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RADIATING
NONUNIFORM
TRANSMISSION-LINE
SYSTEMS AND THE
PARTIAL ELEMENT
EQUIVALENT CIRCUIT
METHOD
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RADIATING
NONUNIFORM
TRANSMISSION-
LINE SYSTEMS AND
THE PARTIAL
ELEMENT
EQUIVALENT
CIRCUIT METHOD
Ju¨rgenNitsch,FrankGronwaldandGu¨nterWollenberg
Otto-von-Guericke-UniversityMagdeburg
A John Wiley and Sons, Ltd., Publication
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Thiseditionfirstpublished2009
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LibraryofCongressCataloging-in-PublicationData
Nitsch,Ju¨rgen.
Radiatingnonuniformtransmission-linesystemsandthepartialelementequivalentcircuitmethod/Ju¨rgen
Nitsch,FrankGronwald,Gu¨nterWollenberg.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-470-84536-3(cloth)
1.Electromagneticcompatibility–Mathematicalmodels. 2.Electriclines–Mathematicalmodels. 3.Electronic
circuitdesign–Dataprocessing. 4.Electronicapparatusandappliances–Designandconstruction–Data
processing. I.Gronwald,Frank. II.Wollenberg,Gu¨nter. III.Title.
TK7867.2.N582009
621.382(cid:2)24–dc22
2009031448
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN 978-0-470-84536-3(H/B)
Typesetin10/12ptTimesbyAptaraInc.,NewDelhi,India
PrintedinGreatBritainbyCPIAntonyRowe,Chippenham,Wiltshire
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To
Ulla,Daniel,Sebastian
andFelicitas
MonikaandWolfgang
Evdokiya,AlexanderandKarin
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“As far as the laws of mathematics refer to reality, they are
not certain, and so far as they are certain, they do not refer to
reality.”
AlbertEinstein(1879–1955)
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Contents
Preface xv
References xvii
Acknowledgments xix
ListofSymbols xxi
Introduction xxxi
References xxxviii
1 FundamentalsofElectrodynamics 1
1.1 MaxwellEquationsDerivedfromConservationLaws–anAxiomaticApproach 1
1.1.1 ChargeConservation 3
1.1.2 LorentzForceandMagneticFluxConservation 5
1.1.3 ConstitutiveRelationsandthePropertiesofSpacetime 11
1.1.4 Remarks 12
1.2 TheElectromagneticFieldasaGaugeField–aGaugeFieldApproach 13
1.2.1 DifferencesofPhysicalFieldsthatareDescribedbyReferenceSystems 14
1.2.2 ThePhaseofMicroscopicMatterFields 15
1.2.3 TheReferenceFrameofaPhase 16
1.2.4 TheGaugeFieldsofaPhase 18
1.2.5 DynamicsoftheGaugeField 21
1.3 TheRelationBetweentheAxiomaticApproachandtheGaugeFieldApproach 24
1.3.1 NoetherTheoremandElectricChargeConservation 24
1.3.2 MinimalCouplingandtheLorentzForce 24
1.3.3 BianchiIdentityandMagneticFluxConservation 26
1.3.4 GaugeApproachandConstitutiveRelations 27
1.4 SolutionsofMaxwellEquations 28
1.4.1 WaveEquations 29
1.4.1.1 DecouplingofMaxwellEquations 29
1.4.1.2 EquationsofMotionfortheElectromagneticPotentials 30
1.4.1.3 MaxwellEquationsintheFrequencyDomainandHelmholtzEquations 31
1.4.1.4 MaxwellEquationsinReciprocalSpace 32
1.4.2 BoundaryConditionsatInterfaces 33
1.4.3 DynamicalandNondynamicalComponentsoftheElectromagneticField 33
1.4.3.1 Helmholtz’sVectorTheorem,LongitudinalandTransverseFields 33
1.4.3.2 NondynamicalMaxwellEquationsasBoundaryConditionsinTime 35
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x Contents
1.4.3.3 LongitudinalPartoftheMaxwellEquations 36
1.4.3.4 TransversePartoftheMaxwellEquations 37
1.4.4 ElectromagneticEnergyandtheSingularitiesoftheElectromagneticField 39
1.4.5 CoulombFieldsandRadiationFields 41
1.4.6 TheGreen’sFunctionMethod 44
1.4.6.1 BasicIdeas 45
1.4.6.2 Self-AdjointnessofDifferentialOperatorsandBoundaryConditions 46
1.4.6.3 GeneralSolutionsofMaxwellEquations 49
1.4.6.4 BasicRelationsBetweenElectromagneticGreen’sFunctions 49
1.5 BoundaryValueProblemsandIntegralEquations 50
1.5.1 SurfaceIntegralEquationsinShort 50
1.5.2 TheStandardElectricFieldIntegralEquationsofAntennaTheoryand
RadiatingNonuniformTransmission-LineSystems 51
1.5.2.1 Pocklington’sEquation 51
1.5.2.2 Halle´n’sEquation 52
1.5.2.3 Mixed-PotentialIntegralEquation 53
1.5.2.4 Schelkunoff’sEquation 55
References 55
2 NonuniformTransmission-LineSystems 57
2.1 MulticonductorTransmissionLines:GeneralEquations 59
2.1.1 GeometricRepresentationofNonuniformTransmissionLines 59
2.1.1.1 LocalCoordinateSystem 59
2.1.1.2 TangentialSurfaceVector 61
2.1.1.3 VolumeandSurfaceIntegrals 61
2.1.2 DerivationofGeneralizedTransmission-LineEquations 62
2.1.2.1 ContinuityEquation 62
2.1.2.2 ReconstructionoftheDensities 63
2.1.3 MixedPotentialIntegralEquation 64
2.1.3.1 Thin-WireApproximation 65
2.1.3.2 RepresentationasMatrixEquations 65
2.1.3.3 CurrentandChargeTrialFunction 65
2.1.3.4 GeneralizedTelegrapherEquationsandTLST 66
2.1.4 ComputationofGeneralizedTransmission-LineParameters 67
2.1.4.1 Parameters 67
2.1.4.2 SourceTerms 69
2.1.4.3 SolutionoftheExtendedTelegrapherEquations 69
2.1.4.4 ReturningtoVoltages? 70
2.1.4.5 DiscussionoftheNewParameters 72
2.1.4.6 AsymmetricParameterMatrices 73
2.1.5 NumericalEvaluationoftheParameters 74
2.1.5.1 StartingValuesfortheIteration 74
2.1.5.2 FirstIteration 75
2.1.5.3 TaylorSeriesExpansionoftheProductIntegral 76
2.1.5.4 EigenvalueDecomposition 78
2.1.5.5 DiscussionoftheNumericalMethods 79
2.2 GeneralCalculationMethodsfortheProductIntegral/Matrizant 79
2.2.1 PicardIteration 80
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Contents xi
2.2.2 Volterra’sMethodandtheProductIntegral 81
2.2.3 RecursionFormulasforLinearInterpolation 83
2.2.4 ApproximationbyPowerSeries 85
2.2.5 InterpolationfromDiagonalization 88
2.2.6 NumericalIntegration 91
2.2.6.1 Euler–CauchyMethod 93
2.2.6.2 Integrationbytrapezoidalrule 93
2.2.6.3 ExplicitRunge–KuttaMethod 94
2.2.6.4 HermiteIntegration 96
2.2.6.5 ImprovingAccuracybytheRombergMethod 98
2.2.6.6 ControllingStepSizeandError 99
2.2.7 RemarksonEfficiencyandtheChoiceofanAppropriateMethod 101
2.3 Semi-AnalyticandNumericalSolutionsforSelectedTransmissionLinesintheTLST 102
2.3.1 TheStraight,FiniteLengthWireAboveGround 102
2.3.2 TheSemi-InfiniteLine 107
2.3.3 FieldCouplingtoanInfiniteLine 109
2.3.4 TheSkewedWireTransmissionLine 111
2.3.5 ThePeriodicTransmissionLine 115
2.3.6 CrossTalkinaNonuniformMulticonductorTransmissionLine 118
2.4 AnalyticApproaches 119
2.4.1 ClassicalTelegrapherEquationsforNonuniformTransmissionLines 119
2.4.1.1 BasicDefinitionsandEquations 119
2.4.2 CalculationMethodsfortheGeneralSolution 122
2.4.2.1 ThePiecewise-ConstantApproximationoftheCharacteristic
ImpedanceMatrix 122
2.4.2.2 TheContinuousApproximationoftheCharacteristicImpedanceMatrix 124
2.4.2.3 CirculantNonuniformMTLs 128
2.4.2.4 GeneralApproachtoCalculatetheMatrizant 130
2.4.3 MatrizantReduction 132
References 135
3 ComplexSystemsandElectromagneticTopology 139
3.1 TheConceptofElectromagneticTopology 139
3.2 TopologicalNetworksandBLTEquations 143
3.2.1 WaveQuantities 144
3.2.2 BLT1Equation 145
3.2.3 BLT2Equation 147
3.2.4 AdmittanceRepresentation 149
3.3 TransmissionLinesandTopologicalNetworks 152
3.3.1 TransformationintoaPropagationMatrix 155
3.3.2 EquivalentScatteringMatrices 156
3.3.3 AdmittanceMatrix 157
3.4 Shielding 157
3.4.1 ModelfortheTransferImpedanceBasedonNMTLT 158
3.4.1.1 TransferParametersofCables 158
3.4.1.2 DeterminationofthePer-unit-lengthParameters 160
3.4.1.3 ComputationoftheTransferImpedance 163
3.4.2 ShieldingofanAnisotropicSphericalShell 168
3.4.2.1 ShieldingofaPlaneWall 171
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xii Contents
3.4.2.2 SphericalShell 171
3.4.2.3 ApplicationtoThinConductiveShells 172
3.4.2.4 Conclusions 177
References 177
4 TheMethodofPartialElementEquivalentCircuits(PEECMethod) 179
4.1 FundamentalEquations 179
4.1.1 MaxwellEquationsandRealMediaforInterconnections 179
4.1.2 MixedPotentialIntegralEquations(MPIE) 183
4.2 DerivationoftheGeneralizedPEECMethodintheFrequencyDomain 186
4.2.1 ThePEECEquationSystemintheFrequencyDomain 186
4.2.2 GeneralizedPartialElementsandCircuitInterpretation 191
4.3 ClassificationofPEECModels 193
4.3.1 ClassificationinDependenceonMedia 193
4.3.1.1 PEECModelsforConductors 193
4.3.1.2 PEECModelsforDielectrics 194
4.3.2 ClassificationinDependenceontheRelationoftheMaximumFrequencyof
InteresttotheDiscretizationLength 194
4.3.2.1 PEECModelswithGeneralizedPartialElements 194
4.3.2.2 PEECModelswithCenter-to-CenterRetardation 195
4.3.2.3 Quasi-StaticPEECModels 196
4.4 PEECModelsforthePlaneHalfSpace 197
4.5 GeometricalDiscretizationinPEECModeling 201
4.5.1 OrthogonalCells 201
4.5.2 NonorthogonalCells 202
4.5.3 TriangularCells 205
4.6 PEECModelsfortheTimeDomainandtheStabilityIssue 209
4.6.1 StandardPEECModelsfortheTimeDomain 210
4.6.2 GeneralRemarksonStabilityofPEECModelSolutions 211
4.6.3 StabilityImprovementofPEECModelswithCenter-to-CenterRetardation 212
4.6.4 StableTimeDomainPEECModelsbyParametricMacromodelingthe
GeneralizedPartialElements 214
4.6.4.1 StableTimeDomainPEECModelsbyFull-SpectrumConvolution
Macromodeling(FSCM) 214
4.6.4.2 StableTimeDomainPEECModelsbyMacromodelingUsing
Foster’sRationalFunctionsandCircuitSynthesis 219
4.7 SkinEffectinPEECModels 220
4.7.1 Cross-SectionalDiscretizationofWires 220
4.7.2 SkinEffectModelingbyMeansofaGlobalSurfaceImpedance 221
4.7.3 SkinEffectModelingbyMeansofaLocalMeanSurfaceImpedance 222
4.8 PEECModelsBasedonDyadicGreen’sFunctionsforConductingStructuresin
LayeredMedia 227
4.8.1 Motivation 227
4.8.2 TheDGFLM–PEECMethod 228
4.8.3 DGFLM–PEECModelfortheStriplineRegion 233
4.8.3.1 Green’sFunctionsfortheStriplineRegion 234
4.8.3.2 DiscussionoftheBehavioroftheGreen’sFunctions 235
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Contents xiii
4.8.3.3 FrequencyDomainDGFLM-PEECModel 236
4.8.3.4 DGFLM–PEECModelsintheTimeDomain 243
4.9 PEECModelsandUniformTransmissionLines 248
4.10 PowerConsiderationsinPEECModels 252
4.10.1 GeneralRemarks 252
4.10.2 PowerAnalysisofMagneticandElectricCouplings 253
4.10.3 PowerAnalysisofPEECModels 255
References 257
AppendixA: TensorAnalysis,IntegrationandLieDerivative 261
A.1 IntegrationOveraCurveandCovariantVectorsasLineIntegrands 261
A.2 IntegrationOveraSurfaceandContravariantVectorDensitiesasSurfaceIntegrands 263
A.3 IntegrationOveraVolumeandScalarDensitiesasVolumeIntegrands 264
A.4 Poincare´Lemma 265
A.5 Stokes’Theorem 266
A.6 LieDerivative 266
References 267
AppendixB: ElementsofFunctionalAnalysis 269
B.1 FunctionSpaces 270
B.1.1 MetricSpaces 270
B.1.2 LinearSpaces,VectorSpaces 272
B.1.3 NormedSpaces 273
B.1.4 InnerProductSpacesandPseudoInnerProductSpaces 274
B.1.5 HilbertSpaces 276
B.1.6 FiniteExpansionsandBestApproximation 277
B.1.7 TheProjectionTheorem 278
B.1.8 BasisofaHilbertSpace 278
B.2 LinearOperators 278
B.2.1 DefinitionofaLinearOperator,DomainandRangeofanOperator 278
B.2.2 BoundedOperatorsandtheNormofanOperator 279
B.2.3 ContinuousOperators 279
B.2.4 LinearFunctionals 279
B.2.5 TheRieszRepresentationTheorem 279
B.2.6 AdjointandPseudoAdjointOperators 280
B.2.7 CompactOperators 280
B.2.8 InvertibleOperators,ResolventOperator 280
B.2.9 Self-Adjoint,NormalandUnitaryOperators 281
B.3 SpectrumofaLinearOperator 281
B.3.1 StandardEigenvalueProblem,SpectrumandResolventSet 281
B.3.2 ClassificationofSpectrabyOperatorProperties 283
B.4 SpectralExpansionsandRepresentations 284
B.4.1 LinearIndependenceofEigenfunctions 284
B.4.2 SpectralTheoremforCompactandSelf-AdjointOperators 285
B.4.3 RemarksontheRelationBetweenDifferentialandIntegralOperators 286
B.4.4 ACommentonSobolevSpaces 287
References 287
