R. W. Erickson

R. W. Erickson Department of Electrical, Computer, and Energy Engineering

University of Colorado, Boulder

ECEN 5807, Spring 2015 1

ECEN 5807!Modeling and Control of Power Electronics Systems

Instructor: Robert Erickson•  Offices: ECOT 356, ECEE 1B55•  Email: [email protected]•  Office hours:

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

D2L course site•  Log into https://learn.colorado.edu with your identikey•  Course lectures, submission of assignments, solutions, grades, discussion forum

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


Power Electronics Courses at the University of Colorado, Boulder

ECEN 2260 Circuits 2


ECEN3170 Energy Conversion

ECEN 4797/5797 Intro to Power Electr.

ECEN 5807 Modeling and Control of

Power Electronics Systems

ECEN 5817 Res. and Soft-Sw. Tech

ECEN 4517/5517 Power Electronics Lab

ECEN 4827/5827 Analog IC Design

ECEN 5837 Mixed-Signal IC Design

ECEN 4167/5737 Energy Conversion 2

ECEN 5737 AC Drives

graduateundergraduate


ECEN 5017 Pwr Elect for Electric

Drive Vehicles

Professional certificate in power electronics

Graduate certificate in electric drivetrain technology

Power management

Coursera (MOOC): Introduction to Power Electronics

Prerequisite for either ECEN 5807 or ECEN 5817: ECEN 5797


Assignments

Weekly homework sets, 50% of total gradeMidterm exam (open book/notes, take home), 17% of gradeFinal exam (comprehensive, open book/notes, take home), 33% of grade All assignments and due dates will be posted on course web siteYou must scan your homework or exam into a pdf

• Black-and-white, no color or grayscale• 200-400 dpi is sufficient• Upload a single pdf file to the Dropbox in D2L• Due dates are typically at the start of Friday lecture; no late assignments will be accepted and the Dropbox will

close automaticallyDue date and time for CAETE students and on-campus students are the same


Required Software

A version of Spice•  You can download LTSpice for free. See course Vitals page for a link•  Examples in class will use LTSpice. PSpice files will also be linked on website.

MATLAB/Simulink•  Available in most department labs•  You can buy the student version from Mathworks•  We will use: Matlab, Simulink, and the Control Systems Toolbox. The student version of Matlab includes all of

these and much more.


Topics

1.  Simulation and averaged switch modeling•  CCM, DCM, and other examples•  Simulation

2.  Techniques of design-oriented analysis, with application to switched-mode converter systems•  Middlebrook’s feedback and extra-element theorems•  Input filter design•  Writing complex transfer function expressions by inspection

3.  Current-programmed control of PWM converters4.  Introduction to digital control of PWM converters5.  Rectifiers

•  Rectifier harmonics in power systems•  Low-harmonic rectifiers and power factor correction converters


1. Simulation and Averaged Switch Modeling

•  Additional notes, Section 7.4, Chapter 11, and Appendix B•  Averaged switch modeling is another approach to derive the averaged model of a PWM

converter.•  Well suited to Spice modeling of PWM converters•  We will use this approach to model CCM, DCM, and current-programmed converters•  Also useful for incorporation of ac losses (switching loss, core loss) into averaged models of PWM converters

•  Computer simulation of small-signal transfer functions•  Objectives of simulation•  Spice models•  Simulink models


Averaged Switch Modeling and Simulation

ECEN5807, Spring 2007


2. Techniques of Design-Oriented Analysis

Chapter 10, Appendix C, and supplementary notes on websiteNull double injection methods for analysis of complex analog systems•  Converter applications

Input filter design

Exact analysis of a fifth-order converter system

•  Middlebrook’s extra element theoremHow to easily determine the effect of an added element on a circuit transfer function, without starting the analysis all over again

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

•  Middlebrook’s feedback theoremHow to easily construct the loop gain and transfer functions of a complex feedback circuit


Middlebrook’s Extra Element Theorem

Appendix CHow a transfer function G(s) is modified by addition of an extra element Z(s):

Transistor voltage:

v

Q1(t) = V

Q1 + v

Q1(t)

where V

Q1 = hvQ1(t)i (dc component)

and hvQ1(t)i = 0 (v

Q1(t) is the ac component)

Similarly, the transistor current is

i

Q1(t) = I

Q1 + i

Q1(t)

The power p

Q

(t) flowing into the transistor is:

p

Q

(t) = v

Q1(t)iQ1(t) = 0 =

�V

Q1 + v

Q1(t)� ⇣

I

Q1 + i

Q1(t)⌘

Now multiply out and average over one period:

0 = V

Q1I

Q1 + hvQ1(t)iQ1(t)i

The transistor “consumes” power at DC,

and “generates” power at the switching

frequency. The transistor functions as an

inverter.

V

Q1I

Q1 = �hvQ1(t)iQ1(t)i

v

out

(s)v

in

(s)=

G(s)

��Z(s)!1

!0BBBBBBBBBBBB@

1 +Z

N

(s)Z(s)

1 +Z

D

(s)Z(s)

1CCCCCCCCCCCCA

1

Simple methods to find ZN(s) and ZD(s) using null double injectionHow to design circuits so that the extra element doesn’t significantly change G(s):

Transistor voltage:

v

Q1(t) = V

Q1 + v

Q1(t)

where V


and hvQ1(t)i = 0 (v



i

Q1(t) = I

Q1 + i

Q1(t)

The power p

Q


p

Q

(t) = v

Q1(t)iQ1(t) = 0 =

�V

Q1 + v

Q1(t)� ⇣

I

Q1 + i

Q1(t)⌘


0 = V

Q1I

Q1 + hvQ1(t)iQ1(t)i




inverter.

V

Q1I


v

out

(s)v

in

(s)=

G(s)

��Z(s)!1

!0BBBBBBBBBBBB@

1 +Z

N

(s)Z(s)

1 +Z

D

(s)Z(s)

1CCCCCCCCCCCCA

kZ( j!)k � kZN

( j!)k kZ( j!)k � kZD

( j!)k

1

Transistor voltage:

v

Q1(t) = V

Q1 + v

Q1(t)

where V


and hvQ1(t)i = 0 (v



i

Q1(t) = I

Q1 + i

Q1(t)

The power p

Q


p

Q

(t) = v

Q1(t)iQ1(t) = 0 =

�V

Q1 + v

Q1(t)� ⇣

I

Q1 + i

Q1(t)⌘


0 = V

Q1I

Q1 + hvQ1(t)iQ1(t)i




inverter.

V

Q1I


v

out

(s)v

in

(s)=

G(s)

��Z(s)!1

!0BBBBBBBBBBBB@

1 +Z

N

(s)Z(s)

1 +Z

D

(s)Z(s)

1CCCCCCCCCCCCA

kZ( j!)k � kZN

( j!)k kZ( j!)k � kZD

( j!)k

1

Design-oriented result: construct Bode plots of above equations, and use to shape Z(s)


Input Filter Design

ECEN 5807 : Introduction� 8�

Input filter design�

•� Filter can seriously degrade converter control system behavior�

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

•� Must design input filter having adequate damping�

+–

Inputfilter

Converter

T(s)

Controller

vgZo(s) Zi(s)

H(s)

d

v

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


Input filter design�

•� Filter can seriously degrade converter control system behavior�

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

•� Must design input filter having adequate damping�

+–

Inputfilter

Converter

T(s)

Controller

vgZo(s) Zi(s)

H(s)

d

v

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

•  Input filter can seriously degrade control system behavior and cause instability

•  Use Extra Element Theorem derive conditions that ensure that input filter does not disrupt dynamics of control system

•  Must design input filter having adequate damping


Design of Input Filters that Do Not Degrade Converter Transfer Functions


Design of damped input filters that don’t degrade converter transfer functions�

-20 dB�

-10 dB�

0 dB�

10 dB�

20 dB�

1 kHz 10 kHz 100 kHz

Section 1

alone

Cascaded

sections 1 and 2

30 dB�

�� ZN ��

�� ZD ��

fo

+–

vg

L1

n1L

1R

1

C1

L2

n2L

2R

2

C2

6.9 �F

31.2 �H

15.6 �H1.9 �0.65 � 2.9 �H

5.8 �H

11.7 �F

Design criteria derived via Extra

Element theorem:�

Two-section damped input filter

design:�

Z( j�) > ZN( j�)

Z( j�) > ZD( j�)

Design criteria derived via Extra Element Theorem:

Two-section damped input filter design:


Write the line-to-output transfer function by inspection


Write the line-to-output transfer functionby inspection�

+–

+–

L

RC

1 : D D' : 1Lf

RfCf

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�Example: buck-boost with input filter

Solution: use n extra element theorem


3. Current-Programmed Control


3. Current Programmed Control�

+–

Buck converter

Current-programmed controller

Rvg(t)

is(t)

+

v(t)

–

iL(t)

Q1

L

CD1

+

–

Analogcomparator

Latch

Ts

0

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 for controlling PWM converters�

•� Transistor turns off when its current is(t) is equal to the

control input ic(t) �

•� Simpler dynamics, more

robust compensator�

Switchcurrentis(t)

Control signalic(t)

m1

t0 dTs Ts

on offTransistor

status:

Clock turnstransistor on

Comparator turnstransistor off


3. Current Programmed Control�

+–

Buck converter

Current-programmed controller

Rvg(t)

is(t)

+

v(t)

–

iL(t)

Q1

L

CD1

+

–

Analogcomparator

Latch

Ts

0

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 for controlling PWM converters�

•� Transistor turns off when its current is(t) is equal to the

control input ic(t) �

•� Simpler dynamics, more

robust compensator�

Switchcurrentis(t)

Control signalic(t)

m1

t0 dTs Ts

on offTransistor

status:

Clock turnstransistor on

Comparator turnstransistor off

•  Chapter 12•  A very popular method for controlling PWM converters•  Transistor turns off when its current is(t) is equal to a

control signal ic(s)•  Simpler dynamics, more robust compensator


Effect of current programming on converter transfer functions!Buck converter example


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

CCM-DCM1

+–

+–

35 �H

100 �F

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 �

10 �

Rf = 1 �

fs = 200 kHz

L = 35 ��

Va = 0.6 V

Xcpm

Xswitch

fs = 200 kHz

L = 35 ��

EiE1 E2

Comparison of control-to-output

transfer functions�

Averaged switch model used in

PSPICE simulations�Comparison of control-to-output transfer functions Averaged switch model used in spice simulations


Digitally Controlled Buck Converter!Simulink ModelCoPEC

5ECEN5807

Digitally Controlled Buck ConverterSimulink Model

• The buck converterblock is the same as in the continuous-time system

• Note the parts of the system that model the digital controller including:

– A/D converter– Discrete-time

compensator, and – Digital PWM

Digital PWM

Discrete-time compensator

A/D converter

•  Buck converter block is same as in continuous-time system

•  Note the parts of the system that model the digital controller, including:•  A/D converter•  Discrete-time compensator•  Digital PWM


4. Modern Rectifiers, Power System Harmonics, and Low-Harmonic Rectifiers


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

Harm

onic

am

pli

tude,

per

cen

t o

f fu

nd

am

enta

l

THD = 136%Distortion factor = 59%

•  The traditional peak-detection rectifier injects very large harmonic currents into the ac power line.

•  At substantial power levels, this type of rectifier is not allowed


The Ideal Rectifier


The Ideal Rectifier�

+–

1 : M(D)

Vg R

+

V

–

Re(vcontrol)

+

–

vac(t)

iac(t)

vcontrol

v(t)

i(t)

+

–

p(t) = vac

2 / Re

Ideal rectifier (LFR)

acinput

dcoutput

Modeling the basic functions of ideal converters�

Dc-dc converter:

dc transformer�

Ac-dc rectifier: “loss-free resistor”�

Modeling the basic functions of ideal converters

DC-DC converter: DC transformer

AC-DC rectifier: “Loss-free resistor”


Controlling a DC-DC Converter to Behave as an Ideal Rectifier


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) to

be proportional to vg(t)�

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


Next Lecture

Begin with circuit averaging and averaged switch modelingAssignment: read Sections 7.4 and 7.5

R. W. Erickson

Documents

Transcript of R. W. Erickson