Ch6 DC-DC Converters6-1 Linear voltage regulators Fig. 6.1

Adjusting base current,

SLLO VRIV ~0

=> linear DC-DC converter or linear regulator The transistor operates in the linear region.

Low efficiency happen under lower output voltage.=> improved by switching converter.

6-2 A basic switching converter. Fig. 6.2

The transistor operates as an electronic switch and in saturation state.

frequtncyswitching:f

ftT

t

tt

tD:ratioduty

DVdtVT

dttvT

V

onon

offon

on

S

DT

S

T

O

0

0 0

1

1

If the ideal switch is assumed, the energy efficiency is 100%

6-3 The Buck converter (Down converter) Fig.6.3, Fig.6.4

→ To produce an output that is purely dc.

Properties:

idealnonlossesP

idealPP

diT

I

dvT

V

tiTti

O

OS

Tt

t cC

Tt

t LL

LL

4

01

3

01

2

1

Assumptions:

(1) In steady state(2) is constant ( positive ) tiL

OV(3) is constant ( C is very large)

(4) Closed time=DT , open time=(1-D)T(5) Components are ideal

Sw. closed :

positiveL

VV

dt

iddt

idLVVv

OSL

LOSL

Li increases linearly

DTL

VVsi

L

VVs

DT

i

t

i

dt

id

OclosedL

OLLL

Sw. open :

)negative(L

V

dt

iddt

idLVv

OL

LOL

Li decrease linearly

TDL

Vi

L

V

TD

i

t

i

OopenL

OLL

1

1

Steady-state operation requires that the inductor current at the end of the switching cycle is the same as that at the beginning, meaning that the net change in inductor current over one period is zero.

closedLi openLi = 0

SSO

OO

VDVV

TDL

VDT

L

VVs

01

Since the average inductor voltage is zero for periodic operation

DVVT

TDVDTVVV

SO

OOSL

01

Average Current statesteadyforIR

VII C

ORL 0

Imax=

Lf

D

RVTD

L

V

R

ViI O

OOLL 2

111

2

1

2

Imin=

Lf

D

RVTD

L

V

R

ViI O

OOLL 2

111

2

1

2

.frestwitchingT

f 1

Since Imin = 0 is the boundary between continuous and discontinuous

Imin = 0 =

Lf

D

RVO 2

11

f

RDL

RDLf

2

1min

2

1min

In practice, the output voltage cannot be kept perfectly constant with a finite capacitance.

Fig. 6.5

2

2

8

1

8

1

188

222

11

LCf

D

V

V

ripplevoltageoutput

rTransformedc

I

I

V

VIVIV,PP

LCf

DV

TDL

V

C

T

C

iT

iT

CC

QV

V C = Q

V C = Q

O

O

O

S

S

OOOSSOS

O

OL

LO

O

O

6-4 Design considerations

Buck Converter:

fs ↑, Lmin↓,Cmin↓, Ploss in switches↑, heat sink ↑

The inductor wire must be rated at the rms current, and the core should not saturate for peak inductor current .The capacitor must be selected to limit the output ripple to the design specifications, to withstand peak output voltage, and to carry the required rms current. The switch and diode must withstand maximum voltage stress when off and maximum current when on. The temperature ratings must not be exceeded, possibly requiring a heat sink.

6-5 The boost converter Fig 6-6

The output voltage is larger than the input

Assumptions: (1) In steady-state.(2) The switch is closed for time DT and open for (1-D)T.(3) The inductor current is continuous (positive).Li

OV

(4) The capacitor is very large, and is held constant.

(5) Components are ideal.

Fig 6-7

When the switch is closed:

L

V

td

id

td

idLVv SLL

SL

Li increases linearly

L

DTVi

L

V

DT

i

t

i

SclosedL

SLL

When the switch is open:

L

VV

td

id

td

idLVVv OSLL

OSL

Li must change linearly

L

VV

TD

i

t

i OSLL

1

L

TDVVi OS

openL

)1)((

For steady-state , the net change in inductor current must be zero.

0 openLclosedL ii

D

VV

DVDDVL

TDVV

L

DTV

SO

OS

OSS

1

011

0)1)((

The average inductor voltage must be zero for periodic operation

So

SO

OSSL

VVD

VV

T

T)D)(VV(DTVV

1

01

Output power: RVP OO /2

Input power

RD

V

R

D

V

RVIVIVP

S

S

OLSSSS

2

2

2

2

)1(

1

/

RD

VI S

L 2)1( -----------Inductor Current

Imax = L

DTV

R)D(

ViI SSL

L 212 2

Imin = L

DTV

R)D(

ViI SSL

L 212 2

The boundary between continuous and discontinuous inductor current is determined form.

f

RDDminL

RDDmin)Lf(

Lf

DV

L

DTV

R)D(

V

L

DTV

R)D(

V Imin

SSS

SS

2

1

2

1

221

210

2

2

2

2

The change in capacitor charge

ripplevoltageoutputRCf

D

V

VRCf

DV

RC

DTVV

VCDTR

VQ

O

O

OOO

OO

f: switching frequency.

6-6 The Buck-Boost converter Fig 6-8

The out voltage can be either higher or lower than the input, and there is a polarity reversal on the output.

Assumptions: (1) In steady-state.(2) Li is continuous.

OV(3) is held constant.(4) The switch is closed for time DT and open for (1-D)T.(5) Components are ideal.

When the switch is closed.

L

V

td

id,

td

idLVv SLL

SL

Li increases linearly.

L

V

DT

i

t

i SLL

L

DTVi S

closedL

When the switch is open:

D

DVV

L

T)D(V

L

DTVii

L

T)D(Vi

L

V

T)D(

i

t

i

L

V

td

id,

td

idLVv

SO

OSopenLclosedL

OopenL

OLL

OLLOL

1

01

0

1

1

The average inductor voltage is zero for periodic operation.

)1

(

0)1(

D

DVV

T

TDVDTVV

SO

OSL

OV can be less than the source or greater than the source.

DIVIVPR

VP LSSSS

OO

2

D

DVV

DR

DV

DV

P

RDV

VI SO

S

S

O

S

OL 11 2

2

Imax = L

DTV

DR

DViI SSL

L 2)1(2 2

Imin = L

DTV

DR

DViI SSL

L 2)1(2 2

For continuous :Li Imin = 0

f

RDL

RDLf

2

1min

2

1min)(

2

2

f : switching fre.

ripplevoltageoutputRCf

D

V

V

RCf

DV

RC

DTVVVCDT

R

VQ

O

O

OOOO

O

6-7 The ‘Cuk converter Fig 6-10

Output voltage magnitude can be either larger or smaller than the input, and there is a polarity reversal on the output .

Assumptions: (1) are very large and the currents in them are constant.21 , LL

(2) are very large and the voltage across them are constant.21 ,CC

(3) In steady-state.(4) The switch is closed for time DT and open for (1-D)T.(5) Switch and diode are ideal.

Average voltage across is1C

OSC VVV 1

With the switch closed, 21 LclosedC Ii

With the switch open , 11 LopenC Ii

SLSLOO PIVIVP 12

Average capacitor current is zero for periodic operation.

D

D

V

V

D

D

V

V

I

I

IVPIVP

D

D

I

IDIDI

T

TDiDTi

S

O

S

O

L

L

LOOLSS

L

LLL

openCclosedC

1

1

101

01

2

1

21

2

112

11

Since the components on the output ( ) are in the same configuration as the Buck converter, the output voltage ripple.

RCL ,, 22

2228

1

fCL

D

V

V

O

O

The ripple in (Sw. is open , )1C

11 CL ii

fRCDV

D

DfRC

VT)D(

C

IdtI

Cv

O

SLT

DT LC

1

2

11

11

11 1

11

R

V

D

D

D

D

R

V

D

DI

D

DI SO

LL

1111 21

When Sw. is closed,

fL

DV

L

DTVi

xd

idLVVVVv

fL

DV

L

DTVi

L

V

DT

i,

td

idLVv

SSL

LSOSOL

SSL

SLLsL

222

222

111

1

1111

For continuous current in the inductors, the average current must be greater than one-half the change in current.

22

11

2

12

1

LL

LL

iI

iI

f

RDL

Df

RDL

2

1min,

2

1min,

2

2

1

f

RDL

fL

DV

D

D

R

V

R

VI

Df

RDL

fL

DV

R

V

D

DI

SSOL

SSL

2

1min,

2

1

1

2

1min,

2

1

1

22

2

2

11

2

1

6-8 Non-ideal effects on converter performance

Switch voltage drops:

For Buck converter:With the switch closed, QOSL VVVv ,

QV : voltage drops across conducting switches.

With the switch open, DoL VVv ,

DV : voltage drop across the diode.

Average voltage across the inductor is zero for the switching period:

0

1

T

TDVVDTVVVV DOQOS

L

DVDVDVV DQSO 1 is lower than DVV SO for the ideal case.

Capacitor resistance : effect on ripple (For Buck converter)

Real capacitor : ESR : equivalent series resistance. . ESL : equivalent series inductance

ESR may produce a ripple greater than that of the capacitance.ESL is not a factor at low switching fre , but may be significant above perhaps 300KHZ .

)(VVV

riV

ESR,Oc,O

CCESR,

最差情況0

0

c,OV is obtained under the ideal capacitor . (6-15)

Inductor resistance ( ): For Boost converter.Lr

.inductoridealforD

VV,

DR

rD

VV

DVDR

rVV

D

R/V

D

II

rIDVV

rIDIVrIIVIV

PPP

SO

L

SO

OLO

S

ODL

LLOS

LLLOLLDOLS

rLOS

1

11

1

1

11

11

1

1

2

22

Fig 6-11

Efficiency of Boost converter:

2

22

2

22

2

11

1

1 DR

rr

D

R/VR/V

R/V

rIR/V

R/V

PP

P

L

LO

O

O

LLO

O

LOSSO

O

Switching losses : In addition to the on-state voltage drops and associated power losses of switches , additional losses occur in the switches as they turn on and off , seeing ( Fig6-12 (a) ) , there are two types of different switch on-off transitions, ( Fig 6-12 (b) ) (power losses) may be closer to actual switching situations.

Higher switching fre.→higher switching losses.Reducing switching losses: making switching occur at zero voltage and/or zero current. (Resonant converter)

6-9 Discontinuous-current operation

A different analysis is required for the discontinuous current ( ) case.Li

Buck converter( Fig 6-13 )

Average inductor voltage is zero for periodic operation.

R

VI

DDaxImTaxDImaxDTImT

I

DD

D

V

V

DVDVVT

TDVDTVV

OR

L

S

O

OOSOOS

11

1

11

2

1

2

1

2

11

0

When switch is closed

L

TDVDT

L

VViax

L

VV

DT

ax

DT

i

t

i

L

VV

td

id

td

idLVVv

OOSL

OSLL

OSLLOSL

1Im

Im

,

)

RT

LDD

D(V

DD

DVV

RT

LDD

D

RT

LDDD

R

VDD

L

TDVDDaxImI

SSO

OOL

8

2

2

8

02

2

1

2

1

21

2

1

12

1

11

1

Discontinuous current occurs when , and Imin will be zero.DD 11

Boost converter: Fig 6-15Average voltage across the inductor:

1

1

1 0

D

DDVV

T

TDVVDTV

SO

OSS

Average diode current:

L

RTD

V

V

L

RTD

V

V

V

V

RDT

L

V

V

RDT

L

V

VD

VV

RDT

L

V

VD

RVDL

DTVI

L

DTViax

axDTaxDT

I

S

O

S

O

S

O

S

O

S

O

SO

S

O

OS

D

SclosedL

D

2

22

1

1

11

211

2

1

02

2

2

2

/2

1

Im

Im2

1Im

2

11 Fig. 6.16

Discontinuous current ( ) occurs when , and Imin will be zero.

LiDD 11