Fundamentals of PWM Dc-to-Dc Power Conversion Transfer Functions of PWM Dc-to-Dc Converters.

Fun

dam

enta

ls o

f PW

M D

c-to

-Dc

Pow

er C

onve

rsio

n

Transfer Functions of PWMDc-to-Dc Converters

2

Graphical Representation of Frequency ResponseG

rap

hic

al R

epr

ese

nta

tion

of F

req

uen

cy R

esp

ons

e

s=js=j0

Re

Im

dB

deg

ω

ω

Polar plotBodeplot

( )( )

( )o

s

v jT j

v j

3

Multiplication FactorsB

ode

Plo

t Co

nstr

uctio

n

dB

20 logK

dB

20 logK

deg

°0

deg

°-180

K 0

ω

ω

ω

ω

K 0

4

Integrations

-20dB/dec or -1

dB

deg

°-90

deg

°-180

( ) IKF s =s

ω

ω

ω

ω

-40dB/dec or -2

dB

2( ) IKF s =

s

Bo

de P

lot C

ons

truc

tion

5

Differentiation

dB

20 dB/dec or +1

dB

deg°90

deg °180

( ) DF s K s

ω

ω

ω

ω

40 dB/dec or +2

2( ) DF s K s

Bo

de P

lot C

ons

truc

tion

6

Single Pole

1 1( ) ( )

1 1p p

F s F js j

dB

deg

ω

°0 ω

Bo

de P

lot C

ons

truc

tion

7

Single Zero

+20dB/dec

deg

ω

ωzω

zω

dB

°0

( ) 1 ( ) 1z z

s jF s F j

Bo

de P

lot C

ons

truc

tion

8

Double Pole

2

2

1( )

1p o o

F ss s

Q

dB

deg

ω

ω

oω

°0

0 dB

Bo

de P

lot C

ons

truc

tion

9

Double Zero

2

2( ) 1

z o o

s sF s

Q

dB

+ 40dB/dec

deg

ω

ωoω

°0

0 dB

Bo

de P

lot C

ons

truc

tion

10

Double Pole

-150

-100

-50

0 -20

-10

0

10

20

Mag

nitu

de[d

B]

Pha

se[d

eg]

Bo

de P

lot C

ons

truc

tion

11

RHP Pole

2

1 1 1( ) ( )

1 1 1

120log | ( ) | 20 log

1

( )

p p p

p

F s F js s j

F j

F j

dB

-20dB/dec

ωpω

deg

ω°0

-3 dB

pωp0.1ω

p10ω

Bo

de P

lot C

ons

truc

tion

12

RHP Zero

2

( ) 1 1 ( ) 1

| ( ) | 20 log 1

( )

z z z

z

s s jF s F j

F j

F j

deg

ω°0

+20dB/dec

ω3 dB

dB

zω

zω

Bo

de P

lot C

ons

truc

tion

13

Example

2 5

1 1( ) 10

1 110 10

T s ss s

0 dB

0.1 1 10 100 1k 10k 100k 1M 10M

0.1 1 10 100 1k 10k 100k 1M 10M

60 dB

100M

100M

°90

0°

°- 90

+1 -1

-1 -1

20log|T(jω)|

T(jω)

Bo

de P

lot C

ons

truc

tion

14

Example

( 10)( )

( 100)( 25)

s sT s

s s

1 10 100 1k

1 10 100 1k

0 dB

90

0

1+

2+

1+

Bo

de P

lot C

ons

truc

tion

15

Examples

°+90

0 dB

1

1

0.1

0.01

°-90

°-90

0 dB

°-90

-1+1 -1

-10.10.01

2

2

2 2

2 2

1800 20050

( ) ( )

1 10.5 4000.1 100

s s

sT s T s

ss s s s

Bo

de P

lot C

ons

truc

tion

16

Non-minimum Phase System

deg

ω°0

deg

°- 90

ω°0

dB

ω0 dB

dB

ωoω

0 dB

zω-2

-1

Regular System Non-minimum Phase System

No

n-m

inim

um P

has

e S

yste

m

The phase variation in the non-minimum phase system is wider than that of the regular system.

17

Identification of Transfer FunctionId

en

tific

atio

n o

f Tra

nsf

er

Fu

nct

ion z

-1

20log mK

+1 1p 2p

18

Identification of Transfer Function

z

-120log hK

+1 1p 2p

1 2

1

( )

1 1

z

p p

s

T s Kss s

Ide

ntif

ica

tion

of T

ran

sfe

r F

un

ctio

n

19

Buck Converter

p

a

cR

CR ov

L Rl

L Rl

C

cRR ˆ

oiˆsv

SV

Li

c

Po

we

r S

tage

Tra

nsf

er

Fun

ctio

ns

20

Small-Signal Transfer FunctionsˆSV d

D 1 : DL Rl

C

cRR ˆ

oiˆ

LI dˆsv

ˆ

ˆ

ˆˆ

ˆˆ

esrvs esr o

esrvd

o

z esrp z

1G (s)

1G (s)

Z (s)

o2

s C2

o o

oS 2

C C2

o o

o2

o2

o o

s1

v R RD

v LC R Rs s1Q

s1

v R RV Q

L CR R R RR Rd s s1Q

s s1 1

vR R

i s s1Q

l

l

l l

l

Bu

ck C

onve

rte

r T

ran

sfe

r F

unct

ions

21

Asymptotic Plot for Frequency Response

esr

o

esr

zesr

| |vdG

| |vsG

|Z |p

2

21

1

11

z o esr

Bu

ck C

onve

rte

r T

ran

sfe

r F

unct

ions

22

Buck Converter Example

0.1 1 10 100-150

-100

-50

0 -80

-60

-40

-20

0

Phas

e [d

eg]

Frequency [kHz]

Mag

nitu

de [

dB]

3.6

3.8

4.0

4.2

12

14

16

18

20

Time [us]V

O(t

)[V

]

VS

(t)[

V]

0 5 10 15 20

3Input voltage: ( ) 16 0.5sin 2 2 10

Output voltage: ( )

S

O

v t t

v t

Input-to-Output Transfer Function Time-Domain Response

Bu

ck C

onve

rte

r E

xam

ple

23

Buck Converter Example

Duty Ratio-to-Output Transfer Function Time-Domain Response

0.1 1 10 100-150

-100

-50

0 -40

-20

0

20

40

Pha

se [

deg]

Frequency [kHz]

Mag

nitu

de [

dB]

0.0 0.5 1.0 1.5 2.03.8

3.9

4.0

4.1

4.20.0

0.2

0.4

0.0

0.5

1.0

v O(t

) [V

]Time [ms]

d(t)

q(t)

3Input voltage: ( ) 0.25 0.5sin 2 5 10

Output voltage: ( )O

d t t

v t

Bu

ck C

onve

rte

r E

xam

ple

24

Boost ConverterB

oos

t Con

vert

er

Tra

nsf

er

Fun

ctio

ns

sv ( s )

1:D

L

CR oi ( s )

lR

cR( )ov s

SV

L lR

cR

CR ov ( t )

Li ( t )

25

Small-Signal Transfer Functions

( )sv s

()

Oˆ

V/Dds

1:D

L

CR ( )oi s

lR

cR( )ov s

( )ˆI d sL

ˆ

ˆ

ˆˆ

ˆˆ

vs o

o

vd

zp

1 1G (s)

( )

G (s)

Z (s)

2esro

esr2s c C

2o o

l

c

esro2

2o o

esro2 2

o2

o o

s1

v RD R1

v D CR LC R Rs s1Q

Q withR 1L C R R

s1

v

d s s1Q

s s1 1

v RR

i D s s1Q

l

l

Bo

ost C

onve

rte

r T

ran

sfe

r F

unct

ions

26

Duty Ratio-to-Output Transfer Function

vdG

vdG

-2

-10

-270

-90

-180

0

-225-180

vdG

vdG

-1

-2

ˆˆ

vdG (s)rhp esro S

2 2

2o o

s s1 1

v V

d D s s1QB

oos

t Con

vert

er

Tra

nsf

er

Fun

ctio

ns

RHP zero at higher frequency RHP zero at lower frequency

27

Duty Ratio-to-Output Transfer FunctionB

oos

t Con

vert

er

Exa

mp

le

0.01 0.1 1 10-300

-200

-100

0

-20

0

20

40

Mag

nitu

de [

dB]

Phas

e [d

eg]

Frequency[kHz]

28

Unit Step Response of Non-minimum Phase SystemR

HP

Ze

ro a

nd

No

n-m

inim

um

Ph

ase

Sys

tem

1 12 2

2 2

2 21 1

2 2

1 1( ) 1

( ) ( ) ( ) ( )( )

1 1

1 1 1 1

11 1

o z zo s

s

o oo o

zo oo o

s sv s

T s v t L v s T s Lv s ss s s s

Q Q

s ss s s sL L sQ Q

t

29

Transient Response of Boost ConverterB

oos

t Con

vert

er

Exa

mp

le

d(t)

initially decreases before the inductor current becomes

sufficiently large.

( )Di t

30

Transient Response of Boost ConverterB

oos

t Con

vert

er

Exa

mp

le

0 5 10 15

0

10

20

30

0

10

20

30

i D(t

)

Time [ms]

v O(t

)

D

initially decreases before the inductor current becomes

sufficiently large.

( )Di t

31

Buck/Boost ConverterB

uck

/Bo

ost

Co

nve

rter

Tra

nsfe

r F

unc

tion

s

( )sv s D :

1

CR ( )oi s

cR

SVL

lR

cR

CR ov ( t )

L

lR

32

Small-Signal Transfer Functions

( )sv s

( )S Oˆ(V V ) / D d s

D :

1

CR ( )oi s

cR

( )ˆI d sL

L

lR

ˆ

ˆ

ˆˆ

ˆˆ

esrvs esr o

vd

z esrp z

1G (s)

G (s)

Z (s)

o2

s c2

o o

e

esro S2 2

2o o

o2 2

o2

o o

s1

v D

v D CRs s1Q

Q with L

s1

v V

d D s s1Q

s s1 1

v R RR

Li D s s1Q

l l

Bu

ck/B

oo

st C

on

vert

er T

rans

fer

Fu

nctio

ns

33

Asymptotic Plot for Frequency ResponseB

uck

/Bo

ost

Co

nve

rter

Tra

nsfe

r F

unc

tion

s

pZz

+1 -1

vdG

vsG

-2

-2

-1

34

Empirical Methods for Small-Signal AnalysisE

mp

iric

al M

eth

ods

Power stage

PWME / A

biasV

( )d t dcV

pvrampV

ov

Two Different Empirical Methods:

35

Duty Ratio-to-Output Transfer Function of Buck ConverterP

ow

er

Sta

ge T

ran

sfe

r F

unct

ion

of B

uck

Co

nve

rter

0.1 1 10

-150

-100

-50

0

-40

-20

0

20

Pha

se [

deg]

Frequency [kHz]

Mag

nitu

de [

dB]

: Small-signal model prediction: Experimental measurement: Computational method