uncoupled converter and extra element...

Uncoupled Converterand Extra Element Theorem

Pulsewidth Modulated Dc-to-Dc Power Conversion. By Byungcho ChoiCopyright © 2013 IEEE, Published by John Wiley & Sons, Inc

2

22

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

3

33

DC Power Distribution System for Computers D

C P

ower

Dis

tribu

tion

Sys

tem

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

Filterstage

Loadconverter Lo

ad

Filterstage

Loadconverter Lo

ad

vF

Filterstage

Loadconverter Lo

ad

Front-endconverter

4

44

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

s

L

O

Z

Z

i

Filterstage

Filterstage

Loadconverter Lo

ad

sZ

Filterstage

Loadconverter Lo

ad

LZ

vF

Dc distribution bus

Filterstage

Loadconverter Lo

ad

Front-endconverter

Source subsystrm

Load subsystrm

Oi

DC

Pow

er D

istri

butio

n S

yste

m

5

55

Equivalent Representation of Front-End ConverterU

ncou

pled

Con

verte

r

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).

S O

s L

s L

v i

Z s Z s

Z s Z

vF

Sv

Oi

LZ

sZ

Front-endconverter

Sourcesubsystem

Loadsubsystem

sZ LZ

Front-endconverter

Source subsystrm

Load subsystrm

Oi

6

66

Uncoupled ConverterU

ncou

pled

Con

verte

r

vF

Sv

Oi

LZ

sZ

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

s L

s L

Z s Z

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

vF

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

(s) = : is an ideal current sinks S

L O

Z vZ i

7

77

Uncoupled Buck Converter

Small-signalmodel

Pow

er S

tage

Dyn

amic

s

Sv

L lR

cR

C

rampVCSN

PWM

2Z

1Z

OI

Iv

conv

Ov

+

-

d̂sv̂

Coı̂

cR

L lR

Lı̂

( )vF s-*

mFconv̂

Iv̂

iR

fk

( )eH s

rk

SV dDˆ

O

DC

Vd

Rˆ

1 :D

ov

+

-

ˆ

: DC load parameterODC

O

VRI

8

88

Control-to-Output Transfer Function

d̂ Coı̂

cR

L lR

Lı̂

*mF conv̂

Iv̂

iR

fk

( )eH s

rk

SV dDˆ

O

DC

Vd

Rˆ

1 :D

ov

+

-

ˆ

( 0.5)1(

with0

1.5)

s cvc pl

ec

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

er S

tage

Dyn

amic

s

2

2

1 1ˆ

( )ˆ

1 1

rhp esrovci vc

con

pl p n n

s svG s K

v s s sQ

9

99

2

2

1 1ˆ

( )ˆ

1 1

rhp esrovci vc

con

pl p n n

s svG s K

v s s sQ

Con

trol D

esig

n P

roce

dure

sCurrent Loop Design

10.3 1.31 0.5

pe

n

QS DS

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

d̂ Coı̂

cR

L lR

Lı̂

*mF conv̂

Iv̂

iR

fk

( )eH s

rk

SV dDˆ

O

DC

Vd

Rˆ

1 :D

ov

+

-

ˆ

10

1010

Voltage Loop Design

2

2

1 1 1( ) ( ) ( )

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

1( )

1

vzc

v

pc

sKF s

ss

| |vciG

| |mT

20log vcK

20log pQ

2

1

2

0 dB

11

1n

rhp esr

zccr

pl

pc

Con

trol D

esig

n P

roce

dure

s

11

1111

Voltage Feedback Compensation

2

2

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

| |mT

0 dB

Con

trol D

esig

n P

roce

dure

s

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

min{ / 2} (0.6 0.8)

v

pzc cr

vv

c rhp esr s op

zcc l

F s

KK

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

12

1212

Uncoupled converter Converter with resistive load

2

2

1 1

( )

1 1

rhp esrvci vs

pl p n n

s s

G s Ks s s

Q

vcK

1

1 0.5si c

RRTR m DL

pl 1 1 0.5sc

RT m DCR L

vc plK 1

iR C

Product Comparison

1( 0.5)i s c

LR T m D

( 0.5)s cT m DLC

1

iR C

vc plK C

ontro

l Des

ign

Pro

cedu

res

13

1313

Control Design Summary

Sv

L lR

cR

C

rampVCSN

PWM

2Z

1Z

OI

Iv

conv

Ov

+

-

Sv

L lR

cR

C

rampVCSN

2Z

1Z

Iv

conv

Ov

+

-

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

.

erte

r topologies.

Con

trol D

esig

n P

roce

dure

s

14

1414

Converter PerformanceC

onve

rter P

erfo

rman

ce

0.01 0.1 1 10 100-300

-240

-180

-120

-60

0 -60

-40

-20

0

20

40

60

80

Ph

ase

(d

eg)

Frequency (kHz)

Magn

itu

de

(dB

)

: uncoupled buck converter: resistor-loaded buck converter

0.01 0.1 1 10 100-60

-40

-20

0

Magn

itude

(dB)

Frequency (kHz)


Loop gain

0.01 0.1 1 10 100-120

-100

-80

-60

-40

Magnit

ude

(dB

)

Frequency (kHz)


Output impedance

Audio-susceptibility

15

1515

Uncoupled Boost Converter

Small-signalmodel

Pow

er S

tage

Dyn

amic

s

: DC load parameterODC

O

VRI

Sv

L lR

cR

C

rampVCSN

PWM

2Z

1Z

OI

Iv

conv

Ov

+

-

OV dDˆ

Coı̂

cRˆO

DC

Vd

R D¢

L lR

1:D

sv̂

d̂

Lı̂Lı̂

( )vF s-*

mFconv̂

Iv̂

iR

fk

( )eH s

rk

ov

+

-

ˆ

16

1616

2

2

1 1ˆ

( )ˆ

1 1

rhp esrovci vc

con

pl p n n

s svG s K

v s s sQ

Control-to-Output Transfer Function

22

2

( 0.5)1

( 0.5)DC D

s cC

DC

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

Pow

er S

tage

Dyn

amic

s

OV dDˆ

Coı̂

cRˆO

DC

Vd

R D¢

L lR

1:D

d̂

Lı̂Lı̂

*mF conv̂

Iv̂

iR

fk

( )eH s

rk

ov

+

-

ˆ

17

1717


vcK

pl

vc plK

Con

trol D

esig

n P

roce

dure

sProduct Comparison

i

DR C

vc plK

2

1

( 0.5)i s cDC

LLR T D m

R D

2( 0.5)s cDC

LT D mR D

LCD

i

DR C

2

12( 0.5)i s c

LLR T D m

R D

2 2( 0.5)s cLT D m

R DLCD

2

2

1 1

( )

1 1

rhp esrvci vs

pl p n n

s s

G s Ks s s

Q

18

1818

Converter PerformanceC

onve

rter P

erfo

rman

ce

Loop gain Output impedance

0.01 0.1 1 10-360

-300

-240

-180

-120

-60

0 -60

-40

-20

0

20

40

60

Phase

(deg

)

Frequency (kHz)

Magn

itude

(dB

)

: theoretical prediction: computational method

0.01 0.1 1 10-40

-30

-20

-10

0

10

Magnitude

(dB

)

Frequency (kHz)

: theoretical prediction: computational method

19

1919

Con

trol D

esig

n P

roce

dure

sBuck/Boost Converter


vcK

pl

vc plK i

DR C

2

1

( 0.5)i s cDC

LDLR T D m

R D

2( 0.5)s cDC

DLT D mR D

LCD

i

DR C

2

1(1 )( 0.5)i s c

LD LR T D m

R D

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

R DLCD

2

2

1 1

( )

1 1

rhp esrvci vs

pl p n n

s s

G s Ks s s

Q

20

2020

Ext

ra E

lem

ent T

heor

emMiddlebrook’s Extra Element Theorem

aa

= o

i

uH

uiu ou

Z

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

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

o

i

u sH su s

Z 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

(

-

1)

sZ sH s H s z sZ s

H s H s

z 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

i

o

u s

s u s

21

2121

Ext

ra E

lem

ent T

heor

emPictorial Illumination

aa

= o

i

uH

uiu ou

Z

iuou

V- +I

Vz

I=

= o

i

uH

u¥iu ou

Vζ

I

¢=

¢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

22

2222

Ext

ra E

lem

ent T

heor

emApplication to MOSFET Amplifier

ivgsC m gsg v

or

gdC

gsv

+

-ov

+

-

iv gsC m gsg v

or

gdC

gsv

+

-ov

+

-

Consider as the extra elementgdC

( )Evaluate the voltage gain using EET( )

o

i

v sv s

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

( )1with ( )

o

i

gd

sv s Z sH s H s z sv s

Z s

Z ssC

23

2323

Ext

ra E

lem

ent T

heor


iv gsC m gsg vor

G D

S

ivgsv

+

-ov

+

-

iv gsC m gsg vor

gdC

gsv

+

-ov

+

-

1MOSFET amplifier with 0 ( )gdgd

C Z ssC

( )

( )( )( )

om o

i Z s

v sH s g rv s

1MOSFET amplifier with 0 ( )gdgd

C Z ssC

( )

( )1( ) ( )

1 (

( ) ( ) ( )( ) 1)

)1 )

(

(gd

m ogd

o

i Z s

sv s Z sH s H s z s

sC

v sZ s

sg r

sC z s

24

2424

Ext

ra E

lem

ent T

heor


gsC 0m gsg v =or

0gsv

+=

-

iv gsC m gsg vor

Tv

Ti

oi

0ov

+

=

-

gsv

+

-

Evaluation of ( )z s( ) 0

( )Evaluation of ( )( )

O

T

T v s

v ssi s

( ) oz s r

i) ( ) 0 0

ii) ( ) 0

1( )

o o T m gs

o T gs

gsT

T m gs m

v s i i g v

v s v v

vvsi g v g

( )

1 ( )( )Tranfer gain ( )( )

1

1

1 ( )gdo

m o

gd

mm o

gd o

i gdZ s

sC sv sH s g

sCgg r

sC r

rv s sC z s

25

2525

Ext

ra E

lem

ent T

heor

emAlternative Form of EET

aa

= o

i

uH

uiu ou

Z

iuou

V- +I

Vz

I=

0=o

i

uH

uiu ou

Vζ

I

¢=

¢0ou =iu¢

V ¢- +I ¢

0

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

Z ssH s H s Z 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

26

2626

Ext

ra E

lem

ent T

heor

emExtension to 2 EET

iu ou

1Z 2Z

iu ou

First adoption of EET

1 2( ) 0 ( )

( ) ( )( )

os

i Z s Z s

u s H su s

iu ou

2Z

12

2

( ) 0 2

2(

2

)

( )1( )( ) (

( )) )

( )( 1

so

Z siZ s

sZ sH s z s

su

Z

us

s

27

2727

Ext

ra E

lem

ent T

heor

emExtension of EET- 2 EET

iu ou

1Z 2Z

Second adoption of EET

iu ou

2Z

12

2

2( ) 0 2( )

2

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

( )

osZ si

Z s

su s Z sH s z su s

Z s

iu ou

2Z1Z

12

( ) 0(

2 1

2 12 1

1)

2

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

( ))

) ( )(

oZ siZ s

s

s Z sZ s su s

u sH s z s Z s

Z s z s

28

2828

Load

Cou

pled

Con

verte

rLoad-Coupled Converter

Uncoupled converter

sv̂

( )vF s-

oUZ

mUT

iUZ

L

ouU

s Z

vA

v =¥=ˆˆ ov

+

-

ˆ oı̂

LZsv̂

( )vF s-

oLZ

mLT

L

ouL

s Z

vA

v ¹¥=ˆˆ ov

+

-

ˆoı̂

iLZ

Load-coupled converter

vF

Sv

Oi

LZ

vF

Sv Oi

29

2929

Performance of Load-Coupled Converter

LZsv̂

( )vF s-

oLZ

mLT

L

ouL

s Z

vA

v ¹¥=ˆˆ ov

+

-

ˆoı̂

iLZ

1( ) ( ) ( )1( )

uL uUoU

L

A s A s Z sZ s

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

mL mUoU

mUL

T s T s Z sT sZ s

1( ) ( ) ( )1( )

oL oUoU

L

Z s Z s Z sZ s

( )

(

1( )( ) ( )

1 )( )

oU

Li

oU

UL i

L

Z sZ

Z ssZ s Z s

Z s

( )vF s-

oUZ ¢

oı̂

: closed-loop input-openoutput impedance

oUZ¢

Load

Cou

pled

Con

verte

r

30

3030

Performance of Load-Coupled Boost Converter

24V

160 Hμ

005. W

470 Fμ

045.iR =

PWM

W.0045

W.003

1250 Fμ

21 Hμ W.0015

Buck converter (92 W)

LZ

46 VOV

+

=

-

2 A

50 kHzeS = ´. 4384 10 V/s

096nF.

217 nF. 28kW

W10k

W4552 V

Load subsytem

Filter stage

0.01 0.1 1 10 100-40

-20

0

20

40

Magnit

ude

(dB

)

Frequency (kHz)

( )LZ s

Load

Cou

pled

Con

verte

r

31

3131

Performance of Load-Coupled Boost Converter

uUA| |

uLA| |oUZ| |

oLZ| |

0.01 0.1 1 10 100-40

-20

0

Magnit

ude

(dB

)

Frequency (kHz)

0.01 0.1 1 10 100-80

-60

-40

-20

Magnit

ude

(dB

)

Frequency (kHz)

: load-coupled converter: uncoupled converter


0.01 0.1 1 10 1000

10

20

30

40

50

Magnit

ude

(dB

)

Frequency (kHz)

| |iUZ

| |iLZ


mUT| |

mLT| |

0.01 0.1 1 10 100-40

-20

0

20

40

60

Magnitude

(dB

)

Frequency (kHz)


Load

Cou

pled

Con

verte

r Loop gain

Output impedanceAudio-susceptibilty

Input impedance

32

3232

Sou

rce-

Cou

pled

Con

verte

rSource-Coupled Converter

Uncoupled converter

sv̂

( )vF s-

oUZ

mUT

iUZ

L

ouU

s Z

vA

v =¥=ˆˆ ov

+

-

ˆ oı̂

Source-coupled converter

vF

Sv Oi

vF

Sv

sZ

Oi sv̂

( )vF s-

oSZ

mST

0

ˆˆ

s

ouS

s Z

vA

v ¹=

sZ

ov

+

-

ˆ oı̂

33

3333

Sou

rce-

Cou

pled

Con

verte

rPerformance of Source-Coupled Converter

1( ) ( ) ( )1( )

uS uUs

iU

A s A s Z sZ s

( )1( )

( ) ( ) ( )1( )

s

imS mU

s

i

Z sZ s

T s T s Z sZ s

sv̂

( )vF s-

oSZ

mST

0

ˆˆ

s

ouS

s Z

vA

v ¹=

sZ

ov

+

-

ˆ oı̂

( )1( )

( ) ( ) ( )1( )

s

ioS oU

s

iU

Z sZ s

Z s Z s Z sZ s

34

3434

Ext

ra E

lem

ent T

heor

emInput Impedance Definition

(s)vF-

sv̂

iZ ¢

: open-loop output-shorted input impedanceiZ ¢

( )vF s-

sv̂

iZ ¢¢

0ov

+=-

ˆ

: output-nullified input impedanceiZ ¢¢

( )vF s-

sv̂

iZ ¢¢¢

: open-loop input impedanceiZ ¢¢¢

sv̂

( )vF s-

iUZ

: closed-loop input impedanceiUZ

35

3535

Sou

rce-

Cou

pled

Con

verte

rPerformance of Source-Coupled Boost Converter

0.01 0.1 1 10-80

-60

-40

-20

0

M

agnitude

(dB

)

Frequency (kHz)

: source coupled converter: uncoupled boost converter

| |uSA | |uUA

0.01 0.1 1 10-80

-60

-40

-20

0

Magnitude

(dB

)

Frequency (kHz)


| |oSZ

| |oUZ

0.01 0.1 1 10-270

-180

-90

0 -40

-20

0

20

40

60

Phase

(deg

)

Frequency (kHz)

Magnit

ude

(dB

)

| |mST

| |mUT

mST

mUT



Audio-susceptibilty Loop gain

Output impedance

36

3636

Sou

rce/

Load

-Cou

pled

Con

verte

rSource/Load-Coupled Converter

sv̂

( )vF s-

mCT

sZ

ov

+

-

ˆ oı̂ LZ

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

( )

mL mUoU

mUL

T s T s Z sT sZ s

( )1( )( )1( )

Source/load coupled converter:

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

s

mC mUoU

mUL

i

s

i

T s T s Z s

Z sZ sZs sT

Z s Z s

37

3737

Feed

back

The

oremMiddlebrook’s Feedback Theorem

01

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

oi m

i m m

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

2 0

0 0

( ) ( ) : feedback-signal nullified transfer gain

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

ou

A

mU

H s H s

H s H sT s A s a s

( ): feedback gain( ): forward gain

A sa s

38

3838

Feed

back

The

orem

Pictorial Illumination

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

1 ( ) 1 ( )m

m 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

( )( )( )

o

i A s

u sH su s

1iu 1ou

2ou2iu

( )a s

( )H s

2

1

1 0

( )( )( )

o

o

i u

u sH su s

1iu 1ou

2 0ou =2iu

( )A s

( )a s

39

3939

Feed

back

The

orem

Single-Output Feedback System

1

10

1

1

1 ( ) 0

10

1

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

( )(s) 0(

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

)o

o m

i m m

o

i

m

u

i

s

o

u s

u sH s H su s T

T sH s H H su s T s T s

u sH

s

u s

1iu

1ou

2iu

( )A s

( )a s

40

4040

Ext

ra E

lem

ent T

heor

emInput Impedance of Uncoupled Converter

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

mUT

iU T i mU i mU

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

( )vF s-

Tv

+

-Ti

mUT

ov

+

-

ˆoı̂

( ): output nullified input impedance

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

i

i

mU

Z s

Z sT s

for frequencies wherefor frequencies where

1

1

( ) | |

( ) ( ) | |( ) i mU

i mUiU

vs

Z s TZ

s s Ts

Z G

æ ¢¢ççç ¢¢¢çè· »

41

4141

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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