1 ECEN5807 Intro to Converter Sampled-Data Modeling Introduction to Converter Sampled-Data Modeling ECEN 5807 Dragan Maksimović

2 ECEN5807 Intro to Converter Sampled-Data Modeling Objectives Better understanding of converter small-signal dynamics, especially at high frequencies Applications –DCM high-frequency modeling –Current mode control –Digital control

3 ECEN5807 Intro to Converter Sampled-Data Modeling Example: A/D and D/A conversion A/D D/A v(t)v(t)vo(t)vo(t) v*(t) Analog-to- digital converter Digital-to- analog converter t t t (n+1)T(n+2)TnT T = sampling period 1/T = sampling frequency v(t)v(t) v*(t) vo(t)vo(t)

4 ECEN5807 Intro to Converter Sampled-Data Modeling Modeling objectives Relationships: v to v* to v o –Time domain: v(t) to v*(t) to v o (t) –Frequency domain: v(s) to v*(s) to v o (s) t t t (n+1)T(n+2)TnT T = sampling period 1/T = sampling frequency v(t)v(t) v*(t) vo(t)vo(t)

5 ECEN5807 Intro to Converter Sampled-Data Modeling Model A/D D/A v(t)v(t)vo(t)vo(t) v*(t) Analog-to- digital converter Digital-to- analog converter v(t)v(t)vo(t)vo(t) v*(t) H SamplerZero-order hold T

6 ECEN5807 Intro to Converter Sampled-Data Modeling Sampling v(t)v(t) v*(t) Sampler T t t v(t)v(t) v*(t) Unit impulse (Dirac)

7 ECEN5807 Intro to Converter Sampled-Data Modeling (t)(t) t tt area = 1 s(t)s(t) Unit impulse Properties unit step Laplace transform

8 ECEN5807 Intro to Converter Sampled-Data Modeling Sampling in frequency domain

9 ECEN5807 Intro to Converter Sampled-Data Modeling Sampling in frequency domain: derivation

10 ECEN5807 Intro to Converter Sampled-Data Modeling Sampling in frequency domain: derivation

11 ECEN5807 Intro to Converter Sampled-Data Modeling Aliasing

12 ECEN5807 Intro to Converter Sampled-Data Modeling Zero-order hold vo(t)vo(t) v*(t) H Zero-order hold t t (n+1)T(n+2)TnT T = sampling period 1/T = sampling frequency v*(t) vo(t)vo(t)

13 ECEN5807 Intro to Converter Sampled-Data Modeling Zero-order hold: time domain vo(t)vo(t) H Zero-order hold (t)(t)

14 ECEN5807 Intro to Converter Sampled-Data Modeling Zero-order hold: frequency domain vo(t)vo(t) H Zero-order hold u(t)u(t)

15 ECEN5807 Intro to Converter Sampled-Data Modeling Sampled-data system example: frequency domain v(t)v(t)vo(t)vo(t) v*(t) H SamplerZero-order hold T Consider only low-frequency signals: System “transfer function” =

16 ECEN5807 Intro to Converter Sampled-Data Modeling Zero-order hold: frequency responses

17 ECEN5807 Intro to Converter Sampled-Data Modeling Zero-order hold: frequency responses f s = 1 MHz MATLAB file: zohfr.m

18 ECEN5807 Intro to Converter Sampled-Data Modeling Zero-order hold: 1 st -order approximation 1 st -order Pade approximation

19 ECEN5807 Intro to Converter Sampled-Data Modeling Zero-order hold: frequency responses f s = 1 MHz MATLAB file: zohfr.m

20 ECEN5807 Intro to Converter Sampled-Data Modeling How does any of this apply to converter modeling?

21 ECEN5807 Intro to Converter Sampled-Data Modeling PWM is a small-signal sampler! PWM sampling occurs at t p (i.e. at dT s, periodically, in each switching period)

22 ECEN5807 Intro to Converter Sampled-Data Modeling 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 ECEN5807 Intro to Converter Sampled-Data Modeling Application to DCM high-frequency modeling TsTs dT s d2Tsd2Ts iLiL c

24 ECEN5807 Intro to Converter Sampled-Data Modeling Application to DCM high-frequency modeling TsTs dT s d2Tsd2Ts iLiL c

25 ECEN5807 Intro to Converter Sampled-Data Modeling 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 ECEN5807 Intro to Converter Sampled-Data Modeling 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 D 2 T s 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 f s /(  D 2 ) is obtained Next: current-mode control (Chapter 12)