The Principle of Equivalent Areas

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1、isastaircaseapproximationtoh(t)asThePrincipleofEquivalentAreasshowninFig.3.Wecanconsiderhp{t)astheoutputofasystemwhichincludesasamplerandazeroorderholdcircuitatRICHARDE.ANDEENtheoutputasshowninFig.4.Later,theASSOCIATEMEMBERAIEEvalidityofthissubstitutionandt

2、hedeter-minationofthesamplingperiodTwillbediscussed.Thequantityhv{t)canbeHENTHESTATEofalinearsys-bemoreeasilyanalyzedbymathematicalWexpressedmathematicallyasfollows:temischangedbytheapplicationorgraphicalmeans.Thisideahasbeen00ofaforcingfunctionr(t),thechan

3、geoc-previouslyusedintheanalysisofpulse-hpW^Y^KnTnu-^t-nT)-curringduringatimeintervalmaybedurationsampled-datasystems,andindeterminedapproximatelybyconsider-connectionwiththeanalysisofnon-u-^t-nT-T)](3)ingonlytheinitialconditionsandtheautonomousnonlinearsys

4、tems.Theob-totalimpulsejectiveofthispaperinintroducingtheIthasbeentacitlyassumed,asisprincipleofequivalentareasistode-customary,thath(t)iszerofornegativer(t)dtscribesomeofthefactorswhichmustbetime.Alsoh(t)isshowninthefigures'-*consideredinitsapplication.Gra

5、phicalasapproachingzeroas/becomeslarge.andmathematicalderivationsaregiven.Thediscussion,however,isgeneralandappliedtothesystemduringtheinterval.TheapproachisintermsoftheconceptsdoesnotonlyapplytosystemshavingthisIfthetimeintervalissmallenough,theofconvoluti

6、on,frequencyspectra,andspecialcharacteristic.approximationwillbeagoodmeasureofsamplingtheory.Fig.5isequivalenttoFig.2,butre-thebehaviorofthesystem.Thisideaisdrawnwithhp(t)replacingh(t).Asbe-exploitedinthispaperasthebasisfortheprincipleofequivalentareas,apri

7、ncipleGraphicalDerivationfore,theareaundertheproductcurveisobtainedinordertodeterminethevalueestablishingtheconditionwhichmustbesatisfiediftwoforcingfunctionswhichTheproblemmayfirstbeapproachedoftheoutputcatthetimet=r.Inthishavedifferentinstantaneouscharact

8、er-byreferringtoFigs.1and2andthecon-case,however,theareaisapproximatelyvolutiontheorem1statedinequations1equaltoC(T).Itisclearthatthesmalleristicsaretobeequivalentasfarastheireffectsonalinearsystemarec