电磁兼容资料第二部分2

1SpectraofEMISignalsOutline•Spectraofimpulse-likesignals•Spectraofperiodicsignals2SpectraofImpulse-LikeSignals•ItisveryimportanttoknowthespectralcontentofsignalsinordertodeterminetheirpotentialityofproducingEMI•ForEMIconsiderationsonlytheworst-caseconditionshouldbeconsideredTheexactspectraofimpulse-likesignalsareapproximatedbasedoninequalitiesrelatedtotheFouriertransform3FourierTransform()()[]()Xjxtxtedtjtωω=ℑ=−∞∞∫()()xtXjedjt=−−∞∞∫12πωωω()()Xjxtdtω≤−∞∞∫()()Xjdxtdtdtnnnωω≤−∞∞∫1Generalinequalities:4DerivationoftheWorst-CaseSpectrumRules:•Incalculatingthederivativestheimpulse-function(Dirac-impulse)mustbeconsidered•Thederivationshouldbecontinuedonlywhilethefirstimpulse-functionappearsinthederivatives5Example:TrapezoidalImpulsex”(t)A1A2tftptrx(t)x’(t)Atr1AAtp21−f2tA−()()Xjxtdtω≤−∞∞∫()XAtAtAAtrfp11212222=+++Firstsegment:6Example:TrapezoidalImpulseA1A2tftptrx(t)x’(t)x”(t)Atr1f2tA−AAtp21−()()Xjdxtdtdtωω≤−∞∞∫1Secondsegment:()XfAf22=π7Example:TrapezoidalImpulseA1A2tftptrx(t)x’(t)x”(t)Atr1AAtp21−f2tA−()()Xjdxtdtdtωω≤−∞∞∫1222Thirdsegment:()XffAtAtrf3221212=+π8ApproximatedSpectrum()XXf121=fAX121=π120dBµV/MHzf[MHz]100800.1110f1f2X1X2(f)X3(f)9ApproximatedSpectrum()()XfXf2232=fAAttrf212121=+π120dBµV/MHzf[MHz]100800.1110f1f2X1X2(f)X3(f)10ExampleA1=1V,A2=1.2V,tr=0.1µs,tf=0.2µs,tp=2.35µs120dBµV/MHzf[MHz]100800.1110f1f2X1X2(f)X3(f)11HFAsymptoticEnvelopeIfthe(k+1)derivativeofawaveformcontainsadiracpulse,thentheHFasymptoticenvelopehasa-(k+1)⋅20dB/decslope12SymmetricTrapezoidalImpulseAτtx(t)A1=A2,tr=tfτ=pulsewidth13SymmetricTrapezoidalImpulsedBµV/MHzf[MHz]AfπAτAftrπ22f11=πτftr21=π14SpectraofPeriodicSignalsComplexexponentialseries:()()xtceccntcnjntnnonno==++∠=−∞+∞=+∞∑∑ωω012cos()cTxtedtnjntttTo=−+∫111ωωππooTf==2215SpectraofPeriodicSignalsThespectrumofpiece-wiselinearperiodicsignalcanbeeasilycalculatedbyexploitingthefollowingproperty:Linearity:()()()()xtcxtcxtxtccnnnn11221212⇒⇒+⇒+αβαβ16SpectraofPeriodicSignalsThespectrumofpiece-wiselinearperiodicsignalcanbeeasilycalculatedbyexploitingthefollowingproperty:Timetranslation:()()xtcxtcennjno⇒−⇒−τωτ17SpectraofPeriodicSignalsThespectrumofpiece-wiselinearperiodicsignalcanbeeasilycalculatedbyexploitingthefollowingproperty:Derivation:()()()xtcdxtdtjncnkkokn⇒⇒ω18SpectraofPeriodicSignalsThespectrumofpiece-wiselinearperiodicsignalcanbeeasilycalculatedbyexploitingthefollowingproperty:Dirac-impulsefunction:()()xttkTcTn=±⇒=δ119SpectraofPiece-WiseLinearPeriodicSignals•Takethederivativesofthesignal•Ifaderivativedoesnotconsistofsoleimpulsefunctions,writeitasasumoftwofunctionsoneofthemmadeupofsoleimpulsefunctions•Continuetheprocessuntiladerivativemadeupofsoleimpulsefunctionsisobtained20ATx(t)Aττt-A()dxtdt=()dxtdt1τTAτ−AτttExamplex1(t)+x2(t)21DerivationofComplexExponentialSeriesCoefficients()()()cjncjnccjnjnccnononnoonn==+=+1111112112ωωωωn≠0cATn2=−()cATATenjno1111=−−ττωτcjAnTjAnTsinnnenoooojno=−−ωωωτωτωτ22222FourierTransformandComplexExponentialSeriesCoefficientsSingle-pulseFourierTransform:X(jω)RepetitivePulseComplexExponentialSeriesCoefficients:cn()onjnXT1cω=periodoftherepetitivesignalTo=2πω23Example:SymmetricTrapezoidalPeriodicSignalAτtx(t)Tduty-cycledT=τ24Example:SymmetricTrapezoidalPeriodicSignal()onnjnXT2c2cω==+dBµVf[MHz]2Affoπ2Ad222Afftorπf11=πτftr21=π25Duty-CycleVariationsd2d1f[MHz]21Adfdoπ1fdoπ222AddBµV26RiseandFallTimeVariationsf[MHz]2Ad11πτr12πτrτr2τr1dBµV27RepetitionFrequencyVariationsf[MHz]2Addf1oπdf2oπdBµVf02f01NOTE:thedistancebetweendiscreteharmonicsnf0changes28Example:AsymmetricTrapezoidalPeriodicSignalA=10,fs=20kHz,τr=800ns,τf=40ns,d=0.1,tp=dTsf3-80-60-40-2002040104105106107108Frequency[Hz]dBVDiscrepancyr11fτπ=f31fτπ=τ+τπ=rf21121ff1f229Example:AsymmetricTrapezoidalPeriodicSignalA=10,fs=20kHz,τr=800ns,τf=40ns,d=0.5,tp=dTs104105106107108-80-60-40-2002040Frequency[Hz]DiscrepancydBV30Example:AsymmetricTrapezoidalPeriodicSignalA=10,fs=20kHz,τr=800ns,τf=40ns,d=0.9,tp=dTs104105106107108-80-60-40-2002040Frequency[Hz]DiscrepancydBV31Example:Drain-SourceVoltageofaSwitchingDeviceUA=600VSwitchoff”voltagetr=tf=100nsriseandfalltimesfs=100kHzSwitchingfrequencyCP=30pFParasiticcapacitancebetweendrainandgroundiCdudtmACppDS==180CP32DrainVoltageandCapacitorCurrentUDS5µs600V0V180mA100nSiCp33Drain-SourceVoltageSpectrumf1f-20dB/dec-40dB/decf2P[dB]PATdBVs==20211510log.6τµfkHz1163==πτMHz183t1ff2.=π=34CapacitorCurrentSpectrumf1f-20dB/dec-40dB/decf2f-20dB/decP[dB][dB]ICpmaxIjCUCppDS=ωForf1ff2IUfCCpAspmax=435CommonModeCurrentftr21=πicmidmipinPowerSourceCPicmicmf1ff2-20dB/dec[dB]ICpmax36Example:ResonantFallIntervalDrainvoltageofaflybackconverteroperatingattheboundarybetweencontinuousanddiscontinuousoperationmodestrtTo/2TSAuDSτResonantinterval37Example:ResonantFallInterval()()Xjxtdtω≤−∞∞∫τ=AXs1Firstsegment:ConsiderfirstthesinglepulsetrtTo/2Aτttx(t)x’(t)x”(t)rtA2Aoω−2A2oω38Example:ResonantFallIntervalConsiderfirstthesinglepulse()()Xjdxtdtdtωω≤−∞∞∫1Secondsegment:()fAfXs2π=trtTo/2Aτttx(t)x’(t)x”(t)rtA2Aoω−2A2oω39Example:ResonantFallIntervalConsiderfirstthesinglepulse()()Xjdx