Analytical-Solutions-to-Eddy-Current-Probe-Coil-Pr

AnalyticalSolutionstoEddyCurrentProbeCoilProblemsC.V.DoddandW.E.DeedsCitation:JournalofAppliedPhysics39,2829(1968);doi:10.1063/1.1656680Viewonline::=pdfcovPublishedbytheAIPPublishingArticlesyoumaybeinterestedinCHARACTERIZINGANDMODELINGARRAYSOFEDDYCURRENTPROBESAIPConf.Proc.1335,341(2011);10.1063/1.3591873MODELDEVELOPMENTFORBROADBANDSPIRALCOILEDDYCURRENTPROBESAIPConf.Proc.1096,335(2009);10.1063/1.3114223SolutiontotheEddyCurrentProblemforTypeIISuperconductorsbyRelaxationMethodAIPConf.Proc.1048,286(2008);10.1063/1.2990914Recentadvancesinmodelingeddy-currentprobesAIPConf.Proc.615,423(2002);10.1063/1.1472829MethodofsolutionofforwardproblemsineddycurrenttestingJ.Appl.Phys.77,4903(1995);10.1063/1.359295[Thisarticleiscopyrightedasindicatedinthearticle.ReuseofAIPcontentissubjecttothetermsat:]IP:117.32.153.155On:Mon,21Apr201414:40:40JOURNALOFAPPLIEDPHYSICSVOLUME39,NUMBER6MAY196RAnalyticalSolutionstoEddy-CurrentProbe-CoilProblems*C.V.DODDMetalsandCeramicsDivision,OakRidgeNationalLaboratory,OakRidge,TennesseeANDW.E.DEEDSConsultantfromtheUniversityofTennessee,KnoX1Jille,Tennessee(Received28September1967;infinalform18January1968)Solutionshavebeenobtainedforaxiallysymmetriceddy-currentproblemsintwoconfigurationsofwideapplicability.Inbothcases,theeddycurrentsareassumedtobeproducedbyacircularcoilofrectangularcrosssection,drivenbyaconstantamplitudealternatingcurrent.Onesolutionisforacoilaboveasemi­infiniteconductingslabwithaplanesurface,coveredwithauniformlayerofanotherconductor.Thissolutionincludesthespecialcasesofacoilaboveasingleinfiniteplaneconductororaboveasheetoffinitethickness,aswellasthecaseofonemetalcladonanother.Theothersolutionisforacoilsurroundinganinfinitelylongcircularconductingrodwithauniformlythickcoatingofanotherconductor.Thisincludesthespecialcasesofacoilaroundaconductingtubeorrod,aswellasonemetalcladonarodofanothermeta!.Thesolutionsareintheformofintegralsoffirst-orderBesselfunctionsgivingthevectorpotential,fromwhichtheotherelectromagneticquantitiesofinterestcanbeobtained.Thecoilimpedancehasbeencalculatedforthecaseofacoilaboveatwo-conductorplane.Theagreementbetweenthecalculatedandexperimentalvaluesisexcellent.I.INTRODUCTIONElectromagneticproblemsareusuallydividedintothreecategories:lowfrequency,intermediatefrequency,andhighfrequency.Atlowfrequencies,staticcon­ditionsareassumed;athighfrequencies,waveequa­tionsareused.Bothoftheseregionshavebeenstudiedextensively.However,intheintermediatefrequencyrange,wherediffusionequationsareused,veryfewproblemshaveactuallybeensolved.Eddy-currentcoilproblemsfallintothisintermediatefrequencyregion.Thispaperpresentsanaccuratetechniqueforanalyzingtheproblemsofeddy-currenttesting.Eddy-currenttestinghasbeenusedinindustryformanyyears.Asearlyas1879,Hughes!usedaninduc­tioncoiltosortmetals.Therehavebeennumerousarticlesonthetestingofmaterialswitheddycurrents.Someofthefirstpapersdealingwithboththetheoryandthepracticalaspectsofeddy-currenttestingarebyForster,2ForsterandStambke,3andForster.4Inthisseriesofpapers,analysesaremadeofacoilaboveaconductingsurface,assumingthecoiltobeamagneticdipole,andofaninfinitecoilencirclinganinfiniterod.Hochschild5alsogivesananalysisofaninfinitecoilincludingsomeeddy-currentdistributionsinthemetal.WaidelichandRenken6madeananalysisofthecoilimpedanceusinganimageapproach.Theirtheoreticalresultsagreedwellwiththeoryforrelativelyhighfrequencies.Libby7presentedatheoryinwhichheassumedthecoilwasatransformerwithanetworktied*ResearchsponsoredbytheU.S.AtomicEnergyCommissionundercontractwiththeUnionCarbideCorporation.1D.E.Hughes,Phi!.Mag.8,50(1879).2FriedrichForster,Z.Metallk.43,163(1952).3FriedrichForsterandKurtStambke,Z.Metallk.45,166(1954).4FriedrichForster,Z.Metallk.45,197(1954).6R.Hochschild,ProgressinNondestructiveTesting(TheMac­millanCompany,NewYork,1959),Vo!'1.6D.L.WaidelichandC.J.Renken,Proc.Nat!.Electron.Conf.12,188(1956).7H.L.Libby,BroadbandElectromagneticTestingMethods,HanfordAtomicProductsOperation,HW-59614(1959).tothesecondary.Thisnetworkrepresentationgavegoodresultswhencomparedtoexperiment.Thediffusionofeddy-currentpulses(AtwoodandLibby8)canberepresentedinthismanner.Russelletal.9gaveananalysisofacupcorecoilwheretheyassumedthefluxwasentirelycoupledintotheconductor.Thesemiempiricalresultsagreedfairlywellwiththeex­perimentalmeasurements.Vein,lOCheng,l1andBur­rowS!2gavetreatmentsbasedondelta-functioncoils,andBurrowscontinuedwiththedevelopmentofaneddy-currentflawtheory.DoddandDeeds,t3Dodd,t4andDodd!5gavearelaxationtheorytocalculatethevectorpotentialofacoilwithafinitecrosssection.Hereweextendaclosed-formsolutiontosuchcoils.Thevectorpotentialisusedasopposedtotheelectricandmagneticfields.ThedifferentialequationsforthevectorpotentialmaybederivedfromMaxwell'sequations,withtheassumptionofcylindricalsym­metry.Thisdifferentialequationwillthenbesolvedtoobtainaclosed-formsolution.Fortheclosed-formsolution,sinus