2009-06-04：【技術專題】right-half-plane (rhp)...

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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

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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


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?

2




3/26


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

3





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

4




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'

5




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

6




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

7




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

8




15/26

Dominant Pole

If p1

9




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

10




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.

11




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

12




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.

13




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.

2009-06-04：【技術專題】right-half-plane (rhp)...

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