Introduction to Converter SampledData Modeling ECEN 5807 Dragan

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

Objectives • Better understanding of converter small-signal dynamics, especially at high frequencies • Applications – DCM high-frequency modeling – Current mode control – Digital control ECEN 5807 Intro to Converter Sampled-Data Modeling 2

Example: A/D and D/A conversion v(t) A/D v*(t) D/A vo(t) Analog-to. Digital-to-analog digital converter v(t) t v*(t) t vo(t) t n. T ECEN 5807 Intro to Converter Sampled-Data Modeling (n+1)T (n+2)T T = sampling period 1/T = sampling frequency 3

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 n. T ECEN 5807 Intro to Converter Sampled-Data Modeling (n+1)T (n+2)T T = sampling period 1/T = sampling frequency 4

Model v(t) A/D v*(t) D/A vo(t) Analog-to. Digital-to-analog digital converter v(t) ECEN 5807 Intro to Converter Sampled-Data Modeling v*(t) T H Sampler Zero-order hold vo(t) 5

Sampling v(t) v*(t) T Sampler v(t) t v*(t) t Unit impulse (Dirac) ECEN 5807 Intro to Converter Sampled-Data Modeling 6

Unit impulse d(t) area = 1 s(t) t Dt Properties Laplace transform unit step ECEN 5807 Intro to Converter Sampled-Data Modeling 7

Sampling in frequency domain ECEN 5807 Intro to Converter Sampled-Data Modeling 8

Sampling in frequency domain: derivation ECEN 5807 Intro to Converter Sampled-Data Modeling 9

Sampling in frequency domain: derivation ECEN 5807 Intro to Converter Sampled-Data Modeling 10

Aliasing ECEN 5807 Intro to Converter Sampled-Data Modeling 11

Zero-order hold v*(t) H vo(t) Zero-order hold v*(t) t vo(t) t n. T ECEN 5807 Intro to Converter Sampled-Data Modeling (n+1)T (n+2)T T = sampling period 1/T = sampling frequency 12

Zero-order hold: time domain d(t) H vo(t) Zero-order hold ECEN 5807 Intro to Converter Sampled-Data Modeling 13

Zero-order hold: frequency domain u(t) H vo(t) Zero-order hold ECEN 5807 Intro to Converter Sampled-Data Modeling 14

Sampled-data system example: frequency domain v(t) v*(t) T Sampler H vo(t) Zero-order hold Consider only low-frequency signals: System “transfer function” = ECEN 5807 Intro to Converter Sampled-Data Modeling 15

Zero-order hold: frequency responses ECEN 5807 Intro to Converter Sampled-Data Modeling 16

Zero-order hold: frequency responses fs = 1 MHz MATLAB file: zohfr. m ECEN 5807 Intro to Converter Sampled-Data Modeling 17

Zero-order hold: 1 st-order approximation 1 st-order Pade approximation ECEN 5807 Intro to Converter Sampled-Data Modeling 18

Zero-order hold: frequency responses fs = 1 MHz MATLAB file: zohfr. m ECEN 5807 Intro to Converter Sampled-Data Modeling 19

How does any of this apply to converter modeling? ECEN 5807 Intro to Converter Sampled-Data Modeling 20

PWM is a small-signal sampler! PWM sampling occurs at tp (i. e. at d. Ts, periodically, in each switching period) ECEN 5807 Intro to Converter Sampled-Data Modeling 21

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 ECEN 5807 Intro to Converter Sampled-Data Modeling 22

Application to DCM high-frequency modeling i. L c d. Ts ECEN 5807 Intro to Converter Sampled-Data Modeling d 2 Ts Ts 23

Application to DCM high-frequency modeling i. L c d. Ts ECEN 5807 Intro to Converter Sampled-Data Modeling d 2 Ts Ts 24

DCM inductor current high-frequency response High-frequency pole due to the inductor current dynamics in DCM, see (11. 77) in Section 11. 3 ECEN 5807 Intro to Converter Sampled-Data Modeling 25

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 Ts 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/(p. D 2) is obtained Next: current-mode control (Chapter 12) ECEN 5807 Intro to Converter Sampled-Data Modeling 26