1. Fundamentals of Power Electronics 1 Chapter 19: Resonant Conversion 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 V oc and I sc in terms of the variables F = f s / f inf, Vg, n = C s / C p, and R inf. There is a square root missing from R inf, i.e. it should read

2. Fundamentals of Power Electronics 2 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:

3. Fundamentals of Power Electronics 3 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 f s = 100 kHz Input voltage V g = 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

4. Fundamentals of Power Electronics 4 Chapter 19: Resonant Conversion Solve for the ellipse which meets requirements

5. Fundamentals of Power Electronics 5 Chapter 19: Resonant Conversion Calculations The required short-circuit current can be found by solving the elliptical output characteristic for I sc : hence Use the requirements to evaluate the above:

6. Fundamentals of Power Electronics 6 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 V oc need not have been given as a requirement, we can solve the elliptical relationship, and therefore find V oc given any two required operating points of ellipse. E.g. I sc could have been a requirement instead of V oc

7. Fundamentals of Power Electronics 7 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

8. Fundamentals of Power Electronics 8 Chapter 19: Resonant Conversion Solving for the tank elements to give required ||Z o0 || and ||H inf || Let’s design an LCC tank network for this example The impedances of the series and shunt branches can be represented by the reactances

9. Fundamentals of Power Electronics 9 Chapter 19: Resonant Conversion Analysis in terms of X s and X p The transfer function is given by the voltage divider equation: The output impedance is given by the parallel combination: Solve for X s and X p :

10. Fundamentals of Power Electronics 10 Chapter 19: Resonant Conversion Analysis in terms of X s and X p

11. Fundamentals of Power Electronics 11 Chapter 19: Resonant Conversion ||H inf ||

12. Fundamentals of Power Electronics 12 Chapter 19: Resonant Conversion ||Z o0 ||

13. Fundamentals of Power Electronics 13 Chapter 19: Resonant Conversion ||Z o0 ||

14. Fundamentals of Power Electronics 14 Chapter 19: Resonant Conversion Analysis in terms of X s and X p

15. Fundamentals of Power Electronics 15 Chapter 19: Resonant Conversion Analysis in terms of X s and X p The transfer function is given by the voltage divider equation: The output impedance is given by the parallel combination: Solve for X s and X p :

16. Fundamentals of Power Electronics 16 Chapter 19: Resonant Conversion Evaluate tank element values

17. Fundamentals of Power Electronics 17 Chapter 19: Resonant Conversion Discussion Choice of series branch elements The series branch is comprised of two elements L and C s, but there is only one design parameter: X s = 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 C s, that satisfies X s = 733 Ω will meet the requirements, but the behavior at switching frequencies other than 100 kHz will differ. Given a choice for C s, L must be chosen according to: For example, C s = 3C p = 3.2 nF leads to L = 1.96 mH

18. Fundamentals of Power Electronics 18 Chapter 19: Resonant Conversion R crit For the LCC tank network chosen, R crit is determined by the parameters of the output ellipse, i.e., by the specification of V g, V oc, and I sc. Note that Z o  is equal to jX p. One can find the following expression for R crit : Since Z o0 and H  are determined uniquely by the operating point requirements, then R crit is also. Other, more complex tank circuits may have more degrees of freedom that allow R crit to be independently chosen. Evaluation of the above equation leads to R crit = 1466 Ω. Hence ZVS for R < 1466 Ω, and the nominal operating point with R = 900 Ω has ZVS.

19. Fundamentals of Power Electronics 19 Chapter 19: Resonant Conversion Ellipse again with R crit, R matched, and R nom Showing ZVS and ZCS

20. Fundamentals of Power Electronics 20 Chapter 19: Resonant Conversion Converter performance For this design, the salient tank frequencies are (note that f s is nearly equal to f m, 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

21. Fundamentals of Power Electronics 21 Chapter 19: Resonant Conversion Extending ZVS range

22. Fundamentals of Power Electronics 22 Chapter 19: Resonant Conversion Extending ZVS range

23. Fundamentals of Power Electronics 23 Chapter 19: Resonant Conversion Extending ZVS range

24. Fundamentals of Power Electronics 24 Chapter 19: Resonant Conversion Dynamic Modeling and Analysis of Resonant Inverters 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 function H(s)? Closed-loop control system to regulate amplitude of ac output (Lamp ballast example shown, but other applications have similar needs) (frequency modulation control shown)

25. Fundamentals of Power Electronics 25 Chapter 19: Resonant Conversion Sinusoidal steady-state resonant inverter behavior

26. Fundamentals of Power Electronics 26 Chapter 19: Resonant Conversion

27. Fundamentals of Power Electronics 27 Chapter 19: Resonant Conversion

28. Fundamentals of Power Electronics 28 Chapter 19: Resonant Conversion Dynamic analysis of resonant inverters

29. Fundamentals of Power Electronics 29 Chapter 19: Resonant Conversion DC gain of control-to-output envelope transfer function

30. Fundamentals of Power Electronics 30 Chapter 19: Resonant Conversion Spectrum of v(t) The control input is varied at modulation frequency f m This leads to sidebands at frequencies f s ± f m (true for both AM and narrowband FM) The spectrum of v(t) contains no component at f = f m. 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 G env (s) depends on the tank transfer function H(s) at the sideband frequencies f s ± f m. It doesn’t depend on H(s) at f = f m.

31. Fundamentals of Power Electronics 31 Chapter 19: Resonant Conversion How frequency modulation of tank input voltage introduces amplitude modulation of output envelope

32. Fundamentals of Power Electronics 32 Chapter 19: Resonant Conversion Poles of G env (s) Example: H(s) has resonant poles at f = f o These poles affect the sidebands when f s ± f m = f o Hence poles are observed in G env (s) at modulation frequencies of f m =  f s – f o 

33. Fundamentals of Power Electronics 33 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 v s1 (t) 2.Passing the frequency-modulated voltage v s1 (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 G env (s) PHASOR TRANSFORMATION APPROACH 1.Equivalent circuit modeling via the phasor transform 2.PSPICE simulation of G env (s) using the phasor transform