uncoupled converter and extra element...
TRANSCRIPT
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)
: uncoupled buck converter: resistor-loaded buck converter
Loop gain
0.01 0.1 1 10 100-120
-100
-80
-60
-40
Magnit
ude
(dB
)
Frequency (kHz)
: uncoupled buck converter: resistor-loaded buck converter
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
Uncoupled converter Converter with resistive load
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
Uncoupled converter Converter with resistive load
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
emApplication to MOSFET Amplifier
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
emApplication to MOSFET Amplifier
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
: load-coupled converter: uncoupled converter
0.01 0.1 1 10 1000
10
20
30
40
50
Magnit
ude
(dB
)
Frequency (kHz)
| |iUZ
| |iLZ
: load-coupled converter: uncoupled converter
mUT| |
mLT| |
0.01 0.1 1 10 100-40
-20
0
20
40
60
Magnitude
(dB
)
Frequency (kHz)
: load-coupled converter: uncoupled converter
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)
: source coupled converter: uncoupled boost converter
| |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
: source coupled converter: uncoupled boost converter
: source coupled converter: uncoupled boost converter
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