1 ECEN5807 Intro to Converter Sampled-Data Modeling.

1ECEN5807 Intro to Converter Sampled-Data Modeling


Introduction to Converter Sampled-Data Modeling

ECEN 5807 Dragan Maksimović


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

A/D D/Av(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)

vo(t)


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)

t

t



v(t)

v*(t)

vo(t)


Model

A/D D/Av(t) vo(t)v*(t)

Analog-to-digital converter

Digital-to-analog converter

v(t) vo(t)v*(t)

H

Sampler Zero-order hold

T


Sampling

v(t)v*(t)

Sampler

T

t

t

v(t)

v*(t)

∑+∞

∞−

−= )()()(* nTttvtv δ

Unit impulse (Dirac)


(t)

tt

area = 1

s(t)

)()( tts →

Unit impulse

∫+∞

∞−

=1)( dttδ

∫+∞

∞−

=− )()()( ss tvdttttv δ

Properties

∫∞−

=t

thd )()( ττδ

unit step

Laplace transform

∫+∞

∞−

− =1)( dtet stδ


Sampling in frequency domain

∑+∞

∞−


∫+∞

∞−

−= dtetvsv st)(*)(*

∑+∞

−∞=

−=k

sjksvT

sv )(1

)(* ω

∫+∞

∞−

−= dtetvsv st)()(


Sampling in frequency domain: derivation

∑∑+∞

−∞=

+∞

∞−

=−k

tjkk

seCnTt ωδ )(ss f

Tππω 2

2==

∑+∞

∞−


∫+∞

∞−

−= dtetvsv st)(*)(*

TdtenTt

TC tjk

T

T

n

nk

s1

)(1 2/

2/

=⎟⎠

⎞⎜⎝

⎛−= −

−

+∞=

−∞=∫ ∑ ω


Sampling in frequency domain: derivation


Aliasing


Zero-order hold

vo(t)v*(t)

H

Zero-order hold

t



v*(t)

vo(t)


Zero-order hold: time domain

vo(t)HZero-order hold

(t)

∫−

=t

Tt

o dtv ττδ )()(


Zero-order hold: frequency domain

vo(t)HZero-order hold

u(t)

∫−

=t

Tt

o dutv ττ )()(

s

eH

sT−−=1


Sampled-data system example: frequency domain

∑+∞

−∞=

−=k

sjksvT

sv )(1

)(* ω

v(t) vo(t)v*(t)

H

Sampler Zero-order hold

T

s

eH

sT−−=1

∑+∞

−∞=

−−

−−

=−

=k

s

sTsT

o jksvsT

esv

s

esv )(

1)(*

1)( ω

)(1

)( svsT

esv

sT

o

−−≈

sT

e

v

v sTo

−−=1

Consider only low-frequency signals:

System “transfer function” =


Zero-order hold: frequency responses

2/2/2/2/

2/ )2/(sinc2/

)2/sin(

2/

1

2

1 TjTjTjTj

TjTj

eTeT

T

Tj

eee

Tj

e ωωωω

ωω

ωωω

ωω−−

−−

−

==−

=−


102 103 104 105 106 107-100

-80

-60

-40

-20

0

20

magnitude [db] Zero-Order Hold magnitude and phase responses

102 103 104 105 106 107

-150

-100

-50

0

frequency [Hz]

phase [deg]


sT

eTH

sT−−=1

/

fs = 1 MHz

MATLAB file: zohfr.m


Zero-order hold: 1st-order approximation

p

sT

ssT

e

ω+

≈− −

1

11

p

psT

s

s

e

ω

ω

+

−≈−

1

1

1st-order Pade approximation

ππs

p

f

Tf ==

1Tp

2=ω


102 103 104 105 106 107-100

-80

-60

-40

-20

0

20

magnitude [db] Zero-Order Hold magnitude and phase responses

102 103 104 105 106 107

-150

-100

-50

0

frequency [Hz]

phase [deg]


fs = 1 MHz

MATLAB file: zohfr.m


How does any of this apply to converter modeling?

+–

L

C R

+

v

–

vg+–

D vg

Vg d

I d

D i

i

+

_

Gc

d1

VM

vrefu


PWM is a small-signal sampler!

pt

sTd

c

( )ps ttTd −δˆ

uu ˆ+u

c

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


General sampled-data model

vref

+

_

Gc(s)u

v

Ts

Equivalent holdGh(s)

d Ts(t − nTs), d = u

• 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


Application to DCM high-frequency modeling

Ts

dTs d2Ts

iL

c


DCM inductor current high-frequency response

∑+∞

−∞=

−−

−−+

=−+

=k

ss

TsD

s

TsD

sL jksdTs

eT

L

VVsd

s

eT

L

VVsi

ss

)(ˆ11)(*ˆ1

)(ˆ22

2121 ω

)(ˆ1)(ˆ

22

212

sdsTD

eTD

L

VVsi

s

TsD

sL

s−−+≈

2

221

1

1

)(ˆ)(ˆ

ωs

TDLVV

sdsi

sL

+

+≈

sTD22

2=ω

22 D

ff s

π=

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


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/(D2) is obtained

• Next: current-mode control (Chapter 12)

1 ECEN5807 Intro to Converter Sampled-Data Modeling.

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Transcript of 1 ECEN5807 Intro to Converter Sampled-Data Modeling.