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 Properties t Laplace transform unit step d(t) area = 1

4. Sampling in frequency domain

5. Sampling in frequency domain: derivation

6. Sampling in frequency domain: derivation

7. Aliasing

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

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

10. Zero-order hold: frequency domain
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 modelingiL c
dTs
d2Ts Ts

20. Application to DCM high-frequency modelingiL 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)