AnnouncementsA series resonant link inverter19.4.4 Design ExampleSolve for the ellipse which meets requirementsCalculationsSolve for the open circuit transfer functionSolve for matched load (magnitude of output impedance )Solving for the tank elements to give required ||Zo0|| and ||Hinf||Analysis in terms of Xs and XpSlide 10||Hinf||||Zo0||Slide 13Slide 14Slide 15Evaluate tank element valuesDiscussion Choice of series branch elementsRcritEllipse again with Rcrit, Rmatched, and Rnom Showing ZVS and ZCSConverter performanceExtending ZVS rangeSlide 22Slide 23Dynamic Modeling and Analysis of Resonant InvertersSinusoidal steady-state resonant inverter behaviorSlide 26Slide 27Dynamic analysis of resonant invertersDC gain of control-to-output envelope transfer functionSpectrum of v(t)How frequency modulation of tank input voltage introduces amplitude modulation of output envelopePoles of Genv(s)Outline of discussionFundamentals of Power Electronics 1Chapter 19: Resonant ConversionAnnouncementsHomework #1 statistics (for on-campus students):Average = 65.2/80 = 82%Homework #3 due Friday, Feb. 8 for on-campus studentsCorrection to Problem 19.6:Part (a) asks you to derive expression for Voc and Isc in terms of the variables F = fs / finf, Vg, n = Cs / Cp, and Rinf. There is a square root missing from Rinf, i.e. it should readpsCCLR Fundamentals of Power Electronics 2Chapter 19: Resonant ConversionA series resonant link inverter+–R+v(t)–Resonant tank networkdcsourcevg(t)Switch networkL Csi(t)Low-passfilternetworkacloadSwitch networkSame as dc-dc series resonant converter, except output rectifiers are replaced with four-quadrant switches:Fundamentals of Power Electronics 3Chapter 19: Resonant Conversion19.4.4 Design ExampleSelect resonant tank elements to design a resonant inverter that meets the following requirements:•Switching frequency fs = 100 kHz•Input voltage Vg = 160 V•Inverter is capable of producing a peak open circuit output voltage of 400 V•Inverter can produce a nominal output of 150 Vrms at 25 WFundamentals of Power Electronics 4Chapter 19: Resonant ConversionSolve for the ellipse which meets requirementsFundamentals of Power Electronics 5Chapter 19: Resonant ConversionCalculationsThe required short-circuit current can be found by solving the elliptical output characteristic for Isc:henceUse the requirements to evaluate the above:Fundamentals of Power Electronics 6Chapter 19: Resonant ConversionSolve for the open circuit transfer functionThe requirements imply that the inverter tank circuit have an open-circuit transfer function of:Note that Voc need not have been given as a requirement, we can solve the elliptical relationship, and therefore find Voc given any two required operating points of ellipse. E.g. Isc could have been a requirement instead of VocFundamentals of Power Electronics 7Chapter 19: Resonant ConversionSolve for matched load (magnitude of output impedance )Matched load therefore occurs at the operating pointHence the tank should be designed such that its output impedance isFundamentals of Power Electronics 8Chapter 19: Resonant ConversionSolving for the tank elementsto give required ||Zo0|| and ||Hinf||Let’s design an LCC tank network for this exampleThe impedances of the series and shunt branches can be represented by the reactancesFundamentals of Power Electronics 9Chapter 19: Resonant ConversionAnalysis in terms of Xs and XpThe transfer function is given by the voltage divider equation:The output impedance is given by the parallel combination:Solve for Xs and Xp:Fundamentals of Power Electronics 10Chapter 19: Resonant ConversionAnalysis in terms of Xs and XpFundamentals of Power Electronics 11Chapter 19: Resonant Conversion||Hinf||Fundamentals of Power Electronics 12Chapter 19: Resonant Conversion||Zo0||Fundamentals of Power Electronics 13Chapter 19: Resonant Conversion||Zo0||Fundamentals of Power Electronics 14Chapter 19: Resonant ConversionAnalysis in terms of Xs and XpFundamentals of Power Electronics 15Chapter 19: Resonant ConversionAnalysis in terms of Xs and XpThe transfer function is given by the voltage divider equation:The output impedance is given by the parallel combination:Solve for Xs and Xp:Fundamentals of Power Electronics 16Chapter 19: Resonant ConversionEvaluate tank element valuesFundamentals of Power Electronics 17Chapter 19: Resonant ConversionDiscussionChoice of series branch elementsThe series branch is comprised of two elements L and Cs, but there is only one design parameter: Xs = 733 Ω. Hence, there is an additional degree of freedom, and one of the elements can be arbitrarily chosen. This occurs because the requirements are specified at only one operating frequency. Any choice of L and Cs, that satisfies Xs = 733 Ω will meet the requirements, but the behavior at switching frequencies other than 100 kHz will differ.Given a choice for Cs, L must be chosen according to:For example, Cs = 3Cp = 3.2 nF leads to L = 1.96 mHFundamentals of Power Electronics 18Chapter 19: Resonant ConversionRcritFor the LCC tank network chosen, Rcrit is determined by the parameters of the output ellipse, i.e., by the specification of Vg, Voc, and Isc. Note that Zo is equal to jXp. One can find the following expression for Rcrit:Since Zo0 and H  are determined uniquely by the operating point requirements, then Rcrit is also. Other, more complex tank circuits may have more degrees of freedom that allow Rcrit to be independently chosen.Evaluation of the above equation leads to Rcrit = 1466 Ω. Hence ZVS for R< 1466 Ω, and the nominal operating point with R = 900 Ω has ZVS.Fundamentals of Power Electronics 19Chapter 19: Resonant ConversionEllipse again with Rcrit, Rmatched, and RnomShowing ZVS and ZCSFundamentals of Power Electronics 20Chapter 19: Resonant ConversionConverter performanceFor this design, the salient tank frequencies are(note that fs is nearly equal to fm, so the transistor current should be nearly independent of load)The open-circuit tank input impedance isSo when the load is open-circuited, the transistor current isSimilar calculations for a short-circuited load lead toFundamentals of Power Electronics 21Chapter 19: Resonant ConversionExtending ZVS rangeFundamentals of Power Electronics 22Chapter 19: Resonant ConversionExtending ZVS rangeFundamentals of Power Electronics 23Chapter 19: Resonant ConversionExtending ZVS rangeFundamentals of Power Electronics 24Chapter 19: Resonant