Field Computation for Accelerator Magnets: Analytical and Numerical Methods for Electromagnetic Design and Optimization

764 Pages·2010·74.19 MB·English

by  Dr.?Ing. Stephan Russenschuck(auth.)

Stephan Russenschuck
Field Computation for
Accelerator Magnets
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Stephan Russenschuck
Field Computation for
Accelerator Magnets

Analytical and Numerical Methods for
Electromagnetic Design and Optimization

WILEY-VCH Verlag GmbH & Co. KGaA
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Dr.-Ing. Stephan Russenschuck
information contained in these books,
CERN, Geneva, Switzerland including this book, to be free of errors.
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Cover
Library of Congress Card No.:
The reduced field from the iron
applied for
magnetization in the MQXA magnets for
the LHC inner triplet
(see also chapter 13) British Library Cataloguing-in-Publication
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from the British Library.
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ISBN: 978-3-527-40769-9
V
Contents
Preface XV
Notation XIX
1 MagnetsforAccelerators 1
1.1 TheLargeHadronCollider 2
1.2 AMagnetMetamorphosis 7
1.3 SuperconductorTechnology 16
1.3.1 CriticalCurrentDensityofSuperconductors 16
1.3.2 Strands 19
1.3.3 Cables 22
1.4 TheLHCDipoleColdmass 27
1.5 SuperfluidHeliumPhysicsandCryogenicEngineering 29
1.6 CryostatDesignandCryogenicTemperatureLevels 32
1.7 VacuumTechnology 33
1.8 PoweringandElectricalQualityAssurance 35
1.9 ElectromagneticDesignChallenges 38
1.9.1 TheCERNFieldComputationProgramROXIE 42
1.9.2 AnalyticalandNumericalFieldComputation 44
References 46
2 AlgebraicStructuresandVectorFields 49
2.1 Mappings 49
2.2 Groups,Rings,andFields 50
2.3 VectorSpace 51
2.3.1 LinearIndependenceandBasis 53
2.4 LinearTransformations 54
2.5 AffineSpace 56
2.5.1 Coordinates 58
2.6 InnerProductSpace 60
FieldComputationforAcceleratorMagnets.StephanRussenschuck
Copyright©2010WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:978-3-527-40769-9
VI Contents
2.6.1 MetricSpace 62
2.6.2 OrthonormalBases 63
2.6.3 TheErhardSchmidtOrthogonalization 64
2.7 Orientation 66
2.8 AGlimpseonTopologicalConcepts 68
2.8.1 Homotopy 69
2.8.2 TheBoundaryOperator 70
2.9 ExteriorProducts 72
2.10 IdentitiesofVectorAlgebra 75
2.11 VectorFields 75
2.12 PhasePortraits 77
2.13 ThePhysicalDimensionSystem 80
References 84
3 ClassicalVectorAnalysis 85
3.1 SpaceCurves 86
3.1.1 TheFrenetFrameofSpaceCurves 88
3.2 TheDirectionalDerivative 93
3.3 Gradient,Divergence,andCurl 94
3.4 IdentitiesofVectorAnalysis 96
3.5 SurfacesinE3 96
3.6 TheDifferential 98
(cid:1)
3.7 DifferentialOperatorsonScalarandVectorFieldsinrandr 102
3.8 ThePathIntegralofaVectorField 103
3.9 Coordinate-FreeDefinitionsoftheDifferentialOperators 104
3.10 IntegralTheorems 106
3.10.1 TheKelvin–StokesTheorem 106
3.10.2 Green’sTheoreminthePlane 107
3.10.3 TheGauss–OstrogradskiDivergenceTheorem 108
3.10.4 AVariantoftheGaussTheorem 108
3.10.5 Green’sFirstIdentity 109
3.10.6 Green’sSecondIdentity(Green’sTheorem) 110
3.10.7 VectorFormofGreen’sTheorem 110
3.10.8 GeneralizationoftheIntegration-by-PartsRule 110
3.10.9 TheStrattonTheorems 111
3.11 CurvilinearCoordinates 111
3.11.1 ComponentsofaVectorField 113
3.11.2 ContravariantCoefficients 114
3.11.3 CovariantCoefficients 115
3.12 IntegrationonSpaceElements 115
3.13 OrthogonalCoordinateSystems 117
3.13.1 DifferentialOperators 119
Contents VII
3.13.2 CylindricalCoordinates 121
3.13.3 SphericalCoordinates 122
3.14 TheLemmataofPoincaré 125
3.15 DeRhamCohomology 126
3.16 FourierSeries 129
References 136
4 Maxwell’sEquationsandBoundaryValueProblemsin
Magnetostatics 137
4.1 Maxwell’sEquations 138
4.1.1 TheGlobalForm 138
4.1.2 TheIntegralForm 139
4.1.3 TheLocalForm 141
4.1.4 Maxwell’sOriginalSetofEquations 142
4.2 Kirchhoff’sLaws 143
4.3 ConversionofEnergyinElectromagneticFields 143
4.4 ConstitutiveEquations 144
4.5 BoundaryandInterfaceConditions 146
4.6 MagneticMaterial 151
4.6.1 Ferromagnetism 152
4.6.2 MeasurementofHysteresisCurves 155
4.6.3 MagneticAnisotropyinLaminatedIronYokes 159
4.6.4 Magnetostriction 160
4.6.5 PermanentMagnets 161
4.6.6 MagnetizationCurrentsandFictitiousMagneticCharges 163
4.7 ClassificationDiagramsforElectromagnetism 165
4.8 FieldLines 167
4.8.1 ClassificationofElectromagneticFieldProblems 167
4.9 BoundaryValueProblems1: Magnetostatic 171
4.9.1 Scalar-PotentialFormulations 171
4.9.2 Vector-PotentialFormulations 173
4.9.3 TheScalarLaplaceEquationin2D 179
4.10 BoundaryValueProblems2: MagneticDiffusionProblems 180
References 184
5 FieldsandPotentialsofLine-Currents 187
5.1 GreenFunctions 188
5.2 PotentialsonBoundedDomains 189
5.3 PropertiesofHarmonicFields 191
5.4 TheBiot–SavartLaw 193
5.5 FieldofaStraightLine-CurrentSegment 197
5.6 FieldofaRingCurrent 200
VIII Contents
5.7 TheMagneticDipoleMoment 203
5.8 TheMagneticDoubleLayer 205
5.8.1 TheSolidAngle 206
5.8.2 ApproximatingtheSolidAngleofaCurrentLoop 208
5.9 TheImage-CurrentMethod 209
5.9.1 PlaneBoundaries 211
5.9.2 CircularBoundaries 213
5.10 StoredEnergyinaMagnetostaticField 216
5.10.1 SelfandMutualInductance 218
5.10.2 TheGeometricMeanDistance 220
5.10.3 MagneticFlux 222
5.11 MagneticEnergyinNonlinearCircuits 224
5.11.1 DifferentialInductance 224
5.12 MagneticForcesandtheMaxwellStressTensor 227
5.13 FieldsandPotentialsofMagnetizationCurrents 230
5.14 MagneticLevitation 232
References 235
6 FieldHarmonics 237
6.1 CircularHarmonics 238
6.1.1 DeterminingtheMultipoleCoefficients 240
6.1.2 MagneticShielding;PermeableCylindricalShellinaUniform
Field 253
6.1.3 IntegratedMultipolesinAcceleratorMagnets 255
6.2 SphericalHarmonics 257
6.2.1 LegendreSeriesExpressionfortheVectorPotential 262
6.2.2 DeterminingtheZonalHarmonics 263
6.3 SeparationinCartesianCoordinates 265
References 268
7 Iron-DominatedMagnets 269
7.1 C-ShapedDipole 270
7.2 Quadrupole 271
7.3 OhmicLossesinDipoleandQuadrupoleCoils 272
7.4 MagneticCircuitwithVaryingYokeWidth 272
7.5 BranchedCircuits 274
7.6 IdealPoleShapesofIron-DominatedMagnets 275
7.6.1 Shimming 277
7.7 RogowskiProfiles 278
7.8 Combined-FunctionMagnets 281
7.9 PermanentMagnetExcitation 282
7.10 CoolingofNormal-ConductingMagnets 287
References 291
Contents IX
8 Coil-DominatedMagnets 293
8.1 AcceleratorMagnets 294
8.1.1 GenerationofPureMultipoleFields 295
8.1.2 SensitivitytoCoil-BlockPositioningErrors 305
8.1.3 ForceDistribution 305
8.1.4 MarginsintheLHCMainDipole 306
8.2 Combined-FunctionMagnetsandtheUnipolarCurrentDipole 309
8.3 RectangularBlock-CoilStructures 310
8.4 FieldEnhancementinCoilEndsofAcceleratorMagnets 311
8.5 MagneticForceDistributionintheLHCDipoleCoilEnds 312
8.6 NestedHelices 314
8.7 Solenoids 315
8.7.1 HelmholtzandMaxwellCoils 315
8.7.2 FabryFactors 317
8.7.3 Off-AxisFields 321
8.7.4 ZonalHarmonics 324
References 325
9 ComplexAnalysisMethodsforMagnetDesign 327
9.1 TheFieldofComplexNumbers 328
9.2 HolomorphicFunctionsandtheCauchy–RiemannEquations 329
9.3 PowerSeries 331
9.4 TheComplexFormoftheDiscreteFourierTransform 333
9.5 ComplexPotentials 335
9.6 ConformalMappings 336
9.7 ComplexRepresentationofFieldQualityinAccelerator
Magnets 338
9.7.1 Feed-Down 338
9.7.2 ReferenceFrameRotation 342
9.7.3 ReflectionabouttheVerticalAxis 343
9.8 ComplexIntegration 344
9.8.1 Cauchy’sTheoremandtheIntegralFormula 345
9.8.2 PropertiesofHolomorphicFunctions 346
9.8.3 TheResidualTheorem 348
9.9 TheFieldandPotentialofaLineCurrent 349
9.9.1 SeriesExpansionoftheLine-CurrentField 350
9.9.2 CircularSectorWindings 351
9.10 MultipolesGeneratedbyaMagneticDipoleMoment 351
9.11 Beth’sCurrent-SheetTheorem 352
9.12 ElectromagneticForcesontheDipoleCoil 354
9.13 TheFieldofaPolygonalConductor 356
9.14 MagneticFluxDensityInsideEllipticalConductors 358
References 362
X Contents
10 FieldDiffusion 363
10.1 TimeConstantsandPenetrationDepths 363
10.2 TheLaplaceTransform 365
10.3 ConductiveSlabinaTime-TransientAppliedField 370
10.3.1 TheStep-ExcitationFunction 371
10.3.2 LinearRampoftheAppliedField 373
10.3.3 SinusoidalExcitation 375
10.4 EddyCurrentsintheLHCColdBoreandBeamScreen 376
References 382
11 ElementaryBeamOpticsandFieldRequirements 383
11.1 TheEquationsofChargedParticleMotioninaMagneticField 383
11.2 MagneticRigidityandtheBendingMagnets 387
11.3 TheLinearEquationsofMotion 389
11.4 WeakFocusing 390
11.5 Thin-LensApproximations 392
11.6 TransferMatrices 393
11.7 StrongFocusingandtheFODOCell 395
11.8 TheBetaFunction,Tune,andTransverseResonances 397
11.9 Off-MomentumParticles 407
11.9.1 Dispersion 408
11.9.2 Chromaticity 410
11.10 FieldErrorSpecifications 412
References 413
12 ReferenceFramesandMagnetPolarities 415
12.1 MagnetPolarityConventions 416
12.1.1 Spool-PieceCorrectors 418
12.1.2 Twin-ApertureandTwo-in-OneMagnets 418
12.2 ReferenceFrames 420
12.3 MultipoleExpansions 421
12.3.1 TheMagnetFrame 421
12.3.2 TheLocalReferenceFrameofBeam1 423
12.3.3 DefinitionofFieldErrorsintheAcceleratorDesignProgram
MAD 424
12.3.4 TransformationbetweentheMagnetandtheBeam1Frames 424
12.4 OrbitCorrectors 426
12.5 PositionoftheConnectionTerminals 426
12.6 TurnedMagnetsandMagnetAssemblies 427
12.7 ElectricalCircuitsintheLHCMachine 429
References 432