Mathematical Models and Numerical Simulation in Electromagnetism PDF
Preview Mathematical Models and Numerical Simulation in Electromagnetism
Alfredo Bermúdez Dolores Gómez Pilar Salgado TT Mathematical Models and Numerical XX Simulation EE in Electromagnetism TT II NN ABC UU UNITEXT – La Matematica per il 3+2 Volume 74 Forfurthervolumes: http://www.springer.com/series/5418 · · Alfredo Bermúdez Dolores Gómez Pilar Salgado Mathematical Models and Numerical Simulation in Electromagnetism AlfredoBermúdez DoloresGómez DepartmentofAppliedMathematics DepartmentofAppliedMathematics UniversidadedeSantiagodeCompostela UniversidadedeSantiagodeCompostela Spain Spain PilarSalgado DepartmentofAppliedMathematics UniversidadedeSantiagodeCompostela Spain UNITEXT–LaMatematicaperil3+2 ISSN2038-5722 ISSN2038-5757(electronic) ISBN978-3-319-02948-1 ISBN978-3-319-02949-8(eBook) DOI10.1007/978-3-319-02949-8 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013949325 ©SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart ofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordis- similarmethodologynowknownorhereafterdeveloped.Exemptedfromthislegalreservationare briefexcerptsinconnectionwithreviewsorscholarlyanalysisormaterialsuppliedspecificallyfor thepurposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaser ofthework.Duplicationofthispublicationorpartsthereofispermittedonlyundertheprovisions oftheCopyrightLawofthePublisher’slocation,initscurrentversion,andpermissionforusemust alwaysbeobtainedfromSpringer.PermissionsforusemaybeobtainedthroughRightsLinkatthe CopyrightClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispub- licationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateof publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibility foranyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied, withrespecttothematerialcontainedherein. CoverDesign:BeatriceB,Milano TypesettingwithLATEX:PTP-Berlin,ProtagoTEX-ProductionGmbH,Germany(www.ptp-berlin.de) SpringerisapartofSpringerScience+BusinessMedia(www.springer.com) Toourstudents Preface Thismonographhasbeendevelopedasabasicsupportforagraduatecourseinelec- tromagnetismorientedtonumericalsimulation.Thus,theemphasisisonthegeneral equations rather that in solving analytically particular problems. The main goal is thatthereaderlearnsabouttheboundary-valueproblemsofpartialdifferentialequa- tionsthatshouldbesolved,ingeneralbyusingnumericalmethods,inordertoper- formcomputersimulationofelectromagneticprocesses.Thebookismoreoriented to electrical engineering applications rather than to wave propagation problems re- latedtotelecommunicationdevices.Unlikemostbooksonthesubject,itgoesfrom thegeneraltothespecific,namely,fromthefullMaxwell’sequationstotheparticu- larcasesofelectrostatics,directcurrent,magnetostaticsandeddycurrentsmodels.It alsocontainsafirstpartdevotedtoelectricalcircuittheorybasedonordinarydiffer- ential equations. Apartfromstandard exercises related toanalytical calculus, some othersorientedtoreal-lifeapplicationsarealsoincluded. Thebookisstructuredinthreeparts.Thefirstoneisdevotedtolumpedparameter modelssoitincludesanelementarycircuittheory.Itconsistsofthreechapters.Chap- ter1isdevotedtorecalltheharmonicoscillator.Theuseofcomplexfunctionstointe- gratethemodelisemphasizedandthenotionofimpedanceisintroduced.InChap.2 themechanical-electricalanalogyisexploitedtostudyseriesRLCcircuits.Chapter3 concernsmoregenerallinearcircuits.Theyaremodelledasdigraphsleadingtolinear systemsofintegro-differentialequations.Numericalsolutionisaddressedbothinthe timeandinthefrequencydomains. The second part concerns distributed parameter models. In Chap. 4 Maxwell’s equationsintheemptyspacearerecalled.Thentheyarereducedtovectorwaveequa- tionsintermsofeithertheelectricfieldorthemagneticinduction.Theequationsfor the harmonic case are also obtained and then plane wave solutions are calculated. Chapter5includesseveralexamplesthatcanbesolvedbyhandandChap.6extends Maxwell’sequationstomaterialmedia.Conceptsaselectricpolarization andmag- netizationaredescribedandtheirinfluenceintheelectromagneticfieldisanalyzed. Next, the particular cases of electrostatics, direct current, magnetostatics and eddy currentsareconsideredfromChaps.7to10.Theweakformulationofeachproblem mostsuitabletoperformcomputersimulationofelectromagneticprocessesischosen viii Preface anddetailed.Moreover,Chap.10includesabriefsectiononcouplingdistributedand lumpedmodels.ThelastchapterofPartIIisdevotedtononlinearmagnetics.Thisis thesubjectofChap.11wherewerecallthehysteresisbehavioranditsmathematical modellingfollowingthePreisachmethodology. Thethirdpartofthebookisdevotedtonumericalsimulationofelectromagnetic problems.ThenumericalanalysisofMaxwell’sequationshasbeenextensivelyde- velopedduringthelastdecadesbymathematiciansandengineers.Thuswecanmen- tionthebooksbyAlonsoandValli[4],Bossavit[23],ChariandSilvester[31],Jin [46],Monk[55],SilvesterandFerrari[64],justtolistarepresentativesamplingof booksdealingwiththissubject. Numerical methods can address situations that otherwise would become totally unmanageable given the complexity or impracticability of its analytical solution. Moreover,withasuitablepostprocessingoftheresults,itispossibletoqualitatively and quantitatively understand what is really happening in the system under study. Among the numerical methods found in the literature to approximate Maxwell’s equations, the finite element method (FEM) is the most extended. Its main advan- tagesareitsgeometricflexibilityandtherichnessintheoreticalmathematical tools usefultoanalyzetheapproximationoftheproblem. In recent years, the proliferation of commercial software packages attests to the growing use of numerical simulation in industry. This is due, in large part, to its usefulnessinthedesignprocessofproductsreducingthe“trialanderror”cycleby evaluatingmultipledesignsinthecomputer. Amongthecommercialprogramsforthesimulationofelectromagneticproblems, wecanmentionFlux2Dand3DfromCedrat[69],AnsysMultiphysics[68],Mag- netfromInfolytica[71],OperafromCobbham[73]orComsolMultiphysics[70],to name the most used in low-frequency applications. Usually, due to their high cost, the use of these packages is mainly restricted to the research field. However, com- puterprogramsandpackagesarealsoincreasinglybeingintegratedintothegraduate andundergraduatecurriculum,servingasvirtuallaboratoriestoteachandreinforce concepts. In this book we will use MaxFEM [72], a free software package developed in theResearchGroupofMathematicalEngineeringfromUniversidadedeSantiagode Compostela(USC),Spain. MaxFEMisanopensoftwarefornumericalsimulationinelectromagnetismbased onthefiniteelementmethod.Itcontainsseveralapplicationscoveringelectrostatics, directcurrent,magnetostaticsandeddycurrentsintwoand/orthreedimensionsand inCartesianand/orcylindricalcoordinates.Inadditiontothecodeitself,MaxFEM includesauserinterfacetofacilitateitshandlinganditisfittedwithanon-linehelp manual.Itcanbedownloadedfromhttp://sourceforge.net/projects/maxfem/. Byintroducingthesecompanioncomputationaltoolsinthetextbook,ourgoalis tostrikeabalancebetweentheclassicalteachingofelectromagnetism,whichempha- sizesitsphysicalandmathematicalfoundations,andanumericalsimulationapproach whichfocusesonsolvingrealisticproblemsusingsoftwarepackages. Preface ix Withthisgoalinmind,andaccordingtoPartsIandII,wehavestructuredPartIII ofthebookinfivechaptersfrom12to16devoted,respectively,toelectricalcircuits, electrostatics,directcurrent,magnetostaticsandeddycurrentsproblems. Finally,somebasicandcomplementarysubjectslikegraphtheory,vectorcalcu- lusorfunctionspacesareincludedinappendicesaswellasMaxwell’sequationsin Lagrangiancoordinates. Most of the research in mathematical electromagnetism of the authors has been developedincollaborationwithprofessorRodolfoRodríguezfromtheUniversityof Concepción (Chile). Along many years we have enjoyed his intelligence, kindness andfriendship.Someofourcommonformerstudents,asBibianaLópezandPablo Venegashavecarefullyreadseveralchaptersofthemanuscriptorwrittencomputer programs now included in MaxFEM. We also express our gratitude to the rest of the team that has written this software: M. Carmen Muñiz, Francisco Pena, Marta Piñeiro,VíctorSandeandRodrigoValiña.Finally,wealsoacknowledgetheexcel- lentworkbyMartaPiñeiroandVíctorSandeforprovidinguswithseveralexamples andfiguresincludedinthebook. SantiagodeCompostelaandLugo AlfredoBermúdez July2013 DoloresGómez PilarSalgado Contents PartI LumpedParameterModels:ElectricCircuitTheory 1 Theharmonicoscillator......................................... 3 1.1 Amechanicalsystem....................................... 3 1.2 Usingcomplexfunctions.Undampedoscillations ............... 5 1.3 Dampedoscillations........................................ 6 1.4 Energybalance............................................ 8 1.5 Forcedoscillations......................................... 10 1.6 Resonance................................................ 13 1.7 Thefirstorderequation ..................................... 14 2 TheseriesRLCcircuit.......................................... 21 2.1 Mathematicalmodel ....................................... 21 2.2 Energybalance............................................ 23 2.3 Harmonicsource.Impedance.Resonance...................... 23 2.4 Activeandreactivepower................................... 25 2.5 Thecircuitwithoutcapacitor ................................ 26 3 Linearelectricalcircuits ........................................ 33 3.1 Mathematicalmodel ....................................... 33 3.1.1 Theincidencematrix.Properties ...................... 34 3.1.2 ThefirstKirchhoff’slaw ............................ 35 3.1.3 Constitutivelaws................................... 35 3.1.4 ThesecondKirchhoff’slaw.......................... 37 3.2 Numericalsolutionofthecircuitequations..................... 38 3.2.1 Existenceofsolutiontothediscreteproblem............ 39 3.2.2 Generatorswithoutinternalresistance ................. 41 3.3 Thecaseofgivenpotentials ................................. 43 3.4 Harmonicregime.Impedancematrix.......................... 44 xii Contents PartII DistributedParameterModels 4 Maxwell’sequationsinfreespace ................................ 53 4.1 Anintroductiontodistributedmodelsforwavepropagation.The one-dimensionalwaveequation .............................. 53 4.1.1 Energyconservation ................................ 55 4.1.2 Harmonicsolutionstotheone-dimensionalwaveequation 56 4.2 Chargesandcurrents ....................................... 56 4.3 Electricandmagneticfields ................................. 57 4.4 Maxwell’sintegralequationsinfreespace ..................... 58 4.5 Maxwell’sequationsindifferentialforminfreespace ........... 58 4.6 Waveequationsforelectricandmagneticfields................. 59 4.7 Harmonicregime .......................................... 60 4.8 Behavioratinfinity:theSilver-Müllerconditions ............... 61 4.9 Planewavesinemptyspace ................................. 61 4.10 Wavepolarization ......................................... 64 5 SomesolutionsofMaxwell’sequationsinfreespace................ 67 5.1 Electrostaticfields ......................................... 67 5.1.1 Electricscalarpotential.............................. 67 5.1.2 PointchargeQattheorigin.Coulomb’slaw ........... 68 5.1.3 Ballofradiusr withauniformchargedensityρ ....... 69 0 V 5.1.4 Alonglinechargewithuniformlinedensityρ.......... 70 l 5.1.5 Infiniteplanarchargewithuniformsurfacedensityρ .... 72 S 5.2 Magnetostaticfields ....................................... 73 5.2.1 Roundtoroidalcoreofcircularcrosssection ............ 74 5.2.2 Infinitelylongsolenoid.............................. 74 6 Maxwell’sequationsinmaterialregions .......................... 77 6.1 Aone-dimensionalmodel................................... 77 6.1.1 Reflectionandtransmissionofwaves .................. 77 6.1.2 Dampedwaves..................................... 79 6.2 Conductorsandinsulators................................... 80 6.2.1 Electricalconductivity.Ohm’slaw .................... 80 6.2.2 Electricpolarization.Dielectrics ...................... 80 6.2.3 Theelectricdipole.................................. 81 6.2.4 Thepolarizationvector.............................. 83 6.2.5 Electricsusceptibilityandelectricpermittivity .......... 84 6.2.6 ElectricGauss’lawformaterials...................... 84 6.3 Magneticmaterials......................................... 85 6.3.1 Themagneticdipole ................................ 85 6.3.2 Themagnetizationvector ............................ 90 6.3.3 Magneticsusceptibility.Magneticpermeability ......... 91 6.3.4 MagneticGauss’lawformaterials .................... 91 6.4 Ampère’slawformaterials.................................. 92
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