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! NASA CR-135174 TRW A72042-RHBE TRW D0zI803-CFCM

I ! I |

MODELING, ANALYSIS AND DESIGN OF SWITCHING CONVERTERS by

I !

Slobodan Cuk and R.D. Middlebrook

I

Electrical Engineering Dept. California

I

Institute

of Technology

Pasadena, California

i

91125

for NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

I

Lewis Research Center

I

Cleveland, Ohio 44135

I

on

I

Subcontracts A72042-RHBE and D04803-CFCM from

I

TRW Defense and Space Systems to Caltech under prime contracts NAS3-19690 and NAS3-20102 from. .....

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NASA CR-135174 TRW A72042-RHBE TRW D04803-CFCM

MODELING, ANALYSIS AND DESIGN OF SWITCHING CONVERTERS by

SlobodanCuk and R.D. Middlebrook Electrical Engineering Dept. California

Institute

of Technology

Pasadena, California

01125

for

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION LewisResearchCenter Cleveland,Ohio 44135 on

Subcontracts A72042-RHBE and DO4803-CFCMfrom TRW Defense and Space Systems to Caltech under prime contracts NAS3-1%90 and NAS3-20P_2 from NASA to TRW Defense and Space Systems

_-'__'_]-G

_AGE

BLANK

NOT

M'_D

Iv

ABSTRACT The

p_nc_p_

analysis

of

a linear

modal

tLon),

switcJ_ng

to

nLnlinear

power

A gen_al

unified conv_rs

de-to-de

mnverter

(continuo_

or

or

terms

pa_c_, averaging,

Pa_t

of

approach n_"

_;_ tl_e s_udy aonv_rter

_e

advantages

over

efficiency,

the

circuit

the

for

analys._ ccnverter

in

Part

in

equation_

_tate-s_

of

of

conue_t_

the

m_od

ea_h

_a_e,

the

a_hieved

in

represent

new

IV to

any

the

such

provide existing

general

s_ate-space

Ill

IV to

and

study buck

of

to

develop-

convair

and

ra_

than is

shown

in

averaged _e

size

in and

models ci_oai_

conv_t_rs

Th_ switching

.:_duc_ive

have

substantial cla_s

in

w_ight.

_d d_signer as

a n_

its

well

their

corresponding with

a_

of

properties

u_ual

to

design

converters. of

the

conv_r_r_ al_o

boost

modellin_

_he

generic

._Jae discovery

converter

and

topologies.

Parts

basic

_apa_i_iue

state-space

of

resu_

In

mode

In is

the

the

conventional

realizations

_alled

made.

from

lead

of

performance, Both

new

If)

The

which

mode.

ex_n_ion

which

through

upon

eiro_it

any

mod_

state-space

condue_on

is

to

configuration. emerge

in

of

analysi_

mod_l_

_t

way

and

tcr_

appropriate

switching

conduction

technique,

and

d_ailed

transl,.

in

of

applicable

current),

_inear

mode

connection

ba_ed

en_gy

II

topologies

paves

two

con_o_

modelling

I and

c_c_de

of

analysis

circuit

[Par_

either

either

the

PaT_t

the

_

directly

des_rip-

accomplished.

analysis

is

obtain

inherently

is

w_ch

to

circuit

the

_nve_ion

inductor

this to

i_igh_s

conv_er

in

is

linear

fo_

and

and

regulators

or

mod@_in_

model

I

and

dc

_orr_ponding

/n

canonical of

to

_onduction

of

The

of

in

discon_in_ou_

rega_dl_s

tool

their

wh//e

ment

the

discontinuous

applied

_ation

wl_ch

developed

linear

on modelling

restrictions,

operating

of

is

op_on, the

is

'work

state-space

appropriate in

thi_

convert_

ough

approach

dynamic

in

_

stage

de-to-de

of

de-to-de

(_Lt_L_

_ubject

a _inal

objective

for

a powerful s_nthesis

V

TABLE OF CONrENTS Page ACKNOWLEDCC4ENTS

ill

ABSTRACT

iv

INTRODUCTION CHAPTER

l

1,I

1 SWITCHING Physical

DC-TO-DC

CONVERTERS

operation

and basic

switching 1.2

Two operating

Switching output

1.4

modes

I I I

of

and their

dc relations

in the 10

regime

ripple

and pulsation

of input

and 17

currents

Dynamic

response

switching

of a switching

converter; 22

regulators

1.5

Generalized

l .6

Review

switching

dc-to-dc

25

converter

28

GENERAL

UNIFIED

PART

I

CONTINUOUS

2

REVIEW

CHAPTER

properties

6

converters

steady-state 1.3

AND REGULATORS

APPROACH

TO MODELLING

CONDUCTION

Brief

2.2

Proposed

2.3

New

2.4

Extension

CONVERTERS

31 31

MODE

OF THE NEW STATE-SPACE

2.1

SWITCHING

_,_DELLING TECHNIQUE

32

of existing

modelling

techniques

33

new state-space

averaging

approach

33

review

canonical

circuit

to complete

37

model regulator

treatment

3g

vi CHAPTER

3

STATE-SPACE CIRCUIT

AVERAGING,

IIYI_RID_%ODEtLING AND 41

AVERAGING

41

3.1

State-space

3.2

Hybri d _w_delIing

55

3.3

Circuit

60

CHAPTER

4

4.1

averaging

averaging

C_ONICAL

CIRCUIT

Derivation

MODEL

64

of the canonical

model

through

state67

space 4.2

Significance related

of the canonical

MODE

5.1

Modulator

stage modelling

circuit

REGULATOR

6

82

and complete

regulator 84

of switching-mode

Input properties

CHAPTER

MODELLING

model

Analysis

II

and 75

SWITCHING

PART

model

generalizations

5

CHAPTER

circuit

DISCONTII_UOUS

of switching

CONDUCTION

REVIEW

OF THE NEW

IN THE

DISCONTINUOUS

6.1

Brief

review

6.2

New state-space switching

regulator regulators

91

MODELLING

CONDUCTION

TECHNIQUE

MODE

modelling

and circuit

converters

88

MODE

STATE-SPACE

of existing

86

92

techniques

averaging

methods

in the discontinuous

New canonical conduction

for

conduction

mode 6.3

93

94 circuit mode

model

for

discontinuous lO0

vii Page 6.4

CHAPTER

7

Extension

to complete

STATE-SPACE CIRCUIT

AVERAGING,

AVERAGIi4G

7.1

State-space

7.2

Hybrid

regulator

lOl

treatment

IIYBRID MODELLING

IN DISCONTINUOUS

AND

CONDUCTION

MODE

averaging

modelling

105

in the discontinuous

conduction

mode 7.3

134

Circuit

averaging

in the discontinuous

conduction

mode

CHAPTER

8

139

CANONICAL

CIRCUIT

MODEL

FOR DISCONTINUOUS

CONDUCTION

MODE 8.1

148

Derivation

of the canonical

discontinuous 8.2

Summary three

8.3

I I I

CHAPTER

9

common

Determination

MODELLING

I I

9.2

models

for 149

mode circuit

model

results

for

converters

153

of the boundary

between

two 15l

verification

OF SWITCHING

of the transfer

REGULATOR

properties

of

167

switching

regulator

in

discontinuous

mode

Input properties discontinuous

162

IN DISCONTINUOUS

MODE

CONDUCTION

9.! Analysis

circuit

modes

Experimental

conduction

I

conduction

of the canonical

conduction 8.4

105

169 of switching

conduction

mode

regulators

in 171

viii Page GENERAL

PART

CHAPTER

III

lO

THEORY

AND DESIGN

CASCADE

GENERIC

OF BUCK-BOOST

CONNECTION

PROPERTIES

177

CONVERTERS

OF BUCK AND BOOST

OF CASCADE

CONVERTERS

CONNECTIONS

OF

177

178

POWER STAGES I0.1

Three

con_on

I0.2

Buck

converter

10.3

Boost

I0.4

Energy

CHAPTER

11

converters

converter transfer

MODELLING CASCADED

revisited

cascaded

180

by a boost

cascaded

by buck

converter

converter

principles

for general

AND EXPERIMENTAL

VERIFICATION

BOOST-BUCK

11 .I

Model]ing

11.2

Experimental

182 IgO

dc conversion

OF

193

197

CONVERTER

of the boost-buck verification

noninverting of

converter

the modelling

Ig8 206

predictions

PART

CHAPTER

IV

NEW

OPTIMUM

12

DISCOVERY

TOPOLOGY

SWITCHING

OF A NEW OPTIMUM

CONVERTER

TOPOLOGY

213

SWITCHING

214

CONVERTER 12.1

Topological

12.2

Physical

reduction

realization

of number and basic

of switches operation

216

of the

221

converter

224

new converter 12.3

Advantages

12.4

General

12.5

Correlation

of the new optimum

theory

topologies

of buck-boost

among

buck,

boost

topology converters

and new converter

227 230

ix

Page 12.6

Modelling

and experimental

verification

of the

233

new converter

CHAPTER

13

COMPARISON

OF THE NEW CONVERTER

BUCK-BOOST

CONVERTER

]3.1

Experimental

13.2

Switching

13.3

Comparison

test

ripple

237

of the two converters

238

comparison

239

of the transistor

and transistor case

circuits

AND CONVENTIONAL

switching

and diode

losses

dc losses

243

for the idealized

(R_I : R_2 : O)

13.4

Comparison

13.5

Real

of the resistive

transistor

switching

and diode

losses

dc losses

only

247

and transistor

250

{R_I, R_2 # O)

13.6

Con_oarison of ESR Tosses

13.7

Size

13.8

Summary

and weight

dc losses

of the output

capacitance

251

reduction

in the new converter

253

of the advantages

of the new switching

254

converter

CHAPTER

i

14

AND 14.1

I I I

FUTURE

Closed

OF THE NEW SWITCHING

AREAS

Implementation power

14.2

I

IMPLEMENTATION

of

CONVERTER

256

with

257

OF INVESTIGATIONS the new converter

VMOS

transistors loop

new converter

switching

regulator

implementing

the

258

X

Page 14.3

Discontinuous

conduction

14.4

Search

new,

toward

mode

innovative

in the new converter

260

converter

261

topologies

CONCLUSION

263

APPENDICES

267

APPENDIX

A

On the linear

approximation

of the fundamental

271

matrix APPENDIX

B

The fundamental averaging

APPENDIX

C

D

Derivation

of the exact under

exponential

matrices

State-space

averaging

with multistructural changes

in the state-space

275

approach

simplification

APPENDIX

approximation

within

each

dc conditions

]inear

and their

approximation

281

of the

step extended

to converters

(three or more)

topologica]

290

period

CO_:PUTER PROGRAMS

297

REFERENCES

307

,

INTRODUCTION

The ever sources

of energy,

conversion energy, new,

increasing as well

to a medium

has provided

range,

a wide

a variety

battery

power

utility

bulk

However, offered

some

the three

of spacecraft

unmatched

amounts flow

1 ! I I

for example), processing

challenges

voltage

a wide

from one voltage

also

ac,

cover

calculator, solar

process

of the

to be

or multiple-phase

including

array

control

Classical

owing

signal

to the user

and

and electric

electronics,

here,

as

in classical

owing

to the relatively

where power

systems,

large

of handling

amounts

of power

substantial energy)

(such as solar

cells,

efficiency

it becomes

coupled

(or electrical

But in distinction

the power

of

electronics,

electronics,

capable

energy

has.

combination

engineering:

the power

of electrical (load).

to its unique

processing

devices

is used to control

raw source

that this new field

of electrical

of semiconductor

of power,

from some

as no surprise

disciplines

owp___, and control. the advent

include

in a hand-held

to industrial

growth

Functions

The applications

systems

efficient

inversion.

it comes

major

of dc source

of ac to dc.

conditioning, power

systems

abundant

as electr,cal

for the recent

Electronics.

processing

supply

and more

use such

of dc to singlephase

from a power

through

of better

environment

conversion

conversion

spectrum,

for new and more

for widespread

of Power

power

to inversion

and controlled

t

field

from efficient

to another,

with

a healthy

by electronic

of society

as for means

suitable

interdisciplinary

performed

demand

with

signal-

is of minor

the major involved.

concern,

issue, Power

efficiency

makes mandatory

transistors tive

and SCR's

switching

the dynamic inherent power

_de,

conversion

further

nature.

feature

of viewpoints

is most

However, achieve

the bringing

rather

requires

the compcnent electronics in either

a revised

engineer

capacitors;

he avoids

used

tion.

he must

example, 9

a different

From

together

engineer mode

avoid

must

resistors path.

about

point of view, regulator

in order

to

in power

accumulation,

interrelations

but from

a signal-processing devices

resistors

in terms

capacitors,

in the

interest

important

circuit

one has only

used

and

On the other

think with

This

and electronics

of active with

together

way of thinking

switching

their

and transformers.

in the power

the system

a dc-to-dc

mode

particularly

innovation

specific

in terms

electronic

disciplines

For example,

thinks

electronics

ing high effic!ency requires

is not merely

inductors

in the switching

and transformers;

of these

the

fruitful.

and consequent

level.

usually

context

control,

the

because

by employing

potentially

look at their

or switching

a power-processing devices

together

systems

to the system

linear

and also

of their

a requirement

arise

It is in this

understanding

electronics

with

naturally

of the power,

necessary

the general

processing

system.

of modelling

instances,

is coupled

obtained

in a repeti-

because

in many

problems

is usually

f_eedbac___kk in a closed-loop

disciplines

function

such as

the problems

circuits

In addition,

"nversion

devices,

rectifiers)

increasing

switching

and stability

self-correcting

that a fusion

controlled

of power

or

for regulation,

(silicon thus

behavior

nonlinear

the use of control

hand,

of active inductors,

of maintain-

distinction

function

realiza-

to recognize,

as a dc, wide-band,

for

nonlinear

sampled-data control constraint),

system (with the ever-present

high-efficiency

to appreciate the challenge of bringing

together these

various disciplines. Hence, the area of systems, more

owing

to their inherent

challenging

analysis

tools

at the disposal

In connection

already

existing

circuit

topologies

The major

tools

with

circuit

with

purposes,

power

regulators.

simple

analysis

processing:

models

vative

converter

provides

which

the necessary

topologies,

offering

to give

through

better

of new

is to provide

dc-to-dc

insight

i'_

one.

in one of the major

this analysis

in this

values

are accurate

to apply

switchin_

working

as the design

of this work

enough

af} even

of adequate

of parameter

as we|]

tools

processing

becomes

designer

a very difficult

analytical

yet

nature,

the choice

an_ purpose

In addition,

circuit

that,

uf power

in view of the lack

topologies,

for design-oriented

electronic

nonlinear

of the circuit

is likewise

thrust

designer

practical

and analysis

task, particularly

field.

circuit

modelling

the

enough

for

him powerful areas

of

converters

and/or

appropriate

linear

which

may lead to inno-

and aear

optimum

performance. The structure yet

firmly

to modelling

of this work

i}_terconnected

major

has been divisions:

and .analysis of switching

in Parts

I and

II, and design

in Parts

Ill and

insights

gained

l, which

is placed

IV, which from

has been

dc-to-dc

directly

methods

and in front

into

_eneral

of new converter

the analysis

outside

divided

two distinct

unified

approach

converters,

presented

tgpologies, presented made

possible

of Parts of these

I and

by the II.

four parts,

Chapter is

intended to familiarize

the reader with the basic switching conversion

concepts and at the sametime to introduce both the analysis difficulties

as _il

as to designate the possible areas of performance

improvements in switching converter design. The principal

objective

of the work on modelling and analysis

of dc-to-dc converters and regulators linear model (either subject power

through

to appropraite

stage

in which

ters operate "continuous to zero

two modes: mode,"

in Part

in which

a dc-to-dc

converter

represents

both

for the first impedance.

in which

inductor

current

the line and duty

transfer

time, cor,-ectly represents

it represents

any such

converter

referred

to as the

do not fall*

mode,

"_iscontinuous (Part

circuit

functions

of the canonical

II).

m_del

mode which

the converter

regardless

Such conver_

to zero

conduction

ratio

advantage

falls

description),

nonlinear

currents

is a canonical

in the continuous

The principal

mode

I), and a three-state

of this work

circuit

is accomplished.

a two-state

an inductor

The culmination

or linear

is to obtain a

for the inherently

the dc conversion

conduction

conduction,"

state-space

restrictions,

in one of

(as modelled

(Parts I and II)

for

properly

and also,

input model

of its detailed

is that configura-

tion. The corresponding converter which

in the discontinuous

not only

tions become

confirms

first-order,

of the continuous correctly

canonical

conduction

that the line in contrast

conduction

represents

circuit

case,

the input

model mode

and duty

for a dc-to-dc is obtained ratio

transfer

to the second-order

but also

for the first

impedance.

4

mm

in Part

func-

functions time

II,

I | I |

moth canonical c_lled

state-space

which

unifies

considered

distinct

i and Part

properties cade

of a new class

Fina]ly,

stantial

ene)_y

transfer,

advantages

performance

and

over

what

Parts

technique

I and If,

had previously

been

methods.

by the state-space in Part

of buck-boost

in Part

The new converter

obtained

by cas-

IV in the discovery rather is shown

converters

also in size and weight.

of

converters.

upon capacitive

conventional

approach

of the generic

converters

and boost

based

averaging

Ill to the study

this study culminates converter

by a powerfu]

in both

in perspective

of basic buck

a new switching inductive

gained

possible

developed

ana]ytic

11 leads

connection

are made

averagin9

and place=

The insights Part

models

of

than the usuai to have sub..

in efficiency,

CHAPTER SWITCHING

1

DC-TO-DC

CONVERTERS

AND REGULATORS

In this converters

introductory

are introduced

explained.

The basic

conversion,

is arrived

on fundamental

chapter and their

property,

Upon associated

this with

initial

any switching developed

and presented

operation

voltage

dc-to-dc

briefly

and current

simplified

level

arguments

te familiarize

exposure

to the nature these

based

the reader

with

converters

nonlinear

of modelling

(even those

in chapters

of the problem

essentially

and coB_p]ete mthod

dc-to-dc

switching

relationships.

the anal)'sis of

the .aenera], unified,

so_

laws in order

some of the basic quantitative

common

physical

dc-to-dc

at following

physical

several

circuits,

and ana]ysls

of

yet to be invented)

to follow will

be more

easily

grasped.

1.1

Physical

operation

We begin with called

power stages

depicted

in Fig.

is shown,

in Fig.

realization also

6

because While

in Fig.

independent

either

switching power

converters

handling

transistor,

double-throw

turned

on

switch

commutating switch

(also

capability)

l.la the topological

l.lb that transistors fully

of switchin 9 converters

of any particular

1 .Ib a bipolar

from Fig.

mode:

common

of their

of the single-pole

evident

switching

the three

l.l.

of these converters

and basic properties

structure realization diode

S is used.

are used

(corresponding

It is

in their to the position

a}

b) buck

power

stage"

k

V

L

V

R

boost

power

stage"

L

V

L

V

i buck-

tin

boost

_.S_

I

power

stage: V

i_u+

lin

V

_iou+

iP

,R

1.1

Fig.

of switch

! i I

S).

This

shown

Three common switching tic-to-de conve_: a) topological configuration independent tian _ b) bipolar t_a_sistar implementation S in Fig. is obtained

in Fig.

repetition

l.la) or fully by bringing

off (the other

a periodic

signal

is defined purposes

The

period

fraction

is defined

of the

complete

as the steady

state

duty

will

drive

rea//zaswitch

signal

which

ratio

D =TN/T s.

as of

frequency

be considered

Tsfor

S.

of switch

The frequency

as the switching

fs = I/Ts' and for discussion

switch of _he

position

switch

1.2 to the base of the transistor.

of this

of

constant.

the transistor The diode

is on in

each converter acts as a switch automatically transistor. biased diode

That

is, when

and effectively is forced

off;

to conduct

stays on as long as there switch

the transistor

is on, the diode

as soon as the transistor by the continuous

is a positive

1. g

the buck converter With stage

current,

and

current.

time

o_ the now more

power stage because

assumption

off, the

drive

Pe__n

Consider

the

is reverse

becomes

inductor

inductor

I

Fig.

synchronous with

pe,,_Lodic

closely

(sometimes

the simplest called

of its property

and diode

represented

of these

the step-down

of reducing

of ideal transistor

can be equivalently

,smE_teh d_/ve. converters,

or chopper

the input

dc voltage).

switches,

the b -.kpower

as in Fig. 1.3. dc voltage

DTs

f

C

R

O

input

low

voli_je

Fig.

1.3

Basic tJ_rough position.

pass

filter

higher

network

dc

conversion harmonic

_v

order

function decomposition

of

buck and

harmonics

pawer

stage

principle

of

ui_ed 6u_er-

Fourier

harmonic

and the principle sists

of superposition

of a dc voltage

the switching chosen

such

Hence,

that

though

of control

its dependence

varying

the switch

dc voltage.

voltage

of the switching dc voltage

on the duty ratio ratio

is capable

fundamental elements

D.

are smaller

ripple.

can be reduced

of filter converter

has been

elements. is that

a

introduced

Therefore,

one is able

at

are sub-

voltage

ripple

choice

con-

simply

to change

by

the output

0 < D < l, it is apparent

only of reducing

that

the dc input

level.

apparent.

However,

very

important

For a properly

negligible,

sistor

output

voltage

value by proper

drive duty

stage

Another

1

to very small

Also, since by definition

the buck power

filter

voltage

(fc 0.5)

0

D

Vg which

V(l-D)Ts=

that

gain

for the buck-boost

the buck-boost

voltage than

(1.2)

l-D

which

the input

power

is either voltage,

converter.

stage

smaller and hence

is capable (for D < 0.5) realizes

a general dc conversion function. hds been accounted for, efficient

case would

Consider during

load before current

the switching

could happen

if the switching

interval change

increased current

inductor

is thus

in which

inductor

current

ended,

last portion

to release

lowering

operating

the name

becomes

waveform

in Fig.

increased,

and hence

to the output.

of the average where

as shown

originates

This

it has shortened Even

the load resistance

in the so-called

clearly

T_

sufficiently

reduced

but instead

to the output

the inauctor

of the period

energy

1.5b to the point current

100%

in the inductor

released

causing

has been

substantially

to cause

stored

is completely

Tshas

has occurred,

I shown on Fig.

instantaneous

mode,"

necessary

the energy

period

has been

sufficiently

converter

cycle

ideal

= D'/D.

in which

zero for the

or if the inductance

if neither

be lout/fin

intervaITsD l -TsD

to become

the time

the dc current gain in this

now the case

the first

Since none of the lossy elements

R is

inductor

i(O) = i(Ts)= in Fig.

O, the

1.6b.

"discontinuous

The

conduction

from the discontinuous

1.6b.

b)

a) inductor

voltage

VL induclor

current

illJ

Ts

I

® I

I D,Ts

_Ts

DsTs

I _ig.

I.6

I

Steady-slate tion mode.

inductor

wavefor_

in

the

cL66corut_uou_

cond._e-

13

I ,m

................

=

....

II llem ,m

, ,

m

I

I ........ II

II

I

I I

I

I|

I

The immediate conduction

mode

configurations

is that there inside

c3) interval

F/g.

consequence

each

has been

vanishes.

to become

in Fig.

switched

reverse

1.7c

is formed.

topologies

particular

and two modes example

take

mode

on,

converter

place within

as displayed that

converter

on the

zero which

sw , tched

the changes

of operation

network

the

topo,ogy

conduction

mode

each period, three

the two properties -- inductive

in nature.

so far known.

causes

for the last

are among

-- are not

two converters

current

but for different

in Fig. 1.7.

example

but are general

to the other

any switching

D3Ts:

converter mode.: of_, diode

Jtored

for the continuous

to emphasize

for the buck-boost

principle

in Fig. I./.

c) interval

nonconducting

the third

As

changes

conduction

becomes

and hence

interval

It is important

14

DaTs:

network

to the load and inductor

voltage

biased

structural

network

released

the inductor

the discontinuous

not only

period Ts as shown

interval TsD 2 ,the energy

completely

Hence,

topological

above

switched

of_, diode od_.

interval TsD3 , for which shown

different

b) interval

the end of the second

diode

in the discontinuous

Three awitched n_t_orks for the buck-booat opew_uting in the discontin_ou_ conduction a) ,transistor 0% diode off, b) transistor

1.7

inductor

are three

switching

DTs:

c) __tor

At

of operation

shown

energy

restricted They

in Fig.

described transfer

to this

are applicable l.l but also

to

Let us now, modes

however,

of operation

state dc voltage Far,day's

complete

for the buck-boost conversion

law and Fig.

ratio

converter

might

be found

between

example. as before

the two

The steady by use of

1.6a as:

vgDT s+ vD2T=o

or

V

-

Vg However, tinuous

the interval conduction

determined.

This

mode

determines

of

ideal

ratio,

converter.

leads

by finding

based From

in the disconis yet to be

an alternative

upon the 100% efficiency

Fig.

= D2Vg2Ts_2L;then, Pout V 2

how deep

is operating,

can be accomplished

property

and so Pin = Vglin

(1.3)

D2

the converter

for the dc voltage the

D

D2Ts, which

relation

which

the comparison

1.6b,

Iin

= V2/Rs°

= DI

= D"Vgls/2L

from Pin = Pout

V2

to

(I .4)

-V2C D or

°

,,g

where K

Comparison

between

=2Lf R

(1.3) and (1.4) gives

O2 so that

s

the dimensionless

immediately

: vi_

parameter

(1.5) K determines

then

the length of

15

the second interval

interval

This

second ratio

It is interesting

D2 is determined

the second R.

D2Ts.

interval

is dependent

not only

that

tinuous

only by the load

function

of duty

conduction

dependent

from Fig.

_de

comparison

conduction

mode

ratio D only

it is a linear

on the dimensionless between

in which

parameter

(I.2)

the dc gain

function

the two modes

the

and (I.4)

is a highly

in the discon-

of duty

ratio

J but

K (I.4).

of operation

is easily

found

1.6b as:

Furthermore,

a criterion

the converter

is operating

inequality

switching

between

(1.2), while

D3 = 0 :>D 2 = 1 - D _D'

an

converter,

resistance

upon K but also upon the duty

converter,

in the continuous

The boundary

modes

converter

D.

nonlinear

also

for a given

for the buck or the boost

For the buck-boost shows

by K so that,

is a corJstant affected

is not true

interval

solely

to note that the second

relationship

frequency

= VIT

to determine

in which

can be established

among circuit

fs' and duty

ratio

(1.6)

parameter

of the two

in the fcrm of values

D of the switching

L,R drive

as

fol lows: continuous

conduction

mode

D' < vi(

(I7) discontinuous

conduction

mode

D' >V_ where

K = 2L/RTsis

a dimensionless

For instance, in the continuous

16

when

parameter.

K >- l the converter

conduction

mode

regardless

will

aljay__sbe operating

of the control--d_ity

ratio

D,

while

tion

mode To

fs

for

K < l

D < 1 -

illustrate

= lOkHz,

operate

for

the

resistance

is

operate

in

this

example

also

also called

to

justifies

mode"

mode

Now that the two distinct

the transition

clearly

understood between

physical

parameters,

features

inherent

and the

conduc-

light

R and heavy

of operation

load

converter

will

mode

._his is sometimes

loading)

while

mode"

currents

of switching

the physical

of conduction

in the switching

the

and output

and the quantitative

we can proceed

if

always

loading).

distinguished,

two modes

will

to as "light

of input

modes

L = lml4,

D < 0.553.

conduction

is referred

and pulsation

have been

converter

mode for

(low resistance

let

However,

the continuous

R and therefore

their appearance

discontinuous

example,

K = 0,2

conduction

why

Switching. ripple

dc converters

the

mode.

R = I00_i,

conduction

resistance

in

a numerical

conduction

discontinuous

the discontinuous

1.3

with

continuous

"heavy

operate

Then, K = 2 and the

increased

the

will

V_.

and R = ]0_I.

in

(higher

it

to expose converters

measure correlated

dc-to-

origin

of

describing with

circuit

some of the undesirable of Fig.

].] in both

conducti on modes. Consider iou t in Fig.

now both

input

and output

l.l) for the buck-boost

conduction

mode.

Even

though

continuous

conduction

transistor

and diode,both

mode,

converter

the converter

owing

currents

currents

(designated

iin and

in the continuo'Js

is operating

to the switching

action

are as illustrated

in the of the

in Fig.

1.8.

17

nput

b)

currel_t

out

purr

current

DTs

Frg.

I ._

Input and operaJ_cng

o(_tp_C in the

It can easily pulsating

be verified

input current

that an input

filter

that

as shown

(usually

the buck-boost conducT_iOn _de.

the buck

a single-section

two converters

current

at the switching

ripple

component That way,

getlerated by the abrupt are reduced,

and contamination

electromagnetic

which

output

current,

is primarily

ripple

of these

the same

smaller

fs'

duty

voltage

nonpulsating very small

the boost

for the much

current

compared

values

ripple

the

problems

(pul_ating

current)

by the undesired

higher

and operating

(similar

_iou

of Fig.

t which

in Fig. 1.8b,

power

conditions conduction

stage

to that can

l.! has the same

output

to the buck

in the buck power

current

from

(EMI)

converter

ratio D, and continuous

ripple

output

converter

as the buck-boost

responsible

element

drawn

interference flow

L,C filter)

alleviated.

two converters

storage

frequency

hand,

frequency

in energy

requires

out the substantial

of the environment

disturbances

On the other pulsating

variation

has the same

invariably

low-pass

to smooth

electromagnetic

conv_er

converter

in Fig. l.8a. This

be put in front of these

line supply.

18

current of continuous

stage

be

with

(switching mode).

is a consequence

shown

easily

voltage

The of the

in Fig. l.Sa) with found

as

r

V = L D'T

Aiout Consequently,

the output

v_Itage

(I._)

ripple

_v is obtained

,%i out

Av(peak-to-peak)

VD'

(1.9)

_

Bfs C and the relative

output

voltage

ripple

from

8L Cf s Av/V

is:

(1.10)

v

2

\?-_s ]

where f

c Here C.

fc is the corner Since

provide

output ments

the ultimate

of the low-pass

requirement

dc voltage

of filter

ripple will

filter

of the dc-to-dc

and output

on the choice

voltage

2_T_

frequency

dc level change

restriction

I

-

elements.

be negligible

only,

formed

converter

by L and is to

this poses

Namely,

from

if the following

a

(l.!O)

require-

are satisfied:

fc = 112:rVt-C

fc Kcrit

mode

(7.52)

K < Kcrit

two conduction

modes (7.53)

K = Kcr it

where

K, as given

R,and

fs" wbile We now

behave

by

investigate

Kcrit

(7.38),

is a function

Kcrit is a function

throughout

insight,

before

how these

the duty

is plotted

a) open-loop

of the duty

ratio

criteria,

range

b)

Kcrit(D) _ discon_i nuous i

!

Drain

I13

(7.51)

of duty

L,

D only. through

(7.53),

To facilitate

this

ratio D in Fig. 7.3a.

closed--loop

consideralion

Kcrit(M)

co nduct i o n

I/

ratio

D ¢[0, I].

as a function

consideration

of parameters

/

2

M-I

"%P"

/! ",K)

as

{ur

th,tee cc,,_ort.. con.fuct,..,:

cZ,_,scJ-/,.,#p

.t.ions. 154

the _le

IV and the

closed-loop

De(D,K)

buck

buck boost

D De

the dimensionless

consideration

9

boost

columns

7 in Section

in which

M(D,K)

I+_

V

K = 2L/RT s = 2Lfs/R.

open-loop

converter

De

circuit model in Fig. of Fig. I. I operating mode.

the last three

generated,

as before

D +D?

D2R

%

Definition of tlte dc three con_non converters discontinuous conduction

With

V

De

D

D + D2

( D+L)2)R

I

D+D 2

D

V

(____-V) D ,_

I

D

i t iQ.s

td .-;,,I, M,,

i --Ue',.//fl

buck

TABLE

qu_tnt

derived

_IGINA_

aS

PAGE

r_X)O_ QUALITY

I$

,,::.,de

,:c,._ Kcrit

discontinuous

modes

7.1 the criteria

conduction

K = Kcr it

continuous

two conduction

R nora

(M +i)2Rnom

Detcm_ina_ion of modes, expressed loop considera_on_.

the for

boundary open-loop

be_een as well

the as

two for

conducCion closed157

In Table

Vlll nominal

resistance

R

nom is

a design

parJ._ter

defined

(8.11)

Rno m = 2Lf s

It has already

_een

converter

that parameter

converter

is always

regardless

holds

operating

conduction

for a portion

in Section

K can be chosen

of the operating

discontinuous only

demopstrated

mode

that

can occur

of the dynamic

range

7.] for the boost

(K > 4/27),

in the continuous point,

such

that the

conduction

mode

is dc duty ratio

D, while

only for K < 4/27, of duty

true for the other two converters,

ratio

D.

the

and then The same

and the following

criteria

can be set: a)

b)

when

K > KM converter

mode

regarC_ess

when

K < KM discontinuous

only

for limited

Parameter dependent

son purposes

of first

listed

in Table

column

158

Sunm_ of unce:_d_Cional convc_tc_

the of

conduction

of duty

mode

ratio

can occur,

of the duty

in Table

VIII,

ratio

D

and is for compari-

buckboost

boost

4 27

parame_erK_ cont_nuo_ Fig. I.I.

det_Lng conduc_on

for

but

D.

IX.

KM

IX

conduction

the maximum

buck

TABLE

in continuous

of D.

KM is actually

function

is always

range

by

the region of three co_J_on

From

Table

converters mode,

listed

and when

conduction and the duty

first

ratio

a)

for

of

a portion in

result

in

the

Fig.

Fig.

b)

.ooy/ _

/

continuous

8.5b

begin

(see for comparison

Tines

mode

to follow

Fig. 7.4).

and the buck-boost modes

for

occurs

three

conduction the

discontinuous

range.

With

gain

as a function

K < 4/27,

conduction

while

mode is

this, of

the

illustrated

K > 1.

discontinuous

conduction

77 _- buck-b°os*

only

o._

uoltage _or the

whereas takes

over

those From

once

the region dotted

of

in the Fig. I.I.

of actual

lines

signify

disthat

and the dc gain

for the continuous

Fig,

converter,

,.o _

gain characteristics common converters

designate

operation,

conduction

characteristics

for

in

the

I .oo,, ./"

the de mod_

heavy

conduction

the continuous

ratio

,.o o oo

Co_arison of _o conduction

In Fig.

operate

8.5b

of

continuous

dc vcltage

8.5a

conduction

8.5

the

duty

continuous

purposes

o.,

in

the

V,

in

when K _ 1 any

them will

of

Table

for

that

operate

each

can be shown as

o.o

conduction

obvious

always

column

comparison

the buck

is

K < 4/27

continuous

Fig.

it

will

mode

corresponding for

IX

8.5b

it is also

the transition

at higher

conduction

duty

ratio

evident

between

mode

that

in

the two

D, and not

1,59

also at the lower end as it does dur'ing initial changes

sfart-up

from zero

of the converter,

to the value

the two conve"ters

in the boo,st converter.

required

when

conduction

mode

ratio

by the steady-state

(buck and buck-boost)

the discontinuous

the duty

Therefore,

gain M,

can be desi_Incd( " to stay

only,

even

in

in this tr_nsitional

period. We now present pictorial

way

and a unique

determination

three

S positioned

elements

natural repeated

frequencies, here

have

three

type •

and the basic

from Fig.

Two of been

of the single

voltage R.

defined

Vg and three

With only

frequencies them, _

sma11-

it is apparent

consist

C, and load

previously

l.l

the source

"inherent"

illuT_inat.__sthe

can be defined

and f c' termed (I.II)

and are

l

another

elements

f '

I

c

"inherent"

(8.12)

2_

frequency

_B can be defined

by these

as R

(8.13)

: 2--C The

dimensionsless

determination

these

for completeness:

2RC

yet

mode

essentially among

different

of the converter

-

However,

converters

in an interesting

interpretation,

Namely,

L, capacitance

three

which

operatfng

differently

inductance

regardless

frequency

requirement.

that the three common

elements,

viewpoint,

of the converter

switching-ripple

switch

another

parameter

K, which

of the conduction

mode,

plays

a cruci_l

role

can now be expressed

in the as

f K =s

160

(8.14)

I

4 =2R c AI

small l

£_I__ S-Ts

'•

switching

high

I

R 2L

,

ripple

ripple

l

_ !

I

I

I

I

I

fsI

I 0 kHz

I k Hz

8.6

Fig.

the position

switching

frequency

=13 < is" each

conduction that _ The

mode

frequencies

the help

fs determines of the three

regardless

contained

summarized

of definitions

the interplay

requirement increase without

converters Also

will

it was

in the position

(8.12)

frequency

can occur,

and

conduction

in switching

but at the price

always shown

of these

before

types,

change

of higher

ripple.

"inherent" frequency

fs

8.6, with

in a convincing switching

ripple For example,

conduction

mode

if inductance

to discontinuous switching

for K > 1

(I.11)

L, C, R and is"

However,

to the

switching

in Fig.

displays

wad

be in continuous

three

to discontinuous ripple.

is reduced,

Hence

for small

mode

values

change

mode.

The diagram

(8.13),

of parameter

respect

to the switching

in Fig. 8.6.

of load R can cause deterioration

respect

mode ¢4pe

=B with

the conduction

of D.

between

and choice

or switching mode

o_ .the conclue_o, requ_eme_.

of this new frequency

_ , _B and fc with

is concisely

[

___