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1
課程講義:【電力電子】 DC-DC Converter - Modeling Techniques交通大學
808-電力電子實驗室 June 4, 2009
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http://pemclab.cn.nctu.edu.tw/Lab-808: Power Electronic Systems
& Chips Lab., NCTU, Taiwan
1/26
2009年6月4日
鄒 應 嶼 教 授
國立交通大學 電機與控制工程研究所
Right-Half-Plane (RHP) Zero
LAB808NCTU
Lab808: 電力電子系統與晶片實驗室Power Electronic Systems & Chips, NCTU,
TAIWAN
台灣新竹•交通大學•電機與控制工程研究所
台灣新竹‧交通大學‧電機與控制工程研究所‧808實驗室電力電子系統晶片、數位電源、DSP控制、馬達與伺服控制
http://pemclab.cn.nctu.edu.tw/Lab-808: Power Electronic Systems
& Chips Lab., NCTU, Taiwan
2/26
Zero in the Boost and Buck/Boost Converters (CCM)
• Boost
( )( )
( )( ( ))
( )'
'v sd s
VD
r Cs sL D Rs
Qs
o i C
o o
2
2
2
1 11 1
Cr
DCr
RDL
DQ
LCD
CLo
o
1
L
C
D
vovi
• Buck/Boost
L C
D
vovi ( )( )
( )( ( ))
( )'
'v sd s
VD
r Cs sDL D Rs
Qs
o i C
o o
2
2
2
1 11 1
C
Dr
Dr
RDL
DQ
LCD
LCo
o
))(
()(
1
2 2
Why the RHP zero will occur? Where is the RHP zero? What is the
effect of the RHP zero? Can the RHP zero be eliminated?
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2
課程講義:【電力電子】 DC-DC Converter - Modeling Techniques交通大學
808-電力電子實驗室 June 4, 2009
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& Chips Lab., NCTU, Taiwan
3/26
Right-Half-Plane (RHP) Zero
A Mathematical Introduction to Control Theory, Shlomo Engelber,
Imperial College Press,pp. 147, The Effect of Zeros in the Right
Half-Plane
Understanding the Right-Half-Plane Zero, May 1, 2009 12:00
PMCHRISTOPHE BASSO, Director, Product Application Engineering, ON
Semiconductor, Phoenix
Tricks of the Trade - understanding the right-half-plane zero in
small-signal DC-DC converter models, IEEE Power Electronics Society
NEWSLETTER, January 2001.
Right-Half-Plane Zero (of DC-DC Converters) - A simplified
explanation, Lloyd Dixon, 1984, (slup084)
4/26
RHP Zero of Boost and Buck-Boost Converters
Right-Half-Plane Zero - A simplified explanation, Lloyd Dixon,
1984, (slup084)
Switchmode power supply handbookKeith H. Billings
Incremental changes in duty ratio.
L C
D
voviIO
IL
DFlyback Circuit, Continuous Inductor Current Mode
IO
IL
d̂
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3
課程講義:【電力電子】 DC-DC Converter - Modeling Techniques交通大學
808-電力電子實驗室 June 4, 2009
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& Chips Lab., NCTU, Taiwan
Right-Half-Plane (RHP) Zero
Frequency response of RHP zero in voltage-mode control
Frequency response of RHP zero in current-mode control
The RHP zero is still present with current mode control!
OOiOiL VDVVDVDVV )()1(
dVVDVdVVV OiOOiL ˆ)()1(ˆˆ)(ˆ
dIdLVjdId
LDVVji LiLOiO ˆˆˆˆ
)1)((
)()1()1( 22
OiO
iOO
O
iOz VVVL
VRDLDR
VLDVR
LOi
LL
Oi
iL
Oi
LLO iVV
ILjiVVVi
VVILjDii ˆ
)(ˆ
)(ˆ
)()1(ˆˆ
fz
40
20
0
-20
-40
dio ˆˆ
(dB)
1 10 100 1K 10K 100(Hz)
0
-90
-180
dio ˆˆ(Deg)
fz
40
20
0
-20
-40
Lo ii ˆˆ
(dB)
1 10 100 1K 10K 100
0
-90
-180
Lo ii ˆˆ(Deg)
6/26
RHP Zero of CCM Buck-Boost (Flyback) Converter
L
iO ILs
Vdi
ˆ
DLDR
LIV OL
iz
2
2)1(
• Buck/Boost
L C
D
vovi ( )( )
( )( ( ))
( )'
'v sd s
VD
r Cs sDL D Rs
Qs
o i C
o o
2
2
2
1 11 1
C
Dr
Dr
RDL
DQ
LCD
LCo
o
))(
()(
1
2 2
1P
2P
CrCz
11
xRQ
20
0
xR
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4
課程講義:【電力電子】 DC-DC Converter - Modeling Techniques交通大學
808-電力電子實驗室 June 4, 2009
台灣新竹‧交通大學‧電機與控制工程研究所‧808實驗室電源系統與晶片、數位電源、馬達控制驅動晶片、DSP/FPGA控制
http://pemclab.cn.nctu.edu.tw/Lab-808: Power Electronic Systems
& Chips Lab., NCTU, Taiwan
7/26
RHP Zero of CCM Boost Converter
The effect of any control action during the ON time is delayed
until the switch is turned FF.
Output response is initially in the opposite direction of the
desired correction.
Brian Lynch, Under the Hood of a DC-DC Boost Converter, 2008-09
Power Supply Design Seminar - SEM1800.
RHP Zero
L
C
D
vovi R
8/26
Compensation of RHP Zero
REF: LM5000 High Voltage Switch Mode Regulator
A current mode control boost regulator has an inherent right
half plane zero (RHP zero). This zero has the effect of a zero in
the gain plot, causing an imposed +20dB/decade on the rolloff, but
has the effect of a pole in the phase, subtracting another 90° in
the phase plot. This can cause undesirable effects if the control
loop is influenced by this zero. To ensure the RHP zero does not
cause instability issues, the control loop should be designed to
have a bandwidth of ½ the frequency of the RHP zero or less. This
zero occurs at a frequency of:
where ILOAD is the maximum load current.
Hz) (inRFPzeroLOAD
OUT
LIDV
2)( 2'
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5
課程講義:【電力電子】 DC-DC Converter - Modeling Techniques交通大學
808-電力電子實驗室 June 4, 2009
台灣新竹‧交通大學‧電機與控制工程研究所‧808實驗室電源系統與晶片、數位電源、馬達控制驅動晶片、DSP/FPGA控制
http://pemclab.cn.nctu.edu.tw/Lab-808: Power Electronic Systems
& Chips Lab., NCTU, Taiwan
Compensation of OP-Amp uA741
Kent H. Lundberg, "Internal and External Op-Amp Compensation: A
Control-Centric Tutorial," American Control Conference, 2004.
Complete Fairchild µA741 schematic. The primary signal path is
comprised of two gain stages. The first gain stage is a
differential pair with active load (transistors Q1–Q6). The second
gain stage is a common-emitter amplifier (transistors Q16and Q17).
The output buffer is a push-pull emitter follower (transistors Q14,
Q20, and Q22).
Compensation for OP-AMP
(a) Simplified schematic of the uncompensated Fairchild µA741 op
amp. (b) Simplified schematic of the Fairchild µA741 op amp with
compensation capacitor.
(d) Frequency response of uncompensated and compensated op amp.
The two low-frequency poles in the uncompensated transfer function
severely degrade the phase margin at crossover.
(d) Equivalent circuit of uncompensated and compensated
OP-Amp.
A1 A2 1
C
Vin V0
A1 A2 1Vin V0
Compensated
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6
課程講義:【電力電子】 DC-DC Converter - Modeling Techniques交通大學
808-電力電子實驗室 June 4, 2009
台灣新竹‧交通大學‧電機與控制工程研究所‧808實驗室電源系統與晶片、數位電源、馬達控制驅動晶片、DSP/FPGA控制
http://pemclab.cn.nctu.edu.tw/Lab-808: Power Electronic Systems
& Chips Lab., NCTU, Taiwan
RHP Zero of a Common-Emitter Amp
(a) Common-emitter amp. (b) Equivalent circuit
2121212
21212211
21
)]([)]([1)(
RRCCCCCsRRRRgCRCRCsRRgsC
IV
fmf
mf
i
o
There is a RHP Zero located at f
m
Cgs
For typical values, for example, in the µA741 of and , VmAgm
/8.6 pFCf 30
MHzCgsf
m 361030108.6
12
3
The RHP zero is usually not a problem in bipolar op amps because
it is at a much higher frequency than the dominant pole, and can be
neglected!
In a CMOS op amp, where transistor transconductances can be much
lower, the right-half plane zero frequency can be quite close to
the unity-gain frequency of the op amp.
fC
2CVgmiI
oV
2R1R 1CV
B CfC
B
C
12/26
3-dB FrequencyPoles and Zeros Can Be Easily Determined
If P1>> P2, P3, , Z1, Z2,then for frequencies near the
midband:
and, Otherwise,
If P1>> P2, P3, , Z1, Z2,then for frequencies near the
midband:
and, Otherwise,
High-Frequency ResponseLow-Frequency Response
)()( sFAsA LM
)())(()())((
)(21
21
L
L
PnPP
ZnZZL sss
ssssF
)()( sFAsA HM
)1()1)(1()1()1)(1(
)(21
21
H
H
PnPP
ZnZZH sss
ssssF
pole)(Dominant 1
)(P
L sssF
.1PL
)(2 2 221
22
21 ZZPPL ωωωω
2
22
12
22
1
112111ZZPP
H ωωωω
pole)(Dominant 11
1)(P
H ssF
.1PH
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7
課程講義:【電力電子】 DC-DC Converter - Modeling Techniques交通大學
808-電力電子實驗室 June 4, 2009
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http://pemclab.cn.nctu.edu.tw/Lab-808: Power Electronic Systems
& Chips Lab., NCTU, Taiwan
13/26
3-dB FrequencyPoles and Zeros Cannot Be Easily Determined
If a dominant pole exists (say, P1), then
Thus,
If a dominant pole exists (say, P1), then
Thus,
High-Frequency ResponseLow-Frequency Response
11
11)( nLnLnLnL
L sessdssF
111 PLPe and
nL
i isiL RC1
1
nL
iioiH RC
11
11
11
PHP
b
and
221
221
11)(
sbsbsasasFH
PnLPPe 211PnHPP
b
11121
1
nL
i isiRCe
11
1
nH
iioiRCb
11
14/26
Dominant Pole
2121212
21212211
21
)]([)]([1)(
RRCCCCCsRRRRgCRCRCsRRgsC
IV
fmf
mf
i
o
If the RHP Zero be neglected,
2121212
21212211
21
)]([)]([1 RRCCCCCsRRRRgCRCRCsRRg
IV
fmf
m
i
o
If the two poles are two separated real poles,
11
21
11pp
m
i
o
ssRRg
IV
1P2Pf
m
Cgs
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8
課程講義:【電力電子】 DC-DC Converter - Modeling Techniques交通大學
808-電力電子實驗室 June 4, 2009
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15/26
Dominant Pole
If p1
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& Chips Lab., NCTU, Taiwan
17/26
Miller Theorem
Miller theorem describes the way to convert a floating load into
two grounded loads, in such way that the voltages and currents are
remained unchanged.
X YZ
I
X Y
Z1
I I
Z2
X
YYX
VVA
ZVVI ,
X
Y
YXX
VVZZ
ZVVZIZV
1111 A
ZZ
11
Y
X
XYY
VVZZ
ZVVZIZV
1222
A
ZZ 112
18/26
Miller Effect on the Compensation Capacitance
(a) Common-emitter amp. (b) Equivalent circuit
fC
2CVgmiI
oV
2R1R 1CV
B CfC
B
C
(c) Equivalent circuit
fC
2CiI
oV
2R1R V
B CVA
2RgA mV 1C 2CiI
oV
2R1R V
B C
fV CA1C fC
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10
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& Chips Lab., NCTU, Taiwan
19/26
Poles Splitting By Miller Effect
High-frequency model of the effective capacitive loading by the
compensation capacitor. When the compensation capacitor C is
removed from the circuit at left, the circuit is transformed into
the circuit at right so that the capacitive loading on each stage
is maintained.
A1 A2 1
C
Vin V0
increasing C
'1 1 2 '2
CV1 V0
C1 C2 C2CC1C
C2
V0V1
20/26
Block Diagram of the OP-AMP Equivalent Circuit
Block diagram of the op-amp equivalent circuit, including the
feedforward current through the compensation capacitor. The
feedforward term causes a right half-plane zero in the op-amp
transfer function.
1222
sCRR
11111
sCRRGM
1MGsC
2MGinV OV
sC
Kent H. Lundberg, "Internal and External Op-Amp Compensation: A
Control-Centric Tutorial," American Control Conference, 2004.
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11
課程講義:【電力電子】 DC-DC Converter - Modeling Techniques交通大學
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http://pemclab.cn.nctu.edu.tw/Lab-808: Power Electronic Systems
& Chips Lab., NCTU, Taiwan
21/26
RHP Zero of a Common-Source CMOS Amp
CMOS: circuit design, layout, and simulation, R. Jacob Baker,
IEEE Press, 2nd Ed., 2005.
22/26
Limitations Imposed by RHP Poles and Zeros
The control handbook,Editor: W. S. Levine
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& Chips Lab., NCTU, Taiwan
23/26
RHP Zero Compensation and Cancellation
[REFERENCES][1] D. M. Sable, B. H. Cho, and R. B. Ridley,
"Elimination of the positive zero in fixed frequency boost and
flyback
converters," IEEE APEC Proc., 1990.[2] Wei-Chung Wu, R. M. Bass,
and J. R. Yeargan, "Eliminating the effects of the right-half plane
zero in fixed frequency
boost converters," IEEE PESC Con. Rec., 1998.[3] Right
half-plane zero compensation and cancellation for switching
regulators, Inventor: David W. Ritter,
Assignee: Micrel, Incorprated, San Jose, CA., Patten No.: US
2007/0252570 A1, Nov. 1, 2007.
24/26
RHP Zero Compensation and CancellationUS 2007/0252570 A1, Nov.
1, 2007.
Abstract—An improved method of canceling a RHPZ of a switching
regulator can include detecting a predetermined error signal
provided to a pulse width modulation (PWM) circuit, wherein the
predetermined error signal is associated with the RHPZ. Once a RHPZ
is detected, a ramp waveform provided to the PWM circuit can be
temporarily lengthened, thereby canceling the RHPZ. Notably,
temporarily lengthening the ramp waveform can be based on adjusting
an RZ*CZ time constant. In one embodiment, the ramp waveform can be
lengthened to create a left half-plane zero (LHPZ), which improves
stability.
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13
課程講義:【電力電子】 DC-DC Converter - Modeling Techniques交通大學
808-電力電子實驗室 June 4, 2009
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& Chips Lab., NCTU, Taiwan
25/26
Summary RHP Zero
Split LC switching converters operating in CCM will have RHP
zero due to the lagging inductor effect! RHP zero CAN NOT be
eliminated with current mode control! EHP zero can be eliminated by
using variable frequency control.
26/26
References: About RHP Zero
[1] Lloyd H. Dixon, Right-Half-Plane Zero - A simplified
explanation, Unitrode Power Supply Design Seminar, 1984.[2] C.
Basso, Understanding the Right-Half-Plane Zero, Power Electronics
Technology, 2009.[3] Brian Lynch, Under the Hood of a DC-DC Boost
Converter, 2008-09 Power Supply Design Seminar - SEM1800.
[4] D. M. Sable, B. H. Cho, and R. B. Ridley, "Elimination of
the positive zero in fixed frequency boost and flyback converters,"
IEEE APEC Proc., 1990.
[5] Wei-Chung Wu, R. M. Bass, and J. R. Yeargan, "Eliminating
the effects of the right-half plane zero in fixed frequency boost
converters," IEEE PESC Con. Rec., 1998.
[6] Right half-plane zero compensation and cancellation for
switching regulators, Inventor: David W. Ritter, Assignee: Micrel,
Incorprated, San Jose, CA., Patten No.: US 2007/0252570 A1, Nov. 1,
2007.