DC-DC Converters: Modeling...

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 April 15, 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

1/38

2009年4月15日

鄒應嶼教授

國立交通大學電機與控制工程研究所

DC-DC Converters: Modeling Techniques

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


2/38

DC-DC Converters: Modeling Techniques

1. Modeling of Switching Converters2. State-Space Averaging Technique3. PWM Switch Modeling Technique4. CIECM Modeling Techniques5. Modeling of Current-Mode DC-DC Converters6. Difficulties with Modeling and Control of SPS

2




3/38

Modeling of Switching Converters

Power Electronic Systems & Chips Lab., NCTU, Taiwan

電力電子系統與晶片實驗室Power Electronic Systems & Chips Lab.交通大學 • 電機與控制工程研究所

4/38

Modeling of Switching Converters

Why Modeling ? Classification of Modeling Techniques Modeling of Switching Power Converters State-Space Averaging Technique Transfer Functions Small-Signal Equivalent Circuit Model

3




5/38

Objective of Modeling

Objective of Modeling

Analysis

Simulation

DesignEfficiency

Output Impedance

Static Characteristics

6/38

Difficulties with Modeling of SPS

Nonsmooth Systems (time and state discontinuity)Nonlinearity due to operating point Concepts of existence, uniqueness, stability not clearly defined for systems with discontinuous right half-plan zero Inherent Nonlinear Dynamic Behavior! Concept of chaotic dynamics relatively new to power electronicsMagneticsNoises and EMI Thermal and Temperature Distribution

4




7/38

Working Profile of a Switching Converter

Power-on Power-off

ov

oi

Wat

ts

% C

PU

time

Dell power edge 2400 (web/SQL server)

Static Characteristics of a DC-DC Converter

Buck

Boost

Buck-Boost L C

D

vovi

L

C

D

vovi

vi vo

L

CD

5




9/38

Operating Region Operating Point Operating Mode?

IN

OUT

VV

)(max,LB

o

II

0 0.5 1.0 1.5 2.0

0

0.25

0.50

0.75

1.0

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCMCRM

VIN = constant

(min)

(max)

IN

OUT

VV

(max)

(min)

IN

OUT

VV

(max)OUTI(min)OUTI

10/38

Operating Point Steady-State Trajectory

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40%20% 110%90%50%

500 R o

testI

testI2A

5msec 100msec

If the components are not ideal, its parasitic parameters will change its static curves. The illustrated example is obtained with the following parameters: RDS(ON) = 50 mMOSFET reverse diode voltage drop = 0.7VDiode voltage drop = 0.7V Inductor ESR = 25mCapacitor ESR = 2m

Vo/Vi = 1.0

0.25

0.5

0.75

0

1.0

6




11/38

Operating Point Small-Signal Perturbations

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40%20% 110%90%

Step Load Change1A

0AOutput Voltage

Inductor Current

B

50%

電流負載的步階切換！Vo/Vi = 1.0

0.25

0.5

0.75

0

1.0

12/38

Operating Point Frequency Response

oR

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40%20% 110%90%50%

YOP XOP

QX = 0.707

QY = 7.070

QB = 3.535

QA = 1.414

QX = 0.707

QY = 7.07

QB = 3.535 QA = 1.414

7




13/38

Operating Region State-Variable Plane

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCM

80% 100%60%40% 110%90%10% 20%

從工作點

到工作點

的軌跡均在連續導

通工作區內，但是從工作點

到工作點

則有部分會進入不連續導通工作區。

Vo/Vi = 1.0

0.25

0.5

0.75

0

1.0

14/38

Modeling of Switching Power Converters

Modeling of Voltage-Mode DC-DC Converters- Power Stage Modeling State-space average model (Middlebrook and C'uk 1977) Discrete time-domain model (Lee 1979) Equivalent circuit model (Chetty 1981) Unified topological model (Pietkiewicz and Tollik, 1987) PWM switch model (Voperian 1988) Injected-absorbed-current model (Kisovski 1991)

- Error Processor Modeling (Chetty 1982)- Pulsewidth Modulator Modeling Describing function model (Lee 1983) Equivalent circuit model (Bello 1981)

- Larger Signal Modeling (Vicua 1992)

Modeling of Current-Mode DC-DC Converters- Equivalent Circuit Model (Chetty 1981)- y-parameter Model (Middlebrook 1989)

8




15/38

Modeling Techniques

State Space Averaging Method[1] R. D. Middlebrook and S. Cuk, “A general unified approach to modeling switching-converter

stages,” IEEE PESC Conf. Rec., pp. 18-34, 1976. [Pioneer Paper][2] S. Cuk and R. D. Middlebrook, “A general unified approach to modeling switching DC-to-DC

converters in discontinuous conduction mode,” IEEE PESC Conf. Rec., pp. 36-57, 1977.

Modeling of Switching Converters in DCM Operation[1] D. Maksimovic and S. Cuk, “A unified analysis of PWM converters in discontinuous modes,” IEEE

Trans. Power Electron., vol. 6, pp. 476–490, May 1991. [2] J. Sun, D. M. Mitchell, M. F. Greuel, P. T. Krein, and R. M. Bass, “Averaged modeling of PWM

converters operating in discontinuous conduction mode,” IEEE Trans. Power Electron., vol. 16, pp. 482-492, July 2001.

R. D. Middlebrook and S. Cuk, “A general unified approach to modeling switching-converter stages,”IEEE PESC Conf. Rec., pp. 18-34, 1976.

16/38

Modeling Techniques ..

PWM Switch Method[1] V. Vorperian, “Simplified Analysis of PWM Converters Using Model of PWM Switch Part I:

Continuous Conduction Mode,” IEEE Trans. on Aero. and Elec. Sys., vol. 26, no. 3, pp. 490-496, May 1990.

[2] V. Vorperian, “Simplified Analysis of PWM Converters Using Model of PWM Switch Part II: Discontinuous Conduction Mode,” IEEE Trans. on Aero. and Electron. Sys., vol. 26, no. 3, pp. 497-505, May 1990.

Fast Analytical Techniques for Electrical and Electronic Circuits,V. Vorperian, Cambridge Press, 2004. Vatche Vorperian, (1984) Analysis of resonant converters. Dissertation (Ph.D.), California Institute of Technology, Advised by S. Cuk.

1. Introduction2. Transfer functions3. The extra element theorem4. The N-extra element theorem5. Electronic negative feedback6. High-frequency and microwave circuits7. Passive filters8. PWM switching dc-to-dc converters

9




17/38


Discrete Time-Domain Method[1] F. C. Lee, R. P. Iwens, Y. Yu, and J. E. Triner, “Generalized computer-aided discrete time-domain

modeling and analysis of DC-DC converters,” IEEE Trans. IECI, vol. 26, pp. 58-69, May 1979.

Equivalent Circuit Method[1] P.R.K. Chetty, Switch-Mode Power Supply Design, TAB BOOKS, Inc., 1986.

Modeling of Current-Programmed Converter[1] R. D. Middlebrook, “Modeling current programmed buck and boost converters,” IEEE Trans. on

Power Electronics, vol. 4, pp. 36-52, January 1989. [2] Teuvo Suntio, “Analysis and modeling of peak-current-mode-controlled buck converter in DICM,”

IEEE Trans. on Ind. Electron., vol. 48, no. 1, pp. 127-135, Feb. 2001.

Unified Topological Method[1] Pietkiewicz, A. and D. Tollik, “Unified topological modeling method of switching dc-dc converters

in duty-ratio programmed mode,” IEEE Trans. on Power Electron., vol. 2, no. 3, pp. 218-226, July 1987.

Injected-Absorbed-Current Method[1] Kislovski, A. S., R. Redl, and N. O. Sokal, Dynamic Analysis of Switching-Mode DC/DC

Converters, Van Nostrand Reinhold, 1991.

18/38


Small-Signal z-Domain Analysis[1] D. M.Van de Sype, K. De Gusseme, F.M.L.L. DeBelie, A. P. Van den Bossche, and J. A. Melkebeek,

“Small-signal z -domain analysis of digitally controlled converters,” IEEE Transactions on Power Electronics, vol. 21, no. 2, pp. 470- 478, March 2006.

[2] D. M.Van de Sype, K. De Gusseme, A. P. Van den Bossche, and J. A. Melkebeek, “Experimental verification of the z-domain model for digitally controlled converters,” IEEE Power Electronics Specialists Conference, vol., no., pp.2164-2170, 16-16 June 2005.

[3] Yu-Cheng Lin; Dan Chen; Yen-Tang Wang; Wei-Hsu Chang; “A novel loop gain correction method for digitally controlled DC-DC power converters,” IEEE Energy Conversion Congress and Exposition(ECCE) pp.3530-3535, 20-24 Sept. 2009.

Digital Current-Mode Control [1]Y. S. Jung, “Small-signal model-based design of digital current-mode control,” IEE Proceedings -

Electric Power Applications, vol.152, no.4, pp. 871- 877, 8 July 2005. [2] S. Chattopadhyay and S. Das, “A digital current-mode control technique for DC–DC converters,”

IEEE Transactions on Power Electronics, vol.21, no.6, pp.1718-1726, Nov. 2006.

10




Control of DC-DC Converters

voltagereference

Pulse-widthmodulator

gate driver

cv)(tcompensator

sGc

refv

+–

tδ

tsTsdT

tv

t

H. P. Forghani-Zadeh and G.A. Rincon-Mora, "Current-sensing techniques for DC-DC converters," The IEEE 45th Midwest Symposium on Circuits and Systems (MWSCAS), Aug. 2002.

J. T. Mossoba and P. T. Krein, "Design and control of sensorless current mode DC-DC converters," IEEE APEC Conf. Rec., 2003.

Timothy Hegarty , Voltage-mode control and compensation - Intricacies for buck regulators, EDN 2008.

Brian Lynch, Current-mode vs. voltage-mode control in synchronous buck converters, TI.

Lloyd Dixon, Average Current Mode Control of Switching Power Supplies, 1990.

Ray Ridley, 30 Years of current-mode control, 2008.

Robert Mammano, SPS Topology -voltage Mode versus current-mode, DN-62 slua119. Lloyd Dixon, Control loop cookbook,

TI-Unitrode slup113a.

BUCK Converter Control Cookbook, Zach Zhang, Alpha & Omega Semiconductor, Inc. PIC-003.

20/38

Modeling and Control of DC-DC Converters

Output voltage feedback only!

PWMModulator

LoopCompensator

v o

digital signal processor analog signal processor

vR

d

load

LR di~

Buck Converter Boost Converter

Buck/Boost Converter C,uk Converter

vi vo

L

CD

L

C

D

vovi

　

L C

D

v ovi

　　　

L1

C2D

C1

vi vo

Switching power converters

oi

Define the Operating Point!

Define the Load Disturbance

vgosZ

sv~

sV

Define the Line

Disturbance

Define the Source Output Impedance!

11




Small-Signal Modeling of a Buck Converter

IN

OUT

VV

)(max,LB

o

II

0 0.5 1.0 1.5 2.0

0

0.25

0.50

0.75

1.0

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCMCRM

Averaged Switch Modeling of Boundary Conduction Mode DC-to-DC ConvertersJ, Chen, R. Erickson, and D. Maksimovic, IECON 2001.

VIN = constant

V. Vorperian, "Simplified Analysis of PWM Converters Using Model of PWM Switch Part II: Discontinuous Conduction Mode," IEEE Trans. on Aero. and Electron. Sys., vol. 26, no. 3, pp. 497-505, May 1990.

V. Vorperian, "Simplified Analysis of PWM Converters Using Model of PWM Switch Part I: Continuous Conduction Mode," IEEE Trans. on Aero. and Elec. Sys., vol. 26, no. 3, pp. 490-496, May 1990.

22/38

Modeling of Switching Converters in DCM Operation

IN

OUT

VV

)(max,LB

o

II

0 0.5 1.0 1.5 2.0

0

0.25

0.50

0.75

1.0

DCM

D = 1.0

0.1

0.3

0.5

0.7

0.9

CCMCRM

VIN = constant

V. Vorperian, "Simplified Analysis of PWM Converters Using Model of PWM Switch Part II: Discontinuous Conduction Mode," IEEE Trans. on Aero. and Electron. Sys., vol. 26, no. 3, pp. 497-505, May 1990.

J. Sun, D. M. Mitchell, M. F. Greuel, P. T. Krein, and R. M. Bass, “Averaged modeling of PWM converters operating in discontinuous conduction mode,”IEEE Trans. Power Electron., vol. 16, pp. 482-492, July 2001.

D. Maksimovic and S. Cuk, “A unified analysis of PWM converters in discontinuous modes,” IEEE Trans. Power Electron., vol. 6, pp. 476–490, May 1991.

12




Systematic View of DC-DC ConvertersEfficiency

Output Impedance

Selection of Switching Frequency and PWM Strategies

Frequency Responses

Time Responses

Current Injection Testing

24/38

PWM DC-DC Power Conversion and Regulation

CLOCK RAMP

Zi

vref

vo

vx

D

vi

T

TOND T

TON

vpvx

Z f

13




25/38

Signal Composition of a PWM DC-DC Converters

comparator

D

d

V c

v c v c

V g

v g vg

igI g

i g

clock ramp

analogamplifier

reference

load

Vv v

modulator-power-stagesubsystem

Ii i

26/38

Frequency Spectrum

f 2f 3f fs - f fs

outputspectrum

bandpassfilter

frequency

fs + f 2fs 3fs

14




27/38

AC and DC Quantities in a PWM Switching DC-DC Converter

I ii i vd ioioIo

vovoVo

C

L

Z f

vref

vc

V vI i

CLOCK RAMP

vd

Vc

vc vc

io

vo

Z i

LoadINPUToR

Analysis of Dynamic Responses

28/38

Small-Signal Modeling of a Switching Power Converter

iIi iIi ˆ

RLC

L

error processor

Zi

Z f

v ref

v c

v V vi I i

duty cyclemodulator

power stage

d D d

open

oOo iIi ˆ v V vo O o

AveragedPower Stage

vi

oîd

vo

v o

The concerned transfer functions under small perturbations can be measured under an open loop condition.

15




29/38

Definition of

v to( )

t

V I VO O I, ,

ioo viv ˆ,ˆ,ˆi to( )

v ti( )

small signal perturbation

t

TtD on

D

d(t)

d t( )

t

IOO VIV ,, Ttd xˆ

power switch gating waveform

xt

Tont

d

30/38

BuckBoost

Buck/Boost

Small-Signal Modeling of Voltage-Mode DC-DC Converters

Power Stage

PulseModulator

ErrorProcessor

vr

vo

vc

io

d

vi

16




31/38

Modeling of Single-Loop DC-DC Converters

vo

vc

i od

vi

;

;

vv

vi

vd

o

i

o

o

o

K PWM A(s)

( , , )v f v i do i o

vi

io

d vc

vo

Zp

Gd

-A(s)

Gv

KPWM

32/38

Small Signal Transfer Functions

G s vvv

o

i d io

( )

,

0 0

: Open-loop input-to-output

G vdd

o

v ii o

,0 0

cvdkˆˆ

PWM

: Open-loop output impedance

: Control to output transfer function

: PWM modulator gain

Z vip

o

o d vi

, 0 0

A s vv

c

o

( )

K

VpPWM

1: Compensator gain: PWM dc gain

17




33/38

Closed-Loop Transfer Functions

Gv,CL (Closed-loop Audio Susceptibility)

v G v G do v i d

d AK vo PWM

vv

GG K A

GT

o

i

v

d

v

1 1PWM

Zp,CL (Closed-loop Output Impedance)

v Z i G do p o d

d AK vo PWMvi

ZT

o

o

p1

Loop Gain: PWMT A s K Gd ( )

vo

vc

i od

vi

;

;

vv

vi

vd

o

i

o

o

o

K PWM A(s)

( , , )v f v i do i o

vi

io

d vc

vo

Zp

Gd

-A(s)

Gv

KPWM

34/38

Modeling of PWM DC-DC Converters

A dc-to-dc switching regulator incorporating a three-port duty ratio programmed modulator-power-stage subsystem whose transfer functions are defined in terms of ratios of small-signal ac quantities (hats) superimposed upon large-signal dc quantities (capitals).

The spectrum of the output signal contains the switching frequency, the control frequency, their respective harmonics, and sidebands.

The modeling objective is to find, as function of frequency, the loop gain and the closed properties of the regulator.

The essential prerequisite is to find the transfer function of the three-port subsystem of the modulator-power-stage.

18




35/38

Concerned Transfer Functions

Control-to-output transfer function Line-to-output transfer function (audio susceptibility) Reference-to-output transfer function Input impedance Output impedance

A voltage sourcing power supply should have a low (zero) output impedance, while a current sourcing power supply should have a high (infinite) output impedance.

Modeling of DC-DC Converters

voltagereference

Pulse-widthmodulator

cv)(t sGc

refv

+–vH

tδ

tsTsdT

tv

t

Complex Behavior of Switching Power Converters, Chi Kong Tse, CRC Press, 2004.

Dynamic Analysis of Switching-Mode DC/DC Converters, Andre'S. Kislovski, Richard Redl and Nathan O. Sokal, Van Nostrand Reinhold, New York, USA, 1991

SMPS Simulation with SPICE 3, Steven M. Sandler, McGraw-Hill Professional, Dec. 1, 1996.

Computer-Aided Analysis and Design of Switch-Mode Power Supplies, Yim-ShuLee, Marcel Dekker, Inc., Feb. 23, 1993.

Switch-Mode Power Supplies -SPICE Simulations and Practical Designs, Christophe Basso, McGraw-Hill, Feb. 1, 2008. Fast Analytical Techniques for Electrical and Electronic Circuits,

V. Vorperian, Cambridge Press, 2004.

Teuvo Suntio, Dynamic Profile of Switched-Mode Converter: Modeling, Analysis and Control,John Wiley, May 2009.

Mikko Hankaniemi, Dynamical Profile of Switched-Mode Converter – Fact or Fiction?, PhD Thesis, Tampere University of Technology, 2007.

19




37/38

PWM Switch Modeling of DC-DC Converters

Advances in Averaged Switch Modeling and SimulationDragan Maksimovic and Robert EricksonColorado Power Electronics Center (CoPEC)IEEE PESC1999 Seminar

The “PWM Switch” in mode transitioning SPICE modelsPCIM Germany 2005.Christophe Basso – On-Semi Application Manager

38/38

Questions inspire effective learning!

Any Questions ???

Power Electronic Systems & Chips Lab., NCTU, Taiwan

電力電子系統與晶片實驗室Power Electronic Systems & Chips Lab.交通大學 • 電機與控制工程研究所

記筆記問問題

學習的關鍵

DC-DC Converters: Modeling...

Documents

Transcript of DC-DC Converters: Modeling...