[PPT] - Modeling approaches for switching converters by Giorgio Spiazzi PowerPoint Presentation

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Modeling approaches for switching converters

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Modeling approaches for switching converters

by

Giorgio Spiazzi

University of Padova – ITALY

Dept. of Information Engineering – DEI

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Modeling approaches for switching converters

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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): first conduction mode (DCM): first-

order model
rder model

PWM converters PWM converters

Summary of the presentation

SLIDE 3

Modeling approaches for switching converters

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Basic DC-DC Converter topologies: 2°order

ug + uo +

D

S L is iL iD Ro

Buck Buck

ug + uo +

D

S L iL iD iS Ro

Boost Boost

ug + uo +

D

S L is iD iL Ro Buck

Buck-

Boost

Boost

SLIDE 4

Modeling approaches for switching converters

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Basic DC-DC Converter topologies: 4°order

ug + uo +

D

S L1 i1 i2 Ro + uC Co L2 C1

uS

+ uD +

Cuk

Cuk

ug + uo +

D

S L1 i1 iD Ro + uC Co L2 C1

uS

+ uD +

i2

SEPIC SEPIC

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Modeling approaches for switching converters

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iD iD iL io iS iL iS ig Ug+Uo Uo Ug Uon+ Uoff Uo Uo-Ug Uo Uoff Ug Ug Ug-Uo Uon Buck-boost Boost Buck

a p c n D S L

+

Uon

is

+

Uoff

iD iL

Commutation Cell for 2°order converters 2 2° ° order converters can be

rder converters can be

described by a unique described by a unique commutation cell: commutation cell:

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Modeling approaches for switching converters

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Averaging

( ) ( )

∫

−

τ τ =

t T t s

s

d x T 1 t x Moving 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

SLIDE 7

Modeling approaches for switching converters

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Average model: CCM

Switching frequency ripples are neglected
Only low-frequency dynamic is investigated

( ) ( )

      − − = = τ τ =

∫ ∫

− − S S L L t i T t i L S t T t L S L

T T t i t i L di T L d u T 1 t u

L S L S

Example: inductors Example: inductors

( ) ( )

dt t di L t u

L L

=

SLIDE 8

Modeling approaches for switching converters

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Average model: CCM

( ) ( )

( ) ( ) ( ) ( )

t z , t y with d , t f z , y , t d , t f t

z y t t

β = α = τ τ = φ = τ τ = φ

∫ ∫

β α

( ) ( )

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

dt t d t , t f dt t d t , t f d dt , t df dt t d

t t

β β + α α − τ τ = φ

∫

β α

( ) ( )

dt t i d L t u

L L

=

( ) ( )

? d i T 1 dt d dt t i d

t T t L S L

S

=         τ τ =

∫

−

( ) ( ) ( )

S S L L L

T T t i t i dt t i d − − =

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Modeling approaches for switching converters

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Averaging approximation

t t dTs iL(t) (1-d)Ts Uon

Uoff

uL(t)

L U m

n

1 =

L U m

ff

2

− = −

iL(Ts) iL(0)

L

u

( ) ( ) ( ) s

ff

s L s L

T d 1 L U dT i T i − − =

( ) ( )

s

n

L s L

dT L U i dT i + =

( ) ( ) ( ) ( )

s L L s

ff

s

n

L s L

T L u i T d 1 L U dT L U i T i + = − − + =

Non steady Non steady-

state

state inductor current inductor current waveform: waveform:

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Modeling approaches for switching converters

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Averaging

Reactive element voltage

Reactive 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.

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Modeling approaches for switching converters

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Continuous conduction mode - CCM

( ) S

ff

S

n

T d 1 U dT U − = d d 1 U U

ff
n

− = At steady At steady-

state:

state:

t t dTs Ts t t iL(t) is(t) id(t) (1-d)Ts Uon

Uoff

uL(t)

L U m

n

1 =

L U m

ff

2

− = −

( )

d 1 i i

L D

− = d i i

L S = L

i

uL = Buck:

d M = Boost: d 1 1 M − = Buck-Boost: d 1 d M − =

( )

d 1 Lf 2 U d Lf 2 U 2 i i

S

ff

S

n

Lpp lim L

− = = ∆ =

Boundary CCM Boundary CCM-

DCM:

DCM:

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Modeling approaches for switching converters

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Switching cell average model: CCM

( ) ( )

D S

ff
n

D

ff
n

S

u d d u u u d u u u d u ′ = ⇒    + = + ′ =

D S L D L S

i d d i i d i i d i ′ = ⇒    ′ = = a p c n D S L

+

uon

iS

+

uoff

iD iL

d’=1-d = complement of duty-cycle

D S iS iD uD

+
uS

+ Non linear components Non linear components Average switch and diode voltages and currents: Average switch and diode voltages and currents:

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Modeling approaches for switching converters

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Switching cell average model: CCM

+

+
d’:d

D

u

S

i

D

i

S

u D S iS iD uD

+
uS

+

( ) ( )

D S

ff
n

D

ff
n

S

u d d u u u d u u u d u ′ = ⇒    + = + ′ =

D S L D L S

i d d i i d i i d i ′ = ⇒    ′ = = + +

D

u d d′

S

i d d′

S

u

D

i

+
D

u

S

i

d’=1-d = complement of duty-cycle

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Modeling approaches for switching converters

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Switching cell average model: CCM

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.

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Modeling approaches for switching converters

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Buck average model: CCM

D g S g D

u d d u u u u ′ − = − =

g D

u d u =

( )

S L D S

i i d d i d d i − ′ = ′ =

L S

i d i = + +

d’:d

L Ro +

+
D

u

S

i

D

i

S

u

L

i

u

g

u C ug + uo +

D

S L iS iL iD Ro ug

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Modeling approaches for switching converters

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Buck average model (alternative approach): CCM

D S ig iL ug uD +

+
Switching

Switching cell cell

Independent variables: Independent variables: u ug

g,

, i iL

L

Dependent variables: Dependent variables: u uD

D,

, i ig

g

ug + uo +

D

S L iS iL iD Ro

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Modeling approaches for switching converters

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Buck average model (alternative approach): CCM

D S ig iL ug uD +

+
+
+
1:d

g

u

L

i

g

i

D

u Averaging Averaging + +

L

i d

g

u d

g

u

L

i

+
D

u

g

i

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Modeling approaches for switching converters

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Buck average model: CCM

       − = = − = =

L

C C

g

L L

R u i i dt u d C u u d u dt i d L L C

L

i d

g

u d

u

g

u Ro

L

i + +

+

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Modeling approaches for switching converters

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Boost average model: CCM

+ +

d’:d

L Ro +

+
D

u

S

i

D

i

S

u

L

i

u

g

u C

S

D
S

u d d u u u u ′ − = − =

S

u d u ′ =

( )

D L S D

i i d d i d d i − ′ = ′ =

L D

i d i ′ = ug + uo +

D

S L iL iD iS Ro

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Modeling approaches for switching converters

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Boost average model (alternative approach): CCM Switching Switching cell cell

Independent variables: Independent variables: u uo

,

, i iL

L

Dependent variables: Dependent variables: u uS

S,

, i iD

D

D S iL iD uS uo +

+
ug

+ uo +

D

S L iL iD iS Ro

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Modeling approaches for switching converters

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Boost average model (alternative approach): CCM

+ +

L

i d′

u

d′

u

L

i

D

S iL iD uS uo +

+
+
+
d’:1
u

L

i

D

i

S

u

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Modeling approaches for switching converters

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Boost average model: CCM

       − ′ = = ′ − = =

L

C C

g

L L

R u i d i dt u d C u d u u dt i d L L C

L

i

u

d′

u

g

u Ro

L

i d′ + +

+
d’:1

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Modeling approaches for switching converters

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Buck-Boost average model: CCM

( )

g

D

u u d u + =

( )

D L S D

i i d d i d d i − ′ = ′ =

L D

i d i ′ =

( )

g

S

u u d u + ′ =

L S

i d i = ug + uo +

D

S L iS iD iL Ro + +

d’:d

L Ro +

+
D

u

S

i

D

i

S

u

L

i

u

g

u C

D
D

S g

u u d 1 u u u u − = − + =

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Modeling approaches for switching converters

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Buck-Boost average model: CCM

+ +

d’:d

L Ro +

+
D

u

S

i

D

i

S

u

L

i

u

g

u C

( )

g

D

u u d u + =        − ′ = = ′ − = − = =

L

C C

g
D

L L

R u i d i dt u d C u d u d u u u dt i d L + +

L

Ro + +

S

i

L

i

u

g

u C

L

i d′

L

i d

g

u d

u

d′

D

i

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Modeling approaches for switching converters

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Buck-Boost equivalent average model: CCM

+ +

L

Ro + +

S

i

L

i

u

g

u C

L

i d′

L

i d

g

u d

u

d′

D

i + +

L

Ro +

S

i

L

i

u

g

u C

D

i 1:d d’:1

Boost

Boost Buck Buck

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Modeling approaches for switching converters

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Cuk average model: CCM

D S iS iD uD

+
uS

+

D S C D C S

u d d u u d u u d u ′ = ⇒    = ′ =

( ) ( )

D S 2 1 D 2 1 S

i d d i i i d i i i d i ′ = ⇒    + ′ = + = ug + uo +

D

S L1 i1 i2 Ro + uC Co L2 C1

uS

+ uD +

Switching cell

Switching cell

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Modeling approaches for switching converters

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Cuk average model: CCM

+ +

Ro

+

1

i

u

g

u

2

i d’:d L1 Co L2 C1

C

u

+

+

S

u

D

u            − = − ′ = − = ′ − =

2
2

1 C 1

C

2 2 C g 1 1

R u i dt u d C i d i d dt u d C u u d dt i d L u d u dt i d L

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Modeling approaches for switching converters

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Cuk average model: CCM

+ +

Ro

+

1

i

u

g

u

2

i L1 Co C1

C

u d L2 +

1

i d′

C

u d′ +

C

u

2

i d + +

Ro

+

1

i

u

g

u

2

i L1 Co C1 L2

1

i d′

C

u

2

i d d’:1 1:d

Boost

Boost Buck Buck

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Modeling approaches for switching converters

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SEPIC average model: CCM

D S is iD uD

+
uS

+

( ) ( )

D S

C

D

C

S

u d d u u u d u u u d u ′ = ⇒    + = + ′ =

( ) ( )

D S 2 1 D 2 1 S

i d d i i i d i i i d i ′ = ⇒    + ′ = + = ug + uo +

D

S L1 i1 iD Ro + uC Co L2 C1

uS

+ uD +

i2

Switching cell Switching cell

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Modeling approaches for switching converters

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SEPIC average model: CCM

+ +

Ro

+

1

i

u

g

u

2

i d’:d L1 Co L2 C1

C

u

D

i

+

+

S

u

D

u

( )

           − + ′ = − ′ = ′ − = + ′ − =

2

1

2

1 C 1

C

2 2

C

g 1 1

R u i i d dt u d C i d i d dt u d C u d u d dt i d L u u d u dt i d L

SLIDE 31

Modeling approaches for switching converters

31/72

SEPIC average model: CCM

+ +

Ro

+

1

i

u

g

u

2

i L1 Co L2 C1

C

u

D

i + + + +

C

u d′

u

d′

1

i d′

C

u d

u

d′

2

i d

1

i d′

2

i d′

Alternative approach Alternative approach

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Modeling approaches for switching converters

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Model perturbation Generic voltage or current: Generic voltage or current: x ˆ X x + = Y x ˆ y ˆ X XY y x + + ≈ ⋅ X x ˆ << Small Small-

signal approximation:

signal approximation: Examples:

( )( )

L L L L L L

I d ˆ i ˆ D DI i ˆ I d ˆ D i d + + ≈ + + =

( )(

)

g g g g g g

U d ˆ u ˆ D DU u ˆ U d ˆ D u d + + ≈ + + =

Product of variables: Product of variables:

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Modeling approaches for switching converters

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General switching cell: DC and small-signal model

+

+
d’:d

D

u

S

i

D

i

S

u

( ) ( )

D D D S S S

U d ˆ u ˆ U D U d ˆ u ˆ U D − + ′ ≈ + +

( )

D D D S S S

u ˆ U D D D U U d ˆ u ˆ U + ′ ≈       + + +

S

u

D

u a p c n D S L

+

uon

iS

+

uoff

iD iL

At steady At steady-

state:

state:

uL =    = = = =

ff

D D

n

S S

U U u U U u D D U D D 1 D U U U 1 D U D U U

S S S D S D S

′ =       ′ + =         + = +

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Modeling approaches for switching converters

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General switching cell: DC and small-signal model

+

+
d’:d

D

u

S

i

D

i

S

u

( ) ( )

D D D S S S

I d ˆ i ˆ I D I d ˆ i ˆ I D + + ≈ − + ′ +

+
D’:D

D

u

S

i

D

i

S

u + d ˆ D D US ′ d ˆ D D ID ′

( )

      + − + ′ ≈ + D I I d ˆ i ˆ I D D i ˆ I

D S S S D D

D

i

S

i

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Modeling approaches for switching converters

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Buck switching cell: DC and small-signal model

   + = + = d ˆ U u D u d ˆ I i D i

g g D L L g

+

+
1:d

g

u

L

i

g

i

D

u d ˆ Ug +

+
1:D

D

u

g

i

L

i

g

u + d ˆ IL

Perturbation and linearization: Perturbation and linearization:

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Modeling approaches for switching converters

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Buck DC and small-signal model

L C

u

g

u Ro

L

i + +

d

ˆ Ug 1:D + d ˆ IL

g

i

( ) ( ) ( )

2

2
g
ud

s Q s 1 U s D ˆ s U ˆ s G ω + ω + = =

L C R Q LC 1

=

= ω

Duty Duty-

cycle to output voltage transfer function:

cycle to output voltage transfer function:

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Modeling approaches for switching converters

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Boost switching cell: DC and small-signal model

   − ′ = − ′ = d ˆ U u D u d ˆ I i D i

S

L L D

d ˆ Uo +

+
D’:1
u

L

i

D

i

S

u + d ˆ IL +

+
d’:1
u

L

i

D

i

S

u

Perturbation and linearization: Perturbation and linearization:

SLIDE 38

Modeling approaches for switching converters

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Boost DC and small-signal model Duty Duty-

cycle to output voltage transfer function:

cycle to output voltage transfer function:

L C

u

g

u Ro

D

i + +

+

d ˆ Uo D’:1

L

i d ˆ IL

( )

2 2 2

2
2

g ud

LCM s M R L s 1 M R L s 1 M U s G + + − = RHP zero

SLIDE 39

Modeling approaches for switching converters

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Boost small-signal model: CCM

1

t

Normalized output voltage response to a Normalized output voltage response to a duty duty-

cycle step change:

cycle step change: The output voltage initially The output voltage initially moves in the wrong direction moves in the wrong direction

SLIDE 40

Modeling approaches for switching converters

40/72

Buck-Boost DC and small-signal model

+

+
D’:D

D

u

S

i

D

i

S

u + d ˆ D D US ′ d ˆ D D ID ′ + +

L

Ro

L

i

u

g

u C

SLIDE 41

Modeling approaches for switching converters

41/72

Buck-Boost DC and small-signal model Duty Duty-

cycle to output voltage transfer function:

cycle to output voltage transfer function:

RHP zero

( ) ( ) ( ) ( ) ( )2

2 2

2

g ud

M 1 LC s M 1 R L s 1 M 1 M R L s 1 M 1 U s G + + + + + − + = + +

L

Ro

S

i

L

i

u

g

u C

D

i + d ˆ Uo D’:1 d ˆ IL d ˆ Ug 1:D + d ˆ IL

SLIDE 42

Modeling approaches for switching converters

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Cuk DC and small-signal model

D’:D

1

i

2

i + d ˆ D D US ′ d ˆ D D ID ′ + +

Ro

+

u

g

u L1 Co C1 L2

C

u + +

Ro

+

1

i

u

g

u

2

i d’:d L1 Co L2 C1

C

u

+

+

S

u

D

u

SLIDE 43

Modeling approaches for switching converters

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Cuk DC and small-signal model

+ +

Ro
u

g

u + d ˆ UC D’:1 d ˆ I1 d ˆ UC 1:D + d ˆ I2

C

u+ C1

1

i

2

i L1 Co L2

Alternative approach Alternative approach

SLIDE 44

Modeling approaches for switching converters

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SEPIC DC and small-signal model

D’:D + d ˆ D D US ′ d ˆ D D ID ′

1

i +

g

u L1 +

Ro
u

Co + C1

C

u

2

i L2

SLIDE 45

Modeling approaches for switching converters

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Discontinuous conduction mode - DCM

t t dTs Ts t t iL(t) is(t) id(t) (d’Ts Uon

Uoff

uL(t)

L U m

n

1 =

L U m

ff

2

− = − D

i

S

i

L

i

S

ff

S

n

T d u dT u ′ = d d u u

ff
n

′ =

( )

        + = ′ + =

ff
n
n

S 2 Lpk L

u u 1 u Lf 2 d d d 2 i i

ff

2

n

S 2 Lpk D

u u Lf 2 d d 2 i i = ′ =

n

S 2 Lpk S

u Lf 2 d d 2 i i = = At steady-state:

uL =

SLIDE 46

Modeling approaches for switching converters

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Discontinuous conduction mode - DCM

Buck: Buck:       − = = 1 M 1 U Lf 2 d i I

g S 2 L

2
N

d I 1 1 M + = Boost: Boost:       − = = 1 M 1 U Lf 2 d i I

g S 2 D

N

2

I d 1 M + = M 1 U Lf 2 d i I

g S 2 D

=

=

N

2

I d M = Buck Buck-

Boost:

Boost:

s g N N

N

Lf 2 U I , I I I = =

t t dTs Ts t t iL(t) is(t) id(t) (d’Ts Uon

Uoff

uL(t)

L U m

n

1 =

L U m

ff

2

− = − D

i

S

i

L

i

SLIDE 47

Modeling approaches for switching converters

47/72

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 generators controlled current generators First order average models - DCM

SLIDE 48

Modeling approaches for switching converters

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First order average models - DCM

a p c n D S L

+

is

+

iD

iL a p c

D

i

S

i

L

i

n

u

ff

u 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)

L U m

n

1 =

L U m

ff

2

− = − D

i

S

i

L

i

SLIDE 49

Modeling approaches for switching converters

49/72

Buck average model: DCM uL L C D S +

ug

uo +

Io

+

Ro

C

L

i

u

g

u

i

g

i

SLIDE 50

Modeling approaches for switching converters

50/72

Buck small-signal model: DCM

d ˆ k u ˆ g u ˆ g d ˆ d f u ˆ u f u ˆ u f i ˆ

i

r

g i

g

g g

+ + = ∂ ∂ + ∂ ∂ + ∂ ∂ = d ˆ k u ˆ g u ˆ g d ˆ d h u ˆ u h u ˆ u h i ˆ

g

f

g

g L

+ − = ∂ ∂ + ∂ ∂ + ∂ ∂ =

( ) ( )

d , u , u h u u u u Lf 2 d i

g
g
g

S 2 L

=         − =

( ) ( )

d , u , u f u u Lf 2 d i i

g
g

S 2 S g

= − = =

{

Average Average quantities: quantities:

{

Perturbation: Perturbation:

SLIDE 51

Modeling approaches for switching converters

51/72

Buck small-signal model: DCM

C

L

i

u

g

u

i

S

i C

g

u ˆ

L

i ˆ

u

ˆ d ˆ ko d ˆ ki

ru

ˆ g

g fu

ˆ g

g

i

g

S

i ˆ

R

First order model First order model

SLIDE 52

Modeling approaches for switching converters

52/72

Boost small-signal model: DCM

C

D

i

u

g

u

i

L

i C

g

u ˆ

D

i ˆ

u

ˆ d ˆ ko d ˆ ki

ru

ˆ g

g fu

ˆ g

g

i

g

L

i ˆ

R

        − =

g

g

S 2 L

u u u u Lf 2 d i

g

2

g S 2 D

u u u Lf 2 d i − =

SLIDE 53

Modeling approaches for switching converters

53/72

Buck-Boost small-signal model: DCM

C

D

i

u

g

u

i

S

i C

g

u ˆ

D

i ˆ

u

ˆ d ˆ ko d ˆ ki

ru

ˆ g

g fu

ˆ g

g

i

g

S

i ˆ

R
2

g S 2 D

u u Lf 2 d i =

g S 2 S

u Lf 2 d i =

SLIDE 54

Modeling approaches for switching converters

54/72

Full order average models: DCM

t

t=0

t t t t t

DTs D’Ts

L

i ˆ

L L

i ˆ i +

s

T d ˆ

s

T d ˆ ′ ′ i ∆

( )

t iL tS tD

S S

t ˆ t +

D D

t ˆ t +

t

S

t ˆ

D

t ˆ Ts

t

( )

{ }

( )

s e 1 T d ˆ L u u dt e i dt e t i ˆ i ˆ s I ˆ

s s

T D s s

ff
n

T D st st L L L ′ − ′ − ∞ + −

− + = ∆ = = =

∫ ∫

( )

{ }

( )

s T d ˆ s T d ˆ st st S S

T d ˆ s e 1 dt e dt e t t ˆ t ˆ s D ˆ

s s

≈ − = = = =

− − ∞ + −

∫ ∫

( )

( ) ( ) ( )

        ′ − ′ + = =

′ − s T D s s

ff
n

L id

T D s e 1 Lf D u u s D ˆ s I ˆ s G

s

Impulsive perturbation: Impulsive perturbation: Response to impulsive perturbation: Response to impulsive perturbation:

SLIDE 55

Modeling approaches for switching converters

55/72

Full order average models: DCM

t

t=0

t t t t t

DTs D’Ts

L

i ˆ

L L

i ˆ i +

s

T d ˆ

s

T d ˆ ′ ′ i ∆

( )

t iL tS tD

S S

t ˆ t +

D D

t ˆ t +

t

S

t ˆ

D

t ˆ Ts

t

First order First order Pad Padé é approximation: approximation:

            ′ + ′ − ≈

′ −

2 T D s 1 2 T D s 1 e

s s T D s

s

( ) ( )

            ′ + ′ + ≈ 2 T D s 1 1 Lf D u u s G

s s

ff
n

id

D f f D f 2

s p s p

′ π = ⇒ ′ = ω

( ) ( ) ( ) ( )

        ′ − ′ + = =

′ − s T D s s

ff
n

L id

T D s e 1 Lf D u u s D ˆ s I ˆ s G

s

SLIDE 56

Modeling approaches for switching converters

56/72

Full order average models: DCM

t

t=0

t t t t t

DTs D’Ts

L

i ˆ

L L

i ˆ i +

s

T d ˆ

s

T d ˆ ′ ′ i ∆

( )

t iL tS tD

S S

t ˆ t +

D D

t ˆ t +

t

S

t ˆ

D

t ˆ Ts

t

Overall d Overall d’ ’ perturbation: perturbation:

d ˆ u u 1 d ˆ u i L T d ˆ

ff
n
ff

s

        + = ′ ′ ⇒ ∆ = ′ ′ d ˆ u u d ˆ d ˆ d ˆ

ff
n

= − ′ ′ = ′ WRONG!

( )

s s s D

T D t T d ˆ ) t ( T d ˆ t ˆ ′ − δ ⋅ ′ ′ + δ ⋅ − =

Dirac function Dirac function

( )

{ }

( ) ( )

s

T D s

ff
n

D

e s D ˆ u u 1 s D ˆ t ˆ s D ˆ

′ −

⋅         + + − = = ′

SLIDE 57

Modeling approaches for switching converters

57/72

Full order average models: DCM

t

t=0

t t t t t

DTs D’Ts

L

i ˆ

L L

i ˆ i +

s

T d ˆ

s

T d ˆ ′ ′ i ∆

( )

t iL tS tD

S S

t ˆ t +

D D

t ˆ t +

t

S

t ˆ

D

t ˆ Ts

t

Inductor current perturbation: Inductor current perturbation:

( ) ( )

s

T D s

ff
n e

u u 1 1 s D ˆ s D ˆ

′ −

        + + − = ′

ff
n

L

u d ˆ u d ˆ dt i ˆ d L ′ − =

( ) ( ) ( ) off

n

L

u s D ˆ u s D ˆ s I ˆ sL ′ − =

( ) ( ) ( ) ( )

        ′ − ′ + = =

′ − s T D s s

ff
n

L id

T D s e 1 Lf D u u s D ˆ s I ˆ s G

s

SLIDE 58

Modeling approaches for switching converters

58/72

Full order average models: DCM

a p c n D S L iL a p c n L iL ia ip a p c n L iL ia 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 generator controlled voltage generator

SLIDE 59

Modeling approaches for switching converters

59/72

Full order average models: DCM

( )

     = ′ + = d i 2 1 i d d i 2 1 i

Lp a Lp L

d d d i i

L a

′ + =

( ) ( )

d d 1 u d u u u

ff
ff
n

D

′ − − + + =

( )

d u d i Lf 2 d d d d Lf 2 u i

n

L s s

n

L

− = ′ ⇒ ′ + =

t iLp iL t Ts dTs (d+d’)Ts uD

ff
n

u u +

ff

u ia a p c n L iL ia uD+

SLIDE 60

Modeling approaches for switching converters

60/72

Example: boost in DCM

a p c n L iL ia uD+

( )(

)

d d 1 u u d u u

g

D

′ − − − + = d u d i Lf 2 d

g L s

− = ′

( )(

)

[ ]

L d u i d f 2 u u 1 d d 1 u u d u u u L 1 dt i d

L

s g

g
g

L

+         − = = ′ − − − + + − =

s

g 2 L

a

L

CR

u LCf 2 u d C i R u i i C 1 dt u d − − =         − − =

( ) (

)(

) ( )

       + − + − + = + + + + +         + − =

s

g 2 L L

L

L s g

L

CR u ˆ U LCf 2 U d ˆ D C i ˆ I dt u ˆ d L d ˆ D u ˆ U i ˆ I d ˆ D f 2 U u ˆ U 1 dt i ˆ d

Duty Duty-

cycle perturbation:

cycle perturbation:

SLIDE 61

Modeling approaches for switching converters

61/72

Example: boost in DCM

( )

       − − = − + − =

s

g L

s
L

L

CR u ˆ d ˆ LCf DU C i ˆ dt u ˆ d u ˆ DR Mf 2 d ˆ L U 2 i ˆ M 1 D f 2 dt i ˆ d

( ) ( ) ( ) ( ) ( )

        ω +         ω +         ω − ≈ − +         − + +       − = =

pAF pBF z B

s

s

2

s s g

ud

s 1 s 1 s 1 K C DR 1 M 2 f 2 D 1 M f 2 C R 1 s s s D f 2 LCf DU s D ˆ s U ˆ s G

      − − = ω 1 M 1 M 2 C R 1

pHF

Small Small-

signal linear model:

signal linear model:

( )

D 1 M f 2 s

pLF

− = ω D f 2 s

z =

ω

( )

k 1 M M 1 M 2 U 2 K

g B

− − =

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

SLIDE 62

Modeling approaches for switching converters

62/72

Example: boost in DCM

a) Full order model b) First order model

10-1 [dB]

60
20
30
40
50

( ) ( ) dB

ud ud

G j G ω

a) b) 10-2 10-3 10-4

10

100 [deg]

180
30
60
90
120

( )

ω ∠ j Gud

a) b)

150

10-1

s

f f

10-2 10-3 10-4 100

s

f f

Control Control-

to

to-

output transfer function
utput transfer function

SLIDE 63

Modeling approaches for switching converters

63/72

State-Space averaging (SSA): CCM

Interval Interval dT dTs

s:

:

   = + = x C y u B x A x

1 1 1

State, input and output

State, input and output variable vector: variable vector:

      =

C L

u i x       =

g

i u u

   ∈ ∈ =

ff
n

t t t t 1 ) t ( q

( ) ( ) [ ] ( )

   ⋅ + = ⋅ − + = ⋅ + + = ⋅ − + − + + = q G x C q x C C x C y q F u B x A q u B B x A A u B x A x

2 2 1 2 2 2 2 1 2 1 2 2

Switching function:

Switching function:

   ⋅ + = ⋅ + + = q G x C y q F u B x A x

2 2 2

Applying moving

Applying moving average operator: average operator:

x x

=

   = + = x C y u B x A x

2 2 2

Interval (1

Interval (1-

d)T

d)Ts

s:

:

      =

g

i

u y

SLIDE 64

Modeling approaches for switching converters

64/72

State-Space averaging (SSA): CCM

Hp: linear ripple approximation Hp: linear ripple approximation

   = + = + = + + = x C d G x C y u B x A d F u B x A x

2 2 2

d

F q F q F ⋅ = ⋅ = ⋅

t iL t Ts dTs t q qiL

L

i

L L

i q qi = d q =

? q G ?, q F = ⋅ = ⋅

t uC t Ts dTs t q quC

C

u

C C

u q qu = d q =

Hp: small ripple approximation Hp: small ripple approximation

( )

d 1 A d A A

2 1

− + =

( )

d 1 B d B B

2 1

− + = d G q G q G ⋅ = ⋅ = ⋅

( )

d 1 C d C C

2 1

− + =

SLIDE 65

Modeling approaches for switching converters

65/72

State-Space averaging (SSA): CCM

Steady Steady-

state solution:

state solution:    = − = ⇒    = + =

−

x C y u B A x x C y u B x A

1

( ) ( ) [ ] ( )

   + = − + = + + = − + − + + = d ˆ G x ˆ C d ˆ X C C x ˆ C y ˆ d ˆ F u ˆ B x ˆ A d ˆ U B B X A A u ˆ B x ˆ A x ˆ

2 1 2 1 2 1

Small

Small-

signal linear model:

signal linear model:

( ) ( ) ( ) ( )

( )

( ) ( ) ( )

   + = + − =

−

s D ˆ G s X ˆ C s Y ˆ s D ˆ F s U ˆ B A sI s X ˆ

1

SLIDE 66

Modeling approaches for switching converters

66/72

Example: boost converter in CCM

      =

C L

u i x

g

u u =

u

y =

Interval Interval dT dTs

s:

: Interval (1 Interval (1-

d)T

d)Ts

s:

:

   = + = x C y u B x A x

1 1 1



  = + = x C y u B x A x

2 2 2

SLIDE 67

Modeling approaches for switching converters

67/72

Example: boost converter in CCM

( ) 

           + − − = C r R 1 L r A

C L 1

( ) ( ) ( ) 

           + − + + − + − = C r R 1 C r R R L r R R L R // r r A

C C C C L 2

      + =

C 1

r R R C       + =

C C 2

r R R r // R C         = = L 1 B B

2 1

Steady Steady-

state

state solution: solution:

g 1 U

B A x x

−

− = ⇒ =



      ′ ′ = =       ′ ′ =       = R R D U U Y R D 1 R U U I X

g

g

C L

( )

R // r D D r R D R

C L 2

′ + + ′ = ′

( )

R // r D D r R D R D D 1 U U M

C L 2 2 g

′

+ + ′ ′ ⋅ ′ = =

Voltage conversion ratio: Voltage conversion ratio:

SLIDE 68

Modeling approaches for switching converters

68/72

Example: boost converter in CCM Small Small-

signal linear model:

signal linear model:

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

d ˆ C r R R L r R r R D R R U u ˆ L 1 u ˆ i ˆ C r R 1 C r R R D L r R R D L R // r D r u ˆ i ˆ

C C C g g C L C C C C L C L

            + − + + ′ ′ +         +                   + − + ′ + ′ − ′ + − =      

(

)

d ˆ R R || r U u ˆ i ˆ r R R R || r D y ˆ

C g C L C C

′ −             + ′ =

SLIDE 69

Modeling approaches for switching converters

69/72

State-Space averaging (SSA): DCM

Applying moving Applying moving average operator: average operator: x x

=

t

iLp

iL

d1Ts d2Ts d3Ts L Uon L Uoff −

t t t

1 1 1 q1 q2 q3

( )

u B q x A q t x

3 1 k k k 3 1 k k k

        +         =

∑ ∑

= =

u

B d x A d x

3 1 k k k 3 1 k k k

        +         ≠

∑ ∑

= =

Why?

State variable vector: State variable vector:       =

C L

u i x

SLIDE 70

Modeling approaches for switching converters

70/72

State-Space averaging (SSA): DCM

Example: Example: i iL

L·

·q q1

1

t

iLp

iL

d1Ts d2Ts d3Ts L Uon L Uoff −

t

1 q1 iLq1

L

i

1 L 1 L

q i q i ≠

( )

2 1 Lpk L

d d 2 i i + =

2 1 2 L 2 1 2 L 2 Lpk D L 2

d d 1 q i d d d i d 2 i i i q + = + = = =

2 1 1 L 2 1 1 L 1 Lpk S L 1

d d 1 q i d d d i d 2 i i i q + = + = = = Hp: linear ripple approximation Hp: linear ripple approximation Corrective term Corrective term

SLIDE 71

Modeling approaches for switching converters

71/72

State-Space averaging (SSA): DCM

u B d x M A d x

3 1 k k k 3 1 k k k

        +         =

∑ ∑

= =

3

, 2 , 1 i u d u q u q

C i C i C i

= = = Hp: small ripple approximation Hp: small ripple approximation         + = 1 d d 1 M

2 1

M is the correction matrix M is the correction matrix

( )

u B q x A q t x

3 1 k k k 3 1 k k k

        +         =

∑ ∑

= =

SLIDE 72

Modeling approaches for switching converters

72/72

State-Space averaging (SSA): DCM

If d If d3

3 = 0, i.e. in CCM, we have:

= 0, i.e. in CCM, we have: u B d x M A d x

3 1 k k k 3 1 k k k

        +         =

∑ ∑

= =



       + = 1 d d 1 M

2 1 1

n

L s 2

d u i Lf 2 d − δ =

2 1 3

d d 1 d − − =       = 1 1 M u B d x A d x

2 1 k k k 2 1 k k k

        +         =

∑ ∑

= =