Transcript Ch 19
Announcements Homework #1 statistics (for on-campus students): Average = 65.2/80 = 82% Homework #3 due Friday, Feb. 8 for on-campus students Correction 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 read R Fundamentals of Power Electronics 1 L Cs C p Chapter 19: Resonant Conversion A series resonant link inverter Same as dc-dc series resonant converter, except output rectifiers are replaced with four-quadrant switches: i(t) + L dc source + – vg (t) Cs v(t) R – Switch network Fundamentals of Power Electronics Resonant tank network 2 Switch network Low-pass ac filter load network Chapter 19: Resonant Conversion 19.4.4 Design Example Select 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 W Fundamentals of Power Electronics 3 Chapter 19: Resonant Conversion Solve for the ellipse which meets requirements Fundamentals of Power Electronics 4 Chapter 19: Resonant Conversion Calculations The required short-circuit current can be found by solving the elliptical output characteristic for Isc: hence Use the requirements to evaluate the above: Fundamentals of Power Electronics 5 Chapter 19: Resonant Conversion Solve for the open circuit transfer function The 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 Voc Fundamentals of Power Electronics 6 Chapter 19: Resonant Conversion Solve for matched load (magnitude of output impedance ) Matched load therefore occurs at the operating point Hence the tank should be designed such that its output impedance is Fundamentals of Power Electronics 7 Chapter 19: Resonant Conversion Solving for the tank elements to give required ||Zo0|| and ||Hinf|| Let’s design an LCC tank network for this example The impedances of the series and shunt branches can be represented by the reactances Fundamentals of Power Electronics 8 Chapter 19: Resonant Conversion Analysis in terms of Xs and Xp The 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 9 Chapter 19: Resonant Conversion Analysis in terms of Xs and Xp Fundamentals of Power Electronics 10 Chapter 19: Resonant Conversion ||Hinf|| Fundamentals of Power Electronics 11 Chapter 19: Resonant Conversion ||Zo0|| Fundamentals of Power Electronics 12 Chapter 19: Resonant Conversion ||Zo0|| Fundamentals of Power Electronics 13 Chapter 19: Resonant Conversion Analysis in terms of Xs and Xp Fundamentals of Power Electronics 14 Chapter 19: Resonant Conversion Analysis in terms of Xs and Xp The 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 15 Chapter 19: Resonant Conversion Evaluate tank element values Fundamentals of Power Electronics 16 Chapter 19: Resonant Conversion Discussion Choice of series branch elements The 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 mH Fundamentals of Power Electronics 17 Chapter 19: Resonant Conversion Rcrit For 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 18 Chapter 19: Resonant Conversion Ellipse again with Rcrit, Rmatched, and Rnom Showing ZVS and ZCS Fundamentals of Power Electronics 19 Chapter 19: Resonant Conversion Converter performance For 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 is So when the load is open-circuited, the transistor current is Similar calculations for a short-circuited load lead to Fundamentals of Power Electronics 20 Chapter 19: Resonant Conversion Extending ZVS range Fundamentals of Power Electronics 21 Chapter 19: Resonant Conversion Extending ZVS range Fundamentals of Power Electronics 22 Chapter 19: Resonant Conversion Extending ZVS range Fundamentals of Power Electronics 23 Chapter 19: Resonant Conversion Dynamic Modeling and Analysis of Resonant Inverters Closed-loop control system to regulate amplitude of ac output (Lamp ballast example shown, but other applications have similar needs) (frequency modulation control shown) Issues for design of closed-loop resonant converter system: • Need open-loop control-to-output transfer function to model loop gain, closed-loop transfer functions • How does control-to-output transfer function depend on the tank transfer functionofH(s)? 24 Fundamentals Power Electronics Chapter 19: Resonant Conversion Sinusoidal steady-state resonant inverter behavior Fundamentals of Power Electronics 25 Chapter 19: Resonant Conversion Fundamentals of Power Electronics 26 Chapter 19: Resonant Conversion Fundamentals of Power Electronics 27 Chapter 19: Resonant Conversion Dynamic analysis of resonant inverters Fundamentals of Power Electronics 28 Chapter 19: Resonant Conversion DC gain of control-to-output envelope transfer function Fundamentals of Power Electronics 29 Chapter 19: Resonant Conversion Spectrum of v(t) The control input is varied at modulation frequency fm Spectrum of v(t) This leads to sidebands at frequencies fs ± fm (true for both AM and narrowband FM) Carrier (s witching) frequency Sideband fs – fm The spectrum of v(t) contains no component at f = fm. Sideband fs fs + f m frequency Effect of the tank transfer function H(s) on the output: • Changing the amplitude of the carrier affects the steady-state output amplitude • Changing the amplitudes of the sidebands affects the ac variations of the output amplitude— i.e., the envelope • The control-to-output-envelope transfer function Genv(s) depends on the tank transfer function H(s) at the sideband frequencies fs ± fm. It doesn’t depend on H(s) at f = fm. Fundamentals of Power Electronics 30 Chapter 19: Resonant Conversion How frequency modulation of tank input voltage introduces amplitude modulation of output envelope Fundamentals of Power Electronics 31 Chapter 19: Resonant Conversion Poles of Genv(s) Example: Spectrum of v(t) Carrier (s witching) frequency H(s) has resonant poles at f = fo Sideband These poles affect the sidebands when fs ± fm = fo fs – fm Sideband fs fs + f m frequency Tank H(s) Hence poles are observed in Genv(s) at modulation frequencies of fm = fs – fo fo Genv(s) fs – fo Fundamentals of Power Electronics fm 32 Chapter 19: Resonant Conversion Outline of discussion DIRECT MODELING APPROACH 1. How small-signal variations in the switching frequency affect the spectrum of the switch network output voltage vs1(t) 2. Passing the frequency-modulated voltage vs1(t) through the tank transfer function H(s) leads to amplitude modulation of the output voltage v(t) 3. How to recover the envelope of the output voltage and determine the small-signal control-to-output-envelope transfer function Genv(s) PHASOR TRANSFORMATION APPROACH 1. Equivalent circuit modeling via the phasor transform 2. PSPICE simulation of Genv(s) using the phasor transform Fundamentals of Power Electronics 33 Chapter 19: Resonant Conversion