Announcements

1. 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. 19.4 Load-dependent propertiesof 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. with 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.

4. A Theorem relating transistor current variations to load resistance R • Theorem 1: If the tank network is purely reactive, then its input impedance || Zi || is a monotonic function of the load resistance R. • So as the load resistance R varies from 0 to , the resonant network input impedance || Zi || varies monotonically from the short-circuit value|| Zi0 || to the open-circuit value || Zi ||. • The impedances || Zi || and || Zi0 || are easy to construct. • If you want to minimize the circulating tank currents at light load, maximize || Zi ||. • Note: for many inverters, || Zi || < || Zi0 || ! The no-load transistor current is therefore greater than the short-circuit transistor current.

5. Example: || Zi|| of LCC • for f < fm, || Zi|| increases with increasing R . • for f > fm, || Zi|| decreases with increasing R . • for f = fm, || Zi || constant for all R . • at a given frequency f, || Zi|| is a monotonic function of R. • It’s not necessary to draw the entire plot: just construct || Zi0|| and || Zi||.

6. 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 Rcrit, 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. LCC example • f > f: ZVS occurs for all R • f < f0: ZCS occurs for all R • f0 < f < f, ZVS occurs forR< Rcrit, and ZCS occurs forR> Rcrit. • Note that R = || Zo0|| corresponds to operation at matched load with maximum output power. The boundary is expressed in terms of this matched load impedance, and the ratio Zi / Zi0.

8. LCC example, continued Typical dependence of Rcrit and matched-load impedance || Zo0 || on frequency f, LCC example. Typical dependence of tank input impedance phase vs. load R and frequency, LCC example.

9. 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

10. Solve for the ellipse which meets requirements

11. 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:

12. 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

13. 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. Solving for the tank elementsto 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

15. 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:

16. Analysis in terms of Xs and Xp

17. ||Hinf||

18. ||Zo0||

19. ||Zo0||

20. Analysis in terms of Xs and Xp

21. 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:

22. Evaluate tank element values

23. DiscussionChoice 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

24. Requirements met at one frequency

25. What if Cs = infinity?

26. DiscussionChoice 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

27. 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.

28. Rcrit

29. Ellipse again with Rcrit, Rmatched, and RnomShowing ZVS and ZCS

30. 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

31. Extending ZVS range

32. Extending ZVS range

33. Extending ZVS range

34. Discussion wrt ZVS and transistor current scalingSeries and parallel tanks • fs above resonance: • No-load transistor current = 0 • ZVS • fs below resonance: • No-load transistor current = 0 • ZCS • fsabove resonance: • No-load transistor current greater than short circuit current • ZVS • fs below resonance but > fm : • No-load transistor current greater than short circuit current • ZCS for no-load; ZVS for short-circuit • fs < fm: • No-load transistor current less than short circuit current • ZCS for no-load; ZVS for short-circuit

35. Discussion wrt ZVS and transistor current scalingLCC tank • fs > finf • No-load transistor current greater than short circuit current • ZVS • fm < fs < finf • No-load transistor current greater than short circuit current • ZCS for no-load; ZVS for short-circuit • f0 < fs < fm • No-load transistor current less than short circuit current • ZCS for no-load; ZVS for short-circuit • fs < f0 • No-load transistor current less than short circuit current • ZCS