Introduction to Converter Sampled-Data ModelingObjectivesExample: A/D and D/A conversionModeling objectivesModelSamplingSlide 7Sampling in frequency domainSampling in frequency domain: derivationSlide 10AliasingZero-order holdZero-order hold: time domainZero-order hold: frequency domainSampled-data system example: frequency domainZero-order hold: frequency responsesSlide 17Zero-order hold: 1st-order approximationSlide 19How does any of this apply to converter modeling?PWM is a small-signal sampler!General sampled-data modelApplication to DCM high-frequency modelingSlide 24DCM inductor current high-frequency responseConclusions1ECEN5807 Intro to Converter Sampled-Data ModelingIntroduction to Converter Sampled-Data Modeling ECEN 5807 Dragan Maksimović2ECEN5807 Intro to Converter Sampled-Data ModelingObjectives•Better understanding of converter small-signal dynamics, especially at high frequencies•Applications–DCM high-frequency modeling–Current mode control–Digital control3ECEN5807 Intro to Converter Sampled-Data ModelingExample: A/D and D/A conversionA/DD/Av(t) vo(t)v*(t)Analog-to-digital converterDigital-to-analog converterttt(n+1)T (n+2)TnTT = sampling period1/T = sampling frequencyv(t)v*(t)vo(t)4ECEN5807 Intro to Converter Sampled-Data ModelingModeling objectives•Relationships: v to v* to vo–Time domain: v(t) to v*(t) to vo(t)–Frequency domain: v(s) to v*(s) to vo(s) ttt(n+1)T (n+2)TnTT = sampling period1/T = sampling frequencyv(t)v*(t)vo(t)5ECEN5807 Intro to Converter Sampled-Data ModelingModelA/DD/Av(t) vo(t)v*(t)Analog-to-digital converterDigital-to-analog converterv(t) vo(t)v*(t)HSampler Zero-order holdT6ECEN5807 Intro to Converter Sampled-Data ModelingSamplingv(t)v*(t)SamplerTttv(t)v*(t) )()()(* nTttvtvUnit impulse (Dirac)7ECEN5807 Intro to Converter Sampled-Data Modeling(t)ttarea = 1s(t))()( ttsUnit impulse1)( dtt )()()(sstvdttttvPropertiestthd )()(unit stepLaplace transform1)( dtetst8ECEN5807 Intro to Converter Sampled-Data ModelingSampling in frequency domain )()()(* nTttvtv dtetvsvst)(*)(*ksjksvTsv )(1)(* dtetvsvst)()(9ECEN5807 Intro to Converter Sampled-Data ModelingSampling in frequency domain: derivationktjkkseCnTt)(ssfT22 )()()(* nTttvtv dtetvsvst)(*)(*TdtenTtTCtjkTTnnks1)(12/2/10ECEN5807 Intro to Converter Sampled-Data ModelingSampling in frequency domain: derivation11ECEN5807 Intro to Converter Sampled-Data ModelingAliasing12ECEN5807 Intro to Converter Sampled-Data ModelingZero-order holdvo(t)v*(t)HZero-order holdtt(n+1)T (n+2)TnTT = sampling period1/T = sampling frequencyv*(t)vo(t)13ECEN5807 Intro to Converter Sampled-Data ModelingZero-order hold: time domainvo(t)HZero-order hold(t)tTtodtv)()(14ECEN5807 Intro to Converter Sampled-Data ModelingZero-order hold: frequency domainvo(t)HZero-order holdu(t)tTtodutv)()(seHsT115ECEN5807 Intro to Converter Sampled-Data ModelingSampled-data system example: frequency domain ksjksvTsv )(1)(*v(t) vo(t)v*(t)HSampler Zero-order holdTseHsT1kssTsTojksvsTesvsesv )(1)(*1)()(1)( svsTesvsTosTevvsTo1Consider only low-frequency signals:System “transfer function” =16ECEN5807 Intro to Converter Sampled-Data ModelingZero-order hold: frequency responses2/2/2/2/2/)2/(sinc2/)2/sin(2/121TjTjTjTjTjTjeTeTTTjeeeTje17ECEN5807 Intro to Converter Sampled-Data Modeling102103104105106107-100-80-60-40-20020magnitude [db]Zero-Order Hold magnitude and phase responses102103104105106107-150-100-500frequency [Hz]phase [deg]Zero-order hold: frequency responsessTeTHsT1/fs = 1 MHzMATLAB file: zohfr.m18ECEN5807 Intro to Converter Sampled-Data ModelingZero-order hold: 1st-order approximationpsTssTe111ppsTsse111st-order Pade approximationspfTf 1Tp219ECEN5807 Intro to Converter Sampled-Data Modeling102103104105106107-100-80-60-40-20020magnitude [db]Zero-Order Hold magnitude and phase responses102103104105106107-150-100-500frequency [Hz]phase [deg]Zero-order hold: frequency responsesfs = 1 MHzMATLAB file: zohfr.m20ECEN5807 Intro to Converter Sampled-Data ModelingHow does any of this apply to converter modeling?+–LC R+v–vg+–D vgVg dI dD ii+_Gcd1VMvrefu21ECEN5807 Intro to Converter Sampled-Data ModelingPWM is a small-signal sampler!ptsTdˆcˆ psttTd ˆuuˆucPWM sampling occurs at tp (i.e. at dTs, periodically, in each switching period)22ECEN5807 Intro to Converter Sampled-Data ModelingGeneral sampled-data modelvref+_Gc(s)uvTsEquivalent holdGh(s)d Ts(t  nTs), d = u•Sampled-data model valid at all frequencies•Equivalent hold describes the converter small-signal response to the sampled duty-cycle perturbations [Billy Lau, PESC 1986]•State-space averaging or averaged-switch models are low-frequency continuous-time approximations to this sampled-data model23ECEN5807 Intro to Converter Sampled-Data ModelingApplication to DCM high-frequency modelingTsdTsd2TsiLc24ECEN5807 Intro to Converter Sampled-Data ModelingApplication to DCM high-frequency modelingTsdTsd2TsiLc25ECEN5807 Intro to Converter Sampled-Data ModelingDCM inductor current high-frequency responsekssTsDsTsDsLjksdTseTLVVsdseTLVVsiss)(ˆ11)(*ˆ1)(ˆ222121)(ˆ1)(ˆ22212sdsTDeTDLVVsisTsDsLs222111)(ˆ)(ˆsTDLVVsdsisLsTD22222DffsHigh-frequency pole due to the inductor current dynamics in DCM, see (11.77) in Section 11.326ECEN5807 Intro to Converter Sampled-Data ModelingConclusions•PWM is a small-signal sampler•Switching converter is a sampled-data system•Duty-cycle perturbations act as a string of impulses•Converter response to the duty-cycle perturbations can be modeled as an equivalent hold•Averaged small-signal models are low-frequency approximations to the equivalent hold•In DCM, at high frequencies, the inductor-current dynamic response is described by an equivalent hold that behaves as zero-order hold of length