On the Modeling and Simulation of Friction

Modeling,controlandsimulationofaPV/FC/UCbasedhybrid

powergenerationsystemforstand-aloneapplications

M.Uzunoglua,b,*,O.C.Onara,b,M.S.Alama

aDepartmentofElectricalandComputerEngineering,UniversityofSouthAlabama,Mobile,AL36688,USAbDepartmentofElectricalEngineering,YildizTechnicalUniversity,Istanbul34349,Turkey

articleinfo

Articlehistory:Received31January2008Accepted5June2008Availableonline25July2008

Keywords:ModelingPVsystemFuelcellHybridpowergenerationUltra-capacitorandrenewableenergyabstract

Differentenergysourcesandconvertersneedtobeintegratedtomeetsustainedloaddemandswhileaccommodatingvariousnaturalconditions.Thispaperfocusesontheintegrationofphotovoltaic(PV),fuelcell(FC)andultra-capacitor(UC)systemsforsustainedpowergeneration.Intheproposedsystem,duringadequateinsolation,thePVsystemfeedstheelectrolyzertoproducehydrogenforfutureuseandtransfersenergytotheloadsideifpossible.WheneverthePVsystemcannotcompletelymeetloaddemands,theFCsystemprovidespowertomeettheremainingload.IftherateofloaddemandincreasestheoutsidelimitsofFCcapability,theUCbankmeetstheloaddemandabovethatwhichisprovidedbyPVandFCsystems.ThemaincontributionofthisworkisthehybridizationofalternateenergysourceswithFCsystemsusinglongandshort-termstoragestrategieswithappropriatepowercontrollersandcontrolstrategiestobuildanautonomoussystem,withapragmaticdesignanddynamicmodelproposedforaPV/FC/UChybridpowergenerationsystem.ThemodelisdevelopedandappliedintheMATLABÒ,SimulinkÒandSimPowerSystemsÒenvironment,basedonthemathematicalandelectricalmodelsdevelopedfortheproposedsystem.Ó2008ElsevierLtd.Allrightsreserved.

Ｊａｎ．２００９，Ｖｏｌｕｍｅ　３，Ｎｏ．１（Ｓｅｒｉａｌ　Ｎｏ．１４）　Ｊｏｕｒｎａｌ　ｏｆ　Ｃｈｅｍｉｓｔｒｙ　ａｎｄ　Ｃｈｅｍｉｃａｌ　Ｅｎｇｉｎ—ｅｅｒｉｎｇ，ＩＳＳＮ１９—３４－７３７５，ＵＳＡ　Ｈｅｔｅｒｏｇｅｎｅｏｕｓ　ｄｙｎａｍｉｃ　ｍｏｄｅｌｉｎｇ　ａｎｄ　ｓｉｍｕｌａｔｉｏｎ　ｏｆ　ａｎ　ｉｎｄｕｓｔｒｉａｌ　ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ　ｒｅａｃｔｏｒ　ｅｘｐｅｒｉｅｎｃｉｎｇ　ｃａｔａｌｙｓｔ　ｄｅａｃｔｉｖａｔｉｏｎ　Ｇ．Ｚａｈｅｄｉ　，Ａ．Ｌｏｈｉ２　ｎ　ＳｉｍｕｌａｔｉｏｎａｎｄＡＩＲｅｓｅａｒｃｈ　Ｃｅｅｎｔｅｒ，Ｄｅｐａｒｔｍｅｎｔ　ｏｆＣｈｅｍｉｃａｌＥｎｇｉｎｅｅｒｉｎｇ，Ｕｎｉｖｅｒｓｉｔｙ　ｏｆＲａｚｉ，Ｋｅｒｍａｎｓｈａｈ　６７１４９６７３４６，１ｒａｎ；　２．Ｄｅｐａｒｔｍｅｎｔ　ｏｆＣｈｅｍｉｃａｌ　Ｅｎｇｉｎｅｅｒｉｎｇ，Ｕｎｉｖｅｒｓｉｔｙ　ｏｆＲｙｅｒｓｏｎ，Ｔｏｒｏｎｔｏ，Ｏｎｔ．Ｍ５Ｂ　２Ｋ３，Ｃａｎａｄａ）　Ａｂｓｔｒａｃｔ：Ｔｈｉｓ　ｐａｐｅｒ　ｉｎｖｅｓｔｉｇａｔｅｓ　ｓｔｅａｄｙ—ｓｔａｔｅ　ａｎｄ　ｄｙｎａｍｉｃ　ｓｉｍｕｌａｔｉｏｎ　ｏｆ　ａｌｌ　ｉｎｄｕｓｔｒｉａｌ　ｆｉｘｅｄ－ｂｅｄ　ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ　ｒｅａｃｔｏｒ．Ａ　ｍａｔｈｅｍａｔｉｃａｌ　ｈｅｔｅｒｏｇｅｎｅｏｕｓ　ｏｎｅ．ｄｉｍｅｎｓｉｏｎａｌ　ｍｏｄｅｌ　ｉｓ　ｄｅｖｅｌｏｐｅｄ　ｆｏｒ　ｓｉｍｕｌａｔｉｏｎ　ｏｆｒｅａｃｔｏｒ　ｐｅｒｆｏｒｍａｎｃｅ　ｉｎ　ｔｈｅ　ｐｒｅｓｅｎｃｅ　ｏｆ　ｌｏｎｇ　ｔｅｒｍ　ｄｅａｃｔｉｖａｔｉｏｎ　ｏｆ　ｓｉｌｖｅｒ／ｑ・ａｌｕｍｉｎａ　ｃａｔａｌｙｓｔ．Ｉｎ　ｔｈｉｓ　ｐａｐｅｒ，ｓｔｅａｄｙ－ｓｔａｔｅ　ｍｏｄｅｌ　ｏｆ　ｔｈｅ　ｒｅａｃｔｏｒ　ｉｓ　ｓｏｌｖｅｄ　ａｎｄ　ｒｅｓｕｌｔｓ　ｏｆ　ｓｔｅａｄｙ　ｓｔａｔｅ　ｓｉｍｕｌａｔｉｏｎ　ａｒｅ　ｆｅｄ　ｔｏ　ｄｙｎａｍｉｃ　ｓｉｍｕｌａｔｏｒ　ａｓ　ｉｎｉｔｉａｌ　ｃｏｎｄｉｔｉｏｎ．Ｗｈｅｎ　ｒｅｓｕｌｔｓ　ｏｆ　ｄｙｎａｍｉｃ　ｓｌｍｕｌａｔｌｏｎ　ａｒｅ　ｃｏｍｐａｒｅｄ　ｗｉｔｈ　ｉｎｄｕｓｔｒｉａｌ　ｒｅａｃｔｏｒ　ｄａｔａ，ｉｔ　ｉｓ　ｆｏｕｎｄ　ｔｈａｔ　ｔｈｅｒｅ　ｗｅｒｅ　ｇｏｏｄ　ａｇｒｅｅｍｅｎｔｓ　ｂｅｔｗｅｅｎ　ｓｉｍｕｌａｔｉｏｎ　ｒｅｓｕｌｔｓ　ａｎｄ　ｉｎｄｕｓｔｒｉａｌ　ｄａｔａ．Ｔｈｅ　ｐｒｏｐｏｓｅｄ　ｍｏｄｅｌ　ｉｓ　ａｌｓｏ　ｖａｌｉｄａｔｅｄ　ｂｙ　ｉｎｄｕｓｔｒｉａｌ　ｐｒｏｃｅｓｓ　ｄａｔａ　ｆｏｒ　ａ　ｐｅｒｉｏｄ　ｏｆ　ｌ　１００　ｏｐｅｒａｔｉｎｇ　ｄａｙｓ．　Ｋｅｙ　ｗｏｒｄｓ：ｓｉｍｕｌａｔｉｏｎ；ｄｙｎａｍｉｃ　ｍｏｄｅｌｉｎｇ；ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ　ｒｅａｃｔｏｒ　１．ＩｎｔＪ　ＯｄｕｃｔｉＯｎ　Ｃｏｎｖｅｒｓｉｏｎ　ｏｆ　ｅｔｈｙｌｅｎｅ　ｔｏ　ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ　ｉｓ　ａｎ　ｉｍｐｏｒｔａｎｔ　ｃａｔａｌｙｔｉｃ　ｒｅａｃｔｉｏｎ　ｉｎ　ｔｈｅ　ｐｅｔｒｏｃｈｅｍｉｃａｌ　ｉｎｄｕｓｔｒｙ　ａｎｄ　ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ　ｉｓ　ｏｎｅ　ｏｆ　ｔｈｅ　ｍｏｓｔ　ｉｍｐｏｒｔａｎｔ　ｐｅｔｒｏｃｈｅｍｉｃａｌ　ｐｒｏｄｕｃｔｓ．Ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ　ｉｓ　ｕｓｅｄ　ｔｏ　ｍａｋｅ　ｅｔｈｙｌｅｎｅ　ｇｌｙｃｏｌ（ｔｈｅ　ｐｒｉｍａｒｙ　ｉｎｇｒｅｄｉｅｎｔ　ｉｎ　ａｎｔｉｆｒｅｅｚｅ），ｐｏｌｙ（ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ），ｇｌｙｃｏｌ　ｅｔｈｅｒｓ，　ｅｔｈａｎｏｌａｍｉｎｅｓ，ｓｕｒｆａｃｔａｎｔｓ，ｅｔｃＩ¨．　Ｃｏｍｍｅｒｃｉａｌ　ｐｒｏｄｕｃｔｉｏｎ　ｏｆ　ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ　ｉｓ　ｋｎｏｗｎ　ｔｏ　ｂｅ　ａ　ｇａｓ　ｐｈａｓｅ　ｒｅａｃｔｉｏｎ　ｏｆ　ｅｔｈｙｌｅｎｅ　ｏｖｅｒ　Ａｇ　ｃａｔａｌｙｓｔｓ【　１　ｗｉｔｈ　ｓｕｆｆｉｃｉｅｎｔ　ａｃｔｉｖｉｔｙ　ａｎｄ　ｓｅｌｅｃｔｉｖｉｔｙ　ｉｎ　Ｆｉｘｅｄ—Ｂｅｄ　Ｒｅａｃｔｏｒ（ＦＢＲ）．Ｔｈｅ　ｍａｉｎ　ｓｔｅｐ　ｏｆ　ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ　ｐｒｏｃｅｓｓ　ｉｓ　ｅｔｈｙｌｅｎｅ　ｏｘｉｄａｔｉｏｎ，ｗｈｉｃｈ　ｉｓ　ｂａｓｅｄ　ｏｎ　ｔｈｅ　ｄｉｒｅｃｔ　ｏｘｉｄａｔｉｏｎ　ｐｒｏｃｅｓｓ．Ｔｈｅｒｅ　ｈａｖｅ　ｂｅｅｎ　ｓｅｖｅｒａｌ　ｒｅｓｅａｒｃｈｅｓ　ｏｎ　ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ　ｐｒｏｃｅｓｓ　ｉｎ　ｔｈｅ　１ｉｔｅｒａｔｕｒｅｓ．　ＣＯｒｒｅｓｐＯｎｄｉｎｇ　ａｕｔｈｏｒ：Ｇ．Ｚａｈｅｄｉ，Ｐｈ．Ｄ．，ａｓｓｉｓｔａｎｔ　ｐｒｏｆｅｓｓｏｒ；　ｒｅｓｅａｒｃｈ　ｆｉｅｌｄｓ：ｐｒｏｃｅｓｓ　ｓｙｓｔｅｍ　ｅｎｇｉｎｅｅｒｉｎｇ，ｓｉｍｕｌａｔｉｏｎ　ａｎｄ　ｏｐｔｉｍｉｚａｔｉｏｎ，ａｒｔｉｆｉｃｉａｌ　ｉｎｔｅｌｌｉｇｅｎｃｅ．Ｅ－ｍａｉｈ　ｇｒｚａｈｅｄｉ＠ｒａｚｉ．ａｃ．ｉｒ．　】８　Ｂａｒａｔｔｉ，ｅｔ　ａｌ【ｊＪ　ｐｒｏｐｏｓｅｄ　ａ　ｓｔｒａｔｅｇｙ　ｔｏ　ｍａｘｉｍｉｚｅ　ｐｅｒｆｏｒｍａｎｃｅ　ｏｆ　ｔｈｅ　ｒｅａｃｔｏｒ　ｂｙ　ｏｐｔｉｍｉｚｉｎｇ　ｄｉｓｔｒｉｂｕｔｉｏｎ　ｏｆ　ｔｈｅ　ｃａｔａｌｙｓｔ　ｉｎ　ｔｈｅ　ｒｅａｃｔｏｒ．Ｂｕｔ　ｔｈｅ　ｗｏｒｋ　ｄｉｄ　ｎｏｔ　ｃｏｎｓｉｄｅｒ　ｅｆｆｅｃｔ　ｏｆ　ｏｐｅｒａｔｉｎｇ　ｃｏｎｄｉｔｉｏｎ　ｏｎ　ｐｒｏｃｅｓｓ　ｐｅｒｆｏｒｍａｎｃｅ．　Ｒｅｆ．【４］ｏｎｌｙ　ｒｅｐｒｅｓｅｎｔｅｄ　ｄｅｓｃｒｉｐｔｉｖｅ　ｏｐｔｉｍｉｚａｔｉｏｎ　ｏｆ　ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ　ｒｅａｃｔｏｒＴＭ．Ｈｗａｎｇ，ｅｔ　ａ／ｓｉｍｕｌａｔｅｄ　ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ　ｒｅａｃｔｏｒ　ｗｈｅｒｅ　ｔｈｅｉｒ　ｐｕｒｐｏｓｅ　ｗａｓ　ｔｏ　ｆｉｎｄ　ａｌｌ　ｏｐｔｉｍｉｚｅｄ　ｔｅｍｐｅｒａｔｕｒｅ　ｆｏｒ　ｔｈｉｓ　ｒｅａｃｔｏｆｌ　－６］。　Ｇｏｎｅｈａｒｏｖａ．ｅｔ　ａ１１７１　ａｎｄ　Ｃａｍｐｂｅｌｌ　ａｎｄ　ｐａｆｆｅｔｔｌｓｌ　ｒｅｐｏｒｔｅｄ　ｔｈａｔ　ａｄｄｉｔｉｏｎ　ｏｆ　ｓｍａｌｌ　ａｍｏｕｎｔ　ｏｆ　ｐｒｏｍｏｔｅｒｓ　ｌｉｋｅ　Ｃ１　ｉｎｃｒｅａｓｅｄ　ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ　ｓｅｌｅｃｔｉｖｉｔｙ．Ｖａｎ　Ｓａｎｔｅｎ［　１　ｒｅｐｏｒｔｅｄ　ｒｅａｃｔｉｏｎ　ｍｅｃｈａｎｉｓｍ　ｏｆ　ｔｈｅ　ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ．Ｈａｒｒｉｏｔｔ　ｓｔｕｄｉｅｄ　ｔｈｅ　ｄｉｓｔｒｉｂｕｔｉｏｎ　ｏｆ　Ａｇ　ｓｕｐｐｏｒｔ　ｏｎ　ａｌｕｍｉｎａ　ａｎｄ　ｒｅｐｏｒｔｅｄ　ｔｈａｔ　ｔｈｅ　３ｃＩ＿＿７　Ａ　ｓｉｌｖｅｒ　ｐａｒｔｉｃｌｅｓ　ｏｎ　ａｌｕｍｉｎａ　ｏｒ　ｓｉｌｉｃａ　ｓｈｏｗｅｄ　ｈｉｇｈｅｒ　ａｃｔｉｖｉｔｉｅｓ／Ｉｏ１．Ｍａｎｙ　ｏｔｈｅｒ　ｗｏｒｋｓ　ｈａｖｅ　ｂｅｅｎ　ｃａｒｒｉｅｄ　ｏｕｔ　ｔｏ　ｉｎｖｅｓｔｉｇａｔｅ　ｔｈｅ　ｅｆｆｅｃｔｓ　ｏｆ　ｔｈｅ　ｐａｒｔｉｃｌｅ　ｓｉｚｅ　ａｎｄ　ｓｈａｐｅｓ，ｔｈｅ　ｓｕｒｆａｃｅ　ｓｔｒｕｃｔｕｒｅｓ　ａｎｄ　ｓｕｐｐｏｒｔ　ｔｙｐｅｓ　ｏｎ　ｔｈｅ　ｃａｔａｌｙｔｉｃ　ａｃｔｉｖｉｔｉｅｓ［　“．Ｃａｍｐｂｅｌｌ　ｓｔｕｄｉｅｄ　ｔｈｅ　ｅｆｆｅｃｔ　ｏｆ　ｔｈｅ　ｓｉｌｖｅｒ　ｐａｒｔｉｃｌｅ　ｓ１。ｚｅｓ　ｏｎ　ｐｒｏｄｕｃｔｉｏｎ　０ｆ　ｅｔｈｙｌｅｎｅ［　】．　Ｗｉｔｈ　ｄｅｅｐ　ｓｔｕｄｙ　ｏｆ　ｔｈｅ　ｐｕｂｌｉｓｈｅｄ　ｐａｐｅｒｓ　ｉｔ　ｉｓ　ｆｏｕｎｄ　ｔｈａｔ　ｔｈｅｒｅ　ｉｓ　ｎｏｔ　ｅｎｏｕｇｈ　ｉｎｆｏｒｍａｔｉｏｎ　ａｂｏｕｔ　ｄｙｎａｍｉｃ　ｍｏｄｅｌｉｎｇ　ｏｆ　ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ　ｒｅａｃｔｏｒ　ｍｏｄｅｌｉｎｇ．　Ｃｕｒｒｅｎｔ　ｒｅｓｅａｒｃｈ　ｆｏｃｕｓｅｓ　ｏｎ　ｔｈｅ　ｓｔｅａｄｙ－ｓｔａｔｅ　ａｎｄ　ｄｙｎａｍｉｃ　ｓｉｍｕｌａｔｉｏｎ　ｏｆ　ｅｔｈｙｌｅｎｅ　ｏｘｉｄａｔｉｏｎ　ｒｅａｃｔｏｒ．Ｔｈｅ　ｐａｐｅｒ　ｉｓ　ｏｒｇａｎｉｚｅｄ　ａｓ　ｆｏｌｌｏｗｉｎｇ：Ｆｉｒｓｔ　ｅｔｈｙｌｅｎｅ　ｏｘｉｄｅ　ｐｒｏｄｕｃｔｉｏｎ　ｐｒｏｃｅｓｓ　ｉｓ　ｄｅｓｃｒｉｂｅｄ．Ｍｏｄｅｌｉｎｇ　ｏｆｒｅａｃｔｏｒ　ｉｓ　ｃｏｎｓｉｄｅｒｅｄ　ｎｅｘｔ．Ｆｏｒ　ｓｏｌｖｉｎｇ　ｔｈｅ　ｇｏｖｅｒｎｉｎｇ　ｅｑｕａｔｉｏｎｓ　ｏｆ　ｔｈｉｓ　ｓｙｓｔｅｍ，ｔｈｅ　ｎｕｍｅｒｉｃａｌ　ｍｅｔｈｏｄｓ　ｔｈａｔ　ａｒｅ　ｕｓｅｄ　ｆｏｒ　ｓｏｌｖｉｎｇ　ｎｏｎｌｉｎｅａｒ　ａｌｇｅｂｒａｉｃ　ａｎｄ　ｄｉｆｆｅｒｅｎｔｉａｌ　ｅｑｕａｔｉｏｎｓ　ａｒｅ　ｄｉｓｃｕｓｓｅｄ　ａｎｄ　ｆｉｎａｌｌｙ　ｓｔｅａｄｙ－ｓｔａｔｅ　ａｎｄ　ｄｙｎａｍｉ

9th International LS-DYNA Users Conference Simulation Technology (1)

3-1 Advanced Modeling and Drop Simulation With New Features of LS-DYNA Jason Wu familyofwu@ Abstract Phone drop simulation is an important application of LS-DYNA in electronics industry. The application has been widely used in all leading companies to produce electronic handsets, like Nokia, Motorola, and Samsung. Since more new features complemented in 970 version of LS-DYNA, phone modeling and drop simulation strategy has changed a lot to obtain more accurate and quick results. This paper introduces new methods, and recent development of phone modeling and drop simulation, based on author’s experiences in recent years. The paper is focused on using right element type in LS-DYNA element library and some new strategies to establish better phone model and drop simulation. The paper provides benchmarks to verify the modeling strategies. The paper still gives an example for advanced application of LS-DYNA in multiply impact simulation. Introduction Since computer simulation technique was applied into cellular phone drop in 1996, computer technology and engineering analysis codes have obtained great progress. Over 100 material models, new contact types, and 10-node tetra elements have been introduced in 970 version of LS-DYNA. Affordable servers equipped with Intel Itanium or AMD Opteron CPU, can finish drop simulation of a complex phone in several hours. The progress of simulation tools gives wide space for modeling upgrade. Now, a model with various element types, consisting of 200 - 300 thousands elements are practicable for drop simulation, instead of the previous model in dozens of thousands elements, with shell as major element type. As computer becomes faster and faster, the model size turns to bigger and bigger. How to efficiently organize a model to obtain more accurate mechanical behavior still is a challenging subject in front of FEA performers. This paper presents some new strategies to establish a phone model, based on author’s practice and study on phone drop simulation in recent years. Element type selection in phone modeling A typical phone model usually has 200K-300K elements divided in 50 – 70 parts. Almost all element types in LS-DYNA will be used into the modeling, based on their geometrical characteristics and deformation modes. Tetrahedral elements are used in modeling of housing and other parts in complex geometry. With the type of elements, user can take advantage of auto-meshing to handle complex geometry. Hexahedral and pentahedral elements are used in the parts in relatively simple geometry, and subjected to 3D complex stress. The brick elements have better performance in explicit codes, but need more manual effort to create mesh. In shell element types, thin shell and thick shell are all recommended. Thin shell element is priority for thin-wall parts in bending dominate deformation, for instance, PCB and thin-wall covers. Thick shell is still useful in phone modeling. It’s suitable for the bending parts attached by another part on surface, for instance LCD glasses with polarizer attachment. Thick shell has good bending performance, and easy to attach another part at its face without the problem of existing a gap to middle plane of a thin shell. Beam element plays important role for simplified solder-joint Simulation Technology (1) 9th International LS-DYNA Users Conference 3-2 modeling, which is the only choice to include the tiny structure into a system level model. And the representation is proven effective and enough accurate. 1D rigid link is necessary in parts attachment and model assembly. It’s a favor method to simplify screw, clip and snap-in attachment with the rigid link. Finally, spring elements are very useful for part connection. It is a reliable alternative of contact in some cases, and replacements of rigid link to a rigid body connection. With pre-defined extension/compression, spring element can supply a pre-force to structures. The performance of different element types in LS-DYNA library is very important for element selection in modeling. The performance includes accuracy and efficiency as well as meshing cost. As CPU time is always a concern when a drop simulation is submitted to run, the efficiency exhibited by each element type is important in performance benchmark. In figure 1, a typical monoblock-type phone model and its element types are demonstrated, where S represents shell element, H means hex/penta element, and T is tetra element.

466 IEEE Transactions on Power Delivery, Vol. 11, No. 1, January 1996

MODELING AND SIMULATION OF THE PROPAGATION OF HARMONICS IN

ELECTRIC POmTER NETWORKS

PART 11 : SAMPLE SYSTEMS AND EXAMPLES

Task Force on Harmonics Modeling and Simulation*

Abstract: This report is the second part of a review on the

nature and modeling of harmonic sources in electrical power

systems and the analysis of harmonics propagation. Existing

IEEE standard test systems are used to develop sample systems

for harmonic studies. Examples of model development and study results are provided. Research needs in the area are identified.

Keywords: Power System Harmonics, Harmonics

Simulation, Harmonics Modeling.

I. INTRODUCTION

Conventional ac electric power systems are designed to operate with sinusoidal voltages and currents. However,