1. Fundamentals of Power Electronics 1 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!)

2. Fundamentals of Power Electronics 2 Chapter 19: Resonant Conversion Analysis of inverter output characteristics – simplifying assumptions Load is resistive –Load does not change resonant frequency –Can include any reactive components in tank Resonant network is purely reactive (neglect losses)

3. Fundamentals of Power Electronics 3 Chapter 19: Resonant Conversion Inverter output characteristics Let H  be the open-circuit ( R→  ) transfer function: and let Z o0 be the output impedance (with v i → short-circuit). Then, The output voltage magnitude is: with This result can be rearranged to obtain Hence, at a given frequency, the output characteristic (i.e., the relation between ||v o || and ||i o | |) of any resonant inverter of this class is elliptical.

4. Fundamentals of Power Electronics 4 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

5. Fundamentals of Power Electronics 5 Chapter 19: Resonant Conversion Matching ellipse to application requirements Electronic ballastElectrosurgical generator

6. Fundamentals of Power Electronics 6 Chapter 19: Resonant Conversion Example of gas discharge lamp ignition and steady-state operation from CoPEC research LCC resonant inverter Vg = 300 V I ref = 5 A

7. Fundamentals of Power Electronics 7 Chapter 19: Resonant Conversion Example of repeated lamp ignition attempts with overvoltage protection LCC resonant inverter Vg = 300 V I ref = 5 A Overvoltage protection at 3500 V

8. Fundamentals of Power Electronics 8 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!)

9. Fundamentals of Power Electronics 9 Chapter 19: Resonant Conversion Input impedance of the resonant tank network Appendix C: Section C.4.4 where Expressing the tank input impedance as a function of the load resistance R:

10. Fundamentals of Power Electronics 10 Chapter 19: Resonant Conversion Z N and Z D Z D is equal to the tank output impedance under the condition that the tank input source v s1 is open-circuited. Z D = Z o  Z N is equal to the tank output impedance under the condition that the tank input source v s1 is short-circuited. Z N = Z o 

11. Fundamentals of Power Electronics 11 Chapter 19: Resonant Conversion Input impedance of the resonant tank network Appendix C: Section C.4.4 where Expressing the tank input impedance as a function of the load resistance R:

12. Fundamentals of Power Electronics 12 Chapter 19: Resonant Conversion Reciprocity

13. Fundamentals of Power Electronics 13 Chapter 19: Resonant Conversion Magnitude of the tank input impedance If the tank network is purely reactive, then each of its impedances and transfer functions have zero real parts, and the tank input and output impedances are imaginary quantities. Hence, we can express the input impedance magnitude as follows:

14. Fundamentals of Power Electronics 14 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.

15. Fundamentals of Power Electronics 15 Chapter 19: Resonant Conversion Proof of Theorem 1  Derivative has roots at: Previously shown:  Differentiate: So the resonant network input impedance is a monotonic function of R, over the range 0 < R < . In the special case || Z i0 || = || Z i  ||, || Z i || is independent of R.

16. Fundamentals of Power Electronics 16 Chapter 19: Resonant Conversion Series resonant tank

17. Fundamentals of Power Electronics 17 Chapter 19: Resonant Conversion Parallel resonant tank

18. Fundamentals of Power Electronics 18 Chapter 19: Resonant Conversion LCC tank

19. Fundamentals of Power Electronics 19 Chapter 19: Resonant Conversion Z i0 and Z i  for 3 common inverters

20. Fundamentals of Power Electronics 20 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. 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  ||.

21. Fundamentals of Power Electronics 21 Chapter 19: Resonant Conversion Discussion: LCC || Z i0 || and || Z i  || both represent series resonant impedances, whose Bode diagrams are easily constructed. || Z i0 || and || Z i  || intersect at frequency f m. For f < f m then || Z i0 || < || Z i  || ; hence transistor current decreases as load current decreases For f > f m then || Z i0 || > || Z i  || ; hence transistor current increases as load current decreases, and transistor current is greater than or equal to short-circuit current for all R LCC example

22. Fundamentals of Power Electronics 22 Chapter 19: Resonant Conversion Discussion —series and parallel No-load transistor current = 0, both above and below resonance. ZCS below resonance, ZVS above resonance Above resonance: no-load transistor current is greater than short-circuit transistor current. ZVS. Below resonance: no-load transistor current is less than short-circuit current (for f <f m ), but determined by || Z i  ||. ZCS.