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SJSU EE 136 - Chapter 7

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Fundamentals of Power Electronics Chapter 7: AC equivalent circuit modeling1Part IIConverter Dynamics and Control7. AC equivalent circuit modeling8. Converter transfer functions9. Controller design10. Input filter design11. AC and DC equivalent circuit modeling of thediscontinuous conduction mode12. Current programmed controlFundamentals of Power Electronics Chapter 7: AC equivalent circuit modeling2Chapter 7. AC Equivalent Circuit Modeling7.1 Introduction7.2The basic AC modeling approach7.3State-space averaging7.4 Circuit averaging and averaged switch modeling7.5The canonical circuit model7.6Modeling the pulse-width modulator7.7 Summary of key pointsFundamentals of Power Electronics Chapter 7: AC equivalent circuit modeling37.1. IntroductionA simple dc-dc regulator system, employing abuck converterObjective: maintain v (t)equal to an accurate,constant value V.There are disturbances:• in vg(t)• in RThere are uncertainties:• in elementvalues• in Vg• in R+–+v(t)–vg(t)Switching converterPowerinputLoad–+RCompensatorGc(s)vrefVoltagereferencevFeedbackconnectionPulse-widthmodulatorvcTransistorgate driverδ(t)δ(t)TsdTsttvc(t)ControllerFundamentals of Power Electronics Chapter 7: AC equivalent circuit modeling4Applications of control in power electronicsDC-DC convertersRegulate dc output voltage.Control the duty cycle d(t) such that v(t) accurately follows a referencesignal vref.DC-AC invertersRegulate an ac output voltage.Control the duty cycle d(t) such that v(t) accurately follows a referencesignal vref (t).AC-DC rectifiersRegulate the dc output voltage.Regulate the ac input current waveform.Control the duty cycle d(t) such that ig (t) accurately follows a referencesignal iref (t), and v(t) accurately follows a reference signal vref.Fundamentals of Power Electronics Chapter 7: AC equivalent circuit modeling5Converter ModelingApplicationsAerospace worst-case analysisCommercial high-volume production: design for reliability and yieldHigh quality designEnsure that the converter works well under worst-case conditions–Steady state (losses, efficiency, voltage regulation)–Small-signal ac (controller stability and transient response)Engineering methodologySimulate model during preliminary design (design verification)Construct laboratory prototype converter system and make it work undernominal conditionsDevelop a converter model. Refine model until it predicts behavior ofnominal laboratory prototypeUse model to predict behavior under worst-case conditionsImprove design until worst-case behavior meets specifications (or untilreliability and production yield are acceptable)Fundamentals of Power Electronics Chapter 7: AC equivalent circuit modeling6Objective of Part IIDevelop tools for modeling, analysis, and design of converter controlsystemsNeed dynamic models of converters:How do ac variations in vg(t), R, or d(t) affect the output voltage v(t)?What are the small-signal transfer functions of the converter?• Extend the steady-state converter models of Chapters 2 and 3, toinclude CCM converter dynamics (Chapter 7)• Construct converter small-signal transfer functions (Chapter 8)•Design converter control systems (Chapter 9)• Design input EMI filters that do not disrupt control system operation(Chapter 10)•Model converters operating in DCM (Chapter 11)• Current-programmed control of converters (Chapter 12)Fundamentals of Power Electronics Chapter 7: AC equivalent circuit modeling7Modeling•Representation of physical behavior by mathematical means•Model dominant behavior of system, ignore other insignificantphenomena•Simplified model yields physical insight, allowing engineer todesign system to operate in specified manner• Approximations neglect small but complicating phenomena•After basic insight has been gained, model can be refined (ifit is judged worthwhile to expend the engineering effort to doso), to account for some of the previously neglectedphenomenaFundamentals of Power Electronics Chapter 7: AC equivalent circuit modeling8Neglecting the switching rippleSuppose the duty cycleis modulatedsinusoidally:where D and Dm areconstants, Dm < D,and the modulationfrequency ωm is muchsmaller than theconverter switchingfrequency ωs = 2πfs.The resulting variations in transistor gatedrive signal and converter output voltage:d(t)=D + DmcosωmtttGatedriveActual waveform v(t)including rippleAveraged waveform 〈v(t)〉Tswith ripple neglectedFundamentals of Power Electronics Chapter 7: AC equivalent circuit modeling9Output voltage spectrumwith sinusoidal modulation of duty cycleContains frequency components at:•Modulation frequency and itsharmonics•Switching frequency and itsharmonics•Sidebands of switching frequencyWith small switching ripple, high-frequency components (switchingharmonics and sidebands) are small.If ripple is neglected, then only low-frequency components (modulationfrequency and harmonics) remain.Spectrumof v(t)ωmωsω{Modulationfrequency and itsharmonics{Switchingfrequency andsidebands{SwitchingharmonicsFundamentals of Power Electronics Chapter 7: AC equivalent circuit modeling10Objective of ac converter modeling•Predict how low-frequency variations in duty cycle inducelow-frequency variations in the converter voltages andcurrents•Ignore the switching ripple•Ignore complicated switching harmonics and sidebandsApproach:•Remove switching harmonics by averaging all waveformsover one switching periodFundamentals of Power Electronics Chapter 7: AC equivalent circuit modeling11Averaging to remove switching ripplewhereAverage over one switchingperiod to remove switchingripple:Note that, in steady-state,vL(t)Ts=0iC(t)Ts=0by inductor volt-secondbalance and capacitor chargebalance.LdiL(t)Tsdt= vL(t)TsCdvC(t)Tsdt= iC(t)Tsx(t)Ts=1Tsx(τ)dτtt + TsFundamentals of Power Electronics Chapter 7: AC equivalent circuit modeling12Nonlinear averaged equationsThe averaged voltages and currents are, in general, nonlinearfunctions of the converter duty cycle, voltages, and currents. Hence,the averaged equationsconstitute a system of nonlinear differential equations.Hence, must linearize by constructing a small-signal converter model.LdiL(t)Tsdt= vL(t)TsCdvC(t)Tsdt= iC(t)TsFundamentals of Power Electronics Chapter 7: AC equivalent circuit modeling13Small-signal modeling of the diodeNonlineardiode, drivenby currentsource havinga DC and smallAC componentLinearization of the diode i-vcharacteristic about a quiescentoperating point+v = V+v–i =


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