Uncoupled Converter and Extra Element...

transcript

Uncoupled Converterand Extra Element Theorem

Chapter Outline

DC Power Distribution System

Uncoupled Converter

Power Stage Dynamics of Uncoupled Converter

Control Design of Uncoupled Converter

Coupled Converters and Middlebrook’s Extra Element Theorem

Load-Coupled Converter

Source-Coupled Converter

Middlebrook’s Feedback Theorem

DC Power Distribution System for Computers D

Cascaded and paralled converters and filter stages for efficient and reliable power conversion

Intermediate line filters to meet regulatory EMI standards

Separate filter stage for each converter

Filterstage

Loadconverter Lo

Filterstage

Loadconverter Lo

Filterstage

Loadconverter Lo

Front-endconverter

DC Power Distribution System for Computers

(s): source impedance or output impedance of source subsystem

(s): load impedance or input impedance of load subsystem

: output current

Filterstage

Loadconverter Lo

Filterstage

Loadconverter Lo

Dc distribution bus

Filterstage

Loadconverter Lo

Front-endconverter

Source subsystrm

Load subsystrm

Equivalent Representation of Front-End ConverterU

and are always known in advance.

( ) and ( ) are unknown or undefined at the design stage of the converter.

Design should be performed without any knowledge about ( ) and (s).

Z s Z s

Front-endconverter

Sourcesubsystem

Loadsubsystem

Front-endconverter

Source subsystrm

Load subsystrm

Uncoupled ConverterU

Front-endconverter

Equivalent representation of converter

Uncoupled converter

The control can be designed independently from the unknown ( )and (s).

The converter performance can be evaluated with an ideal voltage sourceand current sink.

Whenever the information about ( ) and (s) is avai

Z s Z lable, the converter performance can be analyzed using the Extra Element Theorem.

Sv OiFront-endconverter (s) =0: is an ideal voltage source

(s) = : is an ideal current sinks S

Z vZ i

Uncoupled Buck Converter

Small-signalmodel

rampVCSN

d̂sv̂

Coı̂

( )vF s-*

mFconv̂

( )eH s

SV dDˆ

: DC load parameterODC

Control-to-Output Transfer Function

d̂ Coı̂

*mF conv̂

( )eH s

SV dDˆ

( 0.5)1(

s cvc pl

i s c n

T m DLKR T

Smm C SD L

Other parameters are the same as the case of a resistive load.

The DCload parameter does not appear in the transfer function.DCRPow

rhp esrovci vc

pl p n n

s svG s K

v s s sQ

rhp esrovci vc

pl p n n

s svG s K

v s s sQ

trol D

sCurrent Loop Design

10.3 1.31 0.5

Current loop design remains the same as the resistor load case.

d̂ Coı̂

*mF conv̂

( )eH s

SV dDˆ

Voltage Loop Design

1 1 1( ) ( ) ( )

vrhp esr zc

m vci v vc

pl p n pcn

s s sKT s G s F s K

s s s ssQ

| |vciG

20log vcK

20log pQ

rhp esr

trol D

Voltage Feedback Compensation

1 1 1( ) ( ) with ( )

1 1 1 )

vrhp esr zc

m vc v v

pl p n pcn

s s sKT s K F s F s

s s s ssQ

| |vciG

trol D

Selections of ( ) parameters is the same as the case of a resistive load.

min{ / 2} (0.6 0.8)

pzc cr

c rhp esr s op

Voltage loop design becomes the same provided that the product remains unchanged.vc plK

Uncoupled converter Converter with resistive load

rhp esrvci vs

pl p n n

G s Ks s s

1 0.5si c

RRTR m DL

pl 1 1 0.5sc

RT m DCR L

vc plK 1

Product Comparison

1( 0.5)i s c

LR T m D

( 0.5)s cT m DLC

vc plK C

Control Design Summary

rampVCSN

Uncoupled buck converter Resisor-coupled buck converter

Standard design procedures for resist

Control design of the uncoupled convert

ive loads are adoptable to uncoupled co

er remains the same

nverters

as the resistor-loadedconverter.

Conclusions can be extended to other onv

r topologies.

trol D

Converter PerformanceC

rter P

0.01 0.1 1 10 100-300

Frequency (kHz)

: uncoupled buck converter: resistor-loaded buck converter

0.01 0.1 1 10 100-60

Frequency (kHz)

Loop gain

0.01 0.1 1 10 100-120

Magnit

Frequency (kHz)

Output impedance

Audio-susceptibility

Uncoupled Boost Converter

Small-signalmodel

: DC load parameterODC

rampVCSN

OV dDˆ

Coı̂

Lı̂Lı̂

( )vF s-*

mFconv̂

( )eH s

rhp esrovci vc

pl p n n

s svG s K

v s s sQ

Control-to-Output Transfer Function

( 0.5)1

( 0.5)DC D

vc pl rhpi s c

R RLT D m D

D DLK L LCR LTR

D m DD D

The DCload parameter is a key parameter.DCR

OV dDˆ

Coı̂

Lı̂Lı̂

*mF conv̂

( )eH s

vc plK

trol D

sProduct Comparison

vc plK

( 0.5)i s cDC

LLR T D m

2( 0.5)s cDC

LT D mR D

12( 0.5)i s c

LLR T D m

2 2( 0.5)s cLT D m

R DLCD

rhp esrvci vs

pl p n n

G s Ks s s

Converter PerformanceC

rter P

Loop gain Output impedance

0.01 0.1 1 10-360

Frequency (kHz)

: theoretical prediction: computational method

0.01 0.1 1 10-40

Magnitude

Frequency (kHz)

: theoretical prediction: computational method

trol D

sBuck/Boost Converter

vc plK i

( 0.5)i s cDC

LDLR T D m

2( 0.5)s cDC

DLT D mR D

1(1 )( 0.5)i s c

LD LR T D m

2 (1 )( 0.5)s cD LT D m

R DLCD

rhp esrvci vs

pl p n n

G s Ks s s

emMiddlebrook’s Extra Element Theorem

uiu ou

Definitions( )( ) : transfer gain of interest( )

( ): impedance of the circuit component which is desifnated the exta ra elems ent

u sH su s

The Extra Element Theorem:

( ): transfer gain ( ) evaluated with the extra element removed, denoted as the open-circut transfer ga

( )1( )( ) ( ) ( )

in( ) : input impedance looking into a

sZ sH s H s z sZ s

H s H s

a with the input variable ( ) disabled, denoted asthedriving point impedance

( ) : input impedance looking into a-a with the input variable ( ) nullified, denoted asthe null driving point impedance

emPictorial Illumination

uiu ou

u¥iu ou

¢0ou =iu¢

V ¢- +I ¢

( )1( )( ) ( ) ( )1( )

sZ sH s H s z sZ s

( ) ( ) / ( ) with ( )=o iH s u s u s Z s ( ) :H s

( ) :z s ( ) ( ) / ( ) with ( )=0iz s V s I s u s

( ) :s ( ) ( ) / ( ) with ( )=0os V s I s u s

emApplication to MOSFET Amplifier

ivgsC m gsg v

iv gsC m gsg v

Consider as the extra elementgdC

( )Evaluate the voltage gain using EET( )

v sv s

( )1( ) ( )( ) ( ) ( )( ) 1

( )1with ( )

sv s Z sH s H s z sv s

iv gsC m gsg vor

1MOSFET amplifier with 0 ( )gdgd

C Z ssC

( )( )( )

v sH s g rv s

1MOSFET amplifier with 0 ( )gdgd

C Z ssC

( )1( ) ( )

( ) ( ) ( )( ) 1)

sv s Z sH s H s z s

v sZ s

sC z s

gsC 0m gsg v =or

iv gsC m gsg vor

Evaluation of ( )z s( ) 0

( )Evaluation of ( )( )

v ssi s

( ) oz s r

i) ( ) 0 0

ii) ( ) 0

o o T m gs

o T gs

T m gs m

v s i i g v

v s v v

vvsi g v g

1 ( )( )Tranfer gain ( )( )

1 ( )gdo

i gdZ s

sC sv sH s g

sCgg r

rv s sC z s

emAlternative Form of EET

uiu ou

¢0ou =iu¢

V ¢- +I ¢

( )1( )( ) ( ) ( )1( )

Z ssH s H s Z s

0( ) ( ) / ( ) with ( )=0o iH s u s u s Z s0( ) :H s

( ) :z s ( ) ( ) / ( ) with ( )=0iz s V s I s u s

( ) :s ( ) ( ) / ( ) with ( )=0os V s I s u s

emExtension to 2 EET

First adoption of EET

1 2( ) 0 ( )

( ) ( )( )

i Z s Z s

u s H su s

( ) 0 2

( )1( )( ) (

( )) )

( )( 1

Z siZ s

sZ sH s z s

emExtension of EET- 2 EET

Second adoption of EET

2( ) 0 2( )

( )1( ) ( )( ) ( )( ) 1

osZ si

su s Z sH s z su s

( ) 0(

2 12 1

( ) ( )1 1( ) ( )( ) ( ) ( )1 1(

) ( )(

oZ siZ s

s Z sZ s su s

u sH s z s Z s

Z s z s

rLoad-Coupled Converter

Uncoupled converter

( )vF s-

v =¥=ˆˆ ov

ˆ oı̂

LZsv̂

( )vF s-

v ¹¥=ˆˆ ov

ˆoı̂

Load-coupled converter

Performance of Load-Coupled Converter

LZsv̂

( )vF s-

v ¹¥=ˆˆ ov

ˆoı̂

1( ) ( ) ( )1( )

uL uUoU

A s A s Z sZ s

1( ) ( ) ( )1 1 ( )( )

mL mUoU

T s T s Z sT sZ s

1( ) ( ) ( )1( )

oL oUoU

Z s Z s Z sZ s

1( )( ) ( )

1 )( )

Z ssZ s Z s

( )vF s-

oUZ ¢

: closed-loop input-openoutput impedance

Performance of Load-Coupled Boost Converter

160 Hμ

005. W

470 Fμ

045.iR =

W.0045

1250 Fμ

21 Hμ W.0015

Buck converter (92 W)

46 VOV

50 kHzeS = ´. 4384 10 V/s

096nF.

217 nF. 28kW

W4552 V

Load subsytem

Filter stage

0.01 0.1 1 10 100-40

Magnit

Frequency (kHz)

( )LZ s

Performance of Load-Coupled Boost Converter

uUA| |

uLA| |oUZ| |

oLZ| |

0.01 0.1 1 10 100-40

Magnit

Frequency (kHz)

0.01 0.1 1 10 100-80

Magnit

Frequency (kHz)

: load-coupled converter: uncoupled converter

0.01 0.1 1 10 1000

Magnit

Frequency (kHz)

| |iUZ

| |iLZ

mUT| |

mLT| |

0.01 0.1 1 10 100-40

Magnitude

Frequency (kHz)

r Loop gain

Output impedanceAudio-susceptibilty

Input impedance

rSource-Coupled Converter

Uncoupled converter

( )vF s-

v =¥=ˆˆ ov

ˆ oı̂

Source-coupled converter

Oi sv̂

( )vF s-

ˆ oı̂

rPerformance of Source-Coupled Converter

1( ) ( ) ( )1( )

uS uUs

A s A s Z sZ s

( )1( )

( ) ( ) ( )1( )

imS mU

Z sZ s

T s T s Z sZ s

( )vF s-

ˆ oı̂

( )1( )

( ) ( ) ( )1( )

ioS oU

Z sZ s

Z s Z s Z sZ s

emInput Impedance Definition

(s)vF-

: open-loop output-shorted input impedanceiZ ¢

( )vF s-

iZ ¢¢

: output-nullified input impedanceiZ ¢¢

( )vF s-

iZ ¢¢¢

: open-loop input impedanceiZ ¢¢¢

( )vF s-

: closed-loop input impedanceiUZ

rPerformance of Source-Coupled Boost Converter

0.01 0.1 1 10-80

agnitude

Frequency (kHz)

: source coupled converter: uncoupled boost converter

| |uSA | |uUA

0.01 0.1 1 10-80

Magnitude

Frequency (kHz)

| |oSZ

| |oUZ

0.01 0.1 1 10-270

Frequency (kHz)

Magnit

| |mST

| |mUT

Audio-susceptibilty Loop gain

Output impedance

rSource/Load-Coupled Converter

( )vF s-

ˆ oı̂ LZ

1Load-coupled converter: ( ) ( ) ( )1 1 ( )

mL mUoU

T s T s Z sT sZ s

( )1( )( )1( )

Source/load coupled converter:

1( ) ( ) ( )1 (1 ( ))( )

mC mUoU

T s T s Z s

Z sZ sZs sT

Z s Z s

oremMiddlebrook’s Feedback Theorem

( ) ( ) 1( ) = (s) + ( )( ) 1 ( ) 1 ( )

u s T sH s H H su s T s T s

1iu 1ou

2ou2iu

( )A s

( )a s

Two-output feedback controlled system

( ) ( ) : feedback-signal nullified transfer gain

( ) ( ) : open loop transfer gain( ) ( ) ( ) : loop gain

H s H s

H s H sT s A s a s

( ): feedback gain( ): forward gain

A sa s

Pictorial Illumination

0( ) 1( ) (s) + ( )

1 ( ) 1 ( )m

T sH s H H sT s T s

1iu 1ou

2ou2iu

( )A s

( )a s

1 0iu =1ou

2ou2iu

( )A s

( )a s

( )mT s ( ) ( ) ( )mT s A s a s

0( )H s1

01 ( ) 0

( )( )( )

u sH su s

1iu 1ou

2ou2iu

( )a s

( )H s

( )( )( )

u sH su s

1iu 1ou

2 0ou =2iu

( )A s

( )a s

Single-Output Feedback System

1 ( ) 0

( ) ( ) 1( ) = (s) + ( )( ) 1 ( ) 1 ( )

( )(s) 0(

( ) 1( ) = ( )( ) 1 ( )

u sH s H su s T

T sH s H H su s T s T s

( )A s

( )a s

emInput Impedance of Uncoupled Converter

( )( )1 1 1 1+( ) ( ) ( ) 1 ( ) ( ) 1 ( )

iU T i mU i mU

T si sZ s v s Z s T s Z s T s

( )vF s-

ˆoı̂

( ): output nullified input impedance

( ): open-loop input impedance( ): loop gain

Z sT s

for frequencies wherefor frequencies where

( ) | |

( ) ( ) | |( ) i mU

i mUiU

Z s TZ

s s Ts

æ ¢¢ççç ¢¢¢çè· »

Chapter Summary

DC Power Distribution System

Uncoupled Converter

Power Stage Dynamics and Control Design of Uncoupled Converter

Coupled Converters and Middlebrook’s Extra Element Theorem

Load-coupled converter

Source-coupled converter

Source/load-coupled converter

Middlebrook’s Feedback Theorem

Uncoupled Converter and Extra Element...

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