1. Midterm Exam • One-week take-home exam, to cover material of HW#1-6 • Averaged switch modeling • Simulation • Extra element theorem • Input filter design • N-extra element theorem • Feedback theorem • DCM modeling • On-campus: handed out this Friday March 6, due at beginning of class of Friday March 13 • Off-campus: start exam at time of your choosing, after you at least have completed HW#6.

2. Sampling v*(t) v(t) T Sampler v(t) t v*(t) t Unit impulse (Dirac)

3. Unit impulse d(t) area = 1 s(t) t Dt Properties Laplace transform unit step

4. Sampling in frequency domain

5. Sampling in frequency domain: derivation

6. Sampling in frequency domain: derivation

7. Aliasing

8. Zero-order hold v*(t) H vo(t) Zero-order hold v*(t) t vo(t) t nT (n+1)T (n+2)T T = sampling period 1/T = sampling frequency

9. Zero-order hold: time domain H d(t) vo(t) Zero-order hold

10. Zero-order hold: frequency domain H u(t) vo(t) Zero-order hold

11. Sampled-data system example: frequency domain v*(t) H v(t) vo(t) T Sampler Zero-order hold Consider only low-frequency signals: System “transfer function” =

12. Zero-order hold: frequency responses

13. Zero-order hold: frequency responses fs = 1 MHz MATLAB file: zohfr.m

14. Zero-order hold: 1st-order approximation 1st-order Pade approximation

15. Zero-order hold: frequency responses fs = 1 MHz MATLAB file: zohfr.m

16. How does any of this apply to converter modeling?

17. PWM is a small-signal sampler! PWM sampling occurs at tp (i.e. at dTs, periodically, in each switching period)

18. General sampled-data model • Sampled-data model valid at all frequencies • Equivalent hold describes the converter small-signal response to the sampled duty-cycle perturbations [Billy Lau, PESC 1986] • State-space averaging or averaged-switch models are low-frequency continuous-time approximations to this sampled-data model

19. Application to DCM high-frequency modeling iL c dTs d2Ts Ts

20. Application to DCM high-frequency modeling iL c dTs d2Ts Ts

21. DCM inductor current high-frequency response High-frequency pole due to the inductor current dynamics in DCM, see (11.77) in Section 11.3

22. Conclusions • PWM is a small-signal sampler • Switching converter is a sampled-data system • Duty-cycle perturbations act as a string of impulses • Converter response to the duty-cycle perturbations can be modeled as an equivalent hold • Averaged small-signal models are low-frequency approximations to the equivalent hold • In DCM, at high frequencies, the inductor-current dynamic response is described by an equivalent hold that behaves as zero-order hold of length D2Ts • Approximate continuous-time model based on the DCM sampled-data model correlates with the analysis of Section 11.3: the same high-frequency pole at fs/(pD2) is obtained • Next: current-mode control (Chapter 12)