(cid:2) MATHEMATICAL FOUNDATIONS FOR LINEAR CIRCUITS AND SYSTEMS IN ENGINEERING (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) MATHEMATICAL FOUNDATIONS FOR LINEAR CIRCUITS AND SYSTEMS IN ENGINEERING JOHNJ.SHYNK (cid:2) (cid:2) DepartmentofElectricalandComputerEngineering UniversityofCalifornia,SantaBarbara (cid:2) (cid:2) Copyright©2016byJohnWiley&Sons,Inc.Allrightsreserved PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey PublishedsimultaneouslyinCanada Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformor byanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,exceptas permittedunderSection107or108ofthe1976UnitedStatesCopyrightAct,withouteithertheprior writtenpermissionofthePublisher,orauthorizationthroughpaymentoftheappropriateper-copyfeeto theCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,(978)750-8400,fax (978)750-4470,oronthewebatwww.copyright.com.RequeststothePublisherforpermissionshould beaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ 07030,(201)748-6011,fax(201)748-6008,oronlineathttp://www.wiley.com/go/permissions. LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbesteffortsin preparingthisbook,theymakenorepresentationsorwarrantieswithrespecttotheaccuracyor completenessofthecontentsofthisbookandspecificallydisclaimanyimpliedwarrantiesof (cid:2) merchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedorextendedbysales (cid:2) representativesorwrittensalesmaterials.Theadviceandstrategiescontainedhereinmaynotbesuitable foryoursituation.Youshouldconsultwithaprofessionalwhereappropriate.Neitherthepublishernor authorshallbeliableforanylossofprofitoranyothercommercialdamages,includingbutnotlimitedto special,incidental,consequential,orotherdamages. Forgeneralinformationonourotherproductsandservicesorfortechnicalsupport,pleasecontactour CustomerCareDepartmentwithintheUnitedStatesat(800)762-2974,outsidetheUnitedStatesat (317)572-3993orfax(317)572-4002. Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprintmay notbeavailableinelectronicformats.FormoreinformationaboutWileyproducts,visitourwebsiteat www.wiley.com. LibraryofCongressCataloging-in-PublicationDataappliedfor: ISBN:9781119073475 Setin10/12pt,TimesLTStdbySPiGlobal,Chennai,India. PrintedintheUnitedStatesofAmerica 10987654321 1 2016 (cid:2) (cid:2) ToTokie InmemoryofN.andA. (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) CONTENTS Preface xiii NotationandBibliography xvii (cid:2) (cid:2) AbouttheCompanionWebsite xix 1 OverviewandBackground 1 1.1 Introduction, 1 1.2 MathematicalModels, 3 1.3 FrequencyContent, 12 1.4 FunctionsandProperties, 16 1.5 DerivativesandIntegrals, 22 1.6 Sine,Cosine,and𝜋, 33 1.7 Napier’sConstanteandLogarithms, 38 PARTI CIRCUITS,MATRICES,ANDCOMPLEXNUMBERS 51 2 CircuitsandMechanicalSystems 53 2.1 Introduction, 53 2.2 Voltage,Current,andPower, 54 2.3 CircuitElements, 60 2.4 BasicCircuitLaws, 67 2.4.1 Mesh-CurrentandNode-VoltageAnalysis, 69 (cid:2) (cid:2) viii CONTENTS 2.4.2 EquivalentResistiveCircuits, 71 2.4.3 RCandRLCircuits, 75 2.4.4 SeriesRLCCircuit, 78 2.4.5 DiodeCircuits, 82 2.5 MechanicalSystems, 85 2.5.1 SimplePendulum, 86 2.5.2 MassonaSpring, 92 2.5.3 ElectricalandMechanicalAnalogs, 95 3 LinearEquationsandMatrices 105 3.1 Introduction, 105 3.2 VectorSpaces, 106 3.3 SystemofLinearEquations, 108 3.4 MatrixPropertiesandSpecialMatrices, 113 3.5 Determinant, 122 3.6 MatrixSubspaces, 128 3.7 GaussianElimination, 135 3.7.1 LUandLDUDecompositions, 146 3.7.2 BasisVectors, 148 3.7.3 GeneralSolutionof𝐀𝐲=𝐱, 151 3.8 Eigendecomposition, 152 (cid:2) (cid:2) 3.9 MATLABFunctions, 156 4 ComplexNumbersandFunctions 163 4.1 Introduction, 163 4.2 ImaginaryNumbers, 165 4.3 ComplexNumbers, 167 4.4 TwoCoordinates, 169 4.5 PolarCoordinates, 171 4.6 Euler’sFormula, 175 4.7 MatrixRepresentation, 182 4.8 ComplexExponentialRotation, 183 4.9 ConstantAngularVelocity, 189 4.10 Quaternions, 192 PARTII SIGNALS,SYSTEMS,ANDTRANSFORMS 203 5 Signals,GeneralizedFunctions,andFourierSeries 205 5.1 Introduction, 205 5.2 EnergyandPowerSignals, 206 5.3 StepandRampFunctions, 208 5.4 RectangleandTriangleFunctions, 211 (cid:2)