Mathematical Models and Numerical Simulation in Electromagnetism

440 Pages·2014·13.752 MB·English

by  Alfredo Bermúdez, Dolores Gómez, Pilar Salgado (auth.)

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
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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