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AUGUST2017 VOLUME65 NUMBER8 IETMAB (ISSN0018-9480) MINI-SPECIAL ISSUE ON 2016 IEEE MTT-S INTERNATIONAL CONFERENCE ON NUMERICAL ELECTROMAGNETIC AND MULTIPHYSICS MODELING AND OPTIMIZATION (NEMO 2016) GuestEditorial ........................................................................................ E.-P.Li,Z.Chen,andC.-H.Chan 2647 MINI-SPECIALISSUEPAPERS Compact and Passive Time-Domain Models Including Dispersive Materials Based on Order-Reduction in the Frequency Domain .................................................................................. R.Baltes,O.Farle,andR.Dyczij-Edlinger 2650 Communication and Load Balancing Optimization for Finite Element Electromagnetic Simulations Using Multi-GPU Workstation ............................................................... A.Dziekonski,P.Sypek,A.Lamecki,andM.Mrozowski 2661 ModelingofNewSpiralInductorBasedonSubstrateIntegratedSuspendedLineTechnology ........ L.Li,K.Ma,andS.Mou 2672 Dual-ModeDual-BandBandpassCavityFiltersWithWidelySeparatedPassbands ........ U.Naeem,A.Périgaud,andS.Bila 2681 SubstrateIntegratedWaveguideFiltersBasedonaDielectricLayerWithPeriodicPerforations .................................... ..................................................... L.Silvestri,E.Massoni,C.Tomassoni,A.Coves,M.Bozzi,andL.Perregrini 2687 REGULARPAPERS EM Theory and Analysis Techniques FastExplicitandUnconditionallyStableFDTDMethodforElectromagneticAnalysis ........................J.YanandD.Jiao 2698 AnAcceleratedTime-DomainMODMethodtoAnalyzetheEffectoftheShieldBoxonEnclosedMultilayeredStructures .... ......................................................H.Karami,H.Maftooli,P.Dehkhoda,M.Khosravi,andE.Zareian-Jahromi 2711 ATwo-GridVectorDiscretizationSchemefortheResonantCavityProblemWithAnisotropicMedia ............................ .......................................................................................... J.Liu,W.Jiang,F.Lin,N.Liu,andQ.H.Liu 2719 PhysicalMechanismsandDesignPrinciplesinModeFiltersforOversizedRectangularWaveguides .............................. .................................................................................. S.Ceccuzzi,C.Ponti,G.L.Ravera,andG.Schettini 2726 MonteCarloAnalysisofOccurrenceThresholdsofMulticarrierMultipactors ................................... X.Wang,J.Shen, J.Wang,Q.Song,Z.Wang,Y.Li,R.Wang,T.Hu,Y.Xia,Q.Sun,X.Yin,W.Cui,H.Zhang,X.Zhang,C.Liu,C.Li,andL.Ran 2734 Devices and Modeling DesignandSensitivityImprovementofCMOS-MEMSScanningMicrowaveMicroscopes ....... M.AziziandR.R.Mansour 2749 AnEnergy-EfficientandLow-CrosstalkSub-THzI/ObySurfacePlasmonicPolaritonInterconnectinCMOS ................... .................................................................... Y.Liang,H.Yu,G.Feng,A.A.A.Apriyana,X.Fu,andT.J.Cui 2762 (Contents Continued on Back Cover) (Contents Continued from Front Cover) AmmWaveFoldedSubstrateIntegratedWaveguideina130-nmCMOSProcess .................................................... ............................................................................................ M.ShahidzadehMahaniandG.W.Roberts 2775 AWidebandMillimeter-WaveSubstrateIntegratedCoaxialLineArrayforHigh-SpeedDataTransmission ...................... .............................................................................................. Y.Shao,X.-C.Li,L.-S.Wu,andJ.-F.Mao 2789 Matter’sElectromagneticSignatureReproductionbyGraded-DielectricMultilayerAssembly ..................................... ................................................................................. D.Micheli,R.Pastore,A.Vricella,andM.Marchetti 2801 High-PowerTransistor-BasedTunableandSwitchableMetasurfaceAbsorber ........................................................ ........................................................................... A.Li,S.Kim,Y.Luo,Y.Li,J.Long,andD.F.Sievenpiper 2810 TunableMicrowaveAbsorberBasedonPatternedGraphene ......................................... D.Yi,X.-C.Wei,andY.-L.Xu 2819 DifferenceFrequencyResponseofaDetectorDiodeUnderDualW-BandFrequencyIllumination ................................ ................................................................................................................ R.A.SmithandZ.A.Pour 2827 AScalableLarge-SignalMultiharmonicModelofAlGaN/GaNHEMTsandItsApplicationinC-BandHighPowerAmplifier MMIC ................. Y.Xu,C.Wang,H.Sun,Z.Wen,Y.Wu,R.Xu,X.Yu,C.Ren,Z.Wang,B.Zhang,T.Chen,andT.Gao 2836 Passive Circuits TheoryforPseudo-ButterworthFilterResponseandItsApplicationtoBandwidthTuning .......... S.Nam,B.Lee,andJ.Lee 2847 AnAnalyticalApproachtotheDesignofMultipleModeRectangularCavityWaveguideFilters ...... C.KelleciandA.Atalar 2857 AnalysisandDesignofWidebandMicrostrip-to-MicrostripEqualRippleVerticalTransitionsandTheirApplicationtoBandpass Filters ................................................................................... L.Yang,L.Zhu,W.-W.Choi,andK.-W.Tam 2866 Synthesis and Design of Miniaturized Wideband Bandpass Filters With Scaled Transmission Line for Spurious-ResponseSuppression .............................................................R.Zhang,S.Luo,L.Zhu,andL.Yang 2878 Dual-BandHelicalFiltersBasedonNonuniformPitchHelicalResonators ........................... Q.-X.ChuandZ.-C.Zhang 2886 UseofGroupDelayofSub-CircuitsinOptimizationofWidebandLarge-ScaleBandpassFiltersandDiplexers ................. ......................................................................................................... M.S.SorkheriziandA.A.Kishk 2893 ProposalandSynthesisDesignofDifferentialPhaseShiftersWithFilteringFunction ......Y.-P.Lyu,L.Zhu,andC.-H.Cheng 2906 Dual-BandCoupled-LineCouplersWithWideSeparationBetweenBands ......... C.-C.Chang,K.-S.Chin,andY.-C.Chiang 2918 HighlyReconfigurableQuadratureCouplerWithIdealImpedanceMatchingandPortIsolation ....... P.-L.ChiandT.-C.Hsu 2930 AWaveguideMagic-TWithCoplanarArmsforHigh-PowerSolid-StatePowerCombining ........................................ ........................................................... L.Guo,J.Li,W.Huang,H.Shao,T.Ba,T.Jiang,Y.Jiang,andG.Deng 2942 NovelPlanarCompactCoupled-LineSingle-Ended-to-BalancedPowerDivider ..................................................... .............................................. W.Zhang,Y.Liu,Y.Wu,A.Hasan,F.M.Ghannouchi,Y.Zhao,X.Du,andW.Chen 2953 Hybrid and Monolithic RF Integrated Circuits SwitchedSubstrate-Shield-BasedLow-LossCMOSInductorsforWideTuningRangeVCOs ...................................... ....................................P.Agarwal,S.P.Sah,R.Molavi,S.Mirabbasi,P.P.Pande,S.E.Oh,J.-H.Kim,andD.Heo 2964 LinearCMOSLC-VCOBasedonTriple-CoupledInductorsandItsApplicationto40-GHzPhase-LockedLoop ............... .................................................... Z.Chen,M.Wang,J.-X.Chen,W.-F.Liang,P.-P.Yan,J.-F.Zhai,andW.Hong 2977 A Wideband and High-Linearity E-Band Transmitter Integrated in a 55-nm SiGe Technology for Backhaul Point-to-Point 10-Gb/sLinks ................................................................. D.delRio,I.Gurutzeaga,A.Rezola,J.F.Sevillano, I. Velez, S. E. Gunnarsson, N.Tamir, C.E. Saavedra, J. L. Gonzalez-Jimenez, A. Siligaris, C. Dehos, and R. Berenguer 2990 TheInfluenceoftheOutputImpedancesofPeakingPowerAmplifieronBroadbandDohertyAmplifiers ......................... .......................................................................... W.Shi,S.He,F.You,H.Xie,G.Naah,Q.-A.Liu,andQ.Li 3002 Generalized Theory and Design Methodology of Wideband Doherty Amplifiers Applied to the Realization of an Octave-BandwidthPrototype ........... R.Darraji,D.Bhaskar,T.Sharma,M.Helaoui,P.Mousavi,andF.M.Ghannouchi 3014 Investigation of a Class-J Mode Power Amplifier in Presence of a Second-Harmonic Voltage at the Gate Node of the Transistor ...................................................................................................... A.AlizadehandA.Medi 3024 Dual-BandDesignofIntegratedClass-JPowerAmplifiersinGaAspHEMTTechnology ........................................... .................................................................................................. A.Alizadeh,M.Frounchi,andA.Medi 3034 Investigation of Integrated Smooth Transistor’s Switching Transition Power Amplifier—2.4-GHz Realization ofClass-EM ................................................................................................... A.ErshadiandA.Medi 3046 StabilityAnalysisofDigitalMicrowavePowerAmplifiers ................................ A.Suárez,A.Wentzel,andW.Heinrich 3056 DevelopmentofWidebandandHighIIP3Millimeter-WaveMixers .......... T.T.Nguyen,A.Riddle,K.Fujii,andA.-V.Pham 3071 IEEE MICROWAVETHEORYAND TECHNIQUES SOCIETY TheMicrowaveTheoryandTechniquesSocietyisanorganization,withintheframeworkoftheIEEE,ofmemberswithprincipalprofessionalinterestsinthefieldofmicrowave theoryandtechniques.AllmembersoftheIEEEareeligibleformembershipintheSocietyuponpaymentoftheannualSocietymembershipfeeof$24.00,plusanannualsubscription feeof$50.00peryearforelectronicandprintmedia.Forinformationonjoining,writetotheIEEEattheaddressbelow.MembercopiesofTransactions/Journalsareforpersonal useonly. ADMINISTRATIVECOMMITTEE D.WILLIAMS, President T.BRAZIL, PresidentElect J.A.NAVARRO, Secretary A.ABUNJAILEH, Treasurer A.ABUNJAILEH M.BOZZI K.GHORBANI R.HENDERSON P.KHANNA S.KOUL M.MADIHIAN G.PONCHAK J.E.RAYAS-SANCHEZ M.SALAZARPALMA D.SCHREURS D.WILLIAMS S.BARBIN T.BRAZIL R.GUPTA A.JACOB D.KISSINGER G.LYONS D.PASQUET S.RAMAN S.REISING A.SANADA M.STEER HonoraryLifeMembers DistinguishedLecturers PastPresidents J.BARR R.SPARKS 2015–2017 2016–2018 2017–2019 K.WU(2016) T.ITOH P.STAECKER T.-W.HUANG A.MORTAZAWI C.CAMPBELL P.ROBLIN W.Y.ALI-AHMAD T.LEE(2015) M.JARRAHI H.ZIRATH T.NAGATSUMA N.SHINOHARA N.BORGESCARVALHO R.WEIGEL(2014) MTT-SChapterChairs Albuquerque:E.FARR Denver:M.JANEZIC Macau:C.C.PONG PikesPeak:K.HU Sweden:A.RYDBERG Argentina:A.M.HENZE EasternNo.Carolina:T.NICHOLS Madras/India:S.SALIVAHANAN Poland:W.J.KRZYSZTOFIK Switzerland:M.MATTES Atlanta:K.NAISHADHAM Egypt:E.HASHEESH Malaysia:M.K.M.SALLEH Portugal:J.CALDINHASVAZ Syracuse:D.MCPHERSON Austria:A.SPRINGER Finland:V.VIIKARI Malaysia,Penang:B.L.LIM Princeton/CentralJersey:W.CURTICE Taegu:Y.-H.JEONG Baltimore:I.AHMAD FloridaWestCoast:J.WANG Melbourne:R.BOTSFORD Queensland:K.BIALKOWSKI Tainan:H.-H.CHEN Bangalore/India:K.VINOY Foothills:M.CHERUBIN MexicanCouncil: RiodeJaneiro:J.R.BERGMANN Taipei:C.MENG Beijing:Z.FENG France:D.BAJON R.M.RODRIGUEZ-DAGNINO Rochester:M.SIDLEY Thailand: Belarus:S.MALYSHEV Germany:G.BOECK Milwaukee:S.G.JOSHI Romania:T.PETRESCU C.PHONGCHAROENPANICH Benelux:G.VANDENBOSCH Greece:R.MAKRI Monterrey/Mexico: Russia,Moscow:V.A.KALOSHIN Toronto:G.V.ELEFTHERIADES Boston:C.GALBRAITH Gujarat/India:S.CHAKRABARTY R.M.RODRIGUEZ-DAGNINO Russia,Nizhny-Novgorad: Tucson:H.XIN Bombay/India:M.V.PITKE Harbin:Q.WU Morocco:M.ESSAAIDI G.L.PAKHOMOV Tunisia:A.GHARSALLAH Brasilia:J.BEZERRA/ Hawaii:K.MIYASHIRO Montreal:K.WU Russia,Novosibirsk:A.YAROSLAVTSEV Turkey:B.SAKA M.VINICIUSALVESNUNES HongKong:H.WONG Morocco:M.ESSAAIDI Russia,Saratov/Penza:M.D.PROKHOROV TwinCities:C.FULLER Buenaventura:C.SEABURY Houston:S.A.LONG Nagoya:J.BAE Russia,SaintPetersburg:S.P.ZUBKO UK/RI:A.REZAZADEH Buffalo:M.R.GILLETTE Houston,CollegeStation:G.H.HUFF Nanjing:W.HONG Russia,Siberia:V.V.SUHOTIN Ukraine,East:N.K.SAKHNENKO Bulgaria:K.ASPARUHOVA Hungary:L.NAGY Nanjing,Hangzhou:L.SUN Russia,Tomsk:D.ZYKOV Ukraine,Kiev:Y.PROKOPENKO Canada,Atlantic:Z.CHEN Huntsville:H.SCHANTZ NewHampshire:E.H.SCHENK SanDiego:J.TWOMEY Ukraine,Rep.ofGeorgia: CedarRapids/CentralIowa:C.G.XIE Hyderabad/India:S.R.NOOKALA NewJerseyCoast:J.SINSKY SantaClaraValley/SanFrancisco: K.TAVZARASHVILI Central&SouthItaly:L.TARRICONE India:D.BHATNAGER NewSouthWales:Y.RANGA N.SHAMS Ukraine,Vinnitsya: CentralNo.Carolina:Z.XIE India/Kolkata:S.SANKARALINGAM NewZealand:A.WILLIAMSON Seattle:S.EBADI V.M.DUBOVOY CentralTexas:J.PRUITT Indonesia:E.T.RAHARDJO NorthItaly:G.OLIVERI Seoul:C.SEO Ukraine,West:I.IVASENKO Centro-NorteBrasil: Israel:S.AUSTER NorthJersey:A.K.PODDAR SerbiaandMontenegro:B.MILOVANOVIC´ UnitedArabEmirates: M.V.ALVESNUNES Japan:N.SUEMATSU NorthernAustralia:J.MAZIERSKA Shanghai:J.MAO N.K.MALLAT Chengdu:Z.NEI Kansai:T.ISHIZAKI NorthernCanada:M.DANESHMAN Singapore:Z.YANG UttarPradesh/India:M.J.AKHTAR Chicago:D.ERRICOLO Kingston:S.PODILCHAK NorthernNevada:B.S.RAWAT SouthAfrica:A.LYSKO Vancouver:S.MCCLAIN Cleveland:M.SCARDELLETTI Kitchener-Waterloo:R.R.MANSOUR Norway:M.UBOSTAD SouthAustralia:T.KAUFMANN Venezuela:J.B.PENA Columbus:A.O’BRIEN Lebanon:E.NASSAR OrangeCounty: SouthBrazil:J.R.BERGMANN Victoria:K.GHORBANI Connecticut:C.BLAIR Lithuania:B.LEVITAS H.J.DELOSSANTOS SoutheasternMichigan:T.OZDEMIR VirginiaMountain:T.A.WINSLOW Croatia:D.BONEFACIC LongIsland/NewYork: Oregon:K.MAYS SouthernAlberta:E.FEAR WashingtonDC/NorthernVirginia: Czech/Slovakia:J.VOVES S.PADMANABHAN Orlando:K.KARNATI Spain:J.I.ALONSO T.IVANOV Dallas:R.SANTHAKUMAR LosAngeles,Coastal:V.RADISIC Ottawa:Q.ZENG Springfield:P.R.SIQUEIRA WesternSaudiArabia:A.SHAMIM Dayton:A.TERZUOLI LosAngeles,Metro/SanFernando: Philadelphia:A.S.DARYOUSH SriLanka:A.U.A.W.GUNAWARDENA Winnipeg:P.MOJABI Delhi/India:A.BASU T.CISCO Phoenix:S.ROCKWELL St.Louis:D.BARBOUR Xian:X.SHI Editors-In-Chief AssociateEditors ULUniCvA.oPfEPRavRiEaGRINI XSTLE´IPMHANEBILA UPAnTivR.IoCfKNFoAtrYeDame NTZTYUHS-TGHUANGMA UMnAivR.ToIfNEVrlOanSgSIeEnK–Nu¨rnberg Pavia,Italy Limoges,France NotreDame,IN,USA Taipei,Taiwan Erlangen,Germany JOSECARLOSPEDRO XUDONGCHEN ANDREAFERRERO ARUNNATARAJAN JOHNWOOD UniversidadedeAveiro Nat.Univ.ofSingapore KeysightTechnol. 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Allrightsreserved.PeriodicalsPostagePaidatNewYork,NYandatadditionalmailingoffices.Postmaster:SendaddresschangestoIEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,IEEE, 445HoesLane,Piscataway,NJ08854-4141.GSTRegistrationNo.125634188.CPCSalesAgreement#40013087.ReturnundeliverableCanadaaddressesto:PitneyBowesIMEX,P.O.Box4332,StantonRd., Toronto,ONM5W3J4,Canada.IEEEprohibitsdiscrimination,harassmentandbullying.Formoreinformationvisithttp://www.ieee.org/nondiscrimination.PrintedinU.S.A. DigitalObject Identifier10.1109/TMTT.2017.2724958 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 1 Guest Editorial THISMini-SpecialIssueoftheIEEE TRANSACTIONSON to extend their conferencepapers significantly by adding new MICROWAVE THEORY AND TECHNIQUES includes five materials. A total of 14 papers were received and five con- expanded papers from the IEEE International Conference on tributions were accepted for publication in this Mini-Special Numerical Electromagnetic Modeling and Optimization for Issue after a rigorous review process. RF, microwave, and terahertz applications (NEMO), held in The papers included in this Mini-Special Issue take us on Beijing, China, July 27–29, 2016. thetopicsonthemodeldevelopmentofcompacttime-domain NEMO is an annual international conference and was models from frequency-domain finite-element solutions, founded by the IEEE Microwave Theory and Techniques GPU-accelerated finite-element method with multilevel Society (MTT-S) in 2014. It has since been rotated between preconditioners, the modeling and realization of a new class Europe, North America, and Asia. NEMO2016 was held in of perforated substrate-in-waveguide filters, parameter Beijing in 2016 after its previous version of 2015 in Ottawa, de-embedding of spiral inductors on substrate integrated ON, Canada. It aims to provide an ideal forum to bring suspended line, and the design of a compact dual-mode dual- together experts and practitioners to share new ideas on tech- band filter with widely separated passbands for integration niques for electromagnetic modeling, propose efficient design in a TX/RX front-end. These papers present not only new algorithms and tools, and anticipate the modeling/analysis theoretical concepts and ideas, but also interesting designs needs of future technologies and applications. NEMO2016 and novel circuit structures and configurations. is particularly intended to stimulate discussion and explo- As Guest Editors, we would like to express our sincere ration of “disruptive” technologies of EM-CAD in addition gratitudeto Prof. Luca Perregriniand Prof. José Carlos Pedro to traditional topics. It features an exciting technical program (Editors-in-Chief of this TRANSACTION) for the opportu- and invited talks by internationally recognized experts in nity and support to publish a Mini-Special Issue devoted electromagnetic modeling and optimization. The organization to NEMO2016. Special thanks also go to Associate Editor of NEMO clearly shows the willingness and engagement of Prof. Kamran Ghorbani for his editorial help and support. theMTT-Stopromoteandwelcomeresearchincomputational Finally, we wish to thank all the reviewers for devoting electromagnetics,andtoencouragetheauthorstopublishtheir their valuable time and expertise to review the submitted results in MTT-S conferences and journals. manuscripts. NEMO2016 was a success, with 133 submissions from 19countries,108papersacceptedafterthereviewprocess,and 190 attendees. Held in the elegant state-of-the-art new hotel, ER-PING LI, Guest Editor ZJU-UIUC Institute Zhaolin Grand Hotel of Beijing, the conference ran with two paralleloralsessionsonthreeconsecutivedaysandsixplenary Zhejiang University Hangzhou 310027, China session speakers on the advances of the art. Twenty-three poster papers were selected for the Student Paper Contest, of which three best papers were awarded in the banquet. The ZHIZHANGCHEN, Guest Editor technical topics covered a wide range of numerical methods Departmentof Electrical and Computer Engineering and their applications ranging from RF to microwave to Dalhousie University terahertz and from components to systems. It also covered Halifax, NS B3H 4R2, Canada topics in IC packaging and interconnects, multiphysics and modeling, optimization, antennas, and MIMO. CHI-HOU CHAN, Guest Editor Authors of all the papers presented at NEMO2016 were Department of Electronic Engineering invited to submit an expanded version of their work for City University of Hong Kong publication in this Mini-Special Issue. They were requested Hong Kong Digital ObjectIdentifier 10.1109/TMTT.2017.2725098 0018-9480©2017IEEE.Personaluseispermitted, butrepublication/redistribution requires IEEEpermission. Seehttp://www.ieee.org/publications_standards/publications/rights/index.html formoreinformation. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES ∗ Er-PingLi(S’91–M’92–SM’01–F’08)joinedtheSingaporeA STARResearchInstituteofHigh Performance Computing, Singapore, as a Principal Scientist and the Director of the Electronic and PhotonicsDepartmentin 2001. Since 1989,he has been a Research Associate/Fellow with the School of Electronic and Information Technology, Sheffield Hallam University, Sheffield, U.K., and a Senior Research Fellow, the Principal Research Engineer, an Associate Professor, and the Technical Director with the Singapore Research Institute and Industry, Singapore. He is currently a Changjiang-Qianren Distinguished Professor with the School of Electronic Engineering and the Dean of the Joint Institute of Zhejiang University–University of Illinois at Urbana–Champaign,Zhejiang University, Hangzhou,China. He has authoredor co-authored over 400 papers published in international journals and conferences and has authored 2 books published by Wiley–IEEE Press and Cambridge University Press. He holds and has filed a number of U.S. patents. His current research interests include electrical modeling and the design of microscale/nanoscale integrated circuits, 3-D electronic package integration, and nanoplasmonic technology. Dr.LiisaFellowoftheMITElectromagneticsAcademy,USA.Hewasarecipientofthe2015IEEERichardStoddardAward on Electromagnetic Compatibility (EMC), the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY Technical AchievementAward, the SingaporeIES Prestigious EngineeringAchievementAward, and the ChangjiangChair Professorship Award from the Ministry of Education in China, and a number of Best Paper Awards. He was elected for the IEEE EMC Distinguished Lecturer in 2007. He is the founding member of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES(MTT)-RFNanotechnologyCommittee.HehasbeentheGeneralChairandtheTechnicalProgramChairformany internationalconferences.HewasthePresidentforthe2006InternationalZurichSymposiumonElectromagneticCompatibility, the Founding General Chair for the Asia–Pacific EMC Symposium, the General Chair for the 2010 IEEE Symposium on Electrical Design for Advanced Packaging Systems, and the Chairman of the IEEE EMC Singapore Chapter from 2005 to 2006. He served as an Associate Editor of IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS from 2006 to 2008, the Guest Editor of the 2006 and 2010 IEEE EMC Special Issues, and the Guest Editor of the 2010 IEEE MTT Asia–Pacific MicrowaveConferenceSpecialIssue.HeiscurrentlyanAssociateEditoroftheIEEETRANSACTIONSONELECTROMAGNETIC COMPATIBILITY and the IEEE TRANSACTIONS ONCOMPONENTS,PACKAGING, ANDMANUFACTURING TECHNOLOGY. He has been invited to give numerous invited talks and plenary speeches at various international conferences and forums. Zhizhang (David) Chen (S’92–M’92–SM’96–F’10) received the master’s degree in radio engineering from Southeast University, Nanjing, China, and the Ph.D. degree in electrical engineering from the University of Ottawa, Ottawa, ON, Canada. He joined McGill University, Montreal, QC, Canada, as a NSERC Post-Doctoral Fellow in 1993. He was an Adjunct or Visiting Professor with the University of Nottingham, Nottingham,U.K., the ÉcoleNationaleSupérieuredesTélécommunicationsdeBretagne,Brest, France, ShanghaiJiao Tong University,Shanghai,China, the Beijing Universityof Aeronautics andAstronautics,Beijing,China,FuzhouUniversity,Fuzhou,China,theHongKongUniversity of Science and Technology, Hong Kong, and the University of Electronic Science and Technology of China, Chengdu, China. He was one of the originators of the unconditionally stable methods that have been highly cited and used. He and his team have also devel- oped a number of nonlinear ultra-wideband receivers and planar wireless power transfer transmitting and receiving structures. He was the Head of the Department of Electrical and Computer Engineering, Dalhousie University, Halifax, NS, Canada, where he is currently a Professor. He has authored or co-authored over 300 journal and conference papers in computational electromagnetics, RF/microwave electronics, antennas, and wireless technologies. His current research interests include time-domain electromagnetic modeling techniques, ultra-wideband wireless communication systems, and wireless power transfer. Dr. Chen was involved in organizing many international workshops and conferences as the Chair, the Co-Chair, the Session Organizer, and a committee member. He was a recipient of the 2005 Nova Scotia Engineering Award, the 2006 Dalhousie Graduate Teaching Award, the 2007 and 2015 Dalhousie Faculty of Engineering Research Award, the 2013 IEEE Canada FessendenMedal,andtheDalhousieUniversityProfessorship.HeisaFellowoftheCanadianAcademyofEngineeringandthe EngineeringInstituteofCanada.HehasbeenanAssociateEditoroftheIEEEJOURNALONMULTISCALEANDMULTIPHYSICS COMPUTATIONAL TECHNIQUES since 2016 and a Guest Editor of several professional journals and magazines. He has been invited to give many invited talks and plenary speeches at various international meetings, conferences, and forums. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 3 Chi-Hou Chan (S’86–M’86–SM’00–F’02) received the B.S. and M.S. degrees in electrical engineering from The Ohio State University, Columbus, OH, USA, in 1981 and 1982, respectively, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana–Champaign,Champaign, IL, USA, in 1987. From 1987 to 1989, he was a Visiting Assistant Professor with the Department of Electrical andComputerEngineering,UniversityofIllinoisatUrbana–Champaign.From1989to1998,he wasafacultymemberwiththeDepartmentofElectricalEngineering,UniversityofWashington, Seattle,WA,USA.In1996,hejoinedtheDepartmentofElectronicEngineering,CityUniversity of Hong Kong, Hong Kong, and later became a Chair Professor of electronic engineering in 1998. From 1998to 2009,he was initially the Associate Dean and then became the Dean of the College of Science and Engineering. He also served as Acting Provost with the university from 2009 to 2010. He is currently the Director of the State Key Laboratory of Millimeter Waves,PartnerLaboratory,CityUniversityofHongKong.Hiscurrentresearchinterestsinclude computational electromagnetics, millimeter-wave circuits and antennas, and terahertz science and technology. Dr. Chan was a recipient of the U.S. National Science Foundation Presidential Young Investigator Award in 1991, the Joint Research Fund for Hong Kong and Macao YoungScholars, and the National Science Fund for Distinguished Young Scholars, China, in 2004. He was the recipient of the Outstanding Teacher Award from the Department of Electronic Engineering, City UniversityofHongKong,in1998,1999,2000,and2008.HehasbeentheGeneralCo-ChairofISAP2010,iWAT2011,iWEN 2013, ICCEM 2015, ICCEM 2016, and GSMM2017. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 1 Compact and Passive Time-Domain Models Including Dispersive Materials Based on Order-Reduction in the Frequency Domain Rolf Baltes, Ortwin Farle, and Romanus Dyczij-Edlinger Abstract—In this paper, compact time-domain (TD) models function in order to obtain a low-dimensional realization in featuring materials with frequency-dependent electromag- stateordescriptorrepresentation[3].Sincethepassivityofthe netic (EM) properties are derived. The considered frequency- resultingmodelis notguaranteed,extrapassivationsteps may dependentmaterialmodelsincludemultitermDebyeandLorentz be required [4]. Moreover, since all of these transformation modelsfortheelectricpermittivityandthemagneticpermeability and a multiterm Drude model for the electric conductivity. The techniques are based on the system transfer function, it is TD models are based on finite-element systems in the frequency notpossible to reconstructthe transientelectromagnetic(EM) domain(FD).Torenderthemodelcompactandcomputationally fields. efficient,thedimensionof theFD system iscompressed with the In order to overcome these limitations, a new FD-to-TD help of projection-based model-order reduction. In contrast to transformation method was presented in [5]. The idea is to olderapproaches,theTDtransformationisperformedontheFD model itself rather than on thetransfer function.Theresult is a derive an FD description that can be transferred to the TD state-spacerepresentation,whichmayeitherbesolvedbycustom directly. The FD model is based on a finite-element (FE) timeintegrators or importedintocommercial circuitsimulators. scatteringformulation,whichprovidestransparentportbound- The advantages of the new approach include provable passivity aries, and therefore is free of interior resonances. However, of the FD model, provable causality of the TD model, and the FE models are of very high dimension, and direct transfor- ability to reconstruct the transient EM fields. mation to the TD would lead to enormous computational Index Terms—Finite-element (FE) analysis, reduced-order costsforsolvingtheTDmodel.Therefore,model-orderreduc- systems, time-domain (TD) analysis. tion (MOR) is applied. The FE system is projected onto the low-dimensional column space of a well-chosen projection I. INTRODUCTION matrix [6]–[11], which is constructed by the reduced basis COMPLEX microwave systems combine linear compo- approach [6]–[8], that is, its column space is spanned by the nents, such as interconnects or filters, with nonlinear solution vectors of the FE system at certain frequencies. The active devices. In contrast to the nonlinear components, present MOR method thus fits in the general framework of the simulation of the linear subsystem can be carried out in interpolatorymodelreduction[9]–[11]. Theresultingreduced- the frequency domain (FD). This may have some advantages, order model (ROM) preserves the structure, the transparent which include the handlingof frequency-dependentmaterials, port boundaries, and the passivity of the FE system, and it electrically small details, and resonant structures. allows for reconstructing the EM fields. Most importantly, Since the simulation of the whole system hasto be done in ROMevaluationiscomputationallyverycheap,duetoitslow the time domain (TD), efficient techniques for transforming dimension. The subsequent FD-to-TD transformation leads to the FD data to the TD are required. Early methods com- aTDmodelinstate-spacerepresentation,whichinheritsthese puted the transfer function in the FD, converted it to the properties, and which is guaranteed to be causal, thanks to TD, and convoluted it with the input signals [1], [2]. The passivityinthe FD.Theavailabilityofthe TDmodelinstate- main drawback of this procedure is that it is computationally space representation is of great practical utility, because it expensive, especially when the number of time steps is large. allowsforeasyintegrationintoin-houseorcommercialcircuit Laterapproachesusedrationalapproximationsforthetransfer simulators, by techniques like in [12] or [4]. The viability of this approach is demonstrated in Section V, by employing an Manuscript received January 5, 2017; revised April 28, 2017; accepted explicitRunge–Kuttatime-steppingschemeforcomputingthe May 13, 2017. This work was supported by the Deutsche Forschungsge- meinchaft under Grant DY 112/1-2. This paper was presented at the IEEE TD results. MTT-SInternationalConferenceonNumericalElectromagneticModelingand The original method of [5] was limited to frequency- OptimizationforRF,Microwave,andTerahertzApplications, onJuly27–29, independent materials. It was extended in [13] and [14] to 2016,Beijing, China. (Correspondingauthor:RolfBaltes.) R. Baltes and R. Dyczij-Edlinger are with the Chair of Electromag- materials with electric permittivity of single-term Debye type netic Theory, Saarland University, D-66123 Saarbrücken, Germany (e-mail: and electric conductivity of single-term Drude type. The [email protected]). objective of this paper is to extend the method to general O.FarlewaswiththeChairofElectromagneticTheory,SaarlandUniversity, D-66123Saarbrücken, Germany.Heis nowwithCST–Computer Simulation materials, including multiterm Debye and/or Lorentz models Technology AG,D-64289Darmstadt, Germany. for the electric permittivity and the magnetic permeability, Color versions of one or more of the figures in this paper are available andalsomultitermDrudemodelsfortheelectricconductivity. onlineathttp://ieeexplore.ieee.org. Digital ObjectIdentifier 10.1109/TMTT.2017.2708095 This paper covers both the derivation of an appropriate 0018-9480©2017IEEE.Personaluseispermitted, butrepublication/redistribution requires IEEEpermission. Seehttp://www.ieee.org/publications_standards/publications/rights/index.html formoreinformation. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES FD-ROM andthe FD-to-TDtransformation.Thecriticalpoint followingthattheindexesi ,i , j , j ,andi aredefinedin D L D L c with the present setting is that the passivity proof from our the limits of (2). previous works does not apply. Let the boundary of (cid:2) consist of electric walls (cid:9) and E Yeta generalproofof passivity forTD modelsobtainedby the cross section (cid:9) of one feeding waveguide. The outward P interpolatoryMORisavailablefrom[10].However,itexhibits pointing unit normal vector on the boundaryis denoted by nˆ. a high level of abstraction and makes extensive use of the It is assumed that the waveguide supports M modes of theory of matrix polynomials [15]. This paper provides an transverse EM (TEM) type and that higher order modes are alternative proof for the concrete physical setting described rapidly decaying to zero. The excitation of (cid:2) is realized by earlier, which relies on elementary engineering mathematics impressing the tangential componentof the magnetic field H t only. at the portinterface (cid:9) . Thisleads to the followingboundary P value problem in terms of the electric field E: II. FREQUENCY-DOMAIN MODELS ∇×μ−1∇×E+jk η κE−k2ε E = 0 in (cid:2) (3a) r 0 0 0 r A. Problem Statement E×nˆ = 0 on (cid:9) (3b) (cid:3) (cid:4) E Letthe computationaldomain(cid:2) be source-freeand consist of two subdomains (cid:2) and (cid:2) , such that μr−1∇×E ×nˆ=−jk0η0Ht on (cid:9)P (3c) 0 1 (cid:2)=(cid:2)0⊕(cid:2)1. (1) where η0 denotes the characteristic impedance of free-space. The material within (cid:2) is assumed to be linear and isotropic. B. Finite-Element Scattering Formulation The relativeelectric permittivity,the relative magneticperme- ability,andtheelectricconductivityin(cid:2) and(cid:2) aredenoted The scattering formulation takes the modal amplitudes of 0 1 by εr,0, μr,0, and κ0 and εr,1, μr,1, and κ1, respectively. The incidentwavesak asinputandcomputesthemodalamplitudes material properties in (cid:2) are assumed to be constant over of the outgoing waves b. The discrete formulation is based 0 l frequency; εr,1 and μr,1 shall be given by multiterm Debye onthe FE functionspace V ⊂Hcurl; its derivationfollows[5] and Lorentz models, and κ by a multiterm Drude model and leads to the following system: 1 (cid:3) (cid:4) εr,1 = ε∞+εD (2a) R(k0)+jk0η0BBT xE = 2jk0η0bkak (4a) εD = i(cid:2)NDe=D11G+D,jikD0(cid:5)c0ετDe,,iiDD with bl = blTxE −δkl (4b) μr,1 = +μ∞i(cid:2)NL=e+L1ωμe2D,iLG+L,2iLjk(cid:5)0εc0Lδ,ieL,iωLe2−,iLk02c02 ((22bc)) RB == (cid:7)−Sbμk+02(cid:5)Tμ·ε·r−,·1+1(kεb0D)S(k(cid:8)μ0.,)DT+D(cid:6)jk0η0(cid:5)Tκ +κ1(k0)TD(cid:6) ((55ba)) 1 M μD = +jN(cid:2)Dm=(cid:2)NDm1L1F+D,jjkD0F(cid:5)cL0μ,τjDmL,(cid:5),jjDDμL,jLωm2,jL (2d) The enStμr,iie,sj o=f (cid:9)(cid:9)th(cid:2)e0m∇a×triwceis·aμnr−d,01v∇ec×towrsjadr(cid:2)e defined as (6a) jL=1ωm2,jL +2jk0c0δm,jL −k02c02 Tκ,i,j = wi ·κ0wjd(cid:2) (6b) κ1 = i(cid:2)cN=C11+GCjk,i0ccκ00τ,icc,ic. (2e) Tε,i,j = (cid:9)(cid:9)(cid:2)(cid:2)00wi ·εr,0wjd(cid:2)+(cid:9)(cid:2)1wi ·ε∞wjd(cid:2) (6c) Herein, NeD, NmD, NeL, NmL, and NC denote the numberof Sμ,D,i,j = ∇×wi ·∇×w jd(cid:2) (6d) terms in the Debye, Lorentz, and Drude models, respectively; (cid:9)(cid:2)1 k0 and c0 stand for the free-space wavenumber and the TD,i,j = wi ·wjd(cid:2) (6e) free-space speed of light, respectively; ε∞ and μ∞ are the (cid:9)(cid:2)1 permittivity and permeability at infinite frequency, respec- bk,i = (wi ×hˆk)·nˆd(cid:9) (6f) tively; GD,iD, GL,iL, FD,jD, FL,jL, and GC,ic are scaling (cid:9)P factors; and (cid:5)εD,iD and (cid:5)μD,jD, and τe,iD and τm,jD are where w ,w ∈ V, and hˆ denotes the modal magnetic field the Debye relaxation strengths and times, respectively. The i j k ˆ Lorentzresonanceangularfrequenciesanddampingconstants pattern of the kth mode. Since hk may be chosen real-valued aanreddtheneoctehdanbgyesωien,iLp,erωmmit,tjiLv,itδye,aiLn,dapnedrmδmea,bjLi,litryesapbeocutitvetlhye, fsoeernTEfrMomm(o6d)etsh,aatllthmeamtriactersicoesf (S6μ),aSrμe,rDe,aTl-κv,aBluBedT.,ItancdanTbDe respective Lorentz pole by (cid:5)εL,iL and (cid:5)μL,jL, respectively. are symmetric positive semidefinite (SPSD), while the matrix Fthoerstthaeticcornedguimcteiv,iatyn,dththeeκ0τ,ci,cicvvalauluesesarreeptrheesecnotntthriebuctoiollnissioinn STμε,DisasynmdmTeDt,ritchepofsoiltliovweidnegfirneiltaet.ioNnotheotlhdas:t for the kernels of time constants. If not specified otherwise, it is implied in the ker(TD)⊂ker(Sμ,D). (7) This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. BALTESetal.: COMPACTANDPASSIVETDMODELSINCLUDINGDISPERSIVEMATERIALS 3 C. Model-Order Reduction A. Proof of Property P1 Since the FE system (4) may be of very high dimension, It must be shown that the poles of S have nonpositive real direct FD-to-TD transformation of (4) would lead to TD part. Since the matrix B˜ is constant, and because adding the models that are very expensive to solve. To overcome this, unity matrix has no influence on the poles, this is equivalent projection-based MOR techniques are applied to the FE sys- to showing that the eigenvalues λ of tem. The idea is to project the FE system onto a well-chosen subspace of low dimension [6]–[8]. The present method is R˜(λ)v+λμ0B˜B˜Tv=0 (14) based on the self-adaptive multipoint scheme of [8], which have nonpositive real part. Due to the singularity of the leads to a complex-valued projection matrix U. However, it was shown in [5] that preservation of passivity requires a matricesS˜μ,S˜μ,D,T˜κ, and T˜D, a numberof zeroeigenvalues exist. However, λ = 0 clearly features nonpositive real part, projection matrix V that is real-valued. As suggested in [10] sothatthiscasemaybeexcludedfromthefollowinganalysis. and [16], V is obtained from The proof will be in two steps. First, it will be shown ran(V)=ran(ReU)∪ran(ImU). (8) that the eigenvalues of the following generalized eigenvalue problem (EVP) have nonpositive real part: For numerical stability, the columns of V are orthogonalized. Thus, the ROM approximation to the FE solution xE reads M v+λ(M +M )v=0 (15) 0 1 2 x ≈Vx˜ (9) E with M ,M ∈ R being SPSD, M ∈ R being skew- 0 1 2 and the ROM itself is given by symmetric, M2 = −M2T, and (M1 + M2)v being nonzero. (cid:3) (cid:4) In the second step, the EVP (14) will be transformed to (15), R˜(k0)+jk0η0B˜B˜T x˜ = 2jk0η0b˜kak (10a) which completes the proof. ∗ b˜ = b˜Tx˜ −δ (10b) Step 1: Multiplying (15) by v leads to l l lk with m0+λm1+λm2 =0 with mi =v∗Miv. (16) (cid:5) (cid:6) R˜ = S˜μ+(cid:5) μr−,11(k0)S˜μ,D+(cid:6)jk0η0 T˜κ +κ1(k0)T˜D Because of the matrix properties, we have −k02 T˜ε+εD(k0)T˜D (11a) m ,m ≥ 0 (17) 0 1 S˜μ = VTSμV, S˜μ,D =VTSμ,DV (11b) m = m∗ =v∗M∗v=v∗MTv=−v∗M v=−m . (18) T˜κ = VTTκV, T˜D =VTTDV (11c) 2 2 2 2 2 2 T˜ε = VTTεV, b˜k =VTbk, B˜ =VTB. (11d) Thus m2 is purely imaginary and can be expressed as Notethattheprojectionpreservesthesymmetryaswellasthe m2 =jα, α ∈R. (19) definitenesspropertiesofthematrices.Sincethedimensionof Using this and (16), we get theROMismuchsmallerthanthethatoftheFE system,(10) is evaluated much more efficiently than (4). m m m jm α λ=− 0 =− 0 1 + 0 . (20) m +jα |m +jα|2 |m +jα|2 1 1 1 III. PASSIVITY By (17), the real part of λ, given by the first term on the Since nonpassive FD data may lead to noncausal behavior right-hand side, is nonpositive. in the TD, it is important to check the passivity of the ˜ ˜ Step 2: Since Sμ,D and TD are notof fullrank,three cases FD-ROM,beforetransformingittotheTD.Accordingto[17], are needed to be considered to transform (14) to (15): a scattering matrix S in the Laplace domain represents a Case 1: v ∈ ker(T˜D) ⇒ v ∈ ker(S˜μ,D). The EVP (14) passive system if and only if the following hold. reduces to P1: Each element of S(s) is analytic in Re(s)>0. (cid:10) (cid:11) PP23:: IS(−s)S=∗(sS)(Ss()s.)≥0 for all s, with Re(s)>0. S˜μ+λμ0(T˜κ +B˜B˜T)+ λc22 T˜ε v=0. (21) 0 ∗ Herein, s and S denote the complex conjugate of s and Let us introduce the vector z by the complex conjugate transpose of S, respectively. It will be proven that the S matrix of the FD-ROM (10) features the z:=λv. (22) properties P1–P3. In the Laplace domain, S of (10) takes the form Since the matrix T˜ε is regular, (22) may be written as S=2sμ B˜T(R˜(s)+sμ B˜B˜T)−1B˜ −I (12) 0 0 with c12T˜εz−λc12T˜εv=0. (23) (cid:5) (cid:6) 0 0 R˜(s) = S˜μ+μr−,11(s)S˜μ,D +sμ0 T˜κ +κ1(s)T˜D Combining (21) and (23) yields the EVP +cs22(T˜ε +εD(s)T˜D). (13) M0,1x1+λ(M1,1+M2,1)x1 =0 (24) 0 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES with With the help of the abbreviations ⎡ ⎤ M0,1 = ⎡⎣S˜0μ c1020T˜ε⎤⎦, M2,1 =⎢⎢⎣−c102T˜ε c1020T˜ε⎥⎥⎦ (25a) zpzeDDDrr === [[[zpzDeDDrr,,,111 ········· zpzDeDrDr,,N,NNCCe]D]TT]T (((333000bac))) M1,1 = (cid:18)μ0(T˜κ0+B˜B˜T) 00(cid:19), x1 =(cid:18)0vz(cid:19). (25b) pzeeLL == [[(cid:7)zpeeLL,,11 ······ zpeeLL,,NNeeLL]]TT (cid:8) ((3300de)) aBnydcSoPnsStDru,ctwiohnil,ethMe2m,1atirsicreesaMl-v0a,l1ueadn,dsMke1w,1-sayrmemreeatlr-ivca,launedd TTˆDDrr == dμia0g(GμC0,1GκC0,,1icT˜κD0,icT˜.D..) μ0 GC,NCκ0,NCT˜D ((3300gf)) has full rank. The EVP (24) is thus of the same form as (15), TτDr = d(cid:18)iag(μ0 GC,icκ0,icτc,icT˜D) (cid:19) (30h) apnardCt.atsheer2e:fovre∈tkheer(eS˜ige)n,vval∈/ueksero(fT˜ (2)4.)Inhathvies ncaosnep,o(s1it4i)vereraedasl TˆD = GD(cid:20),1c(cid:5)02εD,1T˜D ... (cid:21)GD,NeDc(cid:5)02εD,NeDT˜D (30i) D D (cid:10)S˜μ+λμ0(cid:5)T˜κ +κ1(λ)T˜D +B˜B˜T)+ λc22εD(λ)T˜D(cid:11)v TD = diag(cid:20)GD,iDc(cid:5)02εD,iDT˜D (cid:21) (30j) +0 λT˜εz=0. (26) TτD = diag GD,iD(cid:5)cε2D,iDτe,iDT˜D (30k) c2 (cid:18) 0 (cid:19) 0 Let us introduce some more auxiliary vectors TˆL = GL,1(cid:5)cε2L,1ωe2,1T˜D ... GL,NeL(cid:5)εcL2,NeLωe2,NeLT˜D (30l) (cid:20) 0 (cid:21) 0 ppzeDDLrr,,,iiiLcc === λλλzz+DeLr,,λλiicL2τc,iλcv (((222777bac))) TTδLL == ddiiaagg(cid:20)2GLG,iLL,(cid:5)iLcε(cid:5)02Lε,icLL2,ωiLe2ω,ieL2,iT˜LDδe,iLT˜D(cid:21) ((3300mn)) zeL,iL = ωe2,iL +2λδe,iL +λ2v (27d) Tω = diag(cid:20)GL,iL(cid:5)εL,iL0ωe4,iLT˜ (cid:21) (30o) zeD,iD = 1+λλτe,iDv. (27e) TˆD,L0 = (cid:18)TˆD(cid:19), TˆDr,0c02=(cid:18)TˆDr(cid:19),DTˆL,0 =(cid:18)TˆL(cid:19) (30p) Since all the vectors defined in (27) are parallel to v, they do 0 0 0 ˜ not lie in the kernel of T . Therefore, (27) is equivalent to D (29) and (28) take the form μ0GC,icκ0,icT˜D(cid:5)λzDr,ic +λ(cid:5)τc,icpDr,ic −λv(cid:6)(cid:6) = 0 (28a) M0,2x2+λ(cid:5)M1,2+M2,2(cid:6)x2 =0 (31) μ0GC,iκ0,icτc,icT˜D pDr,ic −λzDr,ic = 0 (28b) with GD,iDc(cid:5)02εD,iDT˜D(cid:5)zeD,iD +λτe,iDzeD,iD −λv(cid:6) = 0 (28c) ⎡⎢⎢M00,1 00 00 00 00 00 ⎤⎥⎥ GL,iL(cid:5)cε02L,iLωe2,iLT˜D(cid:5)ωe2,iLzeL,iL +2λδe,iLzeL,iL(cid:6) M0,2 = ⎢⎢⎢⎢⎣ 00 00 T0τDr T0D 00ω 00 ⎥⎥⎥⎥⎦ (32a) +λpeL,iL −λv = 0 (28d) 0 0 0 0 TL 0 0 0 0 0 0 T GL,iL(cid:5)cε02L,iLωe2,iLT˜D(cid:5)peL,iL −λzeL,iL(cid:6) = 0. (28e) ⎡⎢⎢M01,1 T0Dr 00 00 00 00⎤⎥⎥L With[(S˜2μ7)+, tλhμe0E(TV˜κP+(2B6˜)B˜bTe)c]vom+escλ2T˜εz M1,2 = ⎢⎢⎢⎢⎣ 000 000 000 T00τD T00δL 000⎥⎥⎥⎥⎦ (32b) 0 0 0 0 0 0 0 (cid:2)NC ⎡ ⎤ + ic=1λμ0GC,icκ0,icT˜DzDr,ic ⎢⎢⎢−MTˆ2DT,r1,0 TˆD0rτ,0 T0τDr TˆD0,0 Tˆ0L,0 00 ⎥⎥⎥ + (cid:2)NeD cλ2GD,iD(cid:5)εD,iDT˜DzeD,iD M2,2 = ⎢⎢⎢⎢−ˆT0TD,0 −T0Dr 00 00 00 00 ⎥⎥⎥⎥(32c) iD=1 0 ⎣−TˆT 0 0 0 0 T ⎦ + i(cid:2)NL=eL1cλ02GL,iL(cid:5)εL,iLωe2,iLT˜DzeL,iL =0. (29) x2 = [x1 0zLD,r0 pDr0zeD z0eL pe0L]T. −TL 0L (32d)

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AUGUST2017 VOLUME65 NUMBER8 IETMAB (ISSN0018-9480) MINI-SPECIAL ISSUE ON 2016 IEEE MTT-S INTERNATIONAL CONFERENCE ON NUMERICAL ELECTROMAGNETIC AND MULTIPHYSICS MODELING AND OPTIMIZATION (NEMO 2016) GuestEditorial ........................................................................................ E.-P.Li,Z.Chen,andC.-H.Chan 2647 MINI-SPECIALISSUEPAPERS Compact and Passive Time-Domain Models Including Dispersive Materials Based on Order-Reduction in the Frequency Domain .................................................................................. R.Baltes,O.Farle,andR.Dyczij-Edlinger 2650 Communication and Load Balancing Optimization for Finite Element Electromagnetic Simulations Using Multi-GPU Workstation ............................................................... A.Dziekonski,P.Sypek,A.Lamecki,andM.Mrozowski 2661 ModelingofNewSpiralInductorBasedonSubstrateIntegratedSuspendedLineTechnology ........ L.Li,K.Ma,andS.Mou 2672 Dual-ModeDual-BandBandpassCavityFiltersWithWidelySeparatedPassbands ........ U.Naeem,A.Périgaud,andS.Bila 2681 SubstrateIntegratedWaveguideFiltersBasedonaDielectricLayerWithPeriodicPerforations .................................... ..................................................... L.Silvestri,E.Massoni,C.Tomassoni,A.Coves,M.Bozzi,andL.Perregrini 2687 REGULARPAPERS EM Theory and Analysis Techniques FastExplicitandUnconditionallyStableFDTDMethodforElectromagneticAnalysis ........................J.YanandD.Jiao 2698 AnAcceleratedTime-DomainMODMethodtoAnalyzetheEffectoftheShieldBoxonEnclosedMultilayeredStructures .... ......................................................H.Karami,H.Maftooli,P.Dehkhoda,M.Khosravi,andE.Zareian-Jahromi 2711 ATwo-GridVectorDiscretizationSchemefortheResonantCavityProblemWithAnisotropicMedia ............................ .......................................................................................... J.Liu,W.Jiang,F.Lin,N.Liu,andQ.H.Liu 2719 PhysicalMechanismsandDesignPrinciplesinModeFiltersforOversizedRectangularWaveguides .............................. .................................................................................. S.Ceccuzzi,C.Ponti,G.L.Ravera,andG.Schettini 2726 MonteCarloAnalysisofOccurrenceThresholdsofMulticarrierMultipactors ................................... X.Wang,J.Shen, J.Wang,Q.Song,Z.Wang,Y.Li,R.Wang,T.Hu,Y.Xia,Q.Sun,X.Yin,W.Cui,H.Zhang,X.Zhang,C.Liu,C.Li,andL.Ran 2734 Devices and Modeling DesignandSensitivityImprovementofCMOS-MEMSScanningMicrowaveMicroscopes ....... M.AziziandR.R.Mansour 2749 AnEnergy-EfficientandLow-CrosstalkSub-THzI/ObySurfacePlasmonicPolaritonInterconnectinCMOS ................... .................................................................... Y.Liang,H.Yu,G.Feng,A.A.A.Apriyana,X.Fu,andT.J.Cui 2762 (Contents Continued on Back Cover) (Contents Continued from Front Cover) AmmWaveFoldedSubstrateIntegratedWaveguideina130-nmCMOSProcess .................................................... ............................................................................................ M.ShahidzadehMahaniandG.W.Roberts 2775 AWidebandMillimeter-WaveSubstrateIntegratedCoaxialLineArrayforHigh-SpeedDataTransmission ...................... .............................................................................................. Y.Shao,X.-C.Li,L.-S.Wu,andJ.-F.Mao 2789 Matter’sElectromagneticSignatureReproductionbyGraded-DielectricMultilayerAssembly ..................................... ................................................................................. D.Micheli,R.Pastore,A.Vricella,andM.Marchetti 2801 High-PowerTransistor-BasedTunableandSwitchableMetasurfaceAbsorber ........................................................ ........................................................................... A.Li,S.Kim,Y.Luo,Y.Li,J.Long,andD.F.Sievenpiper 2810 TunableMicrowaveAbsorberBasedonPatternedGraphene ......................................... D.Yi,X.-C.Wei,andY.-L.Xu 2819 DifferenceFrequencyResponseofaDetectorDiodeUnderDualW-BandFrequencyIllumination ................................ ................................................................................................................ R.A.SmithandZ.A.Pour 2827 AScalableLarge-SignalMultiharmonicModelofAlGaN/GaNHEMTsandItsApplicationinC-BandHighPowerAmplifier MMIC ................. Y.Xu,C.Wang,H.Sun,Z.Wen,Y.Wu,R.Xu,X.Yu,C.Ren,Z.Wang,B.Zhang,T.Chen,andT.Gao 2836 Passive Circuits TheoryforPseudo-ButterworthFilterResponseandItsApplicationtoBandwidthTuning .......... S.Nam,B.Lee,andJ.Lee 2847 AnAnalyticalApproachtotheDesignofMultipleModeRectangularCavityWaveguideFilters ...... C.KelleciandA.Atalar 2857 AnalysisandDesignofWidebandMicrostrip-to-MicrostripEqualRippleVerticalTransitionsandTheirApplicationtoBandpass Filters ................................................................................... L.Yang,L.Zhu,W.-W.Choi,andK.-W.Tam 2866 Synthesis and Design of Miniaturized Wideband Bandpass Filters With Scaled Transmission Line for Spurious-ResponseSuppression .............................................................R.Zhang,S.Luo,L.Zhu,andL.Yang 2878 Dual-BandHelicalFiltersBasedonNonuniformPitchHelicalResonators ........................... Q.-X.ChuandZ.-C.Zhang 2886 UseofGroupDelayofSub-CircuitsinOptimizationofWidebandLarge-ScaleBandpassFiltersandDiplexers ................. ......................................................................................................... M.S.SorkheriziandA.A.Kishk 2893 ProposalandSynthesisDesignofDifferentialPhaseShiftersWithFilteringFunction ......Y.-P.Lyu,L.Zhu,andC.-H.Cheng 2906 Dual-BandCoupled-LineCouplersWithWideSeparationBetweenBands ......... C.-C.Chang,K.-S.Chin,andY.-C.Chiang 2918 HighlyReconfigurableQuadratureCouplerWithIdealImpedanceMatchingandPortIsolation ....... P.-L.ChiandT.-C.Hsu 2930 AWaveguideMagic-TWithCoplanarArmsforHigh-PowerSolid-StatePowerCombining ........................................ ........................................................... L.Guo,J.Li,W.Huang,H.Shao,T.Ba,T.Jiang,Y.Jiang,andG.Deng 2942 NovelPlanarCompactCoupled-LineSingle-Ended-to-BalancedPowerDivider ..................................................... .............................................. W.Zhang,Y.Liu,Y.Wu,A.Hasan,F.M.Ghannouchi,Y.Zhao,X.Du,andW.Chen 2953 Hybrid and Monolithic RF Integrated Circuits SwitchedSubstrate-Shield-BasedLow-LossCMOSInductorsforWideTuningRangeVCOs ...................................... ....................................P.Agarwal,S.P.Sah,R.Molavi,S.Mirabbasi,P.P.Pande,S.E.Oh,J.-H.Kim,andD.Heo 2964 LinearCMOSLC-VCOBasedonTriple-CoupledInductorsandItsApplicationto40-GHzPhase-LockedLoop ............... .................................................... Z.Chen,M.Wang,J.-X.Chen,W.-F.Liang,P.-P.Yan,J.-F.Zhai,andW.Hong 2977 A Wideband and High-Linearity E-Band Transmitter Integrated in a 55-nm SiGe Technology for Backhaul Point-to-Point 10-Gb/sLinks ................................................................. D.delRio,I.Gurutzeaga,A.Rezola,J.F.Sevillano, I. Velez, S. E. Gunnarsson, N.Tamir, C.E. Saavedra, J. L. Gonzalez-Jimenez, A. Siligaris, C. Dehos, and R. Berenguer 2990 TheInfluenceoftheOutputImpedancesofPeakingPowerAmplifieronBroadbandDohertyAmplifiers ......................... .......................................................................... W.Shi,S.He,F.You,H.Xie,G.Naah,Q.-A.Liu,andQ.Li 3002 Generalized Theory and Design Methodology of Wideband Doherty Amplifiers Applied to the Realization of an Octave-BandwidthPrototype ........... R.Darraji,D.Bhaskar,T.Sharma,M.Helaoui,P.Mousavi,andF.M.Ghannouchi 3014 Investigation of a Class-J Mode Power Amplifier in Presence of a Second-Harmonic Voltage at the Gate Node of the Transistor ...................................................................................................... A.AlizadehandA.Medi 3024 Dual-BandDesignofIntegratedClass-JPowerAmplifiersinGaAspHEMTTechnology ........................................... .................................................................................................. A.Alizadeh,M.Frounchi,andA.Medi 3034 Investigation of Integrated Smooth Transistor’s Switching Transition Power Amplifier—2.4-GHz Realization ofClass-EM ................................................................................................... A.ErshadiandA.Medi 3046 StabilityAnalysisofDigitalMicrowavePowerAmplifiers ................................ A.Suárez,A.Wentzel,andW.Heinrich 3056 DevelopmentofWidebandandHighIIP3Millimeter-WaveMixers .......... T.T.Nguyen,A.Riddle,K.Fujii,andA.-V.Pham 3071 IEEE MICROWAVETHEORYAND TECHNIQUES SOCIETY TheMicrowaveTheoryandTechniquesSocietyisanorganization,withintheframeworkoftheIEEE,ofmemberswithprincipalprofessionalinterestsinthefieldofmicrowave theoryandtechniques.AllmembersoftheIEEEareeligibleformembershipintheSocietyuponpaymentoftheannualSocietymembershipfeeof$24.00,plusanannualsubscription feeof$50.00peryearforelectronicandprintmedia.Forinformationonjoining,writetotheIEEEattheaddressbelow.MembercopiesofTransactions/Journalsareforpersonal useonly. ADMINISTRATIVECOMMITTEE D.WILLIAMS, President T.BRAZIL, PresidentElect J.A.NAVARRO, Secretary A.ABUNJAILEH, Treasurer A.ABUNJAILEH M.BOZZI K.GHORBANI R.HENDERSON P.KHANNA S.KOUL M.MADIHIAN G.PONCHAK J.E.RAYAS-SANCHEZ M.SALAZARPALMA D.SCHREURS D.WILLIAMS S.BARBIN T.BRAZIL R.GUPTA A.JACOB D.KISSINGER G.LYONS D.PASQUET S.RAMAN S.REISING A.SANADA M.STEER HonoraryLifeMembers DistinguishedLecturers PastPresidents J.BARR R.SPARKS 2015–2017 2016–2018 2017–2019 K.WU(2016) T.ITOH P.STAECKER T.-W.HUANG A.MORTAZAWI C.CAMPBELL P.ROBLIN W.Y.ALI-AHMAD T.LEE(2015) M.JARRAHI H.ZIRATH T.NAGATSUMA N.SHINOHARA N.BORGESCARVALHO R.WEIGEL(2014) MTT-SChapterChairs Albuquerque:E.FARR Denver:M.JANEZIC Macau:C.C.PONG PikesPeak:K.HU Sweden:A.RYDBERG Argentina:A.M.HENZE EasternNo.Carolina:T.NICHOLS Madras/India:S.SALIVAHANAN Poland:W.J.KRZYSZTOFIK Switzerland:M.MATTES Atlanta:K.NAISHADHAM Egypt:E.HASHEESH Malaysia:M.K.M.SALLEH Portugal:J.CALDINHASVAZ Syracuse:D.MCPHERSON Austria:A.SPRINGER Finland:V.VIIKARI Malaysia,Penang:B.L.LIM Princeton/CentralJersey:W.CURTICE Taegu:Y.-H.JEONG Baltimore:I.AHMAD FloridaWestCoast:J.WANG Melbourne:R.BOTSFORD Queensland:K.BIALKOWSKI Tainan:H.-H.CHEN Bangalore/India:K.VINOY Foothills:M.CHERUBIN MexicanCouncil: RiodeJaneiro:J.R.BERGMANN Taipei:C.MENG Beijing:Z.FENG France:D.BAJON R.M.RODRIGUEZ-DAGNINO Rochester:M.SIDLEY Thailand: Belarus:S.MALYSHEV Germany:G.BOECK Milwaukee:S.G.JOSHI Romania:T.PETRESCU C.PHONGCHAROENPANICH Benelux:G.VANDENBOSCH Greece:R.MAKRI Monterrey/Mexico: Russia,Moscow:V.A.KALOSHIN Toronto:G.V.ELEFTHERIADES Boston:C.GALBRAITH Gujarat/India:S.CHAKRABARTY R.M.RODRIGUEZ-DAGNINO Russia,Nizhny-Novgorad: Tucson:H.XIN Bombay/India:M.V.PITKE Harbin:Q.WU Morocco:M.ESSAAIDI G.L.PAKHOMOV Tunisia:A.GHARSALLAH Brasilia:J.BEZERRA/ Hawaii:K.MIYASHIRO Montreal:K.WU Russia,Novosibirsk:A.YAROSLAVTSEV Turkey:B.SAKA M.VINICIUSALVESNUNES HongKong:H.WONG Morocco:M.ESSAAIDI Russia,Saratov/Penza:M.D.PROKHOROV TwinCities:C.FULLER Buenaventura:C.SEABURY Houston:S.A.LONG Nagoya:J.BAE Russia,SaintPetersburg:S.P.ZUBKO UK/RI:A.REZAZADEH Buffalo:M.R.GILLETTE Houston,CollegeStation:G.H.HUFF Nanjing:W.HONG Russia,Siberia:V.V.SUHOTIN Ukraine,East:N.K.SAKHNENKO Bulgaria:K.ASPARUHOVA Hungary:L.NAGY Nanjing,Hangzhou:L.SUN Russia,Tomsk:D.ZYKOV Ukraine,Kiev:Y.PROKOPENKO Canada,Atlantic:Z.CHEN Huntsville:H.SCHANTZ NewHampshire:E.H.SCHENK SanDiego:J.TWOMEY Ukraine,Rep.ofGeorgia: CedarRapids/CentralIowa:C.G.XIE Hyderabad/India:S.R.NOOKALA NewJerseyCoast:J.SINSKY SantaClaraValley/SanFrancisco: K.TAVZARASHVILI Central&SouthItaly:L.TARRICONE India:D.BHATNAGER NewSouthWales:Y.RANGA N.SHAMS Ukraine,Vinnitsya: CentralNo.Carolina:Z.XIE India/Kolkata:S.SANKARALINGAM NewZealand:A.WILLIAMSON Seattle:S.EBADI V.M.DUBOVOY CentralTexas:J.PRUITT Indonesia:E.T.RAHARDJO NorthItaly:G.OLIVERI Seoul:C.SEO Ukraine,West:I.IVASENKO Centro-NorteBrasil: Israel:S.AUSTER NorthJersey:A.K.PODDAR SerbiaandMontenegro:B.MILOVANOVIC´ UnitedArabEmirates: M.V.ALVESNUNES Japan:N.SUEMATSU NorthernAustralia:J.MAZIERSKA Shanghai:J.MAO N.K.MALLAT Chengdu:Z.NEI Kansai:T.ISHIZAKI NorthernCanada:M.DANESHMAN Singapore:Z.YANG UttarPradesh/India:M.J.AKHTAR Chicago:D.ERRICOLO Kingston:S.PODILCHAK NorthernNevada:B.S.RAWAT SouthAfrica:A.LYSKO Vancouver:S.MCCLAIN Cleveland:M.SCARDELLETTI Kitchener-Waterloo:R.R.MANSOUR Norway:M.UBOSTAD SouthAustralia:T.KAUFMANN Venezuela:J.B.PENA Columbus:A.O’BRIEN Lebanon:E.NASSAR OrangeCounty: SouthBrazil:J.R.BERGMANN Victoria:K.GHORBANI Connecticut:C.BLAIR Lithuania:B.LEVITAS H.J.DELOSSANTOS SoutheasternMichigan:T.OZDEMIR VirginiaMountain:T.A.WINSLOW Croatia:D.BONEFACIC LongIsland/NewYork: Oregon:K.MAYS SouthernAlberta:E.FEAR WashingtonDC/NorthernVirginia: Czech/Slovakia:J.VOVES S.PADMANABHAN Orlando:K.KARNATI Spain:J.I.ALONSO T.IVANOV Dallas:R.SANTHAKUMAR LosAngeles,Coastal:V.RADISIC Ottawa:Q.ZENG Springfield:P.R.SIQUEIRA WesternSaudiArabia:A.SHAMIM Dayton:A.TERZUOLI LosAngeles,Metro/SanFernando: Philadelphia:A.S.DARYOUSH SriLanka:A.U.A.W.GUNAWARDENA Winnipeg:P.MOJABI Delhi/India:A.BASU T.CISCO Phoenix:S.ROCKWELL St.Louis:D.BARBOUR Xian:X.SHI Editors-In-Chief AssociateEditors ULUniCvA.oPfEPRavRiEaGRINI XSTLE´IPMHANEBILA UPAnTivR.IoCfKNFoAtrYeDame NTZTYUHS-TGHUANGMA UMnAivR.ToIfNEVrlOanSgSIeEnK–Nu¨rnberg Pavia,Italy Limoges,France NotreDame,IN,USA Taipei,Taiwan Erlangen,Germany JOSECARLOSPEDRO XUDONGCHEN ANDREAFERRERO ARUNNATARAJAN JOHNWOOD UniversidadedeAveiro Nat.Univ.ofSingapore KeysightTechnol. 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Allrightsreserved.PeriodicalsPostagePaidatNewYork,NYandatadditionalmailingoffices.Postmaster:SendaddresschangestoIEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,IEEE, 445HoesLane,Piscataway,NJ08854-4141.GSTRegistrationNo.125634188.CPCSalesAgreement#40013087.ReturnundeliverableCanadaaddressesto:PitneyBowesIMEX,P.O.Box4332,StantonRd., Toronto,ONM5W3J4,Canada.IEEEprohibitsdiscrimination,harassmentandbullying.Formoreinformationvisithttp://www.ieee.org/nondiscrimination.PrintedinU.S.A. DigitalObject Identifier10.1109/TMTT.2017.2724958 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 1 Guest Editorial THISMini-SpecialIssueoftheIEEE TRANSACTIONSON to extend their conferencepapers significantly by adding new MICROWAVE THEORY AND TECHNIQUES includes five materials. A total of 14 papers were received and five con- expanded papers from the IEEE International Conference on tributions were accepted for publication in this Mini-Special Numerical Electromagnetic Modeling and Optimization for Issue after a rigorous review process. RF, microwave, and terahertz applications (NEMO), held in The papers included in this Mini-Special Issue take us on Beijing, China, July 27–29, 2016. thetopicsonthemodeldevelopmentofcompacttime-domain NEMO is an annual international conference and was models from frequency-domain finite-element solutions, founded by the IEEE Microwave Theory and Techniques GPU-accelerated finite-element method with multilevel Society (MTT-S) in 2014. It has since been rotated between preconditioners, the modeling and realization of a new class Europe, North America, and Asia. NEMO2016 was held in of perforated substrate-in-waveguide filters, parameter Beijing in 2016 after its previous version of 2015 in Ottawa, de-embedding of spiral inductors on substrate integrated ON, Canada. It aims to provide an ideal forum to bring suspended line, and the design of a compact dual-mode dual- together experts and practitioners to share new ideas on tech- band filter with widely separated passbands for integration niques for electromagnetic modeling, propose efficient design in a TX/RX front-end. These papers present not only new algorithms and tools, and anticipate the modeling/analysis theoretical concepts and ideas, but also interesting designs needs of future technologies and applications. NEMO2016 and novel circuit structures and configurations. is particularly intended to stimulate discussion and explo- As Guest Editors, we would like to express our sincere ration of “disruptive” technologies of EM-CAD in addition gratitudeto Prof. Luca Perregriniand Prof. José Carlos Pedro to traditional topics. It features an exciting technical program (Editors-in-Chief of this TRANSACTION) for the opportu- and invited talks by internationally recognized experts in nity and support to publish a Mini-Special Issue devoted electromagnetic modeling and optimization. The organization to NEMO2016. Special thanks also go to Associate Editor of NEMO clearly shows the willingness and engagement of Prof. Kamran Ghorbani for his editorial help and support. theMTT-Stopromoteandwelcomeresearchincomputational Finally, we wish to thank all the reviewers for devoting electromagnetics,andtoencouragetheauthorstopublishtheir their valuable time and expertise to review the submitted results in MTT-S conferences and journals. manuscripts. NEMO2016 was a success, with 133 submissions from 19countries,108papersacceptedafterthereviewprocess,and 190 attendees. Held in the elegant state-of-the-art new hotel, ER-PING LI, Guest Editor ZJU-UIUC Institute Zhaolin Grand Hotel of Beijing, the conference ran with two paralleloralsessionsonthreeconsecutivedaysandsixplenary Zhejiang University Hangzhou 310027, China session speakers on the advances of the art. Twenty-three poster papers were selected for the Student Paper Contest, of which three best papers were awarded in the banquet. The ZHIZHANGCHEN, Guest Editor technical topics covered a wide range of numerical methods Departmentof Electrical and Computer Engineering and their applications ranging from RF to microwave to Dalhousie University terahertz and from components to systems. It also covered Halifax, NS B3H 4R2, Canada topics in IC packaging and interconnects, multiphysics and modeling, optimization, antennas, and MIMO. CHI-HOU CHAN, Guest Editor Authors of all the papers presented at NEMO2016 were Department of Electronic Engineering invited to submit an expanded version of their work for City University of Hong Kong publication in this Mini-Special Issue. They were requested Hong Kong Digital ObjectIdentifier 10.1109/TMTT.2017.2725098 0018-9480©2017IEEE.Personaluseispermitted, butrepublication/redistribution requires IEEEpermission. Seehttp://www.ieee.org/publications_standards/publications/rights/index.html formoreinformation. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES ∗ Er-PingLi(S’91–M’92–SM’01–F’08)joinedtheSingaporeA STARResearchInstituteofHigh Performance Computing, Singapore, as a Principal Scientist and the Director of the Electronic and PhotonicsDepartmentin 2001. Since 1989,he has been a Research Associate/Fellow with the School of Electronic and Information Technology, Sheffield Hallam University, Sheffield, U.K., and a Senior Research Fellow, the Principal Research Engineer, an Associate Professor, and the Technical Director with the Singapore Research Institute and Industry, Singapore. He is currently a Changjiang-Qianren Distinguished Professor with the School of Electronic Engineering and the Dean of the Joint Institute of Zhejiang University–University of Illinois at Urbana–Champaign,Zhejiang University, Hangzhou,China. He has authoredor co-authored over 400 papers published in international journals and conferences and has authored 2 books published by Wiley–IEEE Press and Cambridge University Press. He holds and has filed a number of U.S. patents. His current research interests include electrical modeling and the design of microscale/nanoscale integrated circuits, 3-D electronic package integration, and nanoplasmonic technology. Dr.LiisaFellowoftheMITElectromagneticsAcademy,USA.Hewasarecipientofthe2015IEEERichardStoddardAward on Electromagnetic Compatibility (EMC), the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY Technical AchievementAward, the SingaporeIES Prestigious EngineeringAchievementAward, and the ChangjiangChair Professorship Award from the Ministry of Education in China, and a number of Best Paper Awards. He was elected for the IEEE EMC Distinguished Lecturer in 2007. He is the founding member of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES(MTT)-RFNanotechnologyCommittee.HehasbeentheGeneralChairandtheTechnicalProgramChairformany internationalconferences.HewasthePresidentforthe2006InternationalZurichSymposiumonElectromagneticCompatibility, the Founding General Chair for the Asia–Pacific EMC Symposium, the General Chair for the 2010 IEEE Symposium on Electrical Design for Advanced Packaging Systems, and the Chairman of the IEEE EMC Singapore Chapter from 2005 to 2006. He served as an Associate Editor of IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS from 2006 to 2008, the Guest Editor of the 2006 and 2010 IEEE EMC Special Issues, and the Guest Editor of the 2010 IEEE MTT Asia–Pacific MicrowaveConferenceSpecialIssue.HeiscurrentlyanAssociateEditoroftheIEEETRANSACTIONSONELECTROMAGNETIC COMPATIBILITY and the IEEE TRANSACTIONS ONCOMPONENTS,PACKAGING, ANDMANUFACTURING TECHNOLOGY. He has been invited to give numerous invited talks and plenary speeches at various international conferences and forums. Zhizhang (David) Chen (S’92–M’92–SM’96–F’10) received the master’s degree in radio engineering from Southeast University, Nanjing, China, and the Ph.D. degree in electrical engineering from the University of Ottawa, Ottawa, ON, Canada. He joined McGill University, Montreal, QC, Canada, as a NSERC Post-Doctoral Fellow in 1993. He was an Adjunct or Visiting Professor with the University of Nottingham, Nottingham,U.K., the ÉcoleNationaleSupérieuredesTélécommunicationsdeBretagne,Brest, France, ShanghaiJiao Tong University,Shanghai,China, the Beijing Universityof Aeronautics andAstronautics,Beijing,China,FuzhouUniversity,Fuzhou,China,theHongKongUniversity of Science and Technology, Hong Kong, and the University of Electronic Science and Technology of China, Chengdu, China. He was one of the originators of the unconditionally stable methods that have been highly cited and used. He and his team have also devel- oped a number of nonlinear ultra-wideband receivers and planar wireless power transfer transmitting and receiving structures. He was the Head of the Department of Electrical and Computer Engineering, Dalhousie University, Halifax, NS, Canada, where he is currently a Professor. He has authored or co-authored over 300 journal and conference papers in computational electromagnetics, RF/microwave electronics, antennas, and wireless technologies. His current research interests include time-domain electromagnetic modeling techniques, ultra-wideband wireless communication systems, and wireless power transfer. Dr. Chen was involved in organizing many international workshops and conferences as the Chair, the Co-Chair, the Session Organizer, and a committee member. He was a recipient of the 2005 Nova Scotia Engineering Award, the 2006 Dalhousie Graduate Teaching Award, the 2007 and 2015 Dalhousie Faculty of Engineering Research Award, the 2013 IEEE Canada FessendenMedal,andtheDalhousieUniversityProfessorship.HeisaFellowoftheCanadianAcademyofEngineeringandthe EngineeringInstituteofCanada.HehasbeenanAssociateEditoroftheIEEEJOURNALONMULTISCALEANDMULTIPHYSICS COMPUTATIONAL TECHNIQUES since 2016 and a Guest Editor of several professional journals and magazines. He has been invited to give many invited talks and plenary speeches at various international meetings, conferences, and forums. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 3 Chi-Hou Chan (S’86–M’86–SM’00–F’02) received the B.S. and M.S. degrees in electrical engineering from The Ohio State University, Columbus, OH, USA, in 1981 and 1982, respectively, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana–Champaign,Champaign, IL, USA, in 1987. From 1987 to 1989, he was a Visiting Assistant Professor with the Department of Electrical andComputerEngineering,UniversityofIllinoisatUrbana–Champaign.From1989to1998,he wasafacultymemberwiththeDepartmentofElectricalEngineering,UniversityofWashington, Seattle,WA,USA.In1996,hejoinedtheDepartmentofElectronicEngineering,CityUniversity of Hong Kong, Hong Kong, and later became a Chair Professor of electronic engineering in 1998. From 1998to 2009,he was initially the Associate Dean and then became the Dean of the College of Science and Engineering. He also served as Acting Provost with the university from 2009 to 2010. He is currently the Director of the State Key Laboratory of Millimeter Waves,PartnerLaboratory,CityUniversityofHongKong.Hiscurrentresearchinterestsinclude computational electromagnetics, millimeter-wave circuits and antennas, and terahertz science and technology. Dr. Chan was a recipient of the U.S. National Science Foundation Presidential Young Investigator Award in 1991, the Joint Research Fund for Hong Kong and Macao YoungScholars, and the National Science Fund for Distinguished Young Scholars, China, in 2004. He was the recipient of the Outstanding Teacher Award from the Department of Electronic Engineering, City UniversityofHongKong,in1998,1999,2000,and2008.HehasbeentheGeneralCo-ChairofISAP2010,iWAT2011,iWEN 2013, ICCEM 2015, ICCEM 2016, and GSMM2017. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 1 Compact and Passive Time-Domain Models Including Dispersive Materials Based on Order-Reduction in the Frequency Domain Rolf Baltes, Ortwin Farle, and Romanus Dyczij-Edlinger Abstract—In this paper, compact time-domain (TD) models function in order to obtain a low-dimensional realization in featuring materials with frequency-dependent electromag- stateordescriptorrepresentation[3].Sincethepassivityofthe netic (EM) properties are derived. The considered frequency- resultingmodelis notguaranteed,extrapassivationsteps may dependentmaterialmodelsincludemultitermDebyeandLorentz be required [4]. Moreover, since all of these transformation modelsfortheelectricpermittivityandthemagneticpermeability and a multiterm Drude model for the electric conductivity. The techniques are based on the system transfer function, it is TD models are based on finite-element systems in the frequency notpossible to reconstructthe transientelectromagnetic(EM) domain(FD).Torenderthemodelcompactandcomputationally fields. efficient,thedimensionof theFD system iscompressed with the In order to overcome these limitations, a new FD-to-TD help of projection-based model-order reduction. In contrast to transformation method was presented in [5]. The idea is to olderapproaches,theTDtransformationisperformedontheFD model itself rather than on thetransfer function.Theresult is a derive an FD description that can be transferred to the TD state-spacerepresentation,whichmayeitherbesolvedbycustom directly. The FD model is based on a finite-element (FE) timeintegrators or importedintocommercial circuitsimulators. scatteringformulation,whichprovidestransparentportbound- The advantages of the new approach include provable passivity aries, and therefore is free of interior resonances. However, of the FD model, provable causality of the TD model, and the FE models are of very high dimension, and direct transfor- ability to reconstruct the transient EM fields. mation to the TD would lead to enormous computational Index Terms—Finite-element (FE) analysis, reduced-order costsforsolvingtheTDmodel.Therefore,model-orderreduc- systems, time-domain (TD) analysis. tion (MOR) is applied. The FE system is projected onto the low-dimensional column space of a well-chosen projection I. INTRODUCTION matrix [6]–[11], which is constructed by the reduced basis COMPLEX microwave systems combine linear compo- approach [6]–[8], that is, its column space is spanned by the nents, such as interconnects or filters, with nonlinear solution vectors of the FE system at certain frequencies. The active devices. In contrast to the nonlinear components, present MOR method thus fits in the general framework of the simulation of the linear subsystem can be carried out in interpolatorymodelreduction[9]–[11]. Theresultingreduced- the frequency domain (FD). This may have some advantages, order model (ROM) preserves the structure, the transparent which include the handlingof frequency-dependentmaterials, port boundaries, and the passivity of the FE system, and it electrically small details, and resonant structures. allows for reconstructing the EM fields. Most importantly, Since the simulation of the whole system hasto be done in ROMevaluationiscomputationallyverycheap,duetoitslow the time domain (TD), efficient techniques for transforming dimension. The subsequent FD-to-TD transformation leads to the FD data to the TD are required. Early methods com- aTDmodelinstate-spacerepresentation,whichinheritsthese puted the transfer function in the FD, converted it to the properties, and which is guaranteed to be causal, thanks to TD, and convoluted it with the input signals [1], [2]. The passivityinthe FD.Theavailabilityofthe TDmodelinstate- main drawback of this procedure is that it is computationally space representation is of great practical utility, because it expensive, especially when the number of time steps is large. allowsforeasyintegrationintoin-houseorcommercialcircuit Laterapproachesusedrationalapproximationsforthetransfer simulators, by techniques like in [12] or [4]. The viability of this approach is demonstrated in Section V, by employing an Manuscript received January 5, 2017; revised April 28, 2017; accepted explicitRunge–Kuttatime-steppingschemeforcomputingthe May 13, 2017. This work was supported by the Deutsche Forschungsge- meinchaft under Grant DY 112/1-2. This paper was presented at the IEEE TD results. MTT-SInternationalConferenceonNumericalElectromagneticModelingand The original method of [5] was limited to frequency- OptimizationforRF,Microwave,andTerahertzApplications, onJuly27–29, independent materials. It was extended in [13] and [14] to 2016,Beijing, China. (Correspondingauthor:RolfBaltes.) R. Baltes and R. Dyczij-Edlinger are with the Chair of Electromag- materials with electric permittivity of single-term Debye type netic Theory, Saarland University, D-66123 Saarbrücken, Germany (e-mail: and electric conductivity of single-term Drude type. The [email protected]). objective of this paper is to extend the method to general O.FarlewaswiththeChairofElectromagneticTheory,SaarlandUniversity, D-66123Saarbrücken, Germany.Heis nowwithCST–Computer Simulation materials, including multiterm Debye and/or Lorentz models Technology AG,D-64289Darmstadt, Germany. for the electric permittivity and the magnetic permeability, Color versions of one or more of the figures in this paper are available andalsomultitermDrudemodelsfortheelectricconductivity. onlineathttp://ieeexplore.ieee.org. Digital ObjectIdentifier 10.1109/TMTT.2017.2708095 This paper covers both the derivation of an appropriate 0018-9480©2017IEEE.Personaluseispermitted, butrepublication/redistribution requires IEEEpermission. Seehttp://www.ieee.org/publications_standards/publications/rights/index.html formoreinformation. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES FD-ROM andthe FD-to-TDtransformation.Thecriticalpoint followingthattheindexesi ,i , j , j ,andi aredefinedin D L D L c with the present setting is that the passivity proof from our the limits of (2). previous works does not apply. Let the boundary of (cid:2) consist of electric walls (cid:9) and E Yeta generalproofof passivity forTD modelsobtainedby the cross section (cid:9) of one feeding waveguide. The outward P interpolatoryMORisavailablefrom[10].However,itexhibits pointing unit normal vector on the boundaryis denoted by nˆ. a high level of abstraction and makes extensive use of the It is assumed that the waveguide supports M modes of theory of matrix polynomials [15]. This paper provides an transverse EM (TEM) type and that higher order modes are alternative proof for the concrete physical setting described rapidly decaying to zero. The excitation of (cid:2) is realized by earlier, which relies on elementary engineering mathematics impressing the tangential componentof the magnetic field H t only. at the portinterface (cid:9) . Thisleads to the followingboundary P value problem in terms of the electric field E: II. FREQUENCY-DOMAIN MODELS ∇×μ−1∇×E+jk η κE−k2ε E = 0 in (cid:2) (3a) r 0 0 0 r A. Problem Statement E×nˆ = 0 on (cid:9) (3b) (cid:3) (cid:4) E Letthe computationaldomain(cid:2) be source-freeand consist of two subdomains (cid:2) and (cid:2) , such that μr−1∇×E ×nˆ=−jk0η0Ht on (cid:9)P (3c) 0 1 (cid:2)=(cid:2)0⊕(cid:2)1. (1) where η0 denotes the characteristic impedance of free-space. The material within (cid:2) is assumed to be linear and isotropic. B. Finite-Element Scattering Formulation The relativeelectric permittivity,the relative magneticperme- ability,andtheelectricconductivityin(cid:2) and(cid:2) aredenoted The scattering formulation takes the modal amplitudes of 0 1 by εr,0, μr,0, and κ0 and εr,1, μr,1, and κ1, respectively. The incidentwavesak asinputandcomputesthemodalamplitudes material properties in (cid:2) are assumed to be constant over of the outgoing waves b. The discrete formulation is based 0 l frequency; εr,1 and μr,1 shall be given by multiterm Debye onthe FE functionspace V ⊂Hcurl; its derivationfollows[5] and Lorentz models, and κ by a multiterm Drude model and leads to the following system: 1 (cid:3) (cid:4) εr,1 = ε∞+εD (2a) R(k0)+jk0η0BBT xE = 2jk0η0bkak (4a) εD = i(cid:2)NDe=D11G+D,jikD0(cid:5)c0ετDe,,iiDD with bl = blTxE −δkl (4b) μr,1 = +μ∞i(cid:2)NL=e+L1ωμe2D,iLG+L,2iLjk(cid:5)0εc0Lδ,ieL,iωLe2−,iLk02c02 ((22bc)) RB == (cid:7)−Sbμk+02(cid:5)Tμ·ε·r−,·1+1(kεb0D)S(k(cid:8)μ0.,)DT+D(cid:6)jk0η0(cid:5)Tκ +κ1(k0)TD(cid:6) ((55ba)) 1 M μD = +jN(cid:2)Dm=(cid:2)NDm1L1F+D,jjkD0F(cid:5)cL0μ,τjDmL,(cid:5),jjDDμL,jLωm2,jL (2d) The enStμr,iie,sj o=f (cid:9)(cid:9)th(cid:2)e0m∇a×triwceis·aμnr−d,01v∇ec×towrsjadr(cid:2)e defined as (6a) jL=1ωm2,jL +2jk0c0δm,jL −k02c02 Tκ,i,j = wi ·κ0wjd(cid:2) (6b) κ1 = i(cid:2)cN=C11+GCjk,i0ccκ00τ,icc,ic. (2e) Tε,i,j = (cid:9)(cid:9)(cid:2)(cid:2)00wi ·εr,0wjd(cid:2)+(cid:9)(cid:2)1wi ·ε∞wjd(cid:2) (6c) Herein, NeD, NmD, NeL, NmL, and NC denote the numberof Sμ,D,i,j = ∇×wi ·∇×w jd(cid:2) (6d) terms in the Debye, Lorentz, and Drude models, respectively; (cid:9)(cid:2)1 k0 and c0 stand for the free-space wavenumber and the TD,i,j = wi ·wjd(cid:2) (6e) free-space speed of light, respectively; ε∞ and μ∞ are the (cid:9)(cid:2)1 permittivity and permeability at infinite frequency, respec- bk,i = (wi ×hˆk)·nˆd(cid:9) (6f) tively; GD,iD, GL,iL, FD,jD, FL,jL, and GC,ic are scaling (cid:9)P factors; and (cid:5)εD,iD and (cid:5)μD,jD, and τe,iD and τm,jD are where w ,w ∈ V, and hˆ denotes the modal magnetic field the Debye relaxation strengths and times, respectively. The i j k ˆ Lorentzresonanceangularfrequenciesanddampingconstants pattern of the kth mode. Since hk may be chosen real-valued aanreddtheneoctehdanbgyesωien,iLp,erωmmit,tjiLv,itδye,aiLn,dapnedrmδmea,bjLi,litryesapbeocutitvetlhye, fsoeernTEfrMomm(o6d)etsh,aatllthmeamtriactersicoesf (S6μ),aSrμe,rDe,aTl-κv,aBluBedT.,ItancdanTbDe respective Lorentz pole by (cid:5)εL,iL and (cid:5)μL,jL, respectively. are symmetric positive semidefinite (SPSD), while the matrix Fthoerstthaeticcornedguimcteiv,iatyn,dththeeκ0τ,ci,cicvvalauluesesarreeptrheesecnotntthriebuctoiollnissioinn STμε,DisasynmdmTeDt,ritchepofsoiltliovweidnegfirneiltaet.ioNnotheotlhdas:t for the kernels of time constants. If not specified otherwise, it is implied in the ker(TD)⊂ker(Sμ,D). (7) This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. BALTESetal.: COMPACTANDPASSIVETDMODELSINCLUDINGDISPERSIVEMATERIALS 3 C. Model-Order Reduction A. Proof of Property P1 Since the FE system (4) may be of very high dimension, It must be shown that the poles of S have nonpositive real direct FD-to-TD transformation of (4) would lead to TD part. Since the matrix B˜ is constant, and because adding the models that are very expensive to solve. To overcome this, unity matrix has no influence on the poles, this is equivalent projection-based MOR techniques are applied to the FE sys- to showing that the eigenvalues λ of tem. The idea is to project the FE system onto a well-chosen subspace of low dimension [6]–[8]. The present method is R˜(λ)v+λμ0B˜B˜Tv=0 (14) based on the self-adaptive multipoint scheme of [8], which have nonpositive real part. Due to the singularity of the leads to a complex-valued projection matrix U. However, it was shown in [5] that preservation of passivity requires a matricesS˜μ,S˜μ,D,T˜κ, and T˜D, a numberof zeroeigenvalues exist. However, λ = 0 clearly features nonpositive real part, projection matrix V that is real-valued. As suggested in [10] sothatthiscasemaybeexcludedfromthefollowinganalysis. and [16], V is obtained from The proof will be in two steps. First, it will be shown ran(V)=ran(ReU)∪ran(ImU). (8) that the eigenvalues of the following generalized eigenvalue problem (EVP) have nonpositive real part: For numerical stability, the columns of V are orthogonalized. Thus, the ROM approximation to the FE solution xE reads M v+λ(M +M )v=0 (15) 0 1 2 x ≈Vx˜ (9) E with M ,M ∈ R being SPSD, M ∈ R being skew- 0 1 2 and the ROM itself is given by symmetric, M2 = −M2T, and (M1 + M2)v being nonzero. (cid:3) (cid:4) In the second step, the EVP (14) will be transformed to (15), R˜(k0)+jk0η0B˜B˜T x˜ = 2jk0η0b˜kak (10a) which completes the proof. ∗ b˜ = b˜Tx˜ −δ (10b) Step 1: Multiplying (15) by v leads to l l lk with m0+λm1+λm2 =0 with mi =v∗Miv. (16) (cid:5) (cid:6) R˜ = S˜μ+(cid:5) μr−,11(k0)S˜μ,D+(cid:6)jk0η0 T˜κ +κ1(k0)T˜D Because of the matrix properties, we have −k02 T˜ε+εD(k0)T˜D (11a) m ,m ≥ 0 (17) 0 1 S˜μ = VTSμV, S˜μ,D =VTSμ,DV (11b) m = m∗ =v∗M∗v=v∗MTv=−v∗M v=−m . (18) T˜κ = VTTκV, T˜D =VTTDV (11c) 2 2 2 2 2 2 T˜ε = VTTεV, b˜k =VTbk, B˜ =VTB. (11d) Thus m2 is purely imaginary and can be expressed as Notethattheprojectionpreservesthesymmetryaswellasthe m2 =jα, α ∈R. (19) definitenesspropertiesofthematrices.Sincethedimensionof Using this and (16), we get theROMismuchsmallerthanthethatoftheFE system,(10) is evaluated much more efficiently than (4). m m m jm α λ=− 0 =− 0 1 + 0 . (20) m +jα |m +jα|2 |m +jα|2 1 1 1 III. PASSIVITY By (17), the real part of λ, given by the first term on the Since nonpassive FD data may lead to noncausal behavior right-hand side, is nonpositive. in the TD, it is important to check the passivity of the ˜ ˜ Step 2: Since Sμ,D and TD are notof fullrank,three cases FD-ROM,beforetransformingittotheTD.Accordingto[17], are needed to be considered to transform (14) to (15): a scattering matrix S in the Laplace domain represents a Case 1: v ∈ ker(T˜D) ⇒ v ∈ ker(S˜μ,D). The EVP (14) passive system if and only if the following hold. reduces to P1: Each element of S(s) is analytic in Re(s)>0. (cid:10) (cid:11) PP23:: IS(−s)S=∗(sS)(Ss()s.)≥0 for all s, with Re(s)>0. S˜μ+λμ0(T˜κ +B˜B˜T)+ λc22 T˜ε v=0. (21) 0 ∗ Herein, s and S denote the complex conjugate of s and Let us introduce the vector z by the complex conjugate transpose of S, respectively. It will be proven that the S matrix of the FD-ROM (10) features the z:=λv. (22) properties P1–P3. In the Laplace domain, S of (10) takes the form Since the matrix T˜ε is regular, (22) may be written as S=2sμ B˜T(R˜(s)+sμ B˜B˜T)−1B˜ −I (12) 0 0 with c12T˜εz−λc12T˜εv=0. (23) (cid:5) (cid:6) 0 0 R˜(s) = S˜μ+μr−,11(s)S˜μ,D +sμ0 T˜κ +κ1(s)T˜D Combining (21) and (23) yields the EVP +cs22(T˜ε +εD(s)T˜D). (13) M0,1x1+λ(M1,1+M2,1)x1 =0 (24) 0 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES with With the help of the abbreviations ⎡ ⎤ M0,1 = ⎡⎣S˜0μ c1020T˜ε⎤⎦, M2,1 =⎢⎢⎣−c102T˜ε c1020T˜ε⎥⎥⎦ (25a) zpzeDDDrr === [[[zpzDeDDrr,,,111 ········· zpzDeDrDr,,N,NNCCe]D]TT]T (((333000bac))) M1,1 = (cid:18)μ0(T˜κ0+B˜B˜T) 00(cid:19), x1 =(cid:18)0vz(cid:19). (25b) pzeeLL == [[(cid:7)zpeeLL,,11 ······ zpeeLL,,NNeeLL]]TT (cid:8) ((3300de)) aBnydcSoPnsStDru,ctwiohnil,ethMe2m,1atirsicreesaMl-v0a,l1ueadn,dsMke1w,1-sayrmemreeatlr-ivca,launedd TTˆDDrr == dμia0g(GμC0,1GκC0,,1icT˜κD0,icT˜.D..) μ0 GC,NCκ0,NCT˜D ((3300gf)) has full rank. The EVP (24) is thus of the same form as (15), TτDr = d(cid:18)iag(μ0 GC,icκ0,icτc,icT˜D) (cid:19) (30h) apnardCt.atsheer2e:fovre∈tkheer(eS˜ige)n,vval∈/ueksero(fT˜ (2)4.)Inhathvies ncaosnep,o(s1it4i)vereraedasl TˆD = GD(cid:20),1c(cid:5)02εD,1T˜D ... (cid:21)GD,NeDc(cid:5)02εD,NeDT˜D (30i) D D (cid:10)S˜μ+λμ0(cid:5)T˜κ +κ1(λ)T˜D +B˜B˜T)+ λc22εD(λ)T˜D(cid:11)v TD = diag(cid:20)GD,iDc(cid:5)02εD,iDT˜D (cid:21) (30j) +0 λT˜εz=0. (26) TτD = diag GD,iD(cid:5)cε2D,iDτe,iDT˜D (30k) c2 (cid:18) 0 (cid:19) 0 Let us introduce some more auxiliary vectors TˆL = GL,1(cid:5)cε2L,1ωe2,1T˜D ... GL,NeL(cid:5)εcL2,NeLωe2,NeLT˜D (30l) (cid:20) 0 (cid:21) 0 ppzeDDLrr,,,iiiLcc === λλλzz+DeLr,,λλiicL2τc,iλcv (((222777bac))) TTδLL == ddiiaagg(cid:20)2GLG,iLL,(cid:5)iLcε(cid:5)02Lε,icLL2,ωiLe2ω,ieL2,iT˜LDδe,iLT˜D(cid:21) ((3300mn)) zeL,iL = ωe2,iL +2λδe,iL +λ2v (27d) Tω = diag(cid:20)GL,iL(cid:5)εL,iL0ωe4,iLT˜ (cid:21) (30o) zeD,iD = 1+λλτe,iDv. (27e) TˆD,L0 = (cid:18)TˆD(cid:19), TˆDr,0c02=(cid:18)TˆDr(cid:19),DTˆL,0 =(cid:18)TˆL(cid:19) (30p) Since all the vectors defined in (27) are parallel to v, they do 0 0 0 ˜ not lie in the kernel of T . Therefore, (27) is equivalent to D (29) and (28) take the form μ0GC,icκ0,icT˜D(cid:5)λzDr,ic +λ(cid:5)τc,icpDr,ic −λv(cid:6)(cid:6) = 0 (28a) M0,2x2+λ(cid:5)M1,2+M2,2(cid:6)x2 =0 (31) μ0GC,iκ0,icτc,icT˜D pDr,ic −λzDr,ic = 0 (28b) with GD,iDc(cid:5)02εD,iDT˜D(cid:5)zeD,iD +λτe,iDzeD,iD −λv(cid:6) = 0 (28c) ⎡⎢⎢M00,1 00 00 00 00 00 ⎤⎥⎥ GL,iL(cid:5)cε02L,iLωe2,iLT˜D(cid:5)ωe2,iLzeL,iL +2λδe,iLzeL,iL(cid:6) M0,2 = ⎢⎢⎢⎢⎣ 00 00 T0τDr T0D 00ω 00 ⎥⎥⎥⎥⎦ (32a) +λpeL,iL −λv = 0 (28d) 0 0 0 0 TL 0 0 0 0 0 0 T GL,iL(cid:5)cε02L,iLωe2,iLT˜D(cid:5)peL,iL −λzeL,iL(cid:6) = 0. (28e) ⎡⎢⎢M01,1 T0Dr 00 00 00 00⎤⎥⎥L With[(S˜2μ7)+, tλhμe0E(TV˜κP+(2B6˜)B˜bTe)c]vom+escλ2T˜εz M1,2 = ⎢⎢⎢⎢⎣ 000 000 000 T00τD T00δL 000⎥⎥⎥⎥⎦ (32b) 0 0 0 0 0 0 0 (cid:2)NC ⎡ ⎤ + ic=1λμ0GC,icκ0,icT˜DzDr,ic ⎢⎢⎢−MTˆ2DT,r1,0 TˆD0rτ,0 T0τDr TˆD0,0 Tˆ0L,0 00 ⎥⎥⎥ + (cid:2)NeD cλ2GD,iD(cid:5)εD,iDT˜DzeD,iD M2,2 = ⎢⎢⎢⎢−ˆT0TD,0 −T0Dr 00 00 00 00 ⎥⎥⎥⎥(32c) iD=1 0 ⎣−TˆT 0 0 0 0 T ⎦ + i(cid:2)NL=eL1cλ02GL,iL(cid:5)εL,iLωe2,iLT˜DzeL,iL =0. (29) x2 = [x1 0zLD,r0 pDr0zeD z0eL pe0L]T. −TL 0L (32d)

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