intro2005ece-ecen5807/course_material/introduction2005.pdf · • Homework assignments and the final exam may require use of a Spice ... Techniques in PE ECEN5017 Power Electronics

ECEN5807, Spring 2005

ECEN5807Modeling and Control of Power Electronic Systems

• Instructor: Dragan Maksimovic• Office: EE1B71, phone: 303-492-4863, fax: 303-492-2758• E-mail: [email protected]• Office hours: Monday, Tuesday, Friday 9:30-10:30am

• Course web site:• http://ece.colorado.edu/~ecen5807• Announcements, course materials, assignments, solutions

• Textbook:• Erickson and Maksimovic, Fundamentals of Power Electronics, 2nd

edition, Kluwer 2001• On-line course lectures (Tegrity system):

• Accessible through the CAETE site: http://caeteport.colorado.edu


Assignments

• Weekly homeworks (11-12 total), 35% of the grade• Midterm exam (open book/notes, take-home), 25% of the grade• Final exam (comprehensive, open book/notes, take-home), 40% of the

grade• All assignments, due dates, and solutions will be posted on the course

web site• For homework assignments, the due date for off-campus students is

(postmarked) one week after the due date published for on-campus students

• Homework assignments and the final exam may require use of a Spice simulator (free student version of PSpice is sufficient)


Power Electronics Program at CU Boulder

ECEN5797Introduction to Power

Electronics

ECEN5807Modeling and Control of PE

Systems

ECEN5817Resonant and Soft-Switching

Techniques in PE

ECEN5017Power Electronics Lab

Fall semesters

Spring semesters

Alternate Spring semesters (2006)

Alternate Spring semesters (2005) Professional Certificate in

Power Electronics

Power Electronics 2: Introduction 3

Topics

1. Averaged switch modeling and simulation (3 weeks)

CCM, DCM, and other examples. Computer simulation

2. Techniques of design-oriented analysis, with switching converterapplications (4 weeks)Middlebrook’s feedback and extra element theoremsInput filter designWriting complicated transfer functions by inspection

3. Current programmed control of PWM converters (3 weeks)

4. Modern rectifiers, power system harmonics, and low harmonicrectifiers (3 weeks)


1. Averaged switch modeling and simulation

l Section 7.4, Chapter 11, and Appendix B

l This approach has recently become quite popular

l Can be applied to a wide variety of converters

We will use it to model CCM, DCM, and current programmedconverters

Also useful for incorporating switching loss into ac model of CCMconverters

l Computer simulation of small-signal transfer functions

Objectives of simulation

PSPICE examples

Averaged switch modeling

+–

Switching converter circuit

Switchingnetwork +

–+–

Large-signal averaged circuit model

Averagedswitchmodel

d

+–

+–

DC and small-signal averaged circuit model

D+d

2)/(/)/1(1

/1)(

oo

scoc wswsQ

wsGsG

++−=

1D

2S

3K

4A

5

duty

ccm-dcm1+

-

DC, AC and Transient simulation

Model implementation for simulation

simulationmodel

linearization

Analytical results:steady-state characteristicsand small-signal dynamics

averaging

Power Electronics 2: Introduction

Circuit / switch averaging

+–

Time-invariant networkcontaining converter reactive elements

C L

+ vC(t) –iL(t)

R

+

v(t)

–

vg(t)

Power input Load

Switch network

po

rt 1

po

rt 2

d(t)Controlinput

+

v1(t)

–

+

v2(t)

–

i1(t) i2(t)

+–

Averaged time-invariant networkcontaining converter reactive elements

C L

+ ⟨vC(t)⟩Ts –

⟨iL(t)⟩Ts

R

+

⟨v(t)⟩Ts

–

⟨vg(t)⟩Ts

Power input Load

Averagedswitch network

po

rt 1

po

rt 2

d(t)Controlinput

+

⟨v2(t)⟩Ts

–

⟨i1(t)⟩Ts⟨i2(t)⟩Ts

+

⟨v1(t)⟩Ts

–

Separate switch network fromremainder of converter:

Average switch waveforms:


Discontinuous conduction mode:Equivalent circuit and Small-signal Modeling

DC

CCM

DCM

+–

1 : M(D)

Vg R

+

V

–

+–Vg R

+

V

–

+–

+– 1 : M(D) Le

C R

+

–

v(s)

e(s) d(s)

j(s) d(s)

AC

+– Rvg(s)

+

–

v(s)

vg(s)

??

• Chapter 11


Averaged switch modeling: CCM vs. DCM

+

–

1 : d(t)i1(t) Tsi2(t) Ts

+

–

v2(t) Tsv1(t) Ts

Averaged switch modelSwitch network

CCM

+

v2(t)

–

+

v1(t)

–

i1(t) i2(t)

i2(t) Ts

+

–

v2(t) Tsv1(t) Ts

i1(t) Ts

Re(d1)

+

–

DCM

+

v2(t)

–

+

v1(t)

–

i1(t) i2(t)p(t)

Ts


The dependent power source

p(t)

+

v(t)

–

i(t) v(t)i(t) = p(t)

v(t)

i(t)


Example: DCM buck-boost converterAveraged switch model

i2(t) Ts

v2(t) Tsv1(t) Ts

i1(t) Ts

Re(d)

+–

L

C R

+

–

+

–

–

+ v(t)Ts

vg(t) Ts

p(t)Ts


Small-signal ac modelingBuck converter example

+

–

+– v1 r1 j1d g1v2

i1

g2v1 j2d r2

i2

v2

+

–

L

C R

DCM buck switch network small-signal ac model

+

–

vg v

iL


2. Techniques of Design-Oriented Analysis

Chapter 10, Appendix C, and supplementary notes on webNull double injection methods for analysis of complex analog systems

l Converter applicationsInput filter designExact analysis of a fifth-order converter system

l Middlebrook’s extra element theoremHow to easily determine the effect of an extra element on a circuit transfer

function, without starting the analysis all over again

l The n extra element theoremHow to write complicated transfer functions by inspection, in rational form

l Middlebrook’s feedback theoremHow to easily construct the loop gain and closed-loop transfer functions of a

complex feedback circuit


Input filter design

• Filter can seriously degrade converter control system behavior

• Use extra element theorem to derive conditions which ensure that converterdynamics are not affected by input filter

• Must design input filter having adequate damping

• Input EMI filter is required to meet regulationsconcerning electromagnetic emissions

• A well-knownand classicproblem withinthe powerelectronics field

+–

Inputfilter

Converter

T(s)

Controller

vg

Zo(s) Zi(s)

H(s)

d

v


Effect of input filter on Gvd(s)

f

|| Gvd || ∠ Gvd

0˚

– 360˚

– 540˚

0 dB

– 10 dB

20 dB

30 dB

100 Hz

40 dB

1 kHz 10 kHz

– 180˚

10 dB

|| Gvd ||

∠ Gvd

Buck converterexample

Dashed lines: noinput filter. 2 poles.

Solid lines: with LCinput filter. 4 polesand 2 RHP zeroes.

Power Electronics 2, Spring 2003

Use Extra-Element Theorem(Appendix C)

Simple methods to find ZN and ZD

How to design the input filter so that it does not change anything:

D

o

N

o

Zvdvd

Z

ZZ

Z

GsGo

+

+=

=1

1

)(0

, DoNo ZZZZ <<<<


Design of damped input filters that donÕtdegrade converter transfer functions

-20 dBΩ

-10 dBΩ

0 dBΩ

10 dBΩ

20 dBΩ

1 kHz 10 kHz 100 kHz

Section 1alone

Cascadedsections 1 and 2

30 dBΩ

|| ZN |||| ZD ||

fo

+–vg

L1

n1L1R1

C1

L2

n2L2R2

C2

6.9 µF

31.2 µH

15.6 µH1.9 Ω0.65 Ω 2.9 µH

5.8 µH

11.7 µF

Design criteria

derived via

ExtraElement theorem:

Two-section

da

mped input filterdesign:

ZN( jω)> Zo ( jω)

ZD( jω)> Zo ( jω)


Write the line-to-output transfer functionby inspection

+–

+–

L

RC

1 : D D' : 1Lf

Rf

Cf

Cb

vg(t) I d(t)

(Vg – V)d(t)

I d(t)

Solution: use n extra element theorem

Example: buck-boost with input filter


Feedback analysis via null double injectionWriting expressions for loop gains by inspection

–vy(s) or–iy(s)+– +–

Inputui(s)

Outputuo(s)

Loop gain T(s)

Injectionvz(s) or iz(s)

vx(s) orix(s)

Quantity Current injection Voltage injection

T(s)Loop gain

T (s) =iy(s)

ix(s)ui = 0

T (s) =vy(s)

vx(s) ui = 0


3. Current Programmed Control

+–

Buck converter

Current-programmed controller

Rvg(t)

is(t)

+

v(t)

–

iL(t)

Q1

L

CD1

+

–

Analogcomparator

Latch

Ts0

S

R

Q

Clock

is(t)

Rf

Measureswitch

current

is(t)Rf

Controlinput

ic(t)Rf

–+

vref

v(t)Compensator

Conventional output voltage controller

• Chapter 12

• A very popular method forcontrolling PWM converters

• Transistor turns off when itscurrent is(t) is equal to thecontrol input ic(t)

• Simpler dynamics, morerobust compensator

Switchcurrentis(t)

Control signalic(t)

m1

t0 dTs Ts

on offTransistor

status:

Clock turnstransistor on

Comparator turnstransistor off


A first-order CPM modelAveraged switch modeling, buck example

+–

L

C R

+

v(t)

–

vg(t)

iL(t)

+

v2(t)

–

i1(t) i2(t)

Switch network

+

v1(t)

–

+–

L

C R

+

⟨v(t)⟩Ts

–

⟨vg(t)⟩Ts

⟨iL(t)⟩Ts

+

⟨v2(t)⟩Ts

–

⟨i1(t)⟩Ts⟨i2(t)⟩Ts

Averaged switch network

+

⟨v1(t)⟩Ts

–

⟨ic(t)⟩Ts

⟨ p(t)⟩Ts

Originalconverter

Averagedmodel


First-order small-signal model, CPM/CCM

+–

L

C R

+

–

vg ic v– D2

RDR

vic D 1 + sLR

ig iL


A more accurate small-signal model

+–+– –ic Fm

CPMcontrollermodel

d

Gvd(s)

Gid(s)

Gvg(s)

Gig(s)

++

++

v

iL

vg

Converter transfer functions

Fv

Fg


Effect of current programming on transfer functionsBuck converter example

|| Gvd ||

∠ Gvd

f

0˚

–90˚

–180˚

∠ G

–20 dB

–40 dB

0 dB

20 dB

40 dB

10 Hz 100 Hz 10 kHz 100 kHz1 kHz

|| G ||

–60 dB

|| Gvc ||

∠ Gvc

21

345

CC

M-D

CM

1

+–

+–

35 µH

100 µF

Vg

12 V

L

C R

vc

+

v

–

iLOAD

CPM

control current 1 2

d

+–

+–

+–

iL RL1 2 3 4

d

Rf iL v(1)–v(3) v(3)

0.05 Ω

10 Ω

Rf = 1 Ωfs = 200 kHzL = 35 µΗVa = 0.6 V

Xcpm

Xswitch

fs = 200 kHzL = 35 µΗ

EiE1 E2

Comparison of control-to-outputtransfer functions

Averaged switch model used inPSPICE simulations


4. Modern rectifiers, power system harmonics,and low harmonic rectifiers

100%91%

73%

52%

32%

19% 15% 15%13% 9%

0%

20%

40%

60%

80%

100%

1 3 5 7 9 11 13 15 17 19

Harmonic number

Ha

rmo

nic

am

plit

ud

e,

pe

rce

nt o

f fu

nd

am

en

tal

THD = 136%Distortion factor = 59%


Part IV. Modern Rectifiers and Power System Harmonics

Chapter 16. Power and Harmonics in NonsinusoidalSystems

Chapter 17. Line-Commutated Rectifiers

Chapter 18. PWM Rectifiers


The Ideal Rectifier

+–

1 : M(D)

Vg R

+

V

–

Re(vcontrol)

+

–

vac(t)

iac(t)

vcontrol

v(t)

i(t)

+

–

p(t) = vac2 / Re

Ideal rectifier (LFR)

acinput

dcoutput

Modeling the basic functions of ideal converters

Dc-dc converter:dc transformer

Ac-dc rectifier: “loss-free resistor”


Controlling a dc-dc converterto behave as an ideal rectifier

1 : M(d(t))

dc-dc converter

controller

d(t)

Rvac(t)

iac(t)+

vg(t)

–

ig(t)

ig

vg

+

v(t)

–

i(t)

C

Controller varies d(t) as necessary, to cause ig(t) tobe proportional to vg(t)


A complete dc power supply system

boost converter

wide-bandwidth input current controller

vac(t)

iac(t)+

vg(t)

–

ig(t)

ig(t)vg(t)

+

vC(t)

–

i2(t)

Q1

L

C

D1

vcontrol(t)

multiplier X

+–vref1(t)

= kx vg(t) vcontrol(t)

Rsva(t)

Gc(s)

PWM

compensator

verr(t)

DC–DCConverter load

+

v(t)

–

i(t)

d(t)

+–compensatorand modulator

vref3

wide-bandwidth output voltage controller

+–compensatorvref2

low-bandwidth energy-storage capacitor voltage controller

vC(t)

v(t)


Rectifier modeling, design, and control

• Basics of power system harmonics. Power in nonsinusoidalsystems. (Chapter 16)

• Behavior of conventional diode rectifiers. Conventional harmonicmitigation techniques: harmonic trap filters, and polyphasetransformer connections. (Chapter 17)

• Steady-state design of low-harmonic PWM rectifiers. Controltechniques for PWM rectifiers. Use of small-signal models to designcontrol systems of PWM rectifiers. RMS calculations for doubly-modulated waveforms. Energy storage, and low-frequencywaveforms in single-phase low-harmonic rectifier systems. (Chapter18)


Next lecture

l Begin with circuit averaging and averaged switch modeling

l Assignment: Read Section 7.4

Documents

intro2005ece-ecen5807/course_material/introduction2005.pdf · • Homework assignments and the final exam may require use of a Spice ... Techniques in PE ECEN5017 Power Electronics