 Circular Harmonics 5.2 Zonal Harmonics 5.3 Cartesian Coordinates 5.4 The General Case 5.5 Plane Elliptic Coordinates 5.6 Bipolar Coordinates 5.7 Integrated Multipoles in Accelerator Magnets 5.8 3D Field Harmonics - Generalized Gradients 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 The Complex Form of the Discrete Fourier-Series Expansion 6.5 The Fourier Transform of Non-Periodic Functions 6.6 Complex Potentials 6.7 Complex Representation of Field Quality in Accelerator Magnets 6.8 Complex Integration 6.9 The Field and Potential of a Line Current 6.10 Multipoles Generated by a Magnetic Dipole Moment 6.11 Beth?s 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 Electromotive Force 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 Eddy Currents in the LHC Cold Bore and Beam Screen 9 Synchrotron Radiation 9.1 The Wave Equation 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 10 The Theory of the Coil Magnetometer 10.1 Terminology for Magnetic Field Transducers 10.2 Coil-Sensitivity Factors 10.3 Reparametrization to the Arc-Length 10.4 The Sensitivity Factor as a Convolution Kernel 10.5 Polarity Convention and Calibration of Rotating-Coil Magnetometers 10.6 Alternative Calibration Procedures 10.7 The Transversal-Multipole Mapper 10.8 The Harmonic Field Scanner 10.9 The Translating-Coil Magnetometer 10.10 The Solenoidal-Field Transducer 11 Stretched-Wire Field Measurements 11.1 The System Architecture of CERN?s Stretched-Wire Systems 11.2 Caternaries and Sag Parameters of a Taut String 11.3 The Single Stretched-Wire Method 11.4 The Vibrating-Wire Method 11.5 The Oscillating-Wire Technique 11.6 The Frequency-Response Method 11.7 Magnetic-Center Location VOLUME 2: FIELD COMPUTATION FOR MAGNET DESIGN AND OPTIMIZATION 12 Magnets for Accelerators 12.1 The Large Hadron Collider 12.2 A Magnet Metamorphosis 12.3 Superconductor Technology 12.4 The LHC Dipole Coldmass 12.5 Superfluid Helium Physics and Cryogenic Engineering 12.6 Cryostat Design and Cryogenic Temperature Levels 12.7 Vacuum Technology 12.8 Powering and Electrical Quality Assurance 12.9 Electromagnetic Design Challenges 13 Elementary Beam Optics and Field Requirements 13.1 The Equations of Charged Particle Motion in a Magnetic Field 13.2 Magnetic Rigidity and the Bending Magnets 13.3 The Linear Equations of Motion 13.4 Weak Focusing 13.5 Thin-Lens Approximations 13.6 Transfer Matrices 13.7 Strong Focusing and the FODO Cell 13.8 The Beta Function, Tune, and Transverse Resonances 13.9 Off-Momentum Particles 13.10 Field Error Specifications 14 Iron-Dominated Magnets 14.1 C-Shaped Dipoles 14.2 Quadrupoles 14.3 Ohmic Losses in Dipole and Quadrupole Coils 14.4 Magnetic Circuit with Varying Yoke Width 14.5 Ideal Pole Shapes of Iron-Dominated Magnets 14.6 The Mass of the Iron Yoke 14.7 Rogowski Profiles 14.8 Combined-Function Magnets 14.9 Permanent Magnet Excitation 14.10 Wigglers and Undulators 14.11 Cooling of Normal-Conducting Magnets 15 Coil-Dominated Magnets 15.1 Accelerator Magnets 15.2 Combined-Function Magnets and the Unipolar Current Dipole 15.3 Rectangular Block-Coil Structures 15.4 Field Enhancement in Coil Ends of Accelerator Magnets 15.5 Magnetic Force Distribution in the LHC Dipole Coil Ends 15.6 Nested Helices 15.7 Helmholtz and Maxwell Coils 15.8 Solenoids 16 Reference Frames and Magnet Polarities 16.1 Magnet Polarity Conventions 16.2 Reference Frames 16.3 Multipole Expansions 16.4 Orbit Correctors 16.5 Position of the Connection Terminals 16.6 Turned Magnets and Magnet Assemblies 16.7 Electrical Circuits in the LHC Machine 17 Finite-Element Formulations 17.1 One-Dimensional Finite-Element Analysis 17.2 FEM with the Vector-Potential (Curl-Curl) Formulation 17.3 Complementary Formulations 18 Discretization 18.1 Quadrilateral Mesh Generation 18.2 Finite-Element Shape Functions 19 Coupling of Boundary and Finite Elements 19.1 The Boundary-Element Method 19.2 BEM-FEM Coupling 19.3 BEM-FEM Coupling using the Total Scalar-Potential 19.4 The M(B) Iteration 19.5 Applications 20 Superconductor Magnetization 20.1 Superconductor Magnetization 20.2 Critical Surface Modeling 20.3 The Critical State Model 20.4 The Ellipse on a Cylinder Model 20.5 Nested Intersecting Circles and Ellipses 20.6 Hysteresis Modeling 20.7 Magnet Field Errors due to the Superconducting Filament Magnetization 20.8 The M(B) Iteration 20.9 Software Implementation 20.10 Applications to Magnet Design 20.11 Nested Magnets 21 Interstrand Coupling Currents 21.1 Analysis of Linear Networks 21.2 A Network Model for the Interstrand Coupling Currents 21.3 Steady-State Calculations 21.4 Time-Transient Analysis 21.5 The M(B) Iteration Scheme for ISCCs 21.6 Approximation for the Interstrand Coupling Currents 21.7 Interfilament Coupling Currents 21.8 Applications to Magnet Design 22 Quench Simulation 22.1 The Heat Balance Equation 22.2 Electrical Network Models of Superconductors 22.3 Current Sharing 22.4 Winding Schemes and Equivalent Electrical Circuit Diagrams 22.5 Quench Detection 22.6 Magnet Protection 22.7 Numerical Quench Simulation 22.8 The Time-Stepping Algorithm 22.9 Applications 23 Differential Geometry Applied to Coil-End Design 23.1 Constant-Perimeter Coil Ends 23.2 Differential Geometry of the Strip Surfaces 23.3 Discrete Theory of the Strip Surface 23.4 Optimization of the Strip Surface 23.5 Coil-End Transformations 23.6 Corrector Magnet Coil End with Ribbon Cables 23.7 End-Spacer Manufacturing 23.8 Splice Configuration 24 Mathematical Optimization Techniques 24.1 Mathematical Formulation of the Optimization Problem 24.2 Optimality Criteria for Unconstrained Problems 24.3 Karush-Kuhn-Tucker Conditions 24.4 Pareto Optimality 24.5 Methods for Decision Making 24.6 Box Constraints 24.7 Treatment of Nonlinear Constraints 24.8 Deterministic Optimization Algorithms 24.9 Genetic Optimization Algorithms 24.10 Applications A Material-Property Data for Quench Simulations A.1 Mass Density A.2 Electrical Resistivity A.3 Thermal Conductivity A.4 Heat Capacity B The LHC Magnet Zoo B.1 Superconducting Magnets B.2 Normal-Conducting Magnets C Ramping the LHC Dipoles D The Vibration of the Taut String D.1 The Inhomogeneous Wave Equation D.2 Energy in Vibrating Wires D.3 Numerical Solution of the Wave Equation E The Dirac delta F Detailed Calculation of the Field of Moving Charges G Uncertainty in Measurement and Approximation G.1 Sample Mean, Standard Deviation, and Confidence Interval G.2 Random Numbers as a Vector-Space, Error Norms H Orthogonal-Array Testing I SI (MKSA) Units J Glossary