5. dc-dc converters: static...

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5. DC-DC Converters: Static Characteristics

Power Electronic Systems & Chips Lab., NCTU, Taiwan

Power Electronic Systems & Chips Lab.

~

2/252

Contents

1. Introduction2. Definition of DC-DC Converter3. Buck (Step-Down) Converter4. Boost (Step-Up) Converter5. Buck-Boost Converter6. Switch Stress and Switch Utilization 7. Simulation Study of a Buck Converter8. Synchronous Buck Converter9. Summary of Basic DC-DC Converters10.Topologies, Modeling, and Control

3/252

Introduction



~

Pulse-Width Modulated DC-DC Power Converters, 2nd Ed., Marian K. Kazimierczuk, Wiley, 2015.

4/252

The Goal: Synchronous Buck Converter

Static Characteristics Component Determination Modeling & Frequency Responses Control Loop Design

5/252

DC-DC Voltage Regulators

AC line voltage

(1-phase or 3 phase)

Uncontrolleddiode

rectifier filter

capacitor

(unregulated)

DC

(unregulated)

DC DC-DC converter

(regulated)

DCload

battery

A DC-DC converter system.

unregulated dc input regulated dc output

DC-DC converters are the most widely used power converters!

6/252

Functional Block Diagram of a Switching Power Supply

COMP

REF

Line input AC

PFC converterand filter

PWM Controller

Highfrequencyinverter

20-200 KHz Output DC

output rectifierand filter

LoadSource

120 Hz

Feedback Sensing,

Reference, and Isolator

PFC Controller

Input EMI filter

Output EMI filterDC-DC converter

A power supply is a power conversion and control processor.

PWM

OSC

7/252

Two-Stage AC/DC & DC/DC Converter

PFC ConverterDC-DC Converter

Q1

PFC Controller

CboostVLINE

85-260VAC

Vout-3

Vout-2

Vout-1

PWM Controller: Primary & Secondary

CRM PFC IC CCM PFC IC Voltage Mode IC Current Mode IC

8/252

Typical Block Diagram of an ATX Power Supply

PFCcontroller

PFC Diode

SMPScontroller

SMPSregulator

Biasoutput

MOSFET

Output circuitry

12 Vout5 Vout3.3 Vout

Postregulator

Outputrectification

Supe

rvis

ory

EMIfilter

9/252

Power Conversion, Control, and Management

Power Conversion, Control, and Management

AC/DC Battery Charger DC/DCDC/DC

ApplicationsSMPSMonitor / CTVNotebookPC, ServerLamp ballast

Portable ApplicationsNotebookCell PhonePDA

ApplicationsMotherboardNotebookPower Supplies / VRMTelecom

DC

AC85

26

5V

PFC Controller PWM Controller

DC/DCController

SMPS AC/DC

BatteryCharger

DC/DC Converter

DC/DCController

IC

10/252

DC/DC Converters for Mobile Phones

Battery Charger

LDO

Display

Audio

Vibrator

P/DSPcore

D/A

A/DI/O

Antenna

2.5V 2.5V

2.7-5.5V

3.6V 2.5V1.5V

Baseband digital

Power distribution: Vg = 2.85.5V

1-3.6V

Analog/RF

LO

2.5V

Switchingregulators

PA

LNA

LDO

DC-DC

DC-DC LDO

DC-DCDC-DC

LDO

3.6V

DC-DC

REF: Frank De Stasi & Mathem Jacob, Magnetic Buck Converters for Portable Applications, National Semiconductor.

11/252

Low-Power Low-Voltage Power Supplies

Good for the IC, bad for the power supply!

year

5V

3.3V1.5V 0.8V

year

samefunctionality

Increasedfunctionality

ICpowersupplybatV

ccV

cci

ccV cci

12/252

Battery-Based Power Converters for Portable IA

Vo = 1.2 V (+/- 2%)Io = 1 mA (idle)

500 mA (on)

LithiumIon Battery2.8-4.5V1000mAh

SwitchingRegulator

-ProcessorCharger

Battery Protection IC

dcV

dcI sI oI

oVsV

gI

gV

13/252

Linear Voltage Regulator: Basic Principle

RO VRRRV

2

21

Efficiency

Output Impedance

Efficiency Analysis ( = Vout/Vin) Loop Gain of Error Amp for Output Impedance

OCELoss IVP CEL

L

RRR

INVCEV O

I

OV

1R

2R

LRCC

RV

14/252

The Classical Linear Regulator TL431

CATHODE

REF

ANODE

SymbolPackage

Anode Cathode

REF

REF

Cathode

Anode

2.4k

7.2k

3.28k

800

1k

4k

800 800

150

10k

20pF

20pF

TL431 = Reference + OP Amp. + Driver

2.5V REF

15/252

TL431: Circuit Schematics and Device Model

(a)

(b)

(C) TL431 OPEN-LOOP VOLTAGE GAIN VERSUS FREQUENCY

16/252

State of the Art TL431: Schematics and IC Layout

11 x Tr. = Reference + OP + Driver

SymbolPackage

Anode Cathode

REF

17/252

A High Efficiency Step-Down Switching Converter

REF: TL431, A, B Series, NCV431A Programmable Precision References (datasheet, On-Semi)

TL4312.5V REF

Cathode (K)Reference (R)

Anode (A)

2200F

1.0k

4.7k

0.1F2.2k

4.7k 4.7k

TIP115150F @2.0A

0.01F 100k

470F

51k

1N5823

NPSA20

VIN = 10~20V VOUT = 5.0VIOUT = 1.0A

TO-92 (TO-226)LP SUFFIX

case 29

Pin 1. Reference2. Anode3. Cathode

12 3

vo

18/252

A Historic Review of PWM Controller IC

[1] Gene Heftman, PWM - From a Single Chip To a Giant Industry, Power Electronics Technology, October 2005. [2] Bob Mammano, Are We There Yet - power control integration, APEC 2007.

19/252

The First PWM IC: SG1524, Bob Mammano, 1976.

SG1524: -55C ~ 125CSG2524: -25C ~ 85CSG3524: 0C ~ 70C

Bob Mammano is staff technologist and a TI Fellowin Texas Instruments' Power Management Productsgroup. He has more than 50 years of experience inanalog power control technology and is widelyrecognized as the father of the PWM IC industry.Holder of 16 patents in this field, Bob hasparticipated in new product definition, technicalmarketing, and has been a significant part ofUnitrodes and TIs Power Supply Seminarprograms since 1981. He has a degree in physicsfrom the University of Colorado.

20/252

The First Current Mode Control PWM IC: UC1842, Unitrode, 1980s (Bob Mammano)

The First PWM Device, Robert Mammano, Staff Technologist and TI Fellow, Texas Instruments Incorporated - March 1, 2007.

RESETLatch

Output(PWM)

Clock (SET)

Li

eV

sV

Off Line Flyback Regulator Using UC3844

22/252

Power Supplies: Efficiency, Size, Dynamic Response

Topologies

Thermal Management

HarmonicsControlLoss Analysis

EMC Design

SoftSwitching

Reliable, Size, Cost, EasyPackagingDynamicResponse

ControlArchitecture

Control Design

Control IC

PowerManagement

Efficiency Control

23/252

Definition of DC-DC Converter

DC-DC converter is the Gate Way to all other power converters!

~

24/252

Basic Power Converters

DC-AC Converter

DC-DC Converter

AC-AC Converter

AC-DC Converter

25/252

Definition of DC-DC Converter

DC-DC Converter (Chopper)

A dc-to-dc converter is any network that can have as its sole source of energy aconstant dc voltage VIN or a constant dc current IIN and can provide dc outputpower such that VOUT > VIN or IOUT > IIN.

VOUT, IOUTVIN, IIN

E. T. Moore and T. G. Wilson, Basic considerations for dc to dc conversion networks, IEEE Trans. Magn., vol. MAG-2, pp. 620624, Sept. 1966.

According to this definition, A Linear Regulator is NOT A DC-DC Converter!

26/252

Converter Topology

The Issue:

A topology is the arrangement of the power devices and their magnetic elements.Each topology has its own merits within certain applications. Some of the factorswhich determine the suitability of a particular topology to a certain application, suchas isolation, power ratings, component stress, number of output required, utilizationfactor, etc.

vovg

d

27/252

Development of Basic DC-DC Converters

vovg

The Problem:

Configure these four basic elements to devise a dc-dc voltage converter!

d

~

28/252

Two Basic Energy Switching Architectures

Switching Inductor Converter

vovgThe switching inductor as a switching current source!

ovvg

Switching Capacitor Converter

The switching capacitor as a switching voltage source!

~

29/252

Basic DC-DC Converters

Buck Boost Buck-Boost

30/252

Intrinsic Characteristics of Basic DC-DC Converters

vi voBuck

Boost

Buck-Boost vovi

vovi

Switching Inductor

The inductor current must maintain its continuity!

The direction of the inductor current flow can not be changed!

The behavior of the inductor current determines the operating modes of the converter.

The average inductor current is the effective current!

dcV

dcV

dcV

C

L

CL

CL

31/252

Common One-Switch Power Converter Topologies

Buck

Boost

Buck-Boost

Non-Isolated Single-Ended Single-Switch Converter

vo

vo

vo

vi

vi

vi

ControlCircuit

T1

TR1 resetting winding

n : n : 1

Forward Converter

n : 1 oi

Vin

Flyback Converter

vo

vo

C RmL

C

LmL 1D3D

2D

32/252

Basic Topologies of PWM DC-DC Converters

Buck

Boost

Buck-Boost vovi

vovi

vi vo

One Inductor, One CapacitorL

CD

L

C

D

L C

D

C,uk

SEPIC

Zeta

SEPIC: Single-Ended Primary Inductor Converter

Two Inductors, Two Capacitors

vi

vi

vi

vo

vo

vo

1C

2CD

1L 2L

1L

2L

1L

2L

1C

2C

1C

2C

33/252

Switches in the Thee Basic PWM DC-DC Converters

vi vo

Buck Converter

Boost Converter

Buck-boost Converter

vovi

vovi

The switches must keep the continuity of the inductor current!

The buck-boost converter has an inverting output!

L

D C

L D

C

L

D

C

34/252

Basic Circuit Concept

This is not a workable circuit, unless V1 = V2.

This is not a workable circuit.

This is not a workable circuit. 1I

1I 2I

1V 2V

1V 1I This is not a workable circuit.

35/252

Basic Circuit Concept

This is a workable circuit.

This is a workable circuit. 1V 1I

1V



What is the common rule for the judgement?

36/252

At High Freq., The Inductor as A Current Source

The inductor as a current source and the capacitor as a voltage source!

Buck vi vo

Boost vovi

Buck-Boost vovi

L

D C

L D

C

L

D

C

37/252

Switching Energy Transfer in a Cuk Converter

A switching capacitor converter (The CCM and DCM operation is determined by the continuity of the capacitor voltage)

Low input and low output current ripple Optimal DC-DC converter with ripple current free: if the input and

output inductor can be coupled to eliminate the input and output current ripples

C,uk Converter

L1

C2D

L2C1

Svi vo

~

38/252

Control of Basic PWM DC-DC Converters

PWMModulator

LoopCompensator

vg

vo

vR

Efficiency

Boost Converter Buck/Boost Converter Buck Converter

load

RL di~

Switching power converters

Output Impedance

GateDrive

osZ

sv~

sV

Current Injection Method

39/252

Buck Converters

Everett Rogers, Understanding Buck Power Stages in SPS, TI 1999 (slva057)



~

40/252

Step-Down (Buck) Converter

Step-Down (Buck) Converter Spectrum of the PWM Waveforms Switching Current Ripple Analysis Operating Modes of a DC-DC Converter Static Characteristics in CCM Static Characteristics in DCM Output Voltage Ripple Analysis Buck Derivative Converters

~

41/252


1

2

Low-pass LC filter

t0

0 DTs Ts

= DVbat

position 1 position 2

fs = 1/Ts = switching frequency

Characteristics of the Buck Converter:

Larger input current ripples due to the input switch Smaller output voltage ripples due to the output inductor Induce large spiky switch current when duties approach zero

)(tvo

L

C R

)(tiC

)(tiL

)(tvsw

)(tvsw )(tVbat

)(tVbat

)(tvL

( )oi t

42/252


on

on

on0 0

1 1( ) ( 0 )s sT T T

o o dc dc dcTs s s

TV v t dt V dt dt V DVT T T

control controldc

ost

VV v kvV

constantdc

st

VkV

Ts

t0

low-pass filterid

The Simplest DC-DC Converter!

dcV

oi

ov

dcV

oVci

Li

Lvoiv

oiv

offTonT

RC

L

43/252

Buck Converter Ideal Static Characteristic

00

position 1 position 2

00 1

Switch duty ratio

Ideal conversion ratio:

Ideal efficiency:

( ) odc

VM D DV

%100

( )swv t

dcV

oV

D

dcV

sDT sT t

sw dcv DV

44/252

Buck Converter

RC

RC

o iV DV ( )o L AVGI I

Waveforms in Continuous Inductor Current

IQ1 = ia

ICR1 = iP

IL SolidiO Dashed

VC-P SolidVO Dashed

TON TOFF

TS

IL

L

ONState

RL

a c

p

R

Cia RDS(on)

RC

VO

VI

OFFState

RL

a cL

p

R

Cia

RC

VO

VIVd

RL

a cL

p

IL(AVG) = Ioia

VOd s

g CR1

Q1

VI

DriveCircuit

45/252

Step Response of a Buck Converter

http://www.wolfram.com/mathematica/new-in-9/advanced-hybrid-and-differential-algebraic-equations/dc-dc-buck-converter.html

46/252

Pulsewidth Modulator in a Buck Converter

ramp voltage

DT

PWM output

The modulating signal vm compares with the carrier signal vC to generate a pulse width controlled digital vd.

The PWM modulation process can be of the following types:

constant frequency switchingfixed ON-time switchingfixed OFF-time switchingNonlinear carrier PWM/PFM with Skip Cycle PWM with Doube-Edge

2A1A vref

ccv

1R

2R3R

4R

mv

cv cci

ov

dv

iv loadR

LC

mvcv

47/252

Pulse-Width Modulator

Amp comparator

repetitive waveform

switch control signal

vo (desired)

vo (actual)

ton toffTs

on on

off off

switch control signal

stV

(switch frequency fs = 1/Ts)

sts Vv

TtD

controlon

The carrier signal may be a nonlinear function to produce nonlinear PWM control signal.

Modulating signal

Carrier signal

saw-tooth voltage (amplified error)

controlv

controlvstv

stvv control

stvv control

Trailing-Edge PWM

Three Types of PWM Signals

Leading-Edge PWM

Central PWM (usually used in sine-wave inverters)

49/252

RS Flip-Flop in the PWM Modulator

2A1A

clock

R

S

Q

Gate Driver

An RS flip-flop circuit must be added in a practical PWM IC to ensure there is only one state change of its PWM output during a switching period.

vref

ccv

1R

2R3R

4R

cci

ov

iv loadR

LC

cv

mv

dv

50/252

Pulse-Width Modulator in a Voltage Mode Controller

51/252

Pulse-Width Modulator in a Current Mode Controller

OSC

OUT

FB

ISENSE

Error Amp

VREF

Output Driver

Current Comp

Max. Duty limit

SR Latch

Set

Reset

Q

Clock

Osc.

52/252

Buck Converter: Frequency Response of Output Filter

ton toff

Frequency spectrum of voiVo

t0

0 f

VfsV2fs V3fs

fs= 1/Ts 2fs 3fs0

-40 dB

-80 dB

fo 10 fo 100 fo fs

oi

o

vv

10log20

f(log scale)

gain attenuationby the filter

LCfo 2

1

low-pass filter

Ts= 1/fs

Frequency response of the LCR circuit.

L

C Roo Vv dcV

dcV

oiv

oiv

oV

Lv

Li oi

di

53/252

Switching Frequency vs. Resonant Frequency

LC Tank LCf o 2

1

Vo

t0

C Roo Vv dcV

Lv

Lioi

L

dcV

xv

xv

Si

1 /s sT fonT offT

54/252

Switching Frequency vs. Resonant Frequency

fs= 1/Ts(c) PWM Switching

vsvo(c)

vs vo(a)

(a) Low Frequency BehaviorTs

vs vo(b)

(b) Resonant Switching

0

-40 dB

-80 dB

f

s

o

vv

10log20LC

f2

10

-40dB/decade

0f 0100 f010 f

ov loadRL

Csv

55/252

Spectrum of the PWM Waveforms

ton toff

Vot0

do DVAV 0

) cos sin(

3cos 2cos cos 3sin 2sin sin) (

10

321

3210

nnn tnBtnAA

tBtBtBtAtAtAAtf

T

dttfT

A00

)(1 T

n dttntfTA

0 sin)(2

T

n dttntfTB

0 cos)(2

where n = 1, 2, 3, (all positive integers).

2/121211 BAV

2/122 nnn BAV ... ... ...

Ts

dcVoiv

56/252

The n-th Harmonics

Frequency spectrum of voi

0 f

Vfs

V2fsV3fs

fs 2fs 3fs

2cos11| cos1

sin2 sin2 sin)(2

20

000

nDn

Vtnn

V

tdtnVT

dttnVT

dttntfT

A

dDtd

DT

d

DT

d

T

n

2sin1| sin1

cos2 cos2 cos)(2

20

000

nDn

Vtnn

V

tdtnVT

dttnVT

dttntfT

B

dDtd

DT

d

DT

d

T

n

2cos121)2(sin2cos112/1222/122 nD

nVnDnD

nVBAV ddnnn

2cos121 nDn

VV dn 1

ss

fT

,o nV

57/252

The 1-st Harmonics

1 2 1 cos 2dc dcV VV D

Frequency spectrum of voi

0 f

V2fsV3fs

fs 2fs 3fs(= 1/Ts)

D (%)1000 50

2

1 2dcVV

When the duty ratio is D=50%, the 1st harmonics got its maximum value:

2cos12 D

2cos12 D

,o nV

fsV

58/252

Switching Ripple Voltage Analysis

LCRC

ss

LCRsLRLCs

R

RsC

sL

RsC

svsvsH

oi

o

/11/1

//1

//1

)()()(

22

L

C

low-pass filter

Rt0

t

t

22

2

2 2/11/1

)()()(

oo

o

oi

o

ssLCRC

ss

LCsvsvsH

LCo1

CL

R21

1v

222

11 2)(2cos12)(

osos

ods jj

DVjHVv

oivoiv

ovdcV

59/252

Switching Ripple Voltage Analysis

2

1 1 2 2( ) 2 1 cos 2 ( ) 2dc o

ss o s o

Vv V H j Dj j

In general, s >> o, then

2

1 22

12 1 cos 21 (2 )

dc o

s o

s

Vv D

2

1 22 1 cos 2dc o

s

Vv D

If RC

61/252

DC Transformer

od PP dc dc o oV I V Io dc

dc o

V I DV I

The L, C, Q and D are all ideal components, then no loss, therefore:

The step-down converter is equivalent to a dc transformer.

low-pass filter

Lv

Li oi

Si

L

C RovCidcV

oiviC

Cii

dci

62/252

Current Waveforms in CCM

0t

0t

0t

inductor current

switch current

diode current

(max)LI

(min)LI oL Ii iC

low-pass filter

o dc

dc o

V I DV I

LC ov RLv

Li oi

dci

CidcV

Si

Di

Li

Si

Di

sDT

sT

1Ci

1C

63/252

Inductor Current Ripple: A key in selection of inductor

t0

tIL = Io

( )

( )

(1 )

dc o onL

dc o s

dc s

V V TIL

V V DTL

V D DTL

LI

(1 )dc sL

V D DTIL

(max) 4dc s

LV TI

L

0 0.5 1.0

LI

Li

Lvodc VV

oVsT

onT offT

D

64/252

Boundary Between Continuous and Discontinuous Conduction

t

Ts

0

Current at the boundary of continuous-discontinuous conduction: (a) current waveform; (b) ILB versus D keeping Vdc constant.

on,peak

1 ( ) ( )2 2 2

sLB L dc o dc o oB

T DTI i V V V V IL L

At the CCM and DCM boundary condition: peak,21

LLB iI

(a) (b)

D0 0.5 1.0

LTVI sdcLB 8max,

Lv

Li

constant dcV

LB oBI Idc oV V

oV

LBI

offTonT

peak ,Li

65/252

CCM & DCM Boundary

If the average output current becomes less than ILB, then the buck converter willoperate in the DCM mode!

For a constant output voltage Vo, it output power is: ooo IVP For a specified output power Po , the rated output current is:

o

oo V

PI

D0 0.5 1.0

LTVI sdcLB 8max,

oIoP

100%

50%

maxDminD

dc

o

VVD nominalLB oBI I

66/252

Operating Region of the Buck Converter with Load Variations

D0 0.5 1.0

LTVI sdcLB 8max,

oI

oP

100%

50%

maxDminD

dc

o

VVD nominal

The inductor current as a function of load The converter may be designed to operate in either CCM, DCM, CCM/DCM,

or boundary conduction mode according to the selection of inductor and control scheme.

low-pass filter

LC ov RLv

Li oi

di

CidcVSi

Di

oBLB II

67/252

Output Voltage Ripple: selection of capacitor

0 t

0

0

t

t

2sT

PIQ

Output voltage ripple in a step-down converter.

22211 sL

oTI

CCQV

2221 sL TIQ

so

L TDLVI )1(

sos

o TDLV

CTV )1(8

222

))(1(2

)1(81

s

cs

o

o

ffD

LCDT

VV

LCfc 2

1

ss T

f 1

cs ff In general,

LI

LP I21I

Lv

pV

Li

ov

dc oV V

oV

oV

oL II

oV

68/252

Output Voltage Ripple Analysis

%))(1(2

(%)100 22

s

c

o

o

ffD

VVIf output voltage ripple is required to kept below %, then

nff

c

s

D

n

7.155

0.5 1.00

1%

0.5%%)1)(1(

22

2

n

D

Define

100)1(2

22 Dn

2110 Dn

2.2207.7

To reduce the switching ripple, it is required toincrease the switching frequency.The voltage ripple is independent of the load (whenfs >> fc).The voltage ripple is the same wither in CCM orDCM.

69/252

CCM & DCM

Major source of static nonlinearity!

~

70/252

Operating Modes of a DC-DC Converter

CCM: Continuous Conduction Mode

DCM: Discontinuous Conduction Mode

According to the inductor current, a dc-dc converter operates in two modes:

low-pass filter

Lv

Li oi

Si

L

C RovCidcV

oiv

71/252

Continuous-Conduction Mode

0

Step-down converter circuit states (assuming iL flows continuously).

on

on0 00s s

T T T

L L LTv dt v dt v dt

on on( ) ( )dc o o sV V T V T T

on (duty ratio)odc s

V T DV T

dcV

odc VV

oV

oVoVL

C RL

C R

oL II

sT

Li Li

Li

Lv

LI

offTonTt

t

72/252

Buck Converter in Discontinuous Conduction Mode

Li

Li

Lv

oV

odc VV

dcV C R

L

t

t

oI

peak ,Li

sdT2(1 ) sd d T

2 sd T

73/252

Current Ripples in CCM and DCM

LI(max) 4

dc sL

V TIL

0 0.5 1.0

,max 8dc s

LBV TI

L

Li

(1 )2s dc

LBT VI D D

L

)1(4 max, DDII LBLB Average current

D

74/252

Buck Converter in DCM

0

0

1

1 1

1

s

s s

A s

T

o o

DT T

dc oD T

dc A o

V v dtT

V dt V dtT TD V D V

dcA

o VDDV

Adc

oV D

DVVG

t

t

t

odc VV Lv

sv

dcV

LioV

sDT

sDT

sT

sT

A sD T

peakI

sDT sTA sD T

L

C

low-pass filter

dcVsi

Di

Li

Ci

oi

ovLv

75/252

Average Output Voltage in DCM

1Ts 2TsTs

Vo

0 t

Discontinuous conduction in step-down converter.

1( ) ( ) 0dc o s o sV V DT V T

1

odc

V DV D

so

L TLVi 1peak,

2

2,max

1 ( / )4

o

dco LB

V DV D I I

DI

ILB max,

o1 4

,peak 11

,peak

1 1

1 ,max 1

1 ( )T D2T 2

( )2

42

L

o L

o s

dc sLB

i DI i

V T DL

V T D I DL

odc VV

LvLi

peak ,Li

oL II

sDT

76/252

Voltage Gain at CCM and DCM Modes

~

2

2,max

1 ( / )4

o

dco LB

V DV D I I

o

dc

V DV

CCM

DCM

,max 8dc s

LBV TI

L

How to interpret its physical meaning?

low-pass filter

LC ov RLv

Li oi

CidcVSi

Di

77/252

DC Voltage Gain in CCM and DCM

o

dc

V MV

Vdc = constant

max,LB

oDCM I

I

Step-down converter characteristics keeping Vdc constant.

0 0.5 1.0 1.5 2.0

0

0.25

0.50

0.75

1.0

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

2

2,max

1 ( / )4

o

dco LB

V DV D I I

o

dc

V DV

The buck converter has a tendency to increase its voltage gain when operating in DCM.

, max 8s dc

LBTVI

L

( )2

sLB dc o

DTI V VL

Normalization Base

DCM Factor

Voltage Conversion Ratio

78/252

Buck Converter Characteristics in Keeping Vo Constant

)1(2

DLVTI osLB

LVTI osLB 2max,

max,)1( LBLB IDI

,max 1/2/( )1 /

o LBo

dc o dc

I IVDV V V

)(max,LB

o

II

0 0.5 1.0 1.25

0

0.25

0.50

0.75

1.0D

0.25 0.75

1.25dco

VV

2.0dco

VV

1 5.0dco

VV M

Step-down converter characteristics keeping Vo.

If Vo is kept constant, the maximumvalue ILB occurs at D=0:

, max 8s dc

LBTVI

L

Vdc = constant

DCM

CCM

What we can observe from the static characteristics curves of a switching converter?

0 0.5 1.0 1.5 2.0

0

0.25

0.50

0.75

1.0

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

M

DCM

The -M plane is a map for the study the static and dynamic characteristicsof a switching converter.

Boundary curve in the -M plane Locate the OPERATION REGION of the target converter

OPA

What are the differences betweenD=0.9 and D=0.1?

What is the Q-factor at OPA?

80/252

What is the Effect of Filter Inductance?

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40%20% 110%90%

Decreasing InductanceIncreasing Inductance

low-pass filter

LC ov RLv

Li oi

CidcVSi

Di

81/252

Operating Point & Area

Where is the operating point?Where the OP is going to?And, how to go to there?

~

82/252

Buck Converter: Constant Input Voltage, Variable Duty Cycle and Variable Load

http://www.ipes.ethz.ch/ipes/dcdc/e_Buck_1.html

Analysis of the operating point is the first step in the analysis and design of a dc-dc converter!

83/252

Controlled (Constant) Output Voltage, Variable Voltage Transfer Ratio (Input Voltage) / Variable Load

http://www.ipes.ethz.ch/ipes/dcdc/e_Buck_3.html

http://www.ipes.ethz.ch/ipes/index.html

84/252

Characteristics of the Buck Converter

The voltage gain as a function of duty and load In DCM, output voltage higher than expected Highly nonlinear when operating at low duty ratio

~

2

2,max

1 ( / )4

o

dco LB

V DV D I I

o

dc

V DV

CCM

DCM ,max 8dc s

LBV TI

L

low-pass filter

LC ov RLv

Li oi

CidcVSi

Di

85/252

DCM/CCM Boundary

Boundary between constant-frequency CCM and constant-frequencyDCM depends on the circuit parameters and the load

At the CCM/DCM boundary the inductor current ripple equals theoutput load current:

/( )

2dc o o

o L CCM DCMs g

V V VI i ILf V

If Io > ICCM/DCM, the buck converter operates in CCM

If Io < ICCM/DCM, the buck converter operates in DCM

0

2 LioI0

ONT OFFT

ST

2 Li

ST

SDT2 SD T

3 SD T

86/252

Boundary Condition for Discontinuous Inductor Current

Define the critical load current for continuous conduction operation as: Io(crit)During the DCM period, the inductor is disconnected from the output capacitor.

( )o Crit LI I

Inductor Current Boundary Condition

Discontinuous Inductor Current

RL

a c L

P

RC

IL = icia

RC

VOd s

g CR1

Q1

VDC

DriveCircuit

IL SolidiO Dashed = iO(Crit)

TON TOFF

TS

IL SolidiO Dashed

D2TSTS

0

0

D3TSDTS

2 Li

2 Li

87/252

Typical Waveform in Discontinuous Conduction Mode

2

241 1

o DCV V KD

2

S

LKR T

In DCM operation, the voltageconversion relationship is a functionof the input voltage, duty cycle, powerstage inductance, the switchingfrequency and the output loadresistance while for continuousconduction mode.

IQ1

ICR1

IL SolidiO Dashed

VC-P SolidVO Dashed

TS

D3TSD*TS

IPK

IPK

IL

D2*TS

88/252

Voltage Conversion Ratio of a Buck Converter

crit

crit

KK

DK

KKD

M for 411

2for

2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0 K = 0.01

K = 0.1

K = 0.5

K 1

D

M(D,K)

s

L

ss TTRL

RTLK

5.0)2/()/(2

Chap. 5 The Discontinuous Conduction Mode of Fundamentals of Power Electronics, Robert W. Erickson and Dragan Maksimovic, Kluwer Academic Publishers, 2nd Ed., February 2001.

with K = 2L / RTs. DCM occurs for K < Kcrit.

For the buck converter: Kcrit = 1-D

Another Form of the Static Characteristics of CCM/DCM Buck Converter

89/252

Voltage Conversion Ratio of a Buck Converter

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0 K = 0.01

K = 0.1

K = 0.5

K 1

D

M(D,K)

crit

crit

KK

DK

KKD

M for 411

2for

2

The static characteristics of buck converter is linear in CCM, while becomes nonlinear when operating in DCM.

The dynamics of buck converter is a second-order system in CCM, while becomes first-order when operating in DCM.

low-pass filter

LC ov RLv

Li oi

CidcVSi

Di

90/252

Homework Assignments

Load factor (%)

Voltage conversion ratio (%)

Efficiency (%)

LC

low-pass filter

Lr

Calculate and plot the efficiency plane as functions of different loss resistance factor r.

Calculate and plot the crest factor of the inductor current as functions of different loss resistance factor r.

What is the optimal efficiency design strategy?

The (M, D) plane plays a most important role in the analysis of a dc-dc converter. Consider the ideal synchronous buck converter has a resistor in series connection with the inductor.

R

RrL

r

dcV

ov

Lioi

CiSi

Di Lv

91/252

Critical Load Resistance

idlow-pass filter

With a specified constant switching fs and voltage conversion ratio, the critical load resistance for CCM operation is:

2 21

s incritical s

in out

Lf VR LfD V V

in

outin

sscritical V

VVf

Rf

RDL 22

)1(

The critical inductance in designing the Buck converter operating in CCM is:

inV

outV

oi

ciCL

Li

Lv

Si

Di

DCi

R

1ci

1C

92/252

Critical Filter Inductance

RC

C

What is the Critical Inductance to guarantee CCM operation for the critical load current?

( ) ?CritL

(max)min

( )

1

2

oo S

dc

o Crit

VV TV

LI

(min)min

( )

1 ( )2

OFFo dc L L

o Crit

TL V V I R

I

Solving Lmin with Vi(max), we get the Critical Inductance

Li

TON TOFFTS

0DriveCircuit

R

VORL

LQ1

CR1

ca

g

p

d s

dcVLi

Liai

93/252

Where is the Operating Region?

IN

OUT

VV

)(max,LB

o

II

0 0.5 1.0 1.5 2.0

0

0.25

0.50

0.75

1.0

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCMCRM

VIN = constant

(min)

(max)

IN

OUT

VV

(max)

(min)

IN

OUT

VV

(max)OUTI(min)OUTI

94/252

Where is the Working Area?

IN

OUT

VV

)(max,LB

o

II

0 0.5 1.0 1.5 2.0

0

0.25

0.50

0.75

1.0

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCMCRM

VIN = constant

(min)

(max)

IN

OUT

VV

(max)

(min)

IN

OUT

VV

(max)OUTI(min)OUTI

95/252

Required Dynamic Area at Working Boundary

IN

OUT

VV

)(max,LB

o

II

0 0.5 1.0 1.5 2.0

0

0.25

0.50

0.75

1.0

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCMCRM

VIN = constant

(min)

(max)

IN

OUT

VV

(max)

(min)

IN

OUT

VV

(max)OUTI(min)OUTI

96/252

eCircuit Center - SMPS Basics The Buck Converterhttp://www.ecircuitcenter.com/Circuits/smps_buck/smps_buck.htm

FREQUENCY COMPENSATION

97/252

Discontinuous-Input-Voltage Mode Operation

Y. S. Lee, S. J. Wang, and S. Y R, Hui, Modeling, analysis, and application of buck converters in discontinuous-input-voltage mode operation, IEEE Transactions on Power Electronics, vol. 12, no. 2, pp. 350-360, March 1997.

1L

1I2I

2L

1V 2V1C fD LC LR

MISSCO

SW

TD1DT TD1

TD1DT TD1

TD1

ON OFF ON OFF

1

11

CI

dtdv

1

211

CII

dtdv

T2 t

1V PV1

tT2TD1TD1TD1

DT TD1 DT TD1

2V PV1

T

T

ON OFF ON OFF

State of SW

State of SW(b)

(c)

(a)

dcV ov

98/252

Buck Derivative Converters2

3 40

oi

ov

1

2

3 40

oi

ov

1

2

3 4

0

oi

ov

1

(a)

(b) (c)

(d)(e)

More Readings on DCM CharacteristicsChap. 5 The Discontinuous Conduction Modeof Fundamentals of Power Electronics, Robert W. Erickson and Dragan Maksimovic, Kluwer Academic Publishers, 2nd Ed., February 2001.

Control of Boost type Converter in DCM by Controlling the Product of Inductor Voltage-Second, Chongming (Michael) Qiao, Jason Zhang, International Rectifier, USA, PESC 2005.

Designing Flyback Transformer for Discontinuous Mode, Keith Billings, Power Electronics Technology, April 2003.

S. Cuk and R. D. Middlebrook, A general unified approach to modelling switching dc-to-dc converters in discontinuous conduction mode, IEEE PESC Conf. Rec., 1977.

Troy J. Littlefield, Controller circuit for controlling a step down switching regulator operating in DCM, Toko, Inc., US Patent 5,959,443, Sept. 28, 1999.

A. Reatti and Mk. K. Kazimierczuk, Small-signal model of PWM converters for DCM and its application for boost converter, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 50, no. 1, pp. 65-73, Jan. 2003.

The inductor current waveforms determine the operating modes and control scheme of a switching converter!

LiLv

Li

LiLiLi

100/252

Boost Converters



Everett Rogers, Understanding Buck Power Stages in SPS, TI 1999 (slva061)

~

101/252

Boost Converters

Step-Up (Boost) Converter Continuous-Conduction Mode Boundary Between CCM and DCM Boundary Condition of IL Boundary Condition of Io Discontinuous-Conduction Mode Voltage and Current Gain at DCM

102/252

Step-Up (Boost) Converter

Characteristics of the Boost Converter:

Lower input current ripples due to the input inductorLarger output voltage ripples due to pulsing diode current Induce large spiky diode current when duties approach unity

dcV

ov

oi

Li

Lv

di

C

L

R

103/252

Continuous-Conduction Mode

t0

t0

(a) switch on (b) switch off

on off( ) 0dc dc oV T V V T

off

11

o s

dc

V TV T D

dc dc o oV I V I

(1 )odc

I DI

Volts-seconds balance of the inductor:Lv

dcV

dc oV V

dcV dcV

LI

Li

Lv Lv

Li Li

oVoV

sT

onT offT

s (1 )dc o sV T V T D on off off( )dc oV T T V T

104/252

Boundary Between CCM and DCM

t0

D0 0.5 1.0

Vo = constant

Step-up DC-DC converter at the boundary of continuous-discontinuous conduction.

0.25 0.75

LVTI osoB

074.0max,

31

(a)(b)

LVTI osLB 8max,

dcV

LvLi

dc oV V

,L peaki

LBIoBI

sT

onT offT

L LBI I

105/252

Boundary Condition of IL

, on1 1 (1 )2 2 2

dc s oLB L peak

V T VI i t D DL L

LVTI osLB 8max,

max,)1(4 LBLB IDDI

D0 0.5 1.0

Vo = constant

0.25 0.75

LVTI osLB 8max,

t0 ILB reaches its maximum at D=0.5:

In terms of its maximum value, theILB can be expressed as:

Lv

LidcV

dc oV V

,L peakiL LBI I

LBIoBI

106/252

Boundary Condition of Io

LVT

LVTI ososoB 074.027

2max,

max,2)1(

427

oBoB IDDI

The average output current at the theedge of continuous conduction is:

t0

,on

2

1(1 )2 2

12

(1 )2

L peak dcoB

dcs

s o

i D VI D tL

D V DTL

T V D DL

IoB reaches its maximum at D=1/3:

Lv

dcV

dc oV V

Li

di

,L peakiL LBI I

sT

onT offT

,L peakid oBI I

107/252

Boundary Condition of Io

D0 0.5 1.0

Vo = constant

0.25 0.75

31

LVTI osoB

074.0max,

1 1.51

o

dc

VV D

11

o

dc

VV D

1.0 LBI

oBI

108/252

Discontinuous-Conduction Mode

t0 t0

Step-up converter waveforms: (a) at the boundary of continuous-discontinuous conduction; (b) at discontinuous conduction.

LvLi L

vLi

dcV

dc oV V dc oV V

dcV

sT

onT offT

sT

sDT 1 sT 2 sT

109/252

Voltage and Current Gain at DCM

t0

1( ) 0dc s dc o sV DT V V T

1

1

odc

V DV

1

1

(since )o d odc

I P PI D

Volts-seconds balance of the inductor:

1( )2dc

d sVI DT D

L

1( )2s dc

oT VI D

L

The average input current (Idc = IL )

The average output current is:

dcV

dc oV V

Lv

Li

sT

sDT 1 sT 2 sT

110/252

Derivation of D in DCM

In keeping a constant output voltage Vo, It is important to express the duty ratio D as a function of the loading condition.

1

1

odc

V DV

1 ( )2o s dcI T V

D L

We need to replace 1 as a function of the load current.

1( )2s dc

oT VI D

L

2

1

1

( ) ( )2 2 ( ) ( )

2 2

o s dc s dco

o

o s dc s dcdco

I T V T VD I DV D D L LI T V T VV ID L L

2 1

2s dc o

odc

T V VD IL V

max,

1427

2

oB

os

ILVT

1/2

,max

4 ( 1)27

o o o

dc dc oB

V V IDV V I

111/252

Boost Converter Characteristics in Keeping Vo Constant

0 0.5 1.0 1.250

0.25

0.50

0.75

1.0

max,oB

o

II

LVTI osoB 27

2max,

D

0.25 0.75

discontinuous

Vo = constant

0.25dco

VV

0.5dco

VV

0.8dco

VV

Step-up converter characteristics keeping Vo constant.

1/2

,max

4 ( 1)27

o o o

dc dc oB

V V IDV V I

,peak

221 ( ) [W-s]

2 2dc s

LV DTLi

L

Energy stored in the capacitor:

This energy must be transferred to theload during steady state. If the loadbecomes very light, output voltagemay become dangerously high.

0.33

112/252

Example 7.1: Boundary Condition for Boost Converter

ExampleIn a step-up converter, the duty ratio is adjusted to regulate the output voltage Vo at 48 V. The

input voltage varies in a wide range from 12 to 36 V. The maximum power output is 120 W. Forstability reasons, it is required that the converter always operate in a discontinuous-current-conduction mode. The switching frequency is 50 kHz.

Assuming ideal components are C as very large, calculate the maximum value of L that canbe used.

Solution Po = 120 WVo = 48 V

Io = 120 W/48 V = 2.5 A

For the given range of Vd (12-36 V), D is in a range of (0.75-0.25).

48 1 12 1

o

dc

VV D

75.0 D

48 1 36 1

o

dc

VV D

25.0 D

113/252

Solution of Example 7.1

D0 0.5 1.00.25 0.75

IoB has the smallest value at D=0.75.

LVT

LVTI ososoB 074.027

2max,

max,2)1(

427

oBoB IDDI max,oBI

At this boundary condition for CCM and DCM:

2)1(2

DDLVTI osoB

D=0.752

6

)75.01(75.02

481020

LIoB

IoB = Io = 2.5 A

Ts = 20 sVo = 48 V

Duty ratio variation range: 0.25 ~ 0.75

oBILBI

114/252

Solution of Example 7.1 (continued)

H9)75.01(75.05.22

4810020 26

L

Therefore, 5.2)75.01(75.02

481020 26

LIoB

At this boundary condition, this boost converter will operate at the edge of continuous condition with Vdc = 12 V and Po = 120 W.To ensure a DCM operation, the inductance must smaller than 9 H.

115/252

Output Voltage Ripple

t0

0 t

CDT

RV

CDTI

CQV sosoo

current)output constant a (assuming

constant) time (where RCTDRCDT

VV sso

o

ovdcV

Di

Li

oiDi

Lv C

L

RoffTonT

oV

oV

ov

D oI IQ

Q

sDT (1 ) sD T

116/252

Output Voltage Ripple Analysis

To reduce the switching ripple, it is required to increase the switching frequency.

To reduce the ripple factor, we can also increase the output capacitor.

The voltage ripple is also dependent on the load.

The voltage ripple is the same wither in CCM or DCM.

RCTD

VV so

o

Ideal boost converter.

Voltage Ripple Ratio:

ovdcV

C

L

RLv

Li

oiDi

117/252

Effect of Parasitic Elements

Ideal boost converter Practical boost converter

Considering the characteristics of practical components, the boost converter will not have an infinity voltage gain as the duty ratio approach unit.

ovovdcV dcV

LvLv

Li

oi

C

L

R

Dioi

Li

C

L

R

Di

118/252

Effect of Parasitic Elements on Voltage Conversion

D

o

dc

VV D1

1ideal

due to parasitic elements

1.00

5

4.5

4

3

2

1

0

3.5

2.5

1.5

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

RL/R=0

RL/R=0.01

RL/R=0.02

RL/R=0.05

RL/R=0.1

D

2

1 11- 1

(1- )dc L

VV D R

D R

11-dc

VV D

o

dc

VV

119/252

Characteristics of Nonideal Boost Converter in CCM

2

1 11- 1

(1- )

o

in L

VV D R

D R

2

1

1(1- )

LRD R

Inductor winding resistance

2

1

5

4.5

4

3

2

1

0

3.5

2.5

1.5

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

RL/R=0

RL/R=0.01

RL/R=0.02

RL/R=0.05

RL/R=0.1

DV/

V g

100%

90%

80%

60%

40%

20%

0%

70%

50%

30%

10%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.0020.01

0.02

0.05

RL/R=0.1

D

oVinV C

L

RLv

Li

CiLR

in L L oV I R V t

t

(1 ) sD TsDTin L LV I R( )Lv t

/L oI V R(1 ) sD TsDT

( )Ci t

/oV R

120/252

Boost Converter: Power Stage Design

REF: Basic Calculation of a Boost Converter's Power Stage (slva372b, TI 2010)

Define the Specifications: Input Voltage Range: VIN(min) and VIN(max) Nominal Output Voltage: VOUT Maximum Output Current: IOUT(max) Integrated Circuit used to build the boost converter. This is necessary, because some

parameters for the calculations have to be taken out of the data sheet.

The following procedure is to calculate the power stage of a boost converter built with an IC with integrated switch and operating in CCM.

Waveforms of boost converter in CCM operation.

121/252

1. Calculate the Maximum Switch Current

VIN(min) = minimum input voltageVOUT = desired output voltage = efficiency of the converter, e.g. estimated 80%

Calculate the Dmax: VV

1DOUT

IN(min)max

Calculate the inductor current ripple IL: VIN(min) = minimum input voltageDmax = calculated maximum duty cyclefS = minimum switching frequency of the converterL = selected inductor value (L is unknown now)

LfDV

1Is

maxIN(min)L

Calculate the maximum output current be delivered by the selected IC:

ILIM(min) = minimum value of the current limit of the selected integrated switch (given in the data sheet)

IL = calculated inductor ripple current Dmax = calculated maximum duty cycle

)D(1 2III MAXLLIM(min)MAXOUT

LIM(min)I

122/252

1. Calculate the Maximum Switch Current

ISW(max) is the peak current, the inductor, the integrated switch(es) and the external diode has to withstand.

IL = inductor ripple currentDmax = maximum duty cycleIOUT(max) = maximum output current necessary in the application

Calculate the maximum switch current ISW(max): max

OUT(max)LSW(max) D-1

I2II

123/252

2. Inductor Selection

Calculate the required inductance L: VIN = typical input voltageVOUT = desired output voltagefS = minimum switching frequency of the converterIL = estimated inductor ripple current

OUTsL

INOUTIN

VfI)V-(VVL

LINOUT IL)DTV-(V

L

INOUT

I)DTV-(VL

OUTsL

INOUTIN

sL

INOUT

VfI)V-(VVD

fI)V-(VL

IN

OUTOUT(max)L V

VI0.4) to (0.2I Calculate the inductor current ripple IL:

A suggested inductor ripple current is 20% to 40% of its averaged current.

124/252

Optimum Inductor Current Ripple Ratio

What is the optimal ripple ratio?

Note: The figure comes from p. 208 of Switching Power Supply Design & Optimization, Sanjaya Maniktala, McGraw Hill, May 2004.

All parameters normalized to their respective values at r = 0.4

Nor

mal

ized

var

iatio

n

Current ripple ratio r0 0.5 1 21.5

1

2

3

4

5

Energy IOUT_CAP_RMS

IIN_CAP_RMS

ISW_AVGID_AVG

IL_RMSISW_RMS

D = 0.9D = 0.8D = 0.7D = 0.2

o

B

L

P

L

L

II

II

II 22 r

r = 0.4

125/252

CCM Boost Inductor and its B-H Trajectory

cH

satBMinor B-H loop of the filter inductor

coH

B-H loop for large excitation

0

Pc iNH

CCM Boost Converter

Pi

Filter inductor @ CCM: Copper loss is a major concern. Inductor Selection Rules:

1. iP 0.20 iL(AVG) [Define 20% of rated load as the light load for DCM boundary] 2. iP + iL(AVG) iL(SAT) BSAT/Lo (Note: Lo is the inductance at the operating point)

oL

Li

t

H

B

126/252

3. Rectifier Diode Selection

IF(avg) = average forward current of the rectifier diodeIOUT(max) = maximum output current necessary in the application

Averaged diode current IF(avg): OUT(max)F(avg) II

To reduce losses, Schottky diodes should be used. The averaged forward current rating needed is equal to the maximum output current:

Schottky diodes have a much higher peak current rating than average rating. Therefore the higher peak current in the system is not a problem.

Pwoer dissipation of the diode PD: FFD VIP

IF = average forward current of the rectifier diodeVF = forward voltage of the rectifier diode

~

127/252

4. Output Voltage Setting

With the given feedback voltage, VFB, and feedback bias current, IFB, the voltage divider can be calculated. The current through the resistive divider shall be at least 100 times as big as the feedback bias current:

Resistive Divider for Setting the Output Voltage

FBR1/2 I100I

IR1/2 = current through the resistive divider to GNDIFB = feedback bias current from data sheet

128/252

5. Input Capacitor Selection

[1] Jason Arrigo, Input and Output Capacitor Selection, Application Report SLTA055, TI, Feb. 2006.[2] Improve Your Designs with Large Capacitance Value Multi-Layer Ceramic Chip (MLCC) Capacitors, George M. Harayda,

Akira Omi, and Axel Yamamoto, Panasonic. [3] Comparison of Multilayer Ceramic and Tantalum Capacitors by Jeffrey Cain, Ph.D., AVX Corporation

CCM Boost Converter

This minimum value is necessary to stabilize the input voltage due to the peak current requirement of a switching power supply. The best practice is to use low equivalent series resistance (ESR) ceramic capacitors. The dielectric material should be X5R (-55C~+85C, 15%) or better. The input capacitor of a boost converter is less critical than the output capacitor, due to the fact that the inductor is in series with the input, and the input current waveform is continuous. The input voltage source impedance determines the size of the input capacitor, which is typically in the range of 1F to 100F. A low ESR capacitor is recommended, although it is not as critical as for the output capacitor.

129/252

6. Output Capacitor Selection

CCM Boost Converter

Best practice is to use low ESR capacitors to minimize the ripple on the output voltage. Ceramic capacitors are a good choice if the dielectric material is X5R (-55C~+85C, 15%) or better. Contributions of ESR (equivalent series resistance), ESL (equivalent series inductance) and the bulkcapacitance must be considered when choosing the correct output capacitors for a given output ripple voltage. The effect of these three parameters (ESR, ESL and bulk C) on the output voltage ripple waveform for a typical boost converter is illustrated in the top figure.

Averaged diode current IF(avg):

fS = minimum switching frequency of the converterVOUT = desired output voltage ripple

The Output Ripple Waveform of a Boost Converter.

ONt OFFt

COUTV

ESRV

OUTV(AC)

RINGING DUE TOTOTAL INDUCTANCE(BOARD+CAP)

(max)(min)

outOUT

o s

I DC

V f

130/252


0

0

CDT

RV

CDTI

CQV sosoo

current) output constant a (assuming

CCM Boost Converter

Di

(max)o s

o outs

V DT DV IR C f C

(max)(min)out

OUTo s

I DC

V f

)( AVGOV

Di

D oI IQ

Q

ov

t

toffTonT

sDT (1 ) sD T

oV

131/252


The ESR of the output capacitor adds some more ripple, given with the equation:

CCM Boost Converter

DiESRROUTC R

L

VOUT(ESR) = additional output voltage ripple due to capacitors ESRRESR = equivalent series resistance of the used output capacitorIOUT(max) = maximum output current of the applicationD = maximum duty cycleIL = inductor ripple current

(max)( ) 1- 2

out LOUT ESR ESR

I IV RD

0 t

Di

(max)

1-outI

D

2LI

[1] LT3958 High Input Voltage, Boost, Flyback, SEPIC and Inverting Converter.[2] Sanjaya Maniktala, Switching Power Supplies A - Z, 2nd Edition, Chapter 19. Solved Examples, Part 6: Output Capacitor

Selection and Loss.

offTonT

132/252


The output capacitor in a boost regulator experiences high RMS ripple currents, as shown in the top right figure. The RMS ripple current rating of the output capacitor can be determined using the following equation:

CCM Boost Converter

DiESRROUTC R

L

0 t

Di

(max)

1-outI

D

2LI

( ) ( ) 1MAX

RMS COUT OUT MAXMAX

DI ID

Multiple capacitors are often paralleled to meet ESR requirements. Typically, once the ESR requirement is satisfied, the capacitance is adequate for filtering and has the required RMS current rating.

Additional ceramic capacitors in parallel are commonly used to reduce the effect of parasitic inductance in the output capacitor, which reduces high frequency switching noise on the converter output.

offTonT

133/252

Buck-Boost Converters



Everett Rogers, Understanding Buck-Boost Power Stages in SPS, TI 1999 (slva059)

~

134/252

Buck-Boost Converter

Buck-Boost Converter CCM of Buck-Boost Converter Boundary Condition Average Output Current at Boundary Condition Boundary Between CCM and DCM Discontinuous-Conduction Mode Buck-Boost Converter Characteristics in DCM Effect of Parasitic Elements Input Current Ripples and Output Voltage Ripples

135/252

Buck-Boost Converter

1o

dc

V DV D

The inductor operates like a voltagepump, and from the volts-secondsbalancing principle, its pump-in voltagemust be released during pump-out.

Characteristics of the Buck-Boost Converter:

Larger input current ripples due to the input switchLarger output voltage ripples due to pulsing diode current Inverse voltage polarity

dcV

Lv ov

oi

Li

Si Di

RC

DCi

L

inC

cii

136/252

CCM of Buck-Boost Converter

0

0

(a) switch on (b) switch off

( )(1 ) 0dc s o sV DT V D T

1

o

dc

V DV D

1 (assuming )o d odc

I D P PI D

When operating in the continuousconduction mode, from the volts-secondsbalance of the inductor we can obtain:

dcV

Lv

oV

dcV dcVLvLv

Li

L S oI I I

t

toffTonT

sDT (1 ) sD T

oV

oioi

oVLiLi

137/252

Boundary Condition

,peak12 2

s dcLB L

T VI i DL

o L Si i i

)1(2

DLVTI osLB

0

0

At the boundary condition, the average inductor current is:

o L S L dcI I I I I

From the Kirchoffs current law:

Take its average value:

In terms of Vo, the ILB can be expressed as:

1

o

dc

V DV D

2,peak

12 2

s s dcdc L

s

DT T VI i DT L

dcVLv Li

SiSi dc

I

oV

,L peaki

sDT (1 ) sD T

t

t

L LBI I

138/252

Average Output Current at Boundary Condition

2)1(2

DLVTI oso

o L DI I I

2,peak

12 2

s s dcdc L

s

DT T VI i DT L

,peak

12 2

s dcL L

T VI i DL

2 (1 )2 2 2s d s dc s dc

o L DT V T V T VI I I D D D D

L L L

1 dc oDV V

D

At the boundary condition, the average output current is: 2)1(2

DLVTI osoB

139/252

Boundary Between CCM and DCM

0 0.5 1.00

0.25

0.50

0.75

1.0

0.25 0.75

Vo = constant

LVTI osLB 2max,

LVTI osoB 2max,

)1(max, DII LBLB

2max, )1( DII oBoB

The maximum values of ILB and IoB all occur at D=0.

ILB and IoB can be expressed in terms of their maximum values:

In keeping Vo constant, ILB and IoB are functions of the controlled duty ratio D.

LVTII osoBLB 2max,max,

,max/LB LBI I

,max/oB oBI ID

140/252

Discontinuous-Conduction Mode

0

Buck-boost converter waveforms in the discontinuous conduction mode.

1( ) 0dc s o sV DT V T

1

odc

V DV

) (since 1 odd

o PPDI

I

1(D )2dc

L sVI DT

L

The inductor current iL when operating in DCM:Volts-seconds balance of the inductor:

dcV

oV

LI

Lv Li,L peaki

sDT 1 sT 2 sT

t

141/252

Buck-Boost Converter Characteristics in Keeping Vo Constant

0 0.5 1.0 1.250

0.25

0.50

0.75

1.0

max,oB

o

II

LVTI osoB 2max,

0.25 0.75

discontinuous region

Vo = constant

0.33dco

VV

1.0dco

VV

4.0dco

VV

Buck-boost converter characteristics keeping Vo constant.

,max

o o

dc oB

V IDV I

,max

2

2

o o o o

dc oB dc s o

oo

dc s

V I V LIDV I V T V

V L IV T

D

142/252

Buck-Boost Converter in Keeping Vo Constant

,max

2 2oo o o oo

dc oB dc s o dc s

VV I V LI LD IV I V TV V T

dcV oVLv

oi

LiSi

Di

RCL

143/252

Buck-Boost Converter Characteristics in Keeping Vo Constant

Zeta converter normalized output characteristic @ Ls = Lo5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

00 0.1 0.2 0.3 0.4 0.5 0.6

8.0

7.0

6.0

5.0

4.03.0

2.01.0

)/( ino VVM

)/2( sinon TVLII

Zeta Converter: a non-inverted buck-boost converter

inv vo1L

2L1C

2CD

144/252

Buck-Boost Converter in Boundary Condition

Boundary Between Continuous and Discontinuous Mode

a Q1 CR1 p

cDriveCircuit

(max)LI)(CritOO IDashedI

0

dcV

ai

L Ci i CR

oV

RC

L

LR

offTonT

sT

Li

145/252

Buck-Boost Converter in DCM

TS

D3xTSDxTSD2xTS

IQ1

ICR1

iL Solid

IO Dashed

VC-P Solid

VO Dashed

0

LI

a Q1 CR1 p

cDriveCircuitdcV

ai

L Ci i CR

oV

RC

L

LR

146/252

Example 7.2: Duty Ratio at DCM

ExampleIn buck-boost converter operating at 20 kHz, L = 0.05 mH. The output capacitor C is

sufficiently large and Vdc = 15 V. The output is to be regulated at 10 V and the converteris supplying a load of 10 W. Calculate the duty ratio D.

SolutionA1

V 10W 10

o

oo V

PI

Check this buck-boost converter in CCM or RCM?

Calculate the boundary current IoB : 2)1(2

DLVTI osoB

If this buck-boost converter is operating in CCM, then10 0.4

10 15o

o dc

VDV V

147/252

Solution of Example 7.2

3.00.50.1

1510 D

o

o dc

VDV V

CCM:

DCM:,max

o o

dc oB

V IDV I

A8.1)4.01(05.021005.0)1(

222

DLVTI osoBIf D = 0.4, then

However, A8.1 A1 oBo II

Therefore, it is operating in DCM and

dcV

oV

oi

Lv

Li

SiDi

RCL

148/252

Effect of Parasitic Elements

o

dc

VV

DD1ideal

with parasitic elements

1.00

Effect of parasitic elements on voltage conversion ratio in a buck-boost converter

D

149/252

Output Voltage Ripple

0

0

Q

Q

Output voltage ripple in a buck-boost converter.

Assume a constant output current:

C

DTRV

CDTI

CQV

so

soo

ss

o

o TDRCDT

VV

voinv

vo

DiDi

D oI I

offTonT

sDT (1 ) sD T

oV

oV

t

t

150/252

Stress Analysis of DC-DC Converters


Sanjaya Maniktala, Stresses in Wide Input DC-DC Converters, AN-1246 National Semiconductor, Sept. 2002.


~

151/252

Inductor Oriented Design Concept


vovivovivi vo

Optimization of the inductor is the key for the design of a switching converter!

Inductor Current Ripple Factor (r)


vovivovivi vo

The current ripple ratio r is defined as the ratio of the AC to the DC value of the inductor current, with the converter delivering maximum load.

T

DT

LI

PI

LI

)(tiL

)(tis

)(tiD

L

P

L

L

II

II

2r

IL is the average inductor current at maximum load and IL is the peak-to-peak of the current ripple. This definition of r applies only if the converter is in CCM operating mode.

r varies between 0~2. r=2 for critical (boundary) conduction mode. r>2 implies DCM operation. r is the starting point for a converter design!

153/252

Inductor Current Ripple Ratio r

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

The current ripple is a measure of DCM for a CCM converter

T

DT

Io = Output Current at Rated LoadIB = Boundary Output Current at Light Load

Ripple Ratio r = o

B

L

P

L

L

II

II

II 2

2

LI

PI

LI

A: 100%B: 50%C: 20%

D: 1%

r = 0.4

154/252

Worst-Case Inductor Current


vovivovivi vo

DT

LI

PI

LI

)(tiL

Worstcase inductor current depends on the topology.

For magnetic design: A maximum input voltage forms the worst-case

condition for a buck converter. A minimum input voltage forms the worst-case

condition for a boost and buck-boost converter.

The peak current (for the inductor, switch, and diode) is:

21 rII LPK

Inductor current of buck converter with higher input voltage (dash line)

155/252

Peak and RMS Values of Inductor Current


vovivovivi vo

DT

LI

PI

LI

)(tiL

21,

rII LPKLPeak of Inductor Current:

RMS of Inductor Current: 12

12

,rII LRMSL

ESRR

ESRRMSLlossL RIP 2

,,

th

ott

JADAJ ePTT

)(

1

156/252

Do Not Saturate the Inductor

Step Load Change1A

0AOutput Voltage

Inductor Current

B

H (I)

satB

satNI

oI

Keep Io + Io within the linear region

The inductor must not be saturated under rated and transient conditions.

157/252

Peal Energy Stored within a Buck Inductor

T

DT

LI

PI

J

2

128 r

rTVIE Lo

L

P

L

L

II

II

2r

158/252

Peal Energy Stored within a Buck Inductor

J

2

128 r

rTVIE Lo

rrr

rr 1212

22

If r is very small,

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40%20% 110%90%

XOP

satB

satNI

oI


H

Ripple current of the switching inductor

159/252

Inductor Energy

160/252

Energy Stored in a Core

Mean path length l Cross-sectional area Ac

Permeability

I

N: number of turns

lANL 2

The energy stored in the core:

tt

L LIdiLiPdtE 02

0 21''

The energy density (energy/volume) is:

0

22

22

222221

22

211

r

c

ccB

BB

NlB

lAN

lAlALI

The energy stored in the core:

coreBL VLIE 2

21

Vcore: volume of the core

161/252

Typical Energy Density of a Ferrite Core

0

2

2

re

cB

BVE

For a typical ferrite, assuming the relative permeability is about r = 2000, and the saturation flux density Bsat = 0.3 T (3000 G), we get (for most ungapped ferrite cores) a typical power density of

3J/m 9.1710420002

3.02 7

2

0

2

re

cB

BVE

2Newton/A H/m

7

70

104104

3000G)B 2000,( J/cm 18J/m 18 satr

33 e

c

VE

(Ferrite core)18100 kHz50%(CRM)3.63610

162/252

Optimum Inductor Current Ripple Ratio

What is the optimal ripple ratio?

Note: The figure comes from p. 208 of Switching Power Supply Design & Optimization, Sanjaya Maniktala, McGraw Hill, May 2004.

All parameters normalized to their respective values at r = 0.4

Nor

mal

ized

var

iatio

n

Current ripple ratio r0 0.5 1 21.5

1

2

3

4

5

Energy IOUT_CAP_RMS

IIN_CAP_RMS

ISW_AVGID_AVG

IL_RMSISW_RMS

D = 0.9D = 0.8D = 0.7D = 0.2

o

B

L

P

L

L

II

II

II 22 r

r = 0.4

Stress Curves

Sanjaya Maniktala, Stresses in Wide Input DC-DC Converters, AN-1246 National Semiconductor, Sept. 2002.

3.0

2.5

2.0

1.5

1.0

0.5

00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

12

98

510

3,4711

61 2

7

89

10

11

12

All parameters normalized to 1 at D=0.5(for small r)See table 2 for parameters

1 2 34 5 6

HIGH INPUT VOLTAGE LOW INPUT VOLTAGE50_INV

Fact

or b

y w

hich

par

amet

ers

vary

DUTY CYCLE D

164/252

Simulation Study of a Buck Converter



~

165/252

DC Analysis of a DC-DC Converter

Define the Operating Point DC Voltage Gain Steady-State Current and Voltage Ripple Boundary of CCM and DCM Rating Calculation Maximum Value Calculation Design Guarantees for the Worst Case Operating Point Efficiency Analysis Current & Voltage Ripple Factor Q-Factor of Specified Operating Point

166/252

Simulation Study of a Buck Converter

d TTON

TTON

Tf s

1

Vinvao

iL

LvL vo

io

Cout RL

low-pass filteriQ

ico

Input voltage Vin = 10 VNominal output voltage Vout = 5 V

Nominal output power Po = 10 WSwitching frequency fs = 100 kHzInductor L = 62.5 HOutput Capacitor Cout = 5.0 F Input Capacitor Cin = 2.2 FLoad resistor RL = 2.5 Po=10W

id

Cin

ici Q

D

PWM Waveform Generator

iin

A buck dc/dc converter has the given parameters with aPWM control duty set at 50%. The components areassumed as ideal. Make a computer simulation of theconverter to obtain its steady state waveforms of eachcomponent, such as output voltage, inductor current,switch current, etc.

Calculate the RMS value of the inductor current.

DC/DC Converter

167/252

Question: What is the Q-Factor at OPx?

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

IB = Boundary Output Current at Light Load

Inductor Current Ripple Ratio r

o

B

L

P

L

L

II

II

II 22

r

80% 100%60%40%20% 110%90%

Io = Output Current at Rated Load

PI

LI

What is the Q-factor of the buck converter at the specified operating point with specified inductor current ripple factor and capacitor voltage ripple factor?

Quality Factor = ?

XOP

168/252

Selection of Inductor

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM


Ripple Factor = L

P

o

B

II

II

80% 100%60%40%20% 110%90%


PIDefine the inductor current ripple as 10% of its average current, means the converter enters DCM at 10% of its rated load.

LTVI sdL 4(max)

D0 0.5 1.0

LI

LDTDVII sinLP

)1(21

21

RDV

RVII inoutAVGoAVGL )()(

RLTD

II sAVGL

P

/2/)1(

)(

LP I21I

LI

169/252

Selection of Inductor for CCM

RLTD

II sAVGL

P

/2/)1(

)(

2)( s

in

outin

P

AVGL TV

VVRI

IL

Ratio of Current Ripple

Rated Load Resistance

Nominal Input & Output Voltage

Switching Period

For the given example to get a current ripple ratio of 10%:

FVL 5.62sec10)2/10(10

5105.210 6

Sanjaya Maniktala, Selecting Inductors for Buck Converters, AN-1197 National Semiconductor, July 2002.

Chap 9: Selecting Inductors for DC-DC Converters, Sanjaya Maniktala, Switching Power Supply Design & Optimization, McGraw-Hill Int., 2004.

170/252

Selection of Capacitor

Voltage Ripple: 222

)(

))(1(4

)1(81

21

s

cs

AVGo

P

ffD

LCDT

VV

LDT

VV

C sP

AVGo )1(161 2)(

Capacitor Voltage Ripple Ratio: )( AVGo

PV V

V

Inductor Current Ripple Ratio: )( AVGL

PI I

I

oP V21V

LCDT

VV sAVGo

PV

)1(4

2

)(

L

DTC sV

)1(4

1 2

Nominal Input & Output Voltage

LVVVTC

in

outins

V

14

1 2

Ratio of Voltage Ripple Filter Inductance

Switching Period

LP I21I

171/252

Selection of Capacitor: Q-Factor and I & V

CL

RQ V

I

DCL

RQ

11

21

If we define I = 10% and V = 1% at rated load with D=50%, then

707.001.01.0

5.011

21

Q

The Q-factor has a relationship with the inductor current ripple ratio and capacitor voltage ripple ratio.

172/252

Q-Factor and Damping Ratio

L

svi svoC R

21

Impedance sticCharacteri

R

CL

RQ

Resonant frequencyLC

f

2

12

00

LCs

RLssv

svsGi

o

21

1)(

Characteristic Equation 012 LC

sLRs

45

21707.0 Q 707.0

0

s plane

0

j

0

cos

20 1

173/252

Average & I of the Inductor Current

Inductor current

A2L(AVG)I

For the given example, the Q factor is:

707.0

105105.62

5.2

6

6

CL

RQ

A2.0 PI

707.0707.02

121

Q

%10)(

AVGL

PI I

I

Switch current

Output voltage

PWM signal

174/252

Average & RMS Value of the Inductor Current

Inductor current

2

)()( 3

11

AVGL

PAVGLRMS I

III

2.0 LI

2)( 3

11 IAVGLRMS II

A2L(AVG)I

For the given example, the RMS value of the inductor current is: 0033.201.03112 RMSI

The form factor of the inductor current is: 00167.1311 2

)(

IAVGL

RMS

II

The RMS value of a 10% of inductor current ripple in CCM operation can be approximated by its average value with an error about 0.17%.

175/252

Selection of Inductor

Inductance L = 62.5 HSwitching Frequency fs = 100 kHzRated DC Current IR = 2.0 A

H (I)

satB

satNI

oI


Inductor LossCopper Loss (Winding Loss)

Core LossHysteresis Loss

Eddy Current Loss

Ferrite Powdered Iron

5~10% 20~30%

176/252

RMS Value of the Switch (MOSFET) Current

22

)( 311

311 IRMSS DII

IDII

2.0 LI

A2L(AVG)I

RMS Current of MOSFET at CCM Operation:

RMS Current of MOSFET at DCM Operation:

3)(DII pkRMSS L

VVDTI outinspk)(

DS(ON)2S(RMS)cond RIP

177/252

Switching Loss Analysis of the MOSFET

MOSFET Switching Waveforms

DrainVoltage

Drain Current

vDDID

tcr tvf tPS

Switch current

Switch VoltageDSV

DSI

outin VV

LOS(ON) III

LOS(OFF) III ON-SWT

ON-SWS(ON)outinON-SW T)IV-(V21P

OFF-SWS(OFF)outinON-SW T)IV-(V21P

OFF-SWON-SWSW PPP

178/252

Calculated Loss of the MOSFET

Vo/Vi = 1.0

0.2

0.5

0.8

CCM

80% 100%60%40%20% 90%50%

DCM

10%

D

Boundary Condition for DCM )(2 outin

sLB VVL

DTI

For the given example, the load current for the DCM is: A2.0)510(105.62210105.0

6

6

LBI

A2Io(AVG)

OFF-SWON-SWSW PPP

DS(ON)2S(RMS)cond RIP

SWcondMOSFET PPP

SWcondo

o

PPPP

179/252

Loss Analysis of the MOSFET

Consider the MOSFET with a conduction resistance of RDS(ON), what is the theoretical efficiency curve for a constant voltage conversion ratio?

80% 100%60%40%20% 90%50%10%

%

Vo/Vi = 1.0

0.2

0.5

0.8

CCM

80% 100%60%40%20% 90%50%

DCM

10%

D

Note: The x-axis id normalized with the rated load current.

180/252

Loss Analysis of the DIODE

Conduction Loss of DIODE at CCM Operation:

Conduction Loss of DIODE at DCM Operation:

Focond VIP

FSPKcond VDTI21P L

)V(VDTI outinspk

Foutin2

S2

cond VLVVTD

21P

Diode current

Diode VoltageDSV

DSI

outin VV

LOD(ON) III

LOD(OFF) III

OND-D(ON)outinON-SW T)IV-(V21P

OFFD-D(OFF)outinOFF-SW T)IV-(V21P

OFFD-OND-SD PPP

SDcondDIODE PPP

181/252

Dynamics of a Buck Converter

Dynamics describes how the state evolvesThe dynamics of a model is an update rule for the system state that describes how the state evolves, as a function on the current state and any external inputs.

Switching Converter

Linear RLC Network

(5V, 1A)

Output Voltage

Inductor Current

Buck Converter

low-pass filter

LC ov RLv

Li oi

di

CidcVSi

Di

182/252

Static Voltage Conversion Ratio Under Load Variations

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40%20% 110%90%50%

500 Ro

testI

testI2A

5msec 100msec

If the components are not ideal, its parasitic parameters will change its static curves. The illustrated example is obtained with the following parameters: RDS(ON) = 50 mMOSFET reverse diode voltage drop = 0.7VDiode voltage drop = 0.7V Inductor ESR = 25mCapacitor ESR = 2m

Vo/Vi = 1.0

0.25

0.5

0.75

0

1.0

183/252

Buck Converter Characteristics in Keeping Vo Constant

Vo/Vi = 1.0

0.2

0.5

0.8

CCM

80% 100%60%40%20% 110%90%50%

DCM

10%

DHow to calculate the control duty to keep a constant output voltage under load current variations?

500 Ro

testI

1R

2RREFV

CC

testI2A

5msec 100msec

Simulation result

184/252

PowerOn Step Response at OPx?

0

s plane

0

j

0

cos

20 1

LCR Filter

PWM Buck Converter

Step Response with Q=0.707

185/252

PowerOn Transient Trajectory to OPx

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40%20% 110%90%

XOP

Q = 0.707Vo/Vi = 1.0

0.25

0.5

0.75

0

1.0

186/252

PowerOn Transient Trajectory to OPx

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40%20% 110%90%

XOP

Q1.414Q=1.0

140%

Q = 1.0Vo/Vi = 1.0

0.25

0.5

0.75

0

1.0

187/252

Boundary Condition: Q-Factor at OPY?

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM


Ripple Factor = L

P

o

B

II

II

80% 100%60%40%20% 110%90%


PI

LI

What is the Q-factor of the buck converter at the specified operating point with specified inductor current ripple factor and capacitor voltage ripple factor?

Quality Factor = ?

YOP

07.7

105105.62

25

6

6

CL

RQ

XOP

188/252

PowerOn Step Response at OPY?

Step Response with Q=7.07

It can be observed there are quite differences between PWM converter and LCR low-pass filter when operating at the boundary condition OPY.

189/252

PowerOn Transient Trajectory to OPY

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40%20% 110%90%

YOP XOP

Q = 0.707Vo/Vi = 1.0

0.25

0.5

0.75

0

1.0

190/252

Transient Trajectory from Light Load to Rated Load

A2Io

A0.2Io

Inductor Current

Output Voltage

Switch Current

191/252

DCM

D = 1.0

0.1

0.3

0.7

0.9

CCM

80% 100%60%40%20% 110%90%

Transient Trajectory from Light Load to Rated Load

0.5

Transient from Light to Rated Load

Vo/Vi = 1.0

0.25

0.5

0.75

0

1.0

Switching frequency fs = 100 kHz

Power-On Transient to the Light Load

192/252

Time Responses Under Open Loop Control

Power on transients to rated loadSwitched to light (10%) load.

Back to rated load

Steady-state output voltage under CCM

Inductor Current

Output Voltage

10%100%

100%10% 2.5 25Ro

25 2.5Ro

oR

5.8VVRISE

3.0VVFALL

5.0VVnormal

193/252

Dynamic Responses Under Load Variations

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40%20% 110%90%

Transient from Rated to Light Load

(iL, vo)

10%100%

25 2.5Ro

Vo/Vi = 1.0

0.25

0.5

0.75

0

1.0

194/252

Output Voltage Under Step Load Change

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40%20% 110%90%

Step Load Change1A

0AOutput Voltage

Inductor Current

B

50%

Vo/Vi = 1.0

0.25

0.5

0.75

0

1.0

195/252

Load Change Trajectories on Sate-Space Plane

500kHz

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40%20% 110%90%50%

BA

Vo/Vi = 1.0

0.25

0.5

0.75

0

1.0

Switching frequency fs = 500 kHz

196/252

Load Change Trajectories on Sate-Space Plane

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40%20% 110%90%50%

BA

Vo/Vi = 1.0

0.25

0.5

0.75

0

1.0

1MHz

Switching frequency fs = 1 MHz

197/252

Boundary for CCM Operation (Open Loop)

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40% 110%90%10% 20%

Vo/Vi = 1.0

0.25

0.5

0.75

0

1.0

198/252

Transient Response Under Step Load Current Change

5.0VVnormal

20%Q3.5

0.37VVLOW

8.0VVHIGH

QA = 3.535

199/252

Resonant Frequency and Switching Frequency

kHz9105105.622

12

166

LCfo

kHz100sf

222

)(

))(1(4

)1(81

21

s

os

AVGo

Pv f

fDLC

DTV

V

For the given example, the voltage ripple factor is defined as 1% for D=50%:

01.0))(5.01(4

22

)(

s

o

AVGo

P

ff

VV 01.024)( 2

2 s

o

ff

2111

Dff

vs

o

09.01.022 s

o

ff

2111

Dff

vs

o

Optimum Frequency Selection:

200/252

Q-Factor Analysis of the Load Line

oR

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40%20% 110%90%50%

YOP XOP

QX = 0.707QY = 7.07

201/252

Frequency Response of RLC Network

LCR Filter

iv ov

)()()(

svsvsG

i

o

Q = 7.070

Q = 3.535

Q = 1.414

Q = 0.707

202/252

Q-Factor Analysis of the Load Line

oR

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40%20% 110%90%50%

YOP XOP

QX = 0.707

QY = 7.070

QB = 3.535

QA = 1.414

QX = 0.707

QY = 7.07

QB = 3.535 QA = 1.414

203/252

Homework: Effect of Parasitic Elements

Derive the transfer function of the buck converter when operating in CCM and DCM respectively.

Derive the Q-factor as a function of the capacitor ESR?

The ESR of the output filter capacitor will introduce a left-half plane (LHP) zero to the converter, make an analysis of the ESR effect on the frequency response of the buck converter.

204/252

Frequency Response Analysis of the Buck Converter

Define the Operating Points 20%, 50%, 100% Load Current Vin,min and Vin,min

Check the Control Limitations Max. of PWM Duty Max. Load Current Slew Rate Max. Inductor Current Slew Rate

Frequency Response Simulation of Key Operating Points MATLAB and PSIM AC Sweep Simulation Increment Step Responses at Specified Operating Points Verifications of Frequency Response

Small-Signal and Large-Signal Time Response Check Linear Operating Area for Specified OP. Estimate Closed-Loop Bandwidth

205/252

Synchronous Buck Converters



Magnetic Buck Converters for Portable Applications (Frank De Stasi, NS)

~

206/252

Synchronous Buck Converter

Replace the diode with a MOSFET Reduce the voltage drop in low output voltage regulator Reduce the conduction loss in low-voltage high-current applications

207/252

Applications of Synchronous Buck

Vbat vo(t)+

+

Iload1

2R

vi vo

L

CD

BASIC BUCK CONVERTER SYNCHRONOUS BUCK CONVERTER

vi vo

L

C

Lithium Ion Battery Low Power mP

)(tic

C)(tvsw

L)(tiL

)(tvL

208/252

Synchronous Rectifier Reduction of Diode Loss


vovivovivi vo

Synchronous Buck Synchronous Boost Synchronous Buck-Boost

vovivovivi vo

Replace the diode with a synchronous switch (usually a MOSFET) to reduce the conduction loss and voltage drop of the diode.

209/252

A Further Improvement

Synchronous Buck

vi vo

S1

S2

(multi-phase)

S1

S2

The turn-on of S1 and S2 can not be overlapped and the dead-time between the synchronous switching must be kept small. However, small glitch may be resulted and induce large voltage spike due to the switching of inductor current. Therefore, a Schottky diode is needed to be parallelled with the synchronous switch.

210/252

Efficiency vs. Losses

0

100

200

300

400

500

0.8 0.84 0.88 0.92 0.96

Efficiency

Pow

er R

atin

g

W10 Ploss

W20 Ploss

losso(AVG) P P

1

losso(AVG)

o(AVG)

PPP

-1P P o(AVG)loss

211/252

Loss Distribution of DC-DC Converters

(a) Synchronous buck converter topology. (b) Typical power losses in synchronous buck topology.

(a) Isolated forward converter topology. (b) Typical power losses in isolated forward converter topology.

Miscellaneous1%

Control FET36%

Synchronous FET 23%

InputCapacitor

30%

Inductor 10%

(b)

(a)

VoVin RCQ2Q1

L

Vo

Vin

Qctrl QFWD

QFWH RC

LN:1

Misc 8%

Bias 9%

Inductor 9%

Transformer 22% Secondary

MOSFETS 35%

PrimaryMOSFETS

16%

(b)

(a)

212/252

Loss Analysis of a Forward DC/DC Converter

[1] B. E. Taylor, High frequency rectification. Schottky or synchronous rectifier, Power Conversion Proceedings, June 1990.[2] C. Blake, D. Kinzer and P. Wood, Synchronous rectifiers versus Schottky diodes: a comparison of the losses of a synchronous rectifier

versus the losses of a Schottky diode

5. dc-dc converters: static...

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