Use averaged switch modeling technique: apply averaged PWM model, with d replaced by µ

1. AC modeling of quasi-resonant convertersExtension of State-Space Averaging to model non-PWM switches Use averaged switch modeling technique: apply averaged PWM model, with d replaced by µ Buck example with full-wave ZCS quasi-resonant cell: µ = F

4. Equilibrium (dc) state-space averaged model Provided that the natural frequencies of the converter, as well as the frequencies of variations of the converter inputs, are much slower than the switching frequency, then the state-space averaged model that describes the converter in equilibrium is where the averaged matrices are and the equilibrium dc components are

5. Small-signal ac state-space averaged model where So if we can write the converter state equations during subintervals 1 and 2, then we can always find the averaged dc and small-signal ac models

6. Relevant background • State-Space Averaging: see textbook section 7.3 • Averaged Switch Modeling and Circuit Averaging: see textbook section 7.4

7. Circuit averaging and averaged switch modeling • Separate switch elements from remainder of converter • Choose the independent input signals xT to the switch network • The switch network generates dependent output signals xs • Average switch waveforms • Solve for how <xs> depends on <xT>

9. Basic switch networks and their PWM CCM large-signal, nonlinear, averaged switch modelsfor non-isolated converters

10. Basic switch networks and their PWM CCM dc + small-signal averaged switch modelsfor non-isolated converters

11. Averaged Switch Modeling • Separate switch elements from remainder of converter • Remainder of converter consists of linear circuit • The converter applies signals xT to the switch network • The switch network generates output signals xs • We have solved for how <xs> depends on <xT> • Replace switch network with its averaged switch model

12. Block diagram of converter Switch network as a two-port circuit:

13. The linear time-invariant network

14. The circuit averaging step To model the low-frequency components of the converter waveforms, average the switch output waveforms (in xs(t)) over one switching period.

15. Relating the result to previously-derived PWM converter models We can do this if we can express the average xs(t) in the form

16. PWM switch: finding Xs1 and Xs2

17. Finding µ: ZCS example where, from previous slide,

18. Derivation of the averaged system equationsof the resonant switch converter Equations of the linear network (previous Eq. 1): Substitute the averaged switch network equation: Result: Next: try to manipulate into same form as PWM state-space averaged result

19. Conventional state-space equations: PWM converter with switches in position 1 In the derivation of state-space averaging for subinterval 1: the converter equations can be written as a set of linear differential equations in the following standard form (Eq. 7.90): These equations must be equal: Solve for the relevant terms: But our Eq. 1 predicts that the circuit equations for this interval are:

20. Conventional state-space equations: PWM converter with switches in position 2 Same arguments yield the following result: and

21. Manipulation to standard state-space form Eliminate Xs1 and Xs2 from previous equations. Result is: Collect terms, and use the identity µ + µ’ = 1: —same as PWM result, but with d µ

22. Perturbation and Linearization The switch conversion ratio µ is generally a fairly complex function. Must use multivariable Taylor series, evaluating slopes at the operating point:

25. Small signal model Substitute and eliminate nonlinear terms, to obtain: Same form of equations as PWM small signal model. Hence same model applies, including the canonical model of Section 7.5. The dependence of µ on converter signals constitutes built-in feedback.

26. Salient features of small-signal transfer functions, for basic converters

27. Parameters for various resonant switch networks

28. Example 1: full-wave ZCSSmall-signal ac model

29. Low-frequency model

30. Example 2: Half-wave ZCS quasi-resonant buck

31. Small-signal modeling

32. Equivalent circuit model

33. Low frequency model: set tank elements to zero

34. Predicted small-signal transfer functionsHalf-wave ZCS buck