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Introduction to Converter Sampled-Data Modeling . ECEN 5807 Dragan Maksimovi ć. Objectives. Better understanding of converter small-signal dynamics, especially at high frequencies Applications DCM high-frequency modeling Current mode control Digital control. v* ( t ). D/A. A/D.

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Presentation Transcript
objectives
Objectives
  • Better understanding of converter small-signal dynamics, especially at high frequencies
  • Applications
    • DCM high-frequency modeling
    • Current mode control
    • Digital control
example a d and d a conversion

v*(t)

D/A

A/D

v(t)

vo(t)

Analog-to-digital converter

Digital-to-analog converter

v(t)

t

v*(t)

t

vo(t)

t

nT

(n+1)T

(n+2)T

T = sampling period

1/T = sampling frequency

Example: A/D and D/A conversion
modeling objectives
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

model
Model

v*(t)

D/A

A/D

v(t)

vo(t)

Analog-to-digital converter

Digital-to-analog converter

v*(t)

H

v(t)

vo(t)

T

Sampler

Zero-order hold

sampling
Sampling

v*(t)

v(t)

T

Sampler

v(t)

t

v*(t)

t

Unit impulse (Dirac)

slide13

Unit impulse

d(t)

area = 1

s(t)

t

Dt

Properties

Laplace transform

unit step

zero order hold
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

zero order hold time domain
Zero-order hold: time domain

H

d(t)

vo(t)

Zero-order hold

zero order hold frequency domain
Zero-order hold: frequency domain

H

u(t)

vo(t)

Zero-order hold

sampled data system example frequency domain
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” =

zero order hold frequency responses23
Zero-order hold: frequency responses

fs = 1 MHz

MATLAB file: zohfr.m

zero order hold 1 st order approximation
Zero-order hold: 1st-order approximation

1st-order Pade approximation

zero order hold frequency responses25
Zero-order hold: frequency responses

fs = 1 MHz

MATLAB file: zohfr.m

pwm is a small signal sampler
PWM is a small-signal sampler!

PWM sampling occurs at tp (i.e. at dTs, periodically, in each switching period)

general sampled data model
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
dcm inductor current high frequency response
DCM inductor current high-frequency response

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

conclusions
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)