1. Introduction to Converter Sampled-Data Modeling
ECEN Dragan Maksimović

2. Objectives
Better understanding of converter small-signal dynamics, especially at high frequencies
Applications
DCM high-frequency modeling
Current mode control
Digital control

3. Example: A/D and D/A conversion
v(t)
vo(t)
v*(t)
Analog-to-digital converter
Digital-to-analog converter t
(n+1)T
(n+2)T nT
T = sampling period
1/T = sampling frequency
v(t)
v*(t)
vo(t)

4. Modeling objectives Relationships: v to v* to vo
Time domain: v(t) to v*(t) to vo(t)
Frequency domain: v(s) to v*(s) to vo(s)
v(t) t
v*(t) t
vo(t) t nT
(n+1)T
(n+2)T
T = sampling period
1/T = sampling frequency

5. Model H A/D D/A v*(t) v(t) vo(t) v*(t) v(t) vo(t)
Analog-to-digital converter
Digital-to-analog converter
v*(t) H
v(t)
vo(t) T
Sampler
Zero-order hold

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

7. Unit impulse Properties t Laplace transform unit step d(t) area = 1

8. Sampling in frequency domain

9. Sampling in frequency domain: derivation

10. Sampling in frequency domain: derivation

11. Aliasing

12. 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

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

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

15. 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” =

16. Zero-order hold: frequency responses

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

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

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

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

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

22. 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

23. Application to DCM high-frequency modelingiL c
dTs
d2Ts Ts

24. Application to DCM high-frequency modelingiL c
dTs
d2Ts Ts

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

26. 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)