1. 1 Manipulation to standard state-space form Eliminate X s1 and X s2 from previous equations. Result is: Collect terms, and use the identity µ + µ’ = 1: —same as PWM result, but with d  µ

2. 2 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:

3. 3

4. 4

5. 5 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.

6. 6 Salient features of small-signal transfer functions, for basic converters

7. 7 Parameters for various resonant switch networks

9. 9

10. 10 Example 1: full-wave ZCS Small-signal ac model

11. 11 Low-frequency model

12. 12 Example 2: Half-wave ZCS quasi-resonant buck

13. 13 Small-signal modeling

14. 14 Equivalent circuit model

15. 15 Low frequency model: set tank elements to zero

16. 16

17. 17 Predicted small-signal transfer functions Half-wave ZCS buck

18. 18 SEPIC Example 1.Remove switches from rest of converter and define switch network 2.Assign switch network independent inputs and dependent outputs

19. 19 SEPIC Example 3.Express = μX s1 + μ'X s2 a) Find X s1 and X s2 (PWM case) b) Most general: Solve for for the resonant switch case by sketching waveforms (using state plane analysis), and then solve for μ to get desired equation, but may instead be able to leverage previously derived results for the resonant switch cell

20. 20 SEPIC Example Half-wave ZCS-QR version

21. 21 Previously derived solution for half-wave ZCS-QR switch cell

22. 22 SEPIC Example Relating desired result to previously derived solutions for μ

23. 23 SEPIC Example With different independent/dependent port assignments