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ElectromagneticShielding

TheoryandApplications

SecondEdition

SalvatoreCelozzi

RodolfoAraneo

PaoloBurghignoli

GiampieroLovat

ElectricalEngineeringDepartment“LaSapienza”UniversityRome,Italy

EditionHistory

JohnWiley&Sons,Inc.(1e,2008)

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LibraryofCongressCataloging-in-PublicationData

Names:Celozzi,Salvatore,author.

Title:Electromagneticshielding:theoryandapplications/Salvatore Celozzi,RodolfoAraneo,PaoloBurghignoli,GiampieroLovat.

Description:Secondedition.|Hoboken,NJ,USA:Wiley,2023.|Series: Wileyseriesinmicrowaveandopticalengineering|Includes bibliographicalreferencesandindex.

Identifiers:LCCN2022052851(print)|LCCN2022052852(ebook)|ISBN 9781119736288(hardback)|ISBN9781119736295(adobepdf)|ISBN 9781119736301(epub)

Subjects:LCSH:Shielding(Electricity)|Magneticshielding.

Classification:LCCTK454.4.M33C452022(print)|LCCTK454.4.M33 (ebook)|DDC621.34–dc23/eng/20221107

LCrecordavailableathttps://lccn.loc.gov/2022052851

LCebookrecordavailableathttps://lccn.loc.gov/2022052852

CoverDesign:Wiley

CoverImage:©Zita/Shutterstock

Setin9.5/12.5ptSTIXTwoTextbyStraive,Chennai,India

Contents

AbouttheAuthors ix

Preface xiii

1ElectromagneticsBehindShielding 1

1.1Definitions 1

1.2Notation,Symbology,andAcronyms 3

1.3MacroscopicElectromagnetismandMaxwell’sEquations 4

1.4ConstitutiveRelations 6

1.5DiscontinuitiesandSingularities 11

1.6InitialConditions,BoundaryConditions,andCausality 12

1.7Poynting’sTheoremandEnergyConsiderations 13

1.8FundamentalTheorems 16

1.8.1UniquenessTheorem 16

1.8.2ReciprocityTheorem 16

1.8.3EquivalencePrinciple 18

1.8.4Duality 19

1.8.5Symmetry 20

1.8.6ImagePrinciple 21

1.8.7Babinet’sPrinciple 21

1.9WaveEquations,Helmholtz’sEquations,Potentials,andGreen’s Functions 23

1.10BasicShieldingMechanisms 28

1.11SourceInsideorOutsidetheShieldingStructureandReciprocity 29 References 30

2ShieldingMaterials 33

2.1StandardMetallicandFerromagneticMaterials 33

2.2FerrimagneticMaterials 39

2.3FerroelectricMaterials 41

2.4ThinFilmsandConductiveCoatings 43

2.5OtherMaterialsSuitableforEMShieldingApplications 45

2.5.1StructuralMaterials 45

2.5.2ConductivePolymers 45

2.5.3ConductiveGlassesandTransparentMaterials 46

2.5.4Conductive(andFerromagneticorFerrimagnetic)Papers 46

2.6SpecialMaterials 46

2.6.1MetamaterialsandChiralMaterials 46

2.6.2CompositeMaterials 49

2.6.3Graphene 50

2.6.4OtherNanomaterials 53

2.6.5High-TemperatureSuperconductors 54 References 54

3FiguresofMeritforShieldingConﬁgurations 61

3.1(Local)ShieldingEffectiveness 61

3.2TheGlobalPointofView 64

3.3OtherProposalsofFiguresofMerit 65

3.4Energy-Based,Content-OrientedDefinition 69

3.5PerformanceofShieldedCables 69 References 70

4ShieldingEffectiveness:PlaneWaves 73

4.1ElectromagneticPlaneWaves:DefinitionsandProperties 73

4.2UniformPlaneWavesIncidentonaPlanarShield 75

4.2.1Transmission-LineApproach 76

4.2.2TheSinglePlanarShield 79

4.2.3Multiple(orLaminated)Shields 84

4.3PlaneWavesNormallyIncidentonCylindricalShieldingSurfaces 86

4.4PlaneWavesAgainstSphericalShields 93

4.5ExtensionoftheTLAnalogytoNear-FieldSources 94

4.5.1Examples 101 References 106

5ShieldingEffectiveness:Near-FieldSources 109

5.1Spectral-DomainApproach 109

5.1.1Maxwell’sEquationsintheSpectralDomain 110

5.1.2TM/TEDecompositionandEquivalentTransmissionLines 112

5.1.3SpectralDyadicGreen’sFunctions 116

5.1.4FieldEvaluationintheSpatialDomain 119

5.2LFMagneticShieldingofMetalPlates:ParallelLoop 122

5.2.1Spectral-DomainApproach 122

5.2.2VectorMagnetic-PotentialApproach 126

5.2.3ApproximateFormulations 127

5.3LFMagneticShieldingofMetalPlates:PerpendicularLoop 130

5.4LFMagneticShieldingofMetalPlates:ParallelCurrentLine 134 References 137

6TransientShielding 141

6.1PerformanceParameters:DefinitionsandProperties 141

6.2TransientSources:PlaneWavesandDipoles 145

6.2.1TransientUniformPlaneWaves 145

6.2.2TransientDipoles 148

6.3NumericalSolutionsviaInverse-FourierTransform 149

6.4AnalyticalSolutionsinCanonicalConfigurations 150

6.4.1TransientPlaneWavesonaSingle-LayerScreen 151

6.4.2TransientDipoles:TheCagniard–deHoopMethod 155

6.4.2.1ThinConductiveSheet 159

6.4.2.2GrapheneSheet 161

6.4.2.3Generalizations:ThickShields,MultilayeredShields 164 References 164

7NumericalMethodsforShieldingAnalyses 169

7.1Finite-ElementMethod 171

7.2MethodofMoments 187

7.3Finite-DifferenceTime-DomainMethod 208

7.4FiniteIntegrationTechnique 221

7.5Transmission-LineMatrixMethod 226

7.6PartialElementEquivalentCircuitMethod 230

7.7TestCaseforComparingNumericalMethods 239 References 242

8AperturesinPlanarMetalScreens 257

8.1HistoricalBackground 258

8.2StatementoftheProblem 259

8.3Low-FrequencyAnalysis:TransmissionThroughSmall Apertures 260

8.4TheSmallCircularAperture 261

8.4.1Bethe’sTheory 262

8.4.2Spectral-DomainFormulation 267

8.5SmallNoncircularApertures 269

8.6FiniteNumberofSmallApertures 269

8.7AperturesofArbitraryShape:Integral-EquationFormulation 272

8.8RulesofThumb 275 References 277

9Enclosures 283

9.1ModalExpansionofElectromagneticFieldsInsideaMetallic Enclosure 284

9.2OscillationsInsideanIdealSource-FreeEnclosure 287

9.3TheEnclosureDyadicGreenFunction 288

9.4ExcitationofaMetallicEnclosure 291

9.5DampedOscillationsInsideEnclosureswithLossyWallsandQuality Factor 292

9.6AperturesinPerfectlyConductingEnclosures 294

9.6.1Small-ApertureApproximation 295

9.6.2RigorousAnalysis:Integral-EquationFormulation 297

9.6.3Aperture-CavityResonances 299

9.7SmallLoadingEffects 301

9.8TheRectangularEnclosure 302

9.8.1SymmetryConsiderations 306

9.9ShieldingEffectivenessofaRectangularEnclosurewithan Aperture 307

9.9.1NumericalModels 307

9.9.2AnalyticalModels 311

9.10CaseStudy:RectangularEnclosurewithaCircularAperture 315

9.10.1ExternalSources:Plane-WaveExcitation 316

9.10.2InternalSources:ElectricandMagneticDipoleExcitations 321

9.11OverallPerformanceintheFrequencyDomain 326

9.12OverallPerformanceintheTimeDomain 328 References 332

10CableShielding 339

10.1TransferImpedanceinTubularShieldedCablesandAperture Effects 340

10.2RelationshipBetweenTransferImpedanceandShielding Effectiveness 345

10.3ActualCablesandHarnesses 347 References 348

11ComponentsandInstallationGuidelines 351

11.1Gaskets 351

11.2ShieldedWindows 355

11.3ElectromagneticAbsorbers 357

11.4ShieldedConnectors 358

11.5Air-VentilationSystems 358

11.6Fuses,Switches,andOtherSimilarComponents 359 References 359

12FrequencySelectiveSurfaces 363

12.1AnalysisofPeriodicStructures 364

12.1.1FloquetTheoremandSpatialHarmonics 364

12.1.2Plane-WaveIncidenceonaPlanar1DPeriodicStructure 366

12.1.3Plane-WaveIncidenceonaPlanar2DPeriodicStructure 367

12.1.4IntegralEquationFormulationforPlane-WaveIncidenceandPeriodic Green’sFunction 369

12.1.5DipoleExcitationofPlanar2DPeriodicStructure 373

12.2High-andLow-PassFSSs 376

12.3Band-PassandBand-StopFSSs 380

12.3.1Center-ConnectedElementsor N -PoleElements 381

12.3.2Loop-TypeElements 382

12.3.3Solid-Interior-TypeElements 382

12.3.4CombinationsandFractalElements 382

12.4RecentTrendsinFSSs 383

12.4.1MultilayerandCascadedFSSs 383

12.4.23-DFSSs 385

12.4.32.5-DFSSs 387

12.4.4ReconfigurableandActiveFSSs 387

12.5AbsorbingFSSs 388

12.5.1CircuitAnalogAbsorbers 389

12.5.2AbsorptiveFrequencySelectiveReflection/Transmission Structures 390

12.5.2.1AFSRStructures 391

12.5.2.2AFSTStructures(Frequency-SelectiveRasorbers) 391

12.6ModelingandDesignofFSSs 392 References 394

13ShieldingDesignGuidelines 409

13.1EstablishmentoftheShieldingRequirements 410

13.2AssessmentoftheNumberandTypesofFunctional Discontinuities 412

13.3AssessmentofDimensionalConstraintsandNon-Electromagnetic CharacteristicsofMaterials 413

13.4EstimationofShieldingPerformance 413 References 414

14UncommonWaysofShielding 417

14.1ActiveShielding 417

14.2PartialShields 422

14.3ChiralShielding 425

14.4MetamaterialShielding 426 References 432

AppendixAElectrostaticShielding 439

A.1BasicLawsofElectrostatics 440

A.2ElectrostaticTools:ElectrostaticPotentialandGreen’sFunctions 442

A.3ElectrostaticShields 446

A.3.1ConductiveElectrostaticShields 446

A.3.2DielectricElectrostaticShields 450

A.3.3ApertureEffectsinConductiveShields 455 References 457

AppendixBMagneticShielding 459

B.1MagneticShieldingMechanism 460

B.2CalculationMethods 463

B.3Boundary-ValueProblems 465

B.3.1SphericalMagneticConductingShield 465

B.3.2CylindricalMagneticConductingShieldinaTransverseMagnetic Field 471

B.3.3CylindricalMagneticConductingShieldinaParallelMagnetic Field 475

B.4FerromagneticShieldswithHysteresis 477 References 478

AppendixCStatisticalElectromagneticsforShieldingEnclosures 483

C.1StatisticalAnalyses 486

C.2Examples 489 References 495

AppendixDStandardsandMeasurementMethodsforShielding Applications 499

D.1MIL-STD285andIEEESTD-299 501

D.2NSA65-6andNSA94-106 506

D.3ASTME1851 506

D.4ASTMD4935 508

D.5MIL-STD461G 510

D.6CodeofFederalRegulations,Title47,Part15 517

D.7ANSI/SCTE48-3 520

D.8MIL-STD1377 521

D.9IECStandards 522

D.10ITU-TRecommendations 527

D.11AutomotiveStandards 529 References 535

Index 539

AbouttheAuthors

SalvatoreCelozzi wasborninRome,Italy,in1964.Hereceivedthe Laurea (cum laude)andPh.D.degreesfrom LaSapienza UniversityofRomein1988and1994, respectively.

HeisaFullProfessorattheUniversityofRome“LaSapienza”since2005.Hehas authoredmorethan150papersininternationaljournalsorconferenceproceedings,mainlyinthefieldsofelectromagneticshieldingandtransmissionlines.He istheco-authorofthefirsteditionofthebookElectromagneticShielding(Wiley, 2008).

Prof.CelozzihasbeentheChairoftheEMCChapteroftheIEEECentraland SouthItalySectionfrom1997to2006.HewastherecipientoftheBestSymposium PaperAwardin1998and2011attheIEEEEMCConferences.In2002,hewasthe recipientoftheIEEEEMCSocietyAward“CertificateofTechnicalAchievement” foroutstandingcontributionstotheEMCSociety,especiallyinthefieldofshieldingandtransmissionlinetheoryappliedtoprintedcircuitboards.Hewasserving asanAssociateEditorfortheIEEETRANSACTIONSON ELECTROMAGNETIC COMPATIBILITY from1995to2000andisservinginthesamerolesince2016.

RodolfoAraneo wasborninRome,Italy,in1975.Hereceivedthe Laurea (cum laude)andPh.D.degreesinelectricalengineeringfrom LaSapienza Universityof Romein1999and2002,respectively.HeisFullProfessorinthesameUniversity since2021.

In1999,hewasaVisitingStudentattheNationalInstituteofStandardsand Technology,Boulder,CO,USA,wherehewasengagedinTEMcellsandshielding.In2000,hewasaVisitingResearcherattheDepartmentofElectricaland ComputerEngineering,UniversityofMissouri-Rolla(UMR),Rolla,MO,USA, wherehewasengagedinprintedcircuitboardsandfinite-differencetime-domain techniques.HeiscurrentlyaFullProfessorat LaSapienza UniversityofRome. Hehasauthoredmorethan200papersininternationaljournalsandconference proceedings.Heistheco-authorofthefirsteditionofthebookElectromagnetic

Shielding(Wiley,2008).Heservesasareviewerforseveralinternationaljournals. Hisresearchinterestsincludeelectromagneticcompatibility,energyharvesting, piezotronicsbasedonpiezoelectricZnOnanostructures,grapheneelectrodynamics,developmentofnumericalandanalyticaltechniquesformodelinghigh-speed printedcircuitboards,shielding,transmissionlines,periodicstructures,and devicesbasedongraphene.

Dr.AraneowastherecipientofthePastPresident’sMemorialAwardin1999 fromtheIEEEElectromagneticCompatibilitySociety.HeiscurrentlyaGeneral ChairoftheIEEEInternationalConferenceonEnvironmentandElectricalEngineering.In2011,hewastherecipientoftheBestPaperSymposiumAwardatthe 2011IEEEEMC-SInternationalSymposiumonElectromagneticCompatibility.

PaoloBurghignoli wasborninRome,Italy,in1973.Hereceivedthe Laurea degree(cumlaude)inelectronicengineeringandthePh.D.degreeinappliedelectromagneticsfrom LaSapienza UniversityofRome,Rome,Italy,in1997and2001, respectively.

In1997,hejoinedtheDepartmentofElectronicEngineering,SapienzaUniversityofRome,whereheiscurrentlyAssociateProfessorwiththeDepartmentof InformationEngineering,ElectronicsandTelecommunications.In2004,hewasa VisitingResearchAssistantProfessorattheUniversityofHouston,Houston,TX, USA.From2010to2015,hewasanAssistantProfessorat LaSapienza University ofRome,wherehehasbeenanAssociateProfessorsince2015.In2017,hereceived theNationalScientificQualificationfortheroleofaFullProfessorofelectromagneticfieldsatItalianUniversities.Heiscurrentlyteachingcoursesinelectromagneticfields,advancedantennaengineering,andanalyticaltechniquesforwave phenomenaforB.Sc.,M.Sc.,andPh.D.programsintheICTareaat LaSapienza UniversityofRome.Hehasauthoredabout250articlesininternationaljournals, books,andconferenceproceedings.Hisresearchinterestsincludetheanalysisand designofplanarantennasandarrays,leakagephenomenainuniformandperiodicstructures,numericalmethodsforintegralequationsandperiodicstructures, propagationandradiationinmetamaterials,electromagneticshielding,transient electromagnetics,andgrapheneelectromagnetics.

Dr.Burghignoliwasarecipientofthe“GiorgioBarzilai”LaureaPrizein 1996–1997presentedbytheformerInstituteofElectricalandElectronicsEngineers(IEEE)CentralandSouthItalySection,the2003IEEEMTT-SGraduate Fellowship,andthe2005RajMittraTravelGrantforjuniorresearcherspresented attheIEEEAP-SSymposiumonAntennasandPropagation,Washington,DC, USA.HewasaSecretaryofthe12th EuropeanMicrowaveWeekin2009anda memberoftheScientificBoardandtheLocalOrganizingCommitteeofthe41st PhotonicsandElectromagneticsResearchSymposiumin2019.In2016and2020, theIEEEAntennasandPropagationSocietyrecognizedhimasanoutstanding

reviewerfortheIEEETRANSACTIONSON ANTENNASAND PROPAGATION.Heiscurrently anAssociateEditoroftheInstitutionofEngineeringandTechnology(IET) ElectronicsLetters andthe InternationalJournalofAntennasandPropagation (Hindawi).HeisaseniormemberoftheIEEE.

GiampieroLovat wasborninRome,Italy,in1975.Hereceivedthe Laurea degree (cumlaude)inelectronicengineeringandthePh.D.degreeinelectromagneticsin 2001andin2005,respectively,bothfrom LaSapienza UniversityofRome.Since 2010,heisanAssistantProfessorintheAstronautical,Electrical,andEnergetic EngineeringDepartment(DIAEE)atthesameUniversity,andsince2015,hehas beenqualifiedfortheroleofFullProfessorinaNationalScientificCompetition. Hehasbeendoingresearchactivityonleakywaves,periodicstructures,electrodynamicsofgraphene,transientelectromagnetics,andelectromagneticshielding. Heistheco-authorofmorethan170papersoninternationalbooks,journals,and conferenceproceedings.Heistheco-authorofthefirsteditionofthebookElectromagneticShielding(Wiley,2008).

In2005,hewastherecipientoftheYoungScientistAwardattheURSIGeneral AssemblyinNewDelhi,India.Heistheauthorof FastBreakingPapers,October 2007 inEEandCS,aboutmetamaterials(apaperthathadthehighestpercentage increaseincitationsinEssentialScienceIndicators,ESI).In2011,hewastherecipientoftheBestPaperSymposiumAwardatthe2011IEEEEMC-SInternational SymposiumonElectromagneticCompatibility.In2020,hehasbeenincludedin therankingofthetop2%worldscientistsrecentlypublishedonPlosOne(https:// doi.org/10.1371/journal.pbio.3000918).In2021,hehasbeenselectedasadistinguishedreviewerofIEEETRANSACTIONSON ELECTROMAGNETIC COMPATIBILITY.

Preface

ThisisthesecondeditionofourbookElectromagneticShielding.Asforthefirst edition,theassumedbackgroundofthereaderislimitedtostandardundergraduatetopicsinphysicsandmathematics.

Thiseditionhasbeencompletelyupdatedaccordingtotheoreticalandtechnologicalprogressinthefieldofelectromagneticshielding:thebooknowcomprises 14chaptersand4appendices;anoverviewoftheircontentisprovidednext,highlightingthechangeswithrespecttothefirstedition.Chapter1introducesthe basicelectromagnetictheorybehindelectromagneticshielding;thefundamental theoremsarenowalsopresentedinthetimedomain,whilesymmetry,duality, andBabinetprincipleshavebeenaddedfortheirimportanceintreatingclassicalshieldingproblems.Anintroductiontowaveequationsandpotentialsand Greenfunctionsinthetimedomainhasalsobeenaddedtocoverallthebasictools toanalyzetransientshieldingproblems.Chapter2describesthearsenalofconventionalandlessconventionalmaterialsforelectromagneticshielding;anentire newparagraphhasbeendevotedtotheelectromagneticdescriptionofgraphene, amaterialthathasbecomeoneofthemostattractivematerialsinthefieldofelectromagneticlossandabsorption.Chapter3introducesallthefiguresofmeritfor aquantitativeanalysisoftheshieldingperformanceofagivenstructure.Chapter 4coversthesubjectofelectromagneticshieldingofplanar,cylindrical,andsphericalscreensagainstplanewaves;anentirenewparagraphisdevotedtopresenting examplesthathelptounderstandtheextensionofthetransmissionlineanalogy tonear-fieldsources.Chapters5and6areentirelynew.Chapter5considersthe importanttopicofnear-fieldsourcesinthepresenceofplanarscreensandprovides anextensiveintroductiontothespectraldomainapproach.Chapter6introduces thereadertotheemergentareaoftransientshieldinganalysisofferingsuitable figuresofmerit,definingthefundamentaltransientsources,anddescribingthe numericalandanalyticalapproachestoanalyzebasicconfigurations.Chapter7 isathoroughintroductiontotheprincipalnumericalmethodsusedinelectromagneticsandareherepresentedinconnectionwithelectromagneticshielding

problems;moreover,manyillustrativeexampleshavebeenaddedtopresentthe characteristicsofdifferentnumericalapproacheswithparticularreferencetothe analysisoftheshieldingcharacteristicsofenclosures.Chapter8treatstheimportanttopicofelectromagneticpenetrationthroughaperturesinplanarscreens;the circularapertureisnowanalyzedindetail,illustratingtheverylastdevelopments. EnclosuresareconsideredinChapter9:thefundamentalcaseofarectangular enclosurewitharectangularapertureisnowdescribedindetail,consideringboth anumericalsolutionbasedontheMethod-of-Momentsandapproximateanalyticalapproaches.Moreover,twoentirelynewparagraphshavebeenaddedtopresent theoverallperformanceofanenclosureinbothfrequencyandtimedomains. Chapter10consistsinabriefintroductiontocableshielding,whileChapter11 dealswiththemostcommoncomponentsinstalledinmostofshieldingconfigurations.Chapter12introducestheimportanttopicoffrequency-selectivesurfaces; here,theanalysishasbeenextendedtoincludenotonlyaplanewaveexcitation butalsoafinitesource:forthisreason,anintegralequationformulationforthe solutionoftheelectromagneticproblemispresentedtogetherwithanintroduction totheperiodicGreenfunctioncalculation.Theliteratureonthistopicisgrowing almostexponentially,andaplethoraofnewstructureshavebeenproposedsothat somenewparagraphshavebeenaddedtopresentsomeofthemostrecentdevelopments.Finally,Chapter13coverssomeissuesinshieldingdesignprocedures, whileChapter14introducestosomeuncommonwaysofshielding:inparticular, theuseofmetamaterialshasbeenextensivelyreviewed.

Asinthefirstedition,twoappendicesaredevotedtoelectrostaticandmagnetostaticshielding.AnewAppendixChasbeenadded,whichprovidesanintroductiontotheuseofstatisticalelectromagneticsinelectromagneticshielding,while thelastappendix(AppendixD)coversstandardsandmeasurementprocedures andhasbeenobviouslyupdatedwithrespecttothefirstedition.

WewouldliketoacknowledgethesupportprovidedwithinWiley-Interscience byalltheStaffandinparticularthepatientassistancegivenbyTeresaNetzler.

Rome 29June2022

SalvatoreCelozzi

RodolfoAraneo PaoloBurghignoli GiampieroLovat

ElectromagneticsBehindShielding

Shieldinganelectromagneticfieldisacomplexandsometimesformidabletask. Thereasonsaremany,sincetheeffectivenessofanystrategyortechniqueaimed atthereductionoftheelectromagneticfieldlevelsinaprescribedregiondepends largelyuponthesourcecharacteristics,theshieldgeometry,andtheinvolved materials.Moreover,asitoftenhappenswhencommontermsareadoptedina technicalcontext,differentdefinitionsofshieldingexist.Inelectromagneticsthe shieldingeffectiveness (SE)isaconciseparametergenerallyappliedtoquantify shieldingperformance.However,avarietyofstandardsareadoptedforthe measurementortheassessmentoftheperformanceofagivenshieldingstructure. Unfortunately,theyallcallforveryspecificconditionsinthemeasurementsetup. Theresultsthereforeareoftenuselessifthesourceorsystemconfigurations differevenslightly.Lastamongthedifficultiesthatariseinthesolutionof actualshieldingproblemsarethedifficultiesinherentinboththesolutionofthe boundaryvalueproblemandthedescriptionoftheelectromagneticproblemin mathematicalform.

1.1Deﬁnitions

Toestablishacommonground,wewillbeginwithsomeusefuldefinitions.An electromagneticshieldcanbedefinedas[1]:

[A]housing,screen,orotherobject,usuallyconducting,thatsubstantially reducestheeffectofelectricormagneticfieldsononesidethereof,upon devicesorcircuitsontheotherside.

Thisdefinitionisrestrictivebecauseitimplicitlyassumesthepresenceofa “victim.”Thedefinitionisalsobasedonthemisconceptionthatthesourceand observationpointsareinoppositepositionswithrespecttotheshield,andit

1ElectromagneticsBehindShielding

includestheword“substantially”whosemeaningisobscureandintroducesan unacceptablelevelofarbitrariness.

Anotherdefinitionofelectromagneticshieldingevenmorerestrictiveis[2]:

[A]meansofpreventingtwocircuitsfromelectromagneticcouplingby placingatleastoneofthecircuitsinagroundedenclosureofmagnetic conductivematerial.

Themostappropriatedefinitionentailsabroadviewofthephenomenon:

[A]nymeansusedforthereductionoftheelectromagneticfieldina prescribedregion.

Noticethatnoreferencetoshape,material,andgroundingoftheshieldisnecessarytodefineitspurpose.

Ingeneral,electromagneticshieldingrepresentsawaytowardtheimprovement oftheelectromagnetic-compatibility(EMC)(definedasthecapabilityofelectronic equipmentorsystemstobeoperatedintheintendedelectromagneticenvironment atdesignlevelsofefficiency)performanceofsingledevices,apparatus,orsystems. Biologicalsystemsareincluded,forwhichitiscorrecttotalkabouthealthrather thanEMC.Electromagneticshieldingisalsousedtopreventsensitiveinformation frombeingintercepted,thatis,toguaranteecommunicationsecurity.

Electromagneticshieldingisnotimplementedonlyforsuchpurposes.Some sortofelectromagneticshieldingisalmostalwaysusedinelectricalandelectronic systemstoreducetheirelectromagneticemissionsandtoincreasetheirelectromagneticimmunityagainstexternalfields.Incaseswheretheavailablemethodologiesforreducingthesourcelevelsofelectromagneticemissionorstrengthening thevictimimmunityarenotavailableorarenotsufficienttoensurethecorrect operationofdevicesorsystems,areductionofthecouplingbetweenthesource andthevictim(eitherpresentoronlypotentiallypresent)isoftenthepreferred choice.

Theimmunityofthevictimsisgenerallyobtainedbymeansoffiltersthatare analogoustoelectromagneticshieldingwithrespecttoconductedemissionsand immunity.Themainadvantageoffiltersisthattheyare“local”devices.Thus, wherethenumberofsensitivecomponentstobeprotectedislimited,thecost offilteringmaybemuchlowerthanthatofshielding.Themaindisadvantageof usingafilteristhatitisabletoarrestonlyinterferenceswhosecharacteristics (e.g.,levelormodeoftransmission)aredifferentfromthoseofthedevice,sothe correctoperationinthepresenceofsometypesofinterferenceisnotguaranteed. Anotherseriousdisadvantageofthefilterisitsinadequacyoritslowefficiencyfor thepreventionofdatadetection.

Ingeneral,designingafilterismuchsimplerthandesigningashield.Thefilter designerhasonlytoconsiderthewaveformoftheinterference(intermsofvoltage orcurrent)andthevaluesoftheinputandoutputimpedance[3],whereasthe shielddesignermustincludealargeamountofinputinformationandconstraints, asitwillbediscussedthroughoutthebook.

Anyshieldinganalysisbeginsbyanaccurateexaminationoftheshieldgeometry [4–7].Althoughtheidentificationofthecouplingpathsbetweenthemainspace regionsisoftentrivial,sometimesitdeservesmorecare,especiallyincomplexconfigurations.Thecomplexityofashieldisassociatedwithitsshape,apertures,the componentsidentifiedasthemostsusceptible,thesourcecharacteristics,andso forth.Subdividingitsconfigurationintoseveralsubsystems(eachsimplerthanthe originaloneandinteractingwiththeothersinadefiniteway)isalwaysauseful approachtoidentifycriticalproblemsandfindwaystofixandimprovetheoverall performance[5].Thisapproachisbasedontheassumptionthateachsubsystem canbeanalyzed,andhenceitsbehaviorcanbecharacterized,independentlyofthe otherscomponents/subsystems.Forinstance,inthefrequencydomainandfora linearsubsystem,foreachcouplingpathandforeachsusceptibleelement,itispossibletoinvestigatethetransferfunction T (��) relatingtheexternalsourceinput S (��) andthevictimoutput V (��) characteristicsas V (��) = U (��) + T (��) S (��), where U (��) representsthesubsystemoutputintheabsenceofexternal-source excitation.Inthepresenceofmultilevelbarriers,thetransferfunction T (��) may ensuefromtheproductofthetransferfunctionsassociatedwitheachbarrierlevel. Theforegoingapproachcanbegeneralizedforabetterunderstandingofthe shieldingproblemincomplexconfigurations.However,itisoftensufficientto consideronlythemostcriticalsubsystemsandcomponents,ononehand,andthe mostimportantcouplingpaths,ontheotherhand,inordertosolvetheprincipal shieldingproblemsandthusimprovetheoverallperformance[8].Thegeneral approachisobviouslysuitableinadesigncontext.Acompleteanalysisofthe relationsbetweenshieldingandgroundingislefttothespecificliterature[4,9–11].

1.2Notation,Symbology,andAcronyms

Theabbreviationsandsymbolsusedthroughoutthebookarebrieflysummarized hereinordertomakeclearthestandardwehavechosentoadopt.Ofcourse,we willwarnthereaderanytimeanexceptionoccurs.

Scalarquantitiesareshowninitalictype(e.g., V and t),whilevectorsareshown inboldface(e.g., e and H);dyadicsareshowninboldfacewithanunderbar (e.g., �� and G).Aphysicalquantitythatdependsontimeandspacevariablesis indicatedwithalowercaseletter(e.g., e (r, t) fortheelectricfield).TheFourier transformwithrespecttothetimevariableisindicatedwiththecorresponding

1ElectromagneticsBehindShielding

uppercaseletter(e.g., E (r,��) whiletheFouriertransformwithrespecttothe spatialvariablesisindicatedbyatilde(e.g., ̃ e (k, t));whentheFouriertransform withrespecttobothtimeandspatialvariablesisconsidered,thetwosymbologies arecombined(e.g., E (k,��)).

Thesetsofspatialvariablesinrectangular,cylindrical,andsphericalcoordinates aredenotedby (x , y, z), (��,��, z),and (r ,��,�� ),respectively.TheboldfaceLatinletter u isusedtoindicateaunitvectorandasubscriptisusedtoindicateitsdirection: forinstance, (ux , uy , uz ), (u�� , u�� , uz ),and (ur , u�� , u�� ) denotetheunitvectorsin therectangular,cylindrical,andsphericalcoordinatesystem,respectively.

Wewillusethe“del”notation ∇ withthesuitableproducttypetoindicategradient(∇ [⋅]),curl(∇× [⋅])anddivergenceoperators(∇ ⋅ [⋅]);theLaplacianoperatorisindicatedas ∇2 [⋅].Theimaginaryunitisdenotedwithj = √ 1andthe asterisk ∗ asasuperscriptofacomplexquantitydenotesitscomplexconjugate.The realandimaginarypartsofacomplexquantityareindicatedbyRe [⋅] andIm [⋅], respectively,whiletheprincipalargumentisindicatedbythefunctionArg [ ].The base-10logarithmandthenaturallogarithmareindicatedbymeansofthelog (⋅) andln (⋅) functions,respectively.

Finally,throughoutthebook,theinternationalsystemofunitsSIisadopted, electromagneticisabbreviatedasEM,andshieldingeffectivenessasSE.

1.3MacroscopicElectromagnetismandMaxwell’s Equations

Acompletedescriptionofthemacroscopicelectromagnetismisprovidedby Maxwell’sequationswhosevalidityistakenasapostulate.Maxwell’sequations canbeusedeitherinadifferential(local)formorinanintegral(global)form, andtherehasbeenalongdebateoverwhichisthebestrepresentation(e.g., DavidHilbertpreferredtheintegralformwhereasArnoldSommerfeldfound moresuitablethedifferentialform,fromwhichspecialrelativityfollowsmore naturally[12]).Whenstationarymediaareconsidered,themaindifference betweenthetworepresentationsconsistsinhowtheyaccountfordiscontinuities ofmaterialsand/orsources.Basically,ifoneadoptsthedifferentialform,some boundaryconditionsatsurfacediscontinuitiesmustbepostulated;ontheother hand,iftheintegralformsarechosen,onemustpostulatetheirvalidityacross suchdiscontinuities[13,14].

Maxwell’sequationscanbeexpressedinscalar,vector,ortensorform,anddifferentvectorfieldscanbeconsideredasfundamental.Afulldescriptionofallthese detailscanbefound,e.g.,in[12].Inthisbookweassumethefollowingdifferential formoftheMaxwellequations:

Fromtheseequationsthecontinuityequationcanbederivedas

InthisframeworktheEMfield—describedbyvectors e (electricfield,unitof measureV/m), h (magneticfield,unitofmeasureA/m), d (electricdisplacement, unitofmeasureC/m2 ),and b (magneticinduction,unitofmeasureWb/m2 orT)— arisesfromsources j (electriccurrentdensity,unitofmeasureA/m2 )and ��e (electricchargedensity,unitofmeasureC/m3 ).Further,exceptforstaticfields,ifatime canbefoundbeforewhichallthefieldsandsourcesareidenticallyzero,thedivergenceequationsin(1.1)areaconsequenceofthecurlequations[12],sounderthis assumptionthecurlequationscanbetakenasindependent.

ItcanbeusefultomaketheMaxwellequationssymmetricbyintroducingfictitiousmagneticcurrentandchargedensities m and ��m (unitsofmeasureV/m2 andWb/m3 ,respectively),whichsatisfyacontinuityequationsimilarto(1.2)so that(1.1)canberewrittenas ∇×

Asitwillbeshownlater,theequivalenceprincipleindeedrequirestheintroductionofsuchfictitiousquantities.

ItisalsousefultoidentifyinMaxwell’sequationssome“impressed”source terms,whichareindependentoftheunknownfieldsandareinsteadduetoother externalsources(magneticsourcescanbeonlyofthistype).Such“impressed” sourcesareconsideredasknowntermsinMaxwell’sdifferentialequationsand indicatedbythesubscript“i.”Inthisconnection,(1.3)canbeexpressedas

1ElectromagneticsBehindShielding

Theimpressed-sourceconceptiswellknownincircuittheory.Forexample, independentvoltagesourcesarevoltageexcitationsthatareindependentofpossibleloads.

Althoughboththesourcesandthefieldscannothavetruespatialdiscontinuities,fromamodelingpointofview,itisusefultoconsideradditionallysources inoneortwodimensions:surface-andline-sourcedensitiescanbeintroducedin termsoftheDiracdeltadistribution �� ,as(singular)idealizationsofactualcontinuousvolumedensities[12,15].

Finally,inthefrequencydomain,Maxwell’scurlequationsareexpressedas

wheretheuppercasequantitiesindicateeithertheFouriertransformorthephasorsassociatedwiththecorrespondingtime-domainfields.Notethatinthistext thefollowingdefinitionoftemporalFouriertransformwillbeadopted:

withthecorrespondinginverseFouriertransform:

whereasinthephasordomainatime-harmonicdependenceexp (j��t ) isassumed:

where F0 ≥ 0and F = F0 ej�� isthephasorassociatedwith f (t).Thesamedefinitionsalsoapplyforvectorfunctions.

1.4ConstitutiveRelations

BydirectinspectionofMaxwell’scurlequationsin(1.1),itisimmediatelyclear thattheyrepresentsixscalarequationswith15unknownquantities.Withfewer equationsthanunknownsnouniquesolutioncanbeidentified(theproblemis saidtobeindefinite).Theadditionalequationsrequiredtomaketheproblemdefinitearethosethatdescribetherelationsamongthefieldquantities e, h, d, b,and j,enforcedbythemediumfillingtheregionwheretheEMphenomenaoccur.Such relationsarecalled constitutiverelations,andtheydependonthepropertiesofthe mediumsupportingtheEMfield.

Innon-movingmedia,withtheexclusionofmagnetoelectricandchiralmaterials,the d fielddependsonlyonthe e field, b dependsonlyon h,and j depends

onlyon e.Thesedependencesareexpressedasconstitutiverelations,withthe e and h fieldsregardedascausesandthe d, b,and j fieldsaseffects.

Ifalinearcombinationofcauses(withgivencoefficients)producesalinear combinationofeffects(withthesamecoefficients),themediumissaidtobe linear (otherwise nonlinear ).Ingeneral,theconstitutiverelationsaredescribed byasetofconstitutiveparametersandasetofconstitutiveoperatorsthatrelate theabove-mentionedfieldsinsidearegionofspace.Theconstitutiveparameters canbeconstantsofproportionalitybetweenthefields(themediumisthussaid isotropic),ortheycanbecomponentsinatensorrelationship(themediumissaid anisotropic).Iftheconstitutiveparametersareconstantwithinacertainregion ofspace,themediumissaid homogeneous inthatregion(otherwise,themedium isinhomogeneous).Iftheconstitutiveparametersareconstantwithtime,the mediumissaid stationary (otherwise,themediumisnonstationary).

Iftheconstitutiveoperatorsareexpressedintermsoftimeintegrals,themedium issaidtobe temporallydispersive.Iftheseoperatorsinvolvespaceintegrals,the mediumissaidtobe spatiallydispersive.Finally,wenotethattheconstitutive parametersmaydependonothernonelectromagneticpropertiesofthematerial andexternalconditions(temperature,pressure,etc.).

Thesimplestmediumis vacuum.Invacuumthefollowingconstitutiverelations hold:

Thequantities

arethefree-spacemagneticpermeabilityanddielectricpermittivity,respectively. Theirvaluesarerelatedtothespeedoflightinfreespace c0 through c0 = 1∕√��0 ��0 , whoseexactvalueis c0 = 2.99792458 ⋅ 108 m/s;theabovetwovaluesfor ��0 correspondtoapproximating c0 ≃ 3 108 m/sand c0 ≃ 2.998 108 m/s,respectively.

Foralinear,homogeneous,isotropic,andnondispersivematerial,theconstitutiverelationscanbeexpressedas

d (r, t) = ��e (r, t) , b (r, t) = �� h (r, t) , j (r, t) = �� e (r, t) , (1.10) where �� and �� arethemagneticpermeabilityanddielectricpermittivityofthe medium,respectively.Thesequantitiescanberelatedtothecorresponding

1UntilMay20th 2020thiswasanexactvalue.Sincethatdate,aredefinitionoftheSIbaseunits assignedtotheelementarycharge(e.g.,theelectroncharge)anexactvalue: qe = 1.602176634⋅ 10 19 C,therebymakingthereportedvalueofthevacuumpermeability ��0 anapproximate quantity.

1ElectromagneticsBehindShielding

free-spacequantitiesthroughthedimensionlessrelativepermeability ��r and relativepermittivity ��r ,suchthat �� = ��r ��0 and �� = ��r ��0 .Thedimensionless quantities ��m = ��r 1and ��e = ��r 1(knownasmagneticandelectricsusceptibilities,respectively)arealsoused.Thethirdequationof(1.10)expressesthe Ohmlawinlocalform,and �� istheconductivityofthemedium(unitofmeasure S/m).

Forsuchasimplemedium,thanksto(1.10),Maxwell’sequations(1.4)canbe rewrittenas

Ifthemediumisinhomogeneous, ��,��,or �� arequantitiesthatdependonthe vectorposition r.Ifthemediumisanisotropic(butstilllinearandnondispersive) theconstitutiverelationscanbewrittenas d (r, t) = �� ⋅ e (r, t) ,

where ��, ��,and �� arecalledthepermittivitytensor,thepermeabilitytensor,and theconductivitytensor,respectively(theyarespace-dependentquantitiesfor inhomogeneousmedia).

Forlinear,inhomogeneous,anisotropic,stationary,andtemporallydispersive materials,theconstitutiverelationbetween d and e isexpressedbyasuperposition integralas

Theconstitutiverelationsforotherfieldquantitieshavesimilarexpressions. Causalityisimpliedbytheupperlimit t intheintegrals(thismeansthattheeffect cannotdependonfuturevaluesofthecause).Ifthemediumisnonstationary, �� (r, t, t′ ) hastobeusedinsteadof �� (r, t t′ ).Theimportantconceptexpressed by(1.13)isthatthebehaviorof d atthetime t dependsnotonlyonthevalueof e atthesametime t butalsoonitsvaluesatallpasttimes,thusgivingrisetoa time-lagbetweencauseandeffect.Theupperlimitoftheintegralin(1.13)canbe extendedto +∞ byassumingthat �� (r, t, t′ ) = 0 whenever t′ > t,thusobtaining

a convolution integral.Hence,inthefrequencydomaintheconstitutiverelation (1.13)isexpressedas

where,withalittleabuseofnotation, �� (r,��) indicatestheFouriertransformof thecorrespondingquantityinthetimedomain.Theimportantpointtonotehere isthat,inthefrequencydomain,temporaldispersionisassociatedwithcomplex valuesoftheconstitutiveparameters;causalityestablishesarelationshipbetween theirrealandimaginaryparts(knownastheKramers–Kronigrelation)[12]for whichneitherpartcanbeconstantwithfrequency.

Finally,ifthemediumisalsospatiallydispersive(andnonstationary),theconstitutiverelationtakestheform

where V indicatesthewholethree-dimensionalspace;asbefore,similarexpressionsholdfortheconstitutiverelationsofotherfieldquantitiesaswell.Theintegraloverthevolume V in(1.15)expressesthephysicalphenomenonforwhich theeffectatthepoint r dependsonthevalueofthecauseinalltheneighboringpoints r′ .Animportantpointisthatifthemediumisspatiallydispersivebut homogeneous,theconstitutiverelationsinvolveaconvolutionintegralinthespace domain.Thereforetheconstitutiverelationsinalinear,homogeneous,andstationarymediumfortheFouriertransformsofthefieldswithrespecttobothtime andspacecanbewrittenas

Veryoften,inthefrequencydomain,thecontributionsinMaxwell’s equations(1.5)fromtheconductivitycurrentandtheelectricdisplacement arecombinedinauniquetermbyintroducinganequivalentcomplexpermittivity.Forsimplicity,weconsiderisotropicmaterialsforwhichcomplexpermittivity isascalarquantitydefinedas

Thuswecanrewrite(1.5)inadualformas

Alternativelytotheconstitutiveparameters ��c and �� ,themediumcanalsobe describedbythe(possiblycomplex) mediumwavenumber

1ElectromagneticsBehindShielding

(wheretheprincipalbranchofthesquarerootischosensothattheimaginarypart of k isnonpositive)andthe(possiblycomplex) intrinsicimpedance

(wheretheprincipalbranchofthesquarerootischosensothattherealpartof �� isnonnegative)[16].Inparticular,thefree-spacewavenumberandthefree-space impedanceare

and

respectively.Accordingly,thefree-spacewavelengthisdefinedas

Finally,itisimportanttonotethatforthestudyofelectromagnetisminmatter,theEMfieldcanberepresentedbyfourvectorsotherthan e, h, d,and b (providedthatthenewvectorsarealinearmappingofthesevectors).Inparticular,thecommonalternativeistousevectors e, b, p,and m (nottobeconfused withthemagneticcurrentdensity),wherethenewvectors p and m arecalled polarizationandmagnetizationvectors,respectively,andMaxwell’sequationsare consequentlywrittenas

From(1.1)and(1.24),itfollowsthat

or,inthefrequencydomain,