1

Average Model for PWM switch

● Averaging Circuit Equations of PWM Switch

( ) ( ) ( )

( ) ( ) ( )

cp ap cp ap ap

a c a c c

v v q t v v q t v q t

i i q t i i q t i q t

where

● Average Model for PWM Switch

cp ap

a c

( )

( )

v v d t

i i d t

Ave

rag

ing

Po

we

r S

tage

Dyn

am

ics

2

Continuous Duty Ratio

Continuous Duty Ratio

1( ) ( )

s

t

t Tsd t q t dt

T

( )q t

kd

( )d t

( 1) sk T skT ( 1) sk T ( 2) sk T

1

3

1

3

1

0

2

3

2

3

Ave

rag

ing

Po

we

r S

tage

Dyn

am

ics

3

Continuous Duty Ratio

SV 1

Ave

rag

ing

Po

we

r S

tage

Dyn

am

ics

( )d t

( )q t

1

0

0

1

( ) ( )s

t

t Tsd t q t dt

T

4

Buck Converter Example

ci

cp

vap

v

ai

SV

apv

SV

cpvSdT Sd T 0 V SdT Sd T 0 A

ai

Ci

Ave

rag

ing

Po

we

r S

tage

Dyn

am

ics

5

Implication of Average Model

1 : ( )d t

( )q t

( )d t

( )q t

1

0

0

Ave

rag

ing

Po

we

r S

tage

Dyn

am

ics

6

Average Model for Three Basic Converters

SV

SV

SV

Ave

rag

ing

Po

we

r S

tage

Dyn

am

ics

7

Linearization of Nonlinear FunctionL

ine

ariz

ing

Ave

rag

ed

Po

we

r S

tag

e D

ynam

ics

Y

y ( )f xy

X

x

x

y

ˆ

ˆ

ˆ

ˆ

x X

y (x) with x X x with

y: mall signal gain of nonlinear function

x x

x

f

fs

8

Linearization of simple Algebraic Equation

● For simple algebraic nonlinear equations, the small-signal model can be found by

1) Evaluating the nonlinear equation with the variables consisting of the dc and ac components 2) Equating only the ac component of the input and output variables in the resulting expression.

Lin

ear

izin

g A

vera

ge

d P

ow

er

Sta

ge D

yna

mic

s

ˆ ˆ ˆ ˆ

x 2x 2

2 2 2

i) 4x

ii) Y y ( X x ) X 2Xx x

f2x

2

Slope at x 2

● Example: small-signal gain of y=x2 evaluated at x=2

9

Small-signal Model for PWM Switch Model

a a

p p

c cai ci ai ci

ap ˆVd

D 1 : D1 : d

cp

vap

v ˆ

CI d

cp ap

a c

v d v

i d i

a c c

apap ap ap ap( )

ˆ ˆ

ˆ ˆˆ ˆ ˆ

i I d D i

vv v d D Dv v d

D● Small-signal modeling

ˆ ˆ ˆ ˆ ˆ ˆˆ

ˆˆ ˆ

ˆ ˆˆ ˆ

a c a a c c c c c c

cp ap ap cp ap cp

ap ap ap ap

( ) ( )( )

( ) ( )( )

i di I i D d I i DI di Di di

v d v v v D d v v

DV dV Dv dv

line

ariz

ing

Ave

rag

ed

Po

we

r S

tag

e D

ynam

ics

10

ˆapV dD

ˆCI d

Switch model Average model Small-signal model

d

d′

ai ci

apv cpv

( ) ( ) ( )cp apv t q t v t

( ) ( ) ( )a ci t q t i t

( ) ( ) ( )cp apv t d t v t

( ) ( ) ( )a ci t d t i t

1

( ) ( )s

t

t Tsd t q d

T

ˆcpv

ˆapv

cpvapv

ai ciciai 1: d 1:D

Models of PWM Switchlin

ear

izin

g A

vera

ge

d P

ow

er

Sta

ge

Dyn

amic

s

11

SV

1:D

1:D

:1D

Oi

d

SV

SV

d

d

Oi

Oi

Small-signal Model for Three Basic Converterslin

ear

izin

g A

vera

ge

d P

ow

er

Sta

ge

Dyn

amic

s

12

( )Sv s

( )Sv s

( )Sv s

1:D

( )Oi s

( )Oi s

( )Oi s

R

R

R

1

sC

1

sC

1

sC

sL

sL

sL

( )d s

( )d s

( )d s

S-domain Representation of Small-signal Modellin

ear

izin

g A

vera

ge

d P

ow

er

Sta

ge

Dyn

amic

s

13

with

ripple component with

SS S

OO O

v V v

v V v

q

Sv

Ov

SV

OV

Sv

Ov

ˆ| |

ˆ ˆGain response at : Phase response at :ˆ| |O O

s s O SS S

v vv v

v v

Sv

Input-to-Output Frequency ResponseF

requ

enc

y R

esp

onse

of C

on

vert

er

Po

we

r S

tag

e

14

( )d t

Ov

D

OVOv

Gain response at : Phase response at :s s

( )q t

d

1

0

ripple component OO Ov V v

( )q t

SV

Duty ratio-to-Output Frequency ResponseF

requ

enc

y R

esp

onse

of C

on

vert

er

Po

we

r S

tag

e

15

100 1k 10k 100k-30

-20

-10

0

10

20

30

Gai

n [

dB

]

100 1k 10k 100k-120

-100

-80

-60

-40

-20

0

Ph

as

e [

de

g]

Frequency [Hz]

100 1k 10k 100k-30

-20

-10

0

10

20

30

100 1k 10k 100k-120

-100

-80

-60

-40

-20

0

Frequency [Hz]

Bode Plot Representation of Frequency Response F

requ

enc

y R

esp

onse

of C

on

vert

er

Po

we

r S

tag

e

16

( )Sv s ( )Ov s

s-domain Small-signal Model

● Input-to-Output Transfer Function:

● Input-to-Output Frequency Response

Fre

que

ncy

Res

pon

se o

f Co

nve

rte

r P

ow

er

Sta

ge

17

( )Ov s

( )Sv s

1:D

( )Oi sRC

L

( )d s

Li ( )Sv d s

D

| |vdG 2 2

( )( )

( ) 1 / /

O Svd

o o

Vv sG s

d s s Q s

2 2

( )( )

( ) 1 / /

Ovs

S o o

v s DG s

v s s Q s

2

2 2( )

( )( ) 1 / /

Op

O o o

v s sZ s

i s s Q s

| |vsG

| |pZ

Buck Converter ExampleF

requ

enc

y R

esp

onse

of C

on

vert

er

Po

we

r S

tag

e

18

PWMconv

q

sTk sd T

conv

rampv

mvconv

q1

0

rampv

Small-signal Gain of PWM Block

● Control Law

● Approximation

● Linearization

Sm

all-

Sig

nal G

ain

of P

WM

Blo

ck

19

Transfer Function of Voltage Feedback Circuit

Voltage feedback circuit

Power stage

PWME / A

1Z

2Z

conv

refvXR

Ov

Sm

all-

Sig

nal G

ain

of P

WM

Blo

ck

1 2

2

1

( ) 0 0 ( )

( ) ( )

( ) ( )( ) :

( ) ( )Voltage feedback compensation

o con

cv

o

v s v s

Z s Z s

v s Z sF s

v s Z s

20

PWME / A

aX c′ Z

p

Y

1Z

2Z

General Circuit Diagram for Three Basic PWM ConvertersS

ma

ll-S

igna

l Gai

n o

f PW

M B

lock

Buck converter:

Boost converter:

Buck/boost converter:

a X p Y c Z

c X a Y p Z

a X c Y p Z

21

Small-signal Model for Three Basic Converters

ˆsv

X aap ˆVd

D 1 : D

p

Y

Ci

c Z

ˆoi

dmF v ( )F s

Sm

all-

Sig

nal M

ode

ls fo

r P

WM

Dc-

to-D

c C

on

vert

ers

22

Block Diagram Representation

(s)ˆsv

(s)ˆoi

(s)d

vsG (s)

pZ (s)

vdG (s)

mF vF (s)(s)ˆcv

(s)ˆov

Sm

all-

Sig

nal M

ode

ls fo

r P

WM

Dc-

to-D

c C

on

vert

ers