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Modeling approaches for switching converters 1/72

Modeling approaches for switching converters

by

Giorgio Spiazzi

University of Padova – ITALY

Dept. of Information Engineering – DEI


•• Switching cell average model in continuous conduction Switching cell average model in continuous conduction mode (CCM)mode (CCM)

•• Switching cell average model in discontinuous Switching cell average model in discontinuous conduction mode (DCM): firstconduction mode (DCM): first--order modelorder model

PWM convertersPWM converters

Summary of the presentation


Basic DC-DC Converter topologies: 2°order

ug

+uo

+

-

DS L

is iL

iD Ro

BuckBuck

ug

+uo

+

-

DS

L

iL iD

iS Ro

BoostBoost

ug

+uo+

-DS

L

is iD

iL Ro BuckBuck--BoostBoost


Basic DC-DC Converter topologies: 4°order

ug

+uo+

-DS

L1

i1 i2

Ro

+uC

Co

L2C1

-uS

+uD

+

-CukCuk

ug

+uo

+

-

D

SL1

i1 iD

Ro

+uC

CoL2

C1

-uS

+ uD+-

i2SEPICSEPIC


iDiDiLio

iSiLiSig

Ug+UoUoUgUon+ Uoff

UoUo-UgUoUoff

UgUgUg-UoUon

Buck-boostBoostBuck

a p

c

n

DSL

+

-

Uon

is

+

-

Uoff

iD

iL

Commutation Cell for 2°order converters

22°° order converters can be order converters can be described by a unique described by a unique commutation cell: commutation cell:


Averaging

( ) ( )∫−

ττ=t

Ttss

dxT1

txMoving average:

Example: instantaneous and average inductor Example: instantaneous and average inductor current in transient condition current in transient condition

2.8 2.9 3 3.1 [ms]5

6

7

8

9

10

11

[A]

iL

iL


Average model: CCM

• Switching frequency ripples are neglected

• Only low-frequency dynamic is investigated

( ) ( )( )

( ) ( ) ( )

−−==ττ= ∫∫−− S

SLLti

TtiL

S

t

TtL

SL T

TtitiLdi

TL

duT1

tuL

SLS

Example: inductorsExample: inductors ( ) ( )dt

tdiLtu L

L =


Average model: CCM

( ) ( )( )

( )( ) ( ) ( ) ( )tz,tywithd,tfz,y,td,tft

z

y

t

t

β=α=ττ=φ=ττ=φ ∫∫β

α

( ) ( )( )

( )( )( ) ( ) ( )( ) ( )

dttd

t,tfdt

tdt,tfd

dt,tdf

dttd t

t

ββ+αα−ττ=φ∫

β

α

( ) ( )dt

tidLtu L

L =

( ) ( ) ?diT1

dtd

dttid t

TtL

S

L

S

=

ττ= ∫

−

( ) ( ) ( )S

SLLL

TTtiti

dttid −−=


Averaging approximation

t

tdTs

iL(t)

(1-d)Ts

Uon

-Uoff

uL(t)

LU

m on1 =

LU

m off2 −=−

iL(Ts)iL(0)

Lu( ) ( ) ( ) s

offsLsL Td1

LU

dTiTi −−=

( ) ( ) son

LsL dTL

U0idTi +=

( ) ( ) ( )

( ) sL

L

soff

son

LsL

TLu

0i

Td1L

UdT

LU

0iTi

+=

−−+=

Non steadyNon steady--state state inductor current inductor current

waveform:waveform:


Averaging

•• Reactive element voltageReactive element voltage--current relations current relations remain valid also for average quantities;remain valid also for average quantities;

•• for inductors, the current variation in a switching for inductors, the current variation in a switching period can be calculated by integrating their period can be calculated by integrating their average voltage;average voltage;

•• for capacitors, the voltage variation in a for capacitors, the voltage variation in a switching period can be calculated by integrating switching period can be calculated by integrating their average current.their average current.


Continuous conduction mode - CCM

( ) SoffSon Td1UdTU −=d

d1UU

off

on −=

At steadyAt steady--state:state:

t

t

dTs

Ts

t

t

iL(t)

is(t)

id(t)

(1-d)Ts

Uon

-Uoff

uL(t)

LU

m on1 =

LU

m off2 −=−

( )d1ii LD −=

dii LS =

Li

0uL =

Buck: dM =

Boost:d1

1M

−=

Buck-Boost:d1

dM

−=

( )d1Lf2

Ud

Lf2U

2

ii

S

off

S

onLpplimL −==

∆=

Boundary CCMBoundary CCM--DCM:DCM:


Switching cell average model: CCM

( )( ) DS

offonD

offonS udd

uuudu

uudu ′=⇒

+=+′=

DSLD

LS idd

iidi

idi

′=⇒

′==

a p

c

n

DSL

+

-uon

iS

+

-

uoff

iD

iL

d’=1-d = complement of duty-cycle

DS

iS iD

uD

- +

-uS

+

Non linear componentsNon linear components

Average switch and diode voltages and currents:Average switch and diode voltages and currents:



+

- +

-d’:d

Du

Si Di

SuD

S

iS iD

uD

- +

-uS

+

( )( ) DS

offonD

offonS udd

uuudu

uudu ′=⇒

+=+′=

DSLD

LS idd

iidi

idi

′=⇒

′==

+ +

Dudd′

Sidd′

Su

Di

- +

-

Du

Si

d’=1-d = complement of duty-cycle



• The non-linear components (switch and diode) are replaced by controlled voltage and current generators representing the relations between average voltage and currents;

• These controlled voltage and current generators can be substituted by an ideal transformer with a suitable equivalent turn ratio.


Buck average model: CCM

DgSgD udd

uuuu′

−=−=

gD udu =

( )SLDS iidd

idd

i −′

=′

=

LS idi =

+

+

-

d’:d

L

Ro

+

-+

-

DuSi Di

Su

Li

ou

gu

C

ug

+uo

+

-

DS L

iS iL

iD Roug


Buck average model (alternative approach): CCM

D

S

ig iL

ug uD

+

-

+

-

Switching Switching cellcell

Independent variables: Independent variables: uugg, , iiLL

Dependent variables: Dependent variables: uuDD, , iigg

ug

+uo

+

-

DS L

iS iL

iD Ro


Buck average model (alternative approach): CCM

D

S

ig iL

ug uD

+

-

+

-

+

-

+

-

1:d

gu

Ligi

Du

AveragingAveraging + +Lid

gudgu

Li

-

+

-Du

gi


Buck average model: CCM

−==

−==

o

oLC

C

ogLL

Ru

iidt

udC

uududt

idL

L

C

Lid

gudougu Ro

Li

+ +

-

+

-


Boost average model: CCM

+

+

-

d’:d

L

Ro

+

- +

-

DuSi DiSu

Li

ou

gu

C

SoDoS udd

uuuu′

−=−=

oS udu ′=

( )DLSD iidd

idd

i −′

=′

=

LD idi ′=

ug

+uo

+

-

DS

L

iL iD

iS Ro


Boost average model (alternative approach): CCM

Switching Switching cellcell

Independent variables: Independent variables: uuoo, , iiLL

Dependent variables: Dependent variables: uuSS, , iiDD

DS

iL iD

uS uo

+

-

+

-

ug

+uo

+

-

DS

L

iL iD

iS Ro


Boost average model (alternative approach): CCM

++ Lid′

oud′ou

Li

-

DS

iL iD

uS uo

+

-

+

-

+

-

+

-

d’:1

ou

Li Di

Su


Boost average model: CCM

−′==

′−==

o

oLC

C

ogLL

Ru

ididt

udC

uduudt

idL

LC

Li

oud′ougu Ro

Lid′

+ +

-

+

-

d’:1


Buck-Boost average model: CCM

( )ogD uudu +=

( )DLSD iidd

idd

i −′

=′

=

LD idi ′=

( )ogS uudu +′=

LS idi =

ug

+uo+

-DS

L

iS iD

iL Ro

+

+

-d’:d

L

Ro

+

- +

-Du

Si Di

Su

Liou

guC

oDoDSg uud1

uuuu −=−+=


Buck-Boost average model: CCM

+

+

-d’:d

L

Ro

+

- +

-Du

Si Di

Su

Liou

guC

( )ogD uudu +=

−′==

′−=−==

o

oLC

C

ogoDLL

Ru

ididt

udC

ududuuudt

idL

+ +

-

LRo

+ +

Si Li

ouguCLid′

Lid gud oud′

Di


Buck-Boost equivalent average model: CCM

+ +

-

LRo

+ +

Si Li

ouguCLid′

Lid gud oud′

Di

+ +

-

LRo

+

Si Li

ouguC

Di1:d d’:1

}

{

BoostBoost

BuckBuck


Cuk average model: CCM

DS

iS iD

uD

- +

-uS

+DS

CD

CS udd

uudu

udu ′=⇒

=′=

( )( ) DS

21D

21S idd

iiidi

iidi′

=⇒

+′=+=

ug

+uo+

-DS

L1

i1 i2

Ro

+uC

Co

L2C1

-uS

+uD

+

-

Switching cellSwitching cell



+

+

-Ro

+1i

ougu

2i

d’:d

L1

Co

L2

C1

Cu

-

+

+

-

Su Du

−=

−′=

−=

′−=

o

o2

oo

21C

1

oC2

2

Cg1

1

Ru

idt

udC

ididdt

udC

uuddt

idL

ududt

idL



+ +

-Ro

+

1i

ougu

2i

L1

CoC1Cud

L2+1id′

Cud′

+Cu

2id

+ +

-Ro

+

1i

ougu

2i

L1

CoC1

L2

1id′

Cu

2id

d’:1 1:d

}

{

BoostBoost

BuckBuck


SEPIC average model: CCM

DS

is iD

uD

- +

-uS

+

( )( ) DS

oCD

oCS udd

uuudu

uudu ′=⇒

+=+′=

( )( ) DS

21D

21S idd

iiidi

iidi′

=⇒

+′=+=

ug

+uo

+

-

D

SL1

i1 iD

Ro

+uC

CoL2

C1

-uS

+ uD+-

i2

Switching cellSwitching cell



+ +

-Ro

+1i

ougu

2i

d’:d

L1

CoL2

C1

Cu Di

-

+

+

-

Su Du( )

( )

−+′=

−′=

′−=

+′−=

o

o21

oo

21C

1

oC2

2

oCg1

1

Ru

iiddt

udC

ididdt

udC

ududdt

idL

uududt

idL



+ +

-Ro

+

1i

ou

gu

2i

L1Co

L2

C1 Cu

Di

+ +

+

+Cud′

oud′

1id′

Cud oud′

2id

1id′

2id′

Alternative approachAlternative approach


Model perturbation

Generic voltage or current:Generic voltage or current: xXx +=

YxyXXYyx ++≈⋅

Xx <<SmallSmall--signal approximation:signal approximation:

Examples:

( )( ) LLLLLL IdiDDIiIdDid ++≈++=

( )( ) gggggg UduDDUuUdDud ++≈++=

Product of variables:Product of variables:


General switching cell: DC and small-signal model

+

- +

-d’:d

Du

Si Di

Su

( ) ( ) DDDSSS UduUDUduUD −+′≈++

( )DDDS

SS uUDD

DUU

duU +′

≈

+++{

Su{

Du

a p

c

n

DSL

+

-uon

iS

+

-

uoff

iD

iL

At steadyAt steady--state:state:

0uL =

====

offDD

onSS

UUu

UUu

DDU

DD

1DU

UU

1DU

DUU SS

S

DSDS

′=

′

+=

+=+


General switching cell: DC and small-signal model

+

- +

-d’:d

Du

Si Di

Su

( ) ( ) DDDSSS IdiIDIdiID ++≈−+′

+

- +

-

D’:D

Du

Si Di

Su

+d

DDUS

′ dDD

ID′

( )

+−+′

≈+D

IIdiI

DD

iI DSSSDD{

Di{

Si


Buck switching cell: DC and small-signal model

+=+=

dUuDu

dIiDi

ggD

LLg+

-

+

-

1:d

gu

Ligi

Du

dUg

+

-

+

-1:D

Du

gi Li

gu

+

dIL

Perturbation and linearization:Perturbation and linearization:


Buck DC and small-signal model

LC

ougu Ro

Li

+ +

-

dUg

1:D

+

dIL

gi

( ) ( )( )

2o

2

o

goud

sQ

s1

U

sDsU

sG

ω+

ω+

==

LC

RQLC1

oo ==ω

DutyDuty--cycle to output voltage transfer function:cycle to output voltage transfer function:


Boost switching cell: DC and small-signal model

−′=−′=

dUuDu

dIiDi

ooS

LLD

dUo

+

-

+

-D’:1

ou

Li Di

Su

+

dIL

+

-

+

-

d’:1

ou

Li Di

Su

Perturbation and linearization:Perturbation and linearization:


Boost DC and small-signal model

DutyDuty--cycle to output voltage transfer function:cycle to output voltage transfer function:

LC

ougu Ro

Di

+ +

-

+dUo

D’:1

Li

dIL

( )222

o

2

o2gud

LCMsMRL

s1

MRL

s1MUsG

++

−=

RHP zero


Boost small-signal model: CCM

1

0t

Normalized output voltage response to a Normalized output voltage response to a dutyduty--cycle step change:cycle step change:

The output voltage initially The output voltage initially moves in the wrong directionmoves in the wrong direction


Buck-Boost DC and small-signal model

+

- +

-

D’:D

Du

Si Di

Su

+d

DDUS

′ dDD

ID′

+

+

-

L

Ro

Liou

guC


Buck-Boost DC and small-signal model

DutyDuty--cycle to output voltage transfer function:cycle to output voltage transfer function:RHP zero

( ) ( )( )

( ) ( )222

o

o2gud

M1LCsM1RL

s1

M1MRL

s1M1UsG

++++

+−+=

+ +

-

LRo

Si Li

ouguC

Di+

dUo

D’:1dIL

dUg

1:D

+

dIL


Cuk DC and small-signal model

D’:D

1i 2i

+d

DDUS

′ dDD

ID′

+

+

-Ro

+

ouguL1

Co

C1

L2

Cu

+

+

-Ro

+1i

ougu

2i

d’:d

L1

Co

L2

C1

Cu

-

+

+

-

Su Du


Cuk DC and small-signal model

+ +

-Roougu

+dUC

D’:1dI1

dUC

1:D

+

dI2Cu+

C1

1i 2i

L1

Co

L2

Alternative approachAlternative approach


SEPIC DC and small-signal model

D’:D

+d

DDUS

′ dDD

ID′

1i

+gu

L1+

-RoouCo

+C1

Cu

2iL2


Discontinuous conduction mode - DCM

t

t

dTs

Ts

t

t

iL(t)

is(t)

id(t)

(d’Ts

Uon

-Uoff

uL(t)

LU

m on1 =

LU

m off2 −=−

Di

Si

Li

SoffSon TdudTu ′=

dd

uu

off

on ′=

( )

+=′+=

off

onon

S

2Lpk

L uu

1uLf2d

dd2

ii

off

2on

S

2Lpk

D uu

Lf2d

d2

ii =′=

onS

2Lpk

S uLf2d

d2

ii ==

At steady-state:

0uL =


Discontinuous conduction mode - DCM

Buck:Buck:

−== 1M1

ULf2d

iI gS

2

Lo

2oN

dI

1

1M

+=

Boost:Boost:

−==

1M1

ULf2d

iI gS

2

DooN

2

Id

1M +=

M1

ULf2d

iI gS

2

Do ==oN

2

Id

M =BuckBuck--Boost:Boost:

s

gN

N

ooN Lf2

UI,

II

I ==

t

t

dTs

Ts

t

t

iL(t)

is(t)

id(t)

(d’Ts

Uon

-Uoff

uL(t)

LU

m on1 =

LU

m off2 −=−

Di

Si

Li


•• The inductor current is always zero at the The inductor current is always zero at the beginning of each switching period;beginning of each switching period;

•• this loss of the memory effect justifies the this loss of the memory effect justifies the statement that the inductor current is no more a statement that the inductor current is no more a state variable;state variable;

•• switch and diode are replaced by non linear switch and diode are replaced by non linear controlled current generatorscontrolled current generators

First order average models - DCM


First order average models - DCM

a p

c

n

DSL

+

-

is

+

-

iD

iL

a p

c

DiSi

Li

onu offu

Inductor average voltage is always Inductor average voltage is always zero in a switching period! zero in a switching period! (?)(?)

t

t

dTs

Ts

t

t

iL(t)

is(t)

id(t)

(d’Ts

Uon

-Uoff

uL(t)

LU

m on1 =

LU

m off2 −=−

Di

Si

Li


Buck average model: DCM

uL

LCD

S+

-

ug uo

+

-

Io

+ -

Ro

CLi ougu

oi

gi


Buck small-signal model: DCM

dkugugddf

uuf

uuf

i iorgioo

gg

g ++=∂∂+

∂∂+

∂∂=

dkugugddh

uuh

uuh

i ooogfoo

gg

L +−=∂∂+

∂∂+

∂∂=

( ) ( )d,u,uhu

uuu

Lf2d

i ogo

gog

S

2

L =

−=

( ) ( )d,u,ufuuLf2d

ii ogogS

2

Sg =−=={AverageAveragequantities:quantities:

{Perturbation:Perturbation:


Buck small-signal model: DCM

CLi ougu

oi

Si

Cgu

Li

oudko

dk i orug

gfug ogig

Si

oR

First order modelFirst order model


Boost small-signal model: DCM

CDi ougu

oi

Li

Cgu

Di

oudko

dk i orug

gfug ogig

Li

oR

−=

go

og

S

2

L uuu

uLf2d

i

go

2g

S

2

D uu

u

Lf2d

i−

=


Buck-Boost small-signal model: DCM

CDi ougu

oi

Si

Cgu

Di

oudko

dk i orug

gfug ogig

Si

oR

o

2g

S

2

D u

u

Lf2d

i =

gS

2

S uLf2d

i =


Full order average models: DCM

t

t=0

t

t

t

t

t

DTs

D’Ts

Li

LL ii +

sTd

sTd′′

i∆

( )tiL

tS

tD

SS tt +

DD tt +

t

St

Dt

Ts

t

( ) { } ( )

se1

TdL

uu

dteidtetiisI

s

s

TDs

soffon

TD

0

st

0

stLLL

′−

′−

∞+−

−+=

∆=== ∫∫L

( ) { } ( )

s

TdsTd

0

st

0

stSS

Tds

e1dte

dtetttsD

ss

≈−==

==

−−

∞+−

∫

∫L

( ) ( )( )

( )

′−′+==

′−

s

TDs

s

offonLid TDs

e1Lf

DuusD

sIsG

s

Impulsive perturbation:Impulsive perturbation:

Response to impulsive perturbation:Response to impulsive perturbation:



t

t=0

t

t

t

t

t

DTs

D’Ts

Li

LL ii +

sTd

sTd′′

i∆

( )tiL

tS

tD

SS tt +

DD tt +

t

St

Dt

Ts

t

First order First order PadPadééapproximation:approximation:

′+

′−

≈′−

2TDs

1

2TDs

1e

s

s

TDs s

( ) ( )

′+

′+≈

2TDs

1

1Lf

DuusG

ss

offonid

Df

fDf2 s

ps

p ′π=⇒

′=ω

( ) ( )( )

( )

′−′+==

′−

s

TDs

s

offonLid TDs

e1Lf

DuusD

sIsG

s



t

t=0

t

t

t

t

t

DTs

D’Ts

Li

LL ii +

sTd

sTd′′

i∆

( )tiL

tS

tD

SS tt +

DD tt +

t

St

Dt

Ts

t

Overall dOverall d’’ perturbation:perturbation:

duu

1du

iLTd

off

on

offs

+=′′⇒

∆=′′

duu

dddoff

on=−′′=′ WRONG!

( )sssD TDtTd)t(Tdt ′−δ⋅′′+δ⋅−=

Dirac functionDirac function

( ) { } ( ) ( ) sTDs

off

onD esD

uu

1sDtsD ′−⋅

++−==′ L



t

t=0

t

t

t

t

t

DTs

D’Ts

Li

LL ii +

sTd

sTd′′

i∆

( )tiL

tS

tD

SS tt +

DD tt +

t

St

Dt

Ts

t

Inductor current perturbation:Inductor current perturbation:

( )( )

sTDs

off

on euu

11sDsD ′−

++−=

′

offonL udud

dt

idL ′−=

( ) ( ) ( ) offonL usDusDsIsL ′−=

( ) ( )( )

( )

′−′+==

′−

s

TDs

s

offonLid TDs

e1Lf

DuusD

sIsG

s



a

pcn

DS

L

iL

a

pcnL

iL

ia

ip

a

pcnL

iLia

uD+ The switch is replaced by a The switch is replaced by a controlled current generator controlled current generator

while the diode is replaced by a while the diode is replaced by a controlled voltage generatorcontrolled voltage generator



( )

=

′+=

di21

i

ddi21

i

Lpa

LpL

ddd

ii La ′+= ( ) ( )dd1uduuu offoffonD ′−−++=

( ) dud

iLf2dddd

Lf2u

ion

Ls

s

onL −=′⇒′+=

t0

iLpiL

tTs0 dTs (d+d’)Ts

uDoffon uu +

offu

ia

a

pcnL

iLia

uD+


Example: boost in DCM

a

pcnL

iLia

uD+

( )( )dd1uuduu gooD ′−−−+=

dud

iLf2d

g

Ls −=′

( )( )[ ]

Ldu

idf2

uu

1

dd1uuduuuL1

dt

id

oL

s

g

o

gooogL

+

−=

=′−−−++−=

o

o

s

g2

L

o

oaL

o

CRu

LCf2

ud

C

i

Ru

iiC1

dtud −−=

−−=

( ) ( )( )

( )

+−+

−+

=

+++++

+−=

o

oo

s

g2

LLo

ooLL

s

g

ooL

CRuU

LCf2

UdD

C

iI

dtud

LdDuU

iIdD

f2U

uU1

dt

id

DutyDuty--cycle perturbation:cycle perturbation:


Example: boost in DCM( )

−−=

−+−=

o

o

s

gLo

oo

soL

L

CRu

dLCf

DU

C

i

dtud

uDRMf2

dLU2

iM1Df2

dt

id

( ) ( )( ) ( ) ( )

ω+

ω+

ω−

≈−+

−++

−==

pAFpBF

zB

o

ss

o

2

s

s

goud

s1

s1

s1

K

CDR1M2f2

D1Mf2

CR1

ss

sDf2

LCf

DU

sDsU

sG

−−=ω1M1M2

CR1

opHF

SmallSmall--signal linear model:signal linear model:

( )D

1Mf2 spLF

−=ωDf2 s

z =ω( )

k1MM

1M2

U2K g

B−

−=

The same as first order model:The same as first order model:


Example: boost in DCM

a) Full order modelb) First order model

10-1

[dB]

-60

-20

-30

-40

-50

( )( ) dBud

ud

0GjG ω

a)

b)

10-210-310-4

-10

100

[deg]

-180

-30

-60

-90

-120

( )ω∠ jGud

a)

b)

-150

10-1

sff10-210-310-4 100

sff

ControlControl--toto--output transfer functionoutput transfer function


State-Space averaging (SSA): CCM

Interval Interval dTdTss::

=+=

xCy

uBxAx

1

11&

State, input and output State, input and output variable vector:variable vector:

=C

L

u

ix

=o

g

i

uu

∈∈

=off

on

tt0

tt1)t(q

( ) ( )[ ]( )

⋅+=⋅−+=⋅++=⋅−+−++=

qGxCqxCCxCy

qFuBxAquBBxAAuBxAx

2212

22212122&

Switching function:Switching function:

⋅+=⋅++=

qGxCy

qFuBxAx

2

22&Applying moving Applying moving

average operator:average operator: xx && =

=+=

xCy

uBxAx

2

22&Interval (1Interval (1--d)Td)Tss::

=

g

o

i

uy



Hp: linear ripple approximationHp: linear ripple approximation

=+=+=++=

xCdGxCy

uBxAdFuBxAx

2

22&dFqFqF ⋅=⋅=⋅

t

iL

t

Ts0 dTs

t

q

qiL

Li

LL iqqi =

dq =

?qG?,qF =⋅=⋅

t

uC

t

Ts0 dTs

t

q

quC

Cu

CC uqqu =

dq =

Hp: small ripple approximationHp: small ripple approximation

( )d1AdAA 21 −+= ( )d1BdBB 21 −+=dGqGqG ⋅=⋅=⋅

( )d1CdCC 21 −+=



SteadySteady--state solution:state solution:

=−=

⇒

=+= −

xCy

uBAxxCy

uBxA0 1

( ) ( )[ ]( )

+=−+=++=−+−++=

dGxCdXCCxCy

dFuBxAdUBBXAAuBxAx

21

2121&


( ) ( ) ( ) ( )( )( ) ( ) ( )

+=+−= −

sDGsXCsY

sDFsUBAsIsX 1


Example: boost converter in CCM

=C

L

u

ix guu = ouy =

Interval Interval dTdTss:: Interval (1Interval (1--d)Td)Tss::

=+=

xCy

uBxAx

1

11&

=+=

xCy

uBxAx

2

22&



( )

+−

−=

CrR1

0

0Lr

A

C

L

1( )

( ) ( )

+−

+

+−+−

=

CrR1

CrRR

LrRR

LR//rr

A

CC

C

CL

2

+=

C1 rR

R0C

+=

CC2 rR

Rr//RC

==

0L1

BB 21

SteadySteady--state state solution:solution: g

1 UBAx0x −−=⇒=&

′′

==

′′=

=

R

RDUUY

RD

1

R

U

U

IX

go

g

C

L

( )R//rDDrRDR CL2 ′++′=′

( )R//rDDrRD

RDD1

UU

MCL

2

2

g

o

′++′′

⋅′

==

Voltage conversion ratio:Voltage conversion ratio:




( )( )

( ) ( )

( )( )

( )d

CrRR

LrRrRDR

R

Uu

0L1

u

i

CrR1

CrRRD

LrRRD

LR//rDr

u

i

C

C

C

gg

C

L

CC

C

CL

C

L

+−

++′

′+

+

+−

+′

+′

−′+−

=

&

&

( ) dR

R||rU

u

i

rRR

R||rDy Cg

C

L

CC ′

−

+′=


State-Space averaging (SSA): DCM

Applying moving Applying moving average operator:average operator:

xx && =t

iLpiL

d1Ts d2Ts d3Ts

LUon

LUoff−

t

t

t

1

1

1

q1

q2

q3

( ) uBqxAqtx3

1kkk

3

1kkk

+

= ∑∑

==

&

uBdxAdx3

1kkk

3

1kkk

+

≠ ∑∑

==

&

Why?

State variable vector:State variable vector:

=C

L

u

ix



Example:Example: iiLL··qq11t

iLpiL

d1Ts d2Ts d3Ts

LUon

LUoff−

t

1 q1

iLq1

Li

1L1L qiqi ≠

( )21Lpk

L dd2

ii +=

212L

21

2L2

LpkDL2

dd1

qi

ddd

id2

iiiq

+=

+===

211L

21

1L1

LpkSL1 dd

1qi

ddd

id2

iiiq

+=

+===

Hp: linear ripple approximationHp: linear ripple approximation

Corrective termCorrective term



uBdxMAdx3

1kkk

3

1kkk

+

= ∑∑

==

&

3,2,1iuduquq CiCiCi ===

Hp: small ripple approximationHp: small ripple approximation

+=10

0dd

1M

21

M is the correction matrixM is the correction matrix

( ) uBqxAqtx3

1kkk

3

1kkk

+

= ∑∑

==

&



If dIf d33 = 0, i.e. in CCM, we have:= 0, i.e. in CCM, we have:

uBdxMAdx3

1kkk

3

1kkk

+

= ∑∑

==

&

+=10

0dd

1M

21 1on

Ls2 d

u

iLf2d −

δ=

213 dd1d −−=

=10

01M uBdxAdx

2

1kkk

2

1kkk

+

= ∑∑

==

&

modeling approaches for switching converters - unicampantenor/pdffiles/spiazzi.pdf · modeling...

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