1. Fundamentals of Power Electronics 1 Chapter 19: Resonant Conversion Announcements Homework #2 due today for on-campus students. Off-campus students submit according to your own schedule. Homework #3 is posted and is due NEXT Friday for on-campus students (Friday Feb. 8)

2. Fundamentals of Power Electronics 2 Chapter 19: Resonant Conversion 19.4 Load-dependent properties of resonant converters Resonant inverter design objectives: 1. Operate with a specified load characteristic and range of operating points With a nonlinear load, must properly match inverter output characteristic to load characteristic 2. Obtain zero-voltage switching or zero-current switching Preferably, obtain these properties at all loads Could allow ZVS property to be lost at light load, if necessary 3. Minimize transistor currents and conduction losses To obtain good efficiency at light load, the transistor current should scale proportionally to load current (in resonant converters, it often doesn’t!)

3. Fundamentals of Power Electronics 3 Chapter 19: Resonant Conversion Inverter output characteristics General resonant inverter output characteristics are elliptical, of the form This result is valid provided that (i) the resonant network is purely reactive, and (ii) the load is purely resistive. with

4. Fundamentals of Power Electronics 4 Chapter 19: Resonant Conversion A Theorem relating transistor current variations to load resistance R Theorem 1: If the tank network is purely reactive, then its input impedance || Z i || is a monotonic function of the load resistance R. So as the load resistance R varies from 0 to , the resonant network input impedance || Z i || varies monotonically from the short-circuit value || Z i0 || to the open-circuit value || Z i  ||. The impedances || Z i  || and || Z i0 || are easy to construct. If you want to minimize the circulating tank currents at light load, maximize || Z i  ||. Note: for many inverters, || Z i  || < || Z i0 || ! The no-load transistor current is therefore greater than the short-circuit transistor current.

5. Fundamentals of Power Electronics 5 Chapter 19: Resonant Conversion Example: || Z i || of LCC for f < f m, || Z i || increases with increasing R. for f > f m, || Z i || decreases with increasing R. for f = f m, || Z i || constant for all R. at a given frequency f, || Z i || is a monotonic function of R. It’s not necessary to draw the entire plot: just construct || Z i0 || and || Z i  ||.

6. Fundamentals of Power Electronics 6 Chapter 19: Resonant Conversion A Theorem relating the ZVS/ZCS boundary to load resistance R Theorem 2: If the tank network is purely reactive, then the boundary between zero-current switching and zero-voltage switching occurs when the load resistance R is equal to the critical value R crit, given by It is assumed that zero-current switching (ZCS) occurs when the tank input impedance is capacitive in nature, while zero-voltage switching (ZVS) occurs when the tank is inductive in nature. This assumption gives a necessary but not sufficient condition for ZVS when significant semiconductor output capacitance is present.

7. Fundamentals of Power Electronics 7 Chapter 19: Resonant Conversion LCC example f > f  : ZVS occurs for all R f < f 0 : ZCS occurs for all R f 0 R crit. Note that R = || Z o0 || corresponds to operation at matched load with maximum output power. The boundary is expressed in terms of this matched load impedance, and the ratio Z i  / Z i0.

8. Fundamentals of Power Electronics 8 Chapter 19: Resonant Conversion LCC example, continued Typical dependence of R crit and matched-load impedance || Z o0 || on frequency f, LCC example. Typical dependence of tank input impedance phase vs. load R and frequency, LCC example.

9. Fundamentals of Power Electronics 9 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

10. Fundamentals of Power Electronics 10 Chapter 19: Resonant Conversion Solve for the ellipse which meets requirements

11. Fundamentals of Power Electronics 11 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:

12. Fundamentals of Power Electronics 12 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

13. Fundamentals of Power Electronics 13 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

14. Fundamentals of Power Electronics 14 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

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 Analysis in terms of X s and X p

17. Fundamentals of Power Electronics 17 Chapter 19: Resonant Conversion ||H inf ||

18. Fundamentals of Power Electronics 18 Chapter 19: Resonant Conversion ||Z o0 ||

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

20. Fundamentals of Power Electronics 20 Chapter 19: Resonant Conversion Analysis in terms of X s and X p

21. Fundamentals of Power Electronics 21 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 :

22. Fundamentals of Power Electronics 22 Chapter 19: Resonant Conversion Evaluate tank element values

23. Fundamentals of Power Electronics 23 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

24. Fundamentals of Power Electronics 24 Chapter 19: Resonant Conversion Requirements met at one frequency

25. Fundamentals of Power Electronics 25 Chapter 19: Resonant Conversion What if C s = infinity?

26. Fundamentals of Power Electronics 26 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

27. Fundamentals of Power Electronics 27 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.

28. Fundamentals of Power Electronics 28 Chapter 19: Resonant Conversion R crit

29. Fundamentals of Power Electronics 29 Chapter 19: Resonant Conversion Ellipse again with R crit, R matched, and R nom Showing ZVS and ZCS

30. Fundamentals of Power Electronics 30 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

31. Fundamentals of Power Electronics 31 Chapter 19: Resonant Conversion Extending ZVS range

32. Fundamentals of Power Electronics 32 Chapter 19: Resonant Conversion Extending ZVS range

33. Fundamentals of Power Electronics 33 Chapter 19: Resonant Conversion Extending ZVS range

34. Fundamentals of Power Electronics 34 Chapter 19: Resonant Conversion Discussion wrt ZVS and transistor current scaling Series and parallel tanks f s above resonance: No-load transistor current = 0 ZVS fs below resonance: No-load transistor current = 0 ZCS f s above resonance: No-load transistor current greater than short circuit current ZVS f s below resonance but > f m : No-load transistor current greater than short circuit current ZCS for no-load; ZVS for short-circuit f s < f m : No-load transistor current less than short circuit current ZCS for no-load; ZVS for short-circuit

35. Fundamentals of Power Electronics 35 Chapter 19: Resonant Conversion Discussion wrt ZVS and transistor current scaling LCC tank f s > f inf No-load transistor current greater than short circuit current ZVS f m < f s < f inf No-load transistor current greater than short circuit current ZCS for no-load; ZVS for short-circuit f 0 < f s < f m No-load transistor current less than short circuit current ZCS for no-load; ZVS for short-circuit f s < f 0 No-load transistor current less than short circuit current ZCS