Introduction To Small Signal And Averaged Switch Modeling · ©2015 Embedded Power Labs – All...

©2015 Embedded Power Labs – All rights reserved

Introduction To Small SignalAnd Averaged Switch Modeling

Robert V. [email protected]

Seminar Prepared For APEC 2015


Seminar Introduction

• This Is An Introductory Seminar– Why Do We Need Small Signal Modeling?– Small Signal Model Derivation– Deriving Transfer Functions From The

Small Signal Model– Averaged Switch Modeling

• Detailed Examples (All The Algebra)


Seminar Introduction: Not Discussed

• Models For Discontinuous Conduction Mode• Control Loop Design

– Error Amp/Compensator Design– Pole-Zero Placement– Loop Stability



• Much Of This Seminar Is Based On Material From “Fundamentals Of Power Electronics”, 2nd Edition, Erickson & Maksimovic,Kluwer Academic Publishers,ISBN 0-7923-7270-0– Chapter 7, AC Equivalent

Circuit Modeling– Chapter 8, Converter Transfer

Functions



• Timing– 90 Minutes Presentation– 30 Minute Break– 90 Minutes Presentation

• Ask Questions At Any Time• Fill Out The Survey Form!


Controls 101 Review

PhysicalSystem(Plant)

InputU

OutputY

Issue: The Actual OutputIs Not The Desired Output


Feedback

PhysicalSystem(Plant)

InputR Output

YController

ControlU

Goals:• Make the output track the reference input• Reject disturbances

Requirement: Remain stable at all times


Stability

• Bounded Input-Bounded Output (BIBO)– Finite input results in finite output

• Lyapunov– System has an equilibrium point or points– Small disturbance from equilibrium results in

small change in output– Exponential: Output change decays in time


Design And Analysis

PlantP(s)R YController

C(s)U

• Time domain solutions too tedious• Transform to frequency or s domain

+–

Errore

1P s C s

Y s R sP s C s

1

Y s P s C sH s

R s P s C s


Design And Analysis


C(s)U


+–

Errore

1P s C s

Y s R sP s C s

1

Y s P s C sH s

R s P s C s

Stable Means

This Term Can Never Be Equal To Zero

H(s) Has No Poles In RHP

No RHP Poles In P(s) Or C(s)

Cancelled By Matching RHP Zeroes


Design And Analysis


C(s)U


+–

Errore

1P s C s

Y s R sP s C s

1

Y s P s C sH s

R s P s C s

Only Applies ToLinear

Time Invariant(LTI)

Systems!


ErrorAmp/

Compensator

Power Supply Model

Vref Vout+ –PowerStagePWM

VoltageDivider

Not LTI

Not LTI

We Need Circuit ModelsValid In The s-Domain


Small Signal Modeling

Di t

ˆDv t

ActualV-ICurve

vD

iD

ID

VD

DCOperating

Point

ˆD D Di t I i t

ˆD D Dv t V v t


Small Signal Modeling

Di t

ˆDv t

ActualV-ICurve

vD

iD

ID

VD

DCOperating

Point

ˆD D Di t I i t

ˆD D Dv t V v t

Linearized Characteristic

re

ˆˆD e Dv t r i t


DC Model

PowerStage LoadVg VOUT

IOUT

D

Ig

Inputs• Input Voltage, Vg• Control Input, D• Load • Resistance, R

• Current, ILOAD• Voltage, VLOAD

Outputs• Output Voltage, VOUT• Input Current, Ig


DC Model

LoadRVg VOUT

IOUTIg I(Vg, D, R) V(Vg, D, R)


Buck Converter DC Model

vQ1

vQ2

TON TOFF

TS

D·TS (1-D)·TS= D’·TS


Solving The Output VoltageInductor Volt-

Second Balance

Small Ripple Approximation

0

1 0ST

L L LTsS

v t v t v t dtT

0 0

0

1 1 0

0

S S

ON

S

ON

T TTon

L L LS S T

TTon

g C CT

v t dt v t dt v t dtT T

v v t dt v t dt

g g g

g g

g g g g

v t V v t

v t V

v t V v t V

C C C

C C

C C C C

v t V v t

v t V

v t V v t V


Solving The Output Voltage

0

0

0

0

1 0

0

0

S

ON

TTon

g C CT

g C ON C S ON

g C S C S S

g C C

g C C C

g C

C g

V V dt V dt

V V T V T T

V V D T V T D T

V V D V D

D V DvV V D V

D V V

V D V


Solving The Inductor CurrentCapacitor Charge Balance 0C CTs

i t i t

0 0

0

0

0

1 1 0

0

1 1 0

1 0

S S

ON

S

ON

S

ON

S

T TTon

C C CS S T

TTon

C CT

TTon

L C L CT

T

L C

i t dt i t dt i t dtT T

i t dt i t dt

i t v t dt i t v t dtR R

i t v t dtR


Solving The Inductor Current


C C

L L L

L L

L L L L

v t V

i t I i t

i t I

i t I i t I

0

1 0

1 0

1 0

1

ST

L C

L C S

L C

L C

I V dtR

I V TR

I VR

I VR


Solving The Input Current

0

0 0

0 0 0

1

1

1 10

S

ON OFF

ON OFF ON

T

g g g gTsS

T T

g g gS

T T T

g gS S

i t i t I i t dtT

I i t dt i t dtT

i t dt dt i t dtT T

g L Li t i t I From TheSmall Ripple

Approximation1 1 1

g L ON L S L C OUTS S

I I T I D T D I D V D IT T R


Buck Converter DC Model

LoadRVg VOUT

IOUTIg D·IOUT D·Vg

VgLoad

R

Ig 1:D

VOUT

IOUT


AC (Small Signal) Modeling

PowerStage Loadˆg g gv V v

ˆg g gi I i

Ô O Ov V v

Ô O Oi I i

ˆd D d ˆ

ˆ

ˆ

ˆ

ˆ

g g

g g

O O

O O

v V

i I

v V

i I

d D

SmallSignalModeling

How Small is Small?Small Enough TheSystem Remains

Linear


AC Modeling

PowerStage Loadˆgv

gi

Ôv

Oi

d

Control To OutputTransfer Function ˆ 0

ˆˆ

g

Ovd

v

vGd


AC Modeling

PowerStage Loadˆgv

gi

Ôv

Oi

dAudio-

susceptibility Input ImpedanceOutput

Impedance

ˆ 0

ˆÔ

vgg d

vGv

ˆ 0

ˆˆg

ig d

vZ

i

ˆ ˆ0, 0

ˆˆ

g

OO

O d v

vZi


Averaging

0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

1.2000

1.4000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

d Gate

I Iave


Averaging

0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

1.2000

1.4000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

d Gate

I Iave

1 S

S

LLL L

CCC C

t T

T tS

d i tdi tL v t L v t

dt dtd v tdv t

C i t C i tdt dt

x t x t x t dtT


Another View Of Averaging

0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

1.2000

1.4000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

d Gate

I Iave 1 S

S

t T

T tS

x t x t x t dtT

0 1 105 2 105 3 105 4 105 5 105 6 105 7 105 8 105 9 105 1 1060

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Hma fk

fk

Moving AverageFilter Response

+


Modeling The Buck Converter:On Time

1 1

LL g O g C

CC L C L C

di tv t L v t v t v t v t

dtdv t

i t C i t v t i t v tdt R R



Modeling The Buck Converter:Off Time

1 1

LL C C

CC L C L C

di tv t L v t v t

dtdv t

i t C i t v t i t v tdt R R


Averaging The Inductor Voltage

1

'

1

St T

L LtS

g C C

g C C

g C C C

g C

LL g C

v t v dT

d t v t v t d t v t

d t v t v t d t v t

d t v t d t v t v t d t v t

d t v t v t

d i tv t L d t v t v t

dt


Averaging The Capacitor Current

Averaging The Input Current

1 1'

1

1

C L C L C

L C

CC L C

i t d t i t v t d t i t v tR R

i t v tR

d v ti t C i t v t

dt R

' 0g L

L

i t d t i t d t

d t i t


Perturb And Linearize

DC Operating Point

Input Perturbation

ˆ

ˆg g gv t V v t

d t D d t

Resulting Output Perturbation

ˆ

ˆ

ˆ

L L L

C C C

g g g

i t I i t

v t V v t

i t I i t

Small Signal Constraint

ˆ

ˆg gv t V

d t D

ˆ

ˆ

ˆ

L L

C C

g g

i t I

v t V

i t I

gV C gV D V

1L CI V

R g LI D I



Averaged Differential Equation

Substitute DC Plus Perturbation For Average Values:

Expand:

L

g C

d i tL d t v t v t

dt

ˆˆ ˆ ˆL L

g g C C

d I i tL D d t V v t V v t

dt

ˆ ˆ ˆˆ ˆ

ˆ

LLg g g g

C C

di tdIL L D V D v t V d t d t v tdt dt

V v t



Collect Terms: DC Terms Equal Zero

Discard 2nd Order Terms

This Leaves A First Order Equation:

ˆ

ˆˆ ˆˆ ˆ

LLg C

g g C

g

di tdIL L D V Vdt dt

D v t V d t v t

d t v t

ˆ ˆˆ ˆL

g g C

di tL D v t V d t v t

dt



Averaged Differential Equation

Substitute DC Plus Perturbation For Average Values:

Expand: DC Terms Equal Zero

1C

L C

d v tC i t v t

dt R

ˆ 1ˆ ˆC C

L L C C

d V v tC I i t V v t

dt R

ˆ 1 1ˆ ˆCC

L C L C

dv tdVC C I V i t v tdt dt R R

ˆ 1ˆ ˆC

L C

dv tC i t v t

dt R



ˆˆ ˆ

ˆ ˆˆ ˆ

ˆ ˆˆ ˆ ˆ

g L

g g L L

L L L L

g L L L

i t d t i t

I i t D d t I i t

D I D i t I d t d t i t

i t D i t I d t d t i t

ˆˆ ˆg L Li t D i t I d t


Construct The Model

ˆ ˆˆ ˆL

g g C

di tL D v t V d t v t

dt

ˆgD v t

gi t ˆ

gV d t L

Ldi tL

dt

Li t

C ˆCv t


Construct The Model

1 ˆCv tR

R ˆCv t ˆCdv t

Cdt Li tC

ˆ 1ˆ ˆC

L C

dv tC i t v t

dt R


Construct The Model


ˆgv t

Li t

LD i t ˆLI d t


Construct The Model

ˆgv t

gi t

ˆLI d t LD i t ˆgD v t

ˆgV d t L

Ldi tL

dt

Li t

C ˆCv t Li tC

ˆCv t ˆCdv t

Cdt R

1 ˆCv tR

ˆgv t

gi t

ˆLI d t

ˆgV d t

1: DL Li t

C ˆCv t R


Control-To-Output Transfer Function

ˆ 0

ˆˆ

g

Cvd

v

v sG s

d s

0 ˆgv t

gi t

ˆLI d t

ˆgV d t

1: DL Li t

C ˆCv t R


gi t

ˆLI d t

ˆgV d t

1: DL Li t

C ˆCv t R


ˆgV d t L Li t

C ˆCv t R



ˆgV d t L Li t

C ˆCv t R

1||ˆˆ

1||C g

Rs Cv s V d s

s L Rs C

2

2

0 0

ˆ 1 1ˆ 11 1

Cvd g g

v sG s V VLd s s ss L C s

R Q

20

1L C

12

CQ RL


ˆgv t

gi t

ˆLI d t

ˆgV d t

1: DL Li t

C ˆCv t R

Input To Output Transfer Function

ˆ 0

ˆˆC

vgg d

v sG s

v s

“Audiosusceptibility”


Input To Output Transfer Function

L Li t

C ˆCv t R

gi t

ˆgD v t

ˆgv t

gi t 1: DL Li t

C ˆCv t R


Input To Output Transfer FunctionL Li t

C ˆCv t R

gi t

ˆgD v t

1||ˆ ˆ

1||C g

Rs Cv s D v

s L Rs C

2

2

0 0

ˆˆ 11 1

Cvd

g

v s D DG s Lv s ss L C sR Q


ˆgv t

gi t

ˆLI d t

ˆgV d t

1: DL Li t

C ˆCv t R

Input Impedance

ˆ 0

ˆˆg

ig d

v sZ s

i s


Input Impedance

ˆgv t

gi t2

1 LD

2D C 2

1 RD

ˆgv t

gi t 1: DL Li t

C ˆCv t R


Input Impedance

ˆgv t

gi t2

1 LD

2D C 2

1 RD

2 2 2

ˆˆ1||

gg

v si s L Rs

D D s D C

2 2 2

ˆ 1||ˆg

ig

v s L RZ s sD D s D Ci s

2

2

0 02 2

111

1 1i

s sLs s L C QR RRZ sD s R C D s R C


Boost And Buck-Boost Models

Re-drawn from “Fundamentals of Power Electronics”, 2nd ed., Erickson and Maksimovic, Figure 7.17

ˆgv t

gi t L ˆCV d t

' :1D

ˆLI d t ˆCv tC R

ˆgv t

gi t

ˆLI d t

1: D Li t L ˆg CV V d t

' :1D

ˆLI d t C ˆCv t R


Flyback Example

Magnetizing Inductance: L

D1 ForwardVoltage: VD

Q1 On Resistance: RON


On-Time Circuit

C

g

g

L

L C


Off Time Circuit


Sketching The Inductor Waveforms

d·TS TS

Lv t

0

00

Lv t

Li t Li t

g ON Lv t R i t

D CN V v t


Sketching The Capacitor Waveforms

S S

Cv t

Ci t

Cv t

Ci t

1Cv t

R

1L CN i t v t

R


Averaging The Circuit Equations

On Time Off Time

1L g ON L

C C

g L

v t v t R i t

i t v tR

i t i t

1L g ON L

C C

g L

v t v t R i t

i t v tR

i t i t

1

0

L D c

C L C

g

v t N V v t

i t N i t v tR

i t

1

0

L D c

C L C

g

v t N V v t

i t N i t v tR

i t



'

' '

L g ON L

D c

g ON L

c D

v t d t v t R i t

d t N V v t

d t v t d t R i t

d t N v t d t N V

' '

LL

g ON L

c D

d i tL v t

dtd t v t d t R i t

d t N v t d t N V


Averaging The Capacitor Current

1

1'

1'

C C

L C

L C

i t d t v tR

d t N i t v tR

d t N i t v tR

1'

cC

L C

d v tC i t

dt

d t N i t v tR


Averaging The Input Current/Perturbation

Perturbing The Inputs Perturbed Circuit Variables

Substitute Expressions Into Averaged Differential Equations

' 0g L

L

i t d t i t d t

d t i t

ˆ

ˆ' 1 1

ˆg g g

d t D d t

d t d t D d t

V t V v t

ˆ

ˆ

ˆ

L L L

C C C

g g g

i t I i t

v t V v t

i t I i t



' '

LL

g ON L

c D

d i tL v t

dtd t v t d t R i t

d t N v t d t N V

ˆˆ ˆ ˆˆ

ˆ ˆˆ1 1

L Lg g ON L L

C C D

d I i tL D d t V v t D d t R I i t

dtD d t N V v t D d t N V

!



ˆˆ ˆˆ ˆLL

g g g g

d i td IL L D V D v t V d t d t v t

dt dt

ˆ ˆˆ ˆ

ˆ ˆˆ ˆ ˆˆ

ON L ON L ON L ON L

C C C C C C

D D D

D R I D R i t R vI d t R d t i t

N V N v t D N V D N v t N V d t N v t d t

N V D N V N V d t

!


Averaging The Inductor Voltage !

ˆ

ˆˆ ˆ ˆ ˆ

ˆ

ˆ ˆˆ ˆ ˆˆˆ ˆ

LLg ON L C C D D

g

g ON L C D

ON L

C C

g ON L C

d i td IL L D V D R I N V D N V N V D N V

dt dtD v t

V d t R I d t N V d t N V d t

D R i t

N v t D N v t

d t v t R d t i t N v t d t


Averaging The Inductor Voltage !

Get The Same Result By InductorVolt-Second Balance Calculation

0

1 1

' '

1'

L

g ON L C C D D

g ON L C D

C g ON L D

C g ON L D

d IL

dtD V D R I N V D N V N V D N V

D V R I D N V D N V

D N V D V R I D N V

DV V R I VD N



ˆ

ˆ ˆ ˆ ˆ

ˆ

ˆ ˆ

ˆˆ

ˆ

ˆ1

Lg

g ON L C D

ON L

C C

g

g ON L C D

ON L

C

d i tL D v t

dtV d t R I d t N V d t N V d t

D R i t

N v t D N v t

D v t

V R I N V V d t

D R i t

D N v t


Construct The Model

ˆˆ

ˆ ˆ'

Lg g ON L C D

ON L C

d i tL D v t V R I N V V d t

dtD R i t D N v t

ˆgD v t Ld i tL

dt

OND RL ˆg ON L C DV R I N V V d t

ˆ' CD N v t

Li t


ˆ ˆ ˆ

ˆ ˆˆ

1 1 ˆ

CCL L L L

L L

C C

d v td VC C N I N i t D N I D N i t

dt dtN I d t N i t d t

V v tR R

Averaging The Capacitor Current !

1'c

L C

d v tC d t N i t v t

dt R

ˆ 1ˆ ˆ ˆ1C C

L L C C

d V v tC D d t N I i t V v t

dt R



ˆ 1ˆˆ ˆ'C

L L C

d v tC D N i t N I d t v t

dt R

ˆ 1

ˆ ˆ

ˆ

1 ˆ

CCL L C

L L

L

C

d v td VC C N I D N I V

dt dt RN i t D N i t

N I d t

v tR



1' 0

1'

1 1'

CL

L C

L C

d VC D N I V

dt R

D N I VR

I VD N R

Get The Same Result By CapacitorCharge Balance Calculation


Constructing The Model

ˆ 1ˆˆ ˆ'C

L L C

d v tC D N i t N I d t v t

dt R

ˆCv t ˆCd v t

Cdt ˆ' LD N i t ˆ

LN I d t

1 ˆCv tR

RC


Averaging The Input Current

ˆˆ ˆ

ˆ ˆˆ ˆ

ˆˆ ˆ

g L

g g L L

L L L L

g L

g L L

i t d t i t

I i t D d t I i t

D I D i t I d t d t i tI D I

i t D i t I d t


Constructing The Model


gi t

ˆgv t ˆLI d t LD i t


Completing The Flyback Model

ˆLI d t

LD i t ˆgD v t L

Ld i tL

dt

Li t OND R ˆ

g ON L C DV R I N V V d t

ˆ' CD N v t ˆgv t

gi t

ˆgv t

gi t

ˆLI d t

Li t

Ld i tL

dt

LOND R

ˆg ON L C DV R I N V V d t

ˆ' CD N v t

1: D


Completing The Flyback Model

ˆ' CD N v t ˆ' LD N i t

ˆLN I d t

C

ˆCd v tC

dt ˆCv t

1 ˆCv tR

R

R

1 ˆCv tR

ˆCv t ˆCd v t

Cdt ˆ

LN I d t

':1N D

C


Complete Flyback Model


gi t

ˆLI d t

1: D Li t

Ld i tL

dt

LOND R ':1N D

R ˆCv t ˆCd v t

Cdt ˆ

LN I d t C



gi t

ˆLI d t

1: D

+



1: D Li t

Ld i tL

dt

L

+




Li t

Ld i tL

dt

LOND R ':1N D

ˆLN I d t

+



Li t

Ld i tL

dt

LOND R ':1N D

+



R ˆCv t ˆCd v t

Cdt ˆ

LN I d t C

+



ˆ 0

ˆˆ

g

Cvd

v

v sG s

d s

0


gi t

ˆLI d t

1: D Li t

Ld i tL

dt

LOND R ':1N D

R ˆCv t ˆCd v t

Cdt ˆ

LN I d t C



Still Have Two dhat Terms

Solve By Using Superposition


Li t

Ld i tL

dt

LOND R ':1N D

R ˆCv t ˆCd v t

Cdt ˆ

LN I d t C




Li t

Ld i tL

dt

LOND R ':1N D

R ˆCv t ˆCd v t

CdtC

Li t

Ld i tL

dt

LOND R ':1N D

R ˆCv t ˆCd v t

Cdt ˆ

LN I d t C



Li t

Ld i tL

dt

LOND R ':1N D

R ˆCv t ˆCd v t

CdtC

Control-To-Output Transfer Function ' g ON L C DV V R I N V V

Li t LOND R ˆ'V d t

'C 'R ˆ' CN D v t

2' 'R N D R

2''

CCN D



Li t LOND R ˆ'V d t

'C 'R ˆ' CN D v t

1

1' ||' ˆˆ' '1' ||

'1' ||ˆ ' '

ˆ 1' ' ||'

C

ON

Cvd

ON

Rs CN D v s V d t

s L D R Rs C

Rv s V s CG sN Dd t s L D R R

s C



2

2

1'1 ' ''' || 1' ' 1 ' ''

'1 '' ||

1 ' '' 1 ' '1 ' 1 ' '' ||

' 1 ' ''

' ' ' ' '

'' ' '

ON ON

ON ON

ON ON

Rs C Rs CR

s C s C s R CRs CRR

s R Cs C s R CR s R Cs L D R R s L D R

s C s R CR

s L s R C L D R s D R R C R

RR D R s L s D R R C s

2

1'

1' ''

1

1 ' '' 'ON

ON

RR C L

R

D R Ls D R C s C LR R



2

1' ||1'

1' || 1 ' '' ' '

ONON ON

Rs C

D R Ls L D R R s D R C s C Ls C R R

2' 'R N D R 2''

CCN D

2

2 2 2 2

1

1' ' ' 'ON

OND R L C Cs D R s L

N D R N D R N D N D



' g ON L C DV V R I N V V

1

22 2 2 2

' 1'

1' ' ' '

vd

ON ON

G sV

N D D R D R CL L Cs sN D R N D R N D N D


ˆCv t ˆLN I d t

' :1N DOND R Li t


ˆCv tRC ˆLN I d t

2''

OND RRN D

2'

'LL

N D

Li t



ˆCv tRC ˆLN I d t

2''

OND RRN D

2'

'LL

N D

Li t

1ˆˆ ' ' || ||

1ˆ1 1 1

1' '

C L

L

v t N I d t s L R Rs C

N I d t

s L R Rs C



2

1 11 1 1 1 1

1' ' ' '

11 1' '

1' ' 1

' '

' '' ' ' '

s Cs L R R s L R R

s C

s R Cs L R R

R s L R s R CR s L R

R s L RR s L R s R C R s R L C



2

2

1' '1

1 1 1 1' ' ' '1' '

' '' '1 ' '

R s L R RR R s L s R C R s R L C

s L R R Rs C

s L RR Ls R C s L CR R

2''

LLN D

2'

'OND RR

N D



2 2

2 22

2 2

' '

' '1

' '

ON

ON

ON

D RLsN D N D

D R LN D N D D R Ls C s C

R R N D N D

2

22 2 2 2

1'

1' ' ' '

ON

ON ON

s L D RN D

D R D R CL L Cs sN D R N D R N D N D



2

ˆˆC

vd

v tG s

d t

2

22 2 2 2

1'

1' ' ' '

ON

L

ON ON

s L D RN D

N ID R D R CL L Cs s

N D R N D R N D N D



1 2vd vd vdG s G s G s

1 2

1 2

vd vd vdvd

vd vd vd

num G s num G s num G sG s

den G s den G s den G s

1 2vd vd vdden G s den G s den G s

1 2vd vd vdvd

vd vd

num G s num G s num G sG s

den G s den G s



1

2 2

''

1'

vd

vd L ON

Vnum G sN D

num G s N I s L D RN D

1 2 2

' 1' '

vd vd L ONVnum G s num G s N I s L D R

N D N D



1 2 2 2

' '' '

L ONvd vd

N I s L D RV N Dnum G s num G sN D N D

2

2

2

' ''

' ' 1' ''

' ' 1' ' '

ON L L

ON L L

ON L

ON L L

ON L

V N D N D R I s N I LN D

V N D N D R I N I LsV N D N D R IN D

V D D R I N I LsN D V N D N D R I

!


Control To Output Transfer Function !

1' '

1

Dimensionless

L

ON L

N I LsV N D N D R I

v sA vs Av A v

Quick check on the algebra: Are the units correct?



' g ON L C DV V R I N V V

2

22 2 2 2

' ' 1' ' '

1' ' ' '

ON L L

ON Lvd

ON ON

V D D R I N I LsN D V N D N D R I

G sD R D R CL L Cs s

N D R N D R N D N D

!


Averaging Summary

• Write Circuit Differential Equations For Each State In Terms Of Averaged Values

• Average Over One Switching Cycle– Inductor Current– Capacitor Voltages

• Perturb And Linearize– DC Terms Are Zero– Discard 2nd And Higher Order Terms

• Construct The Circuit Model


Averaged Switch Modeling

g O

1

1

2

2

Two Port Switch Network Includes All Nonlinear

And Time Varying Elements

EverythingElse Is

LTI


Boost And Buck-BoostSwitch Networks

g O

1

1

22







Large SignalModel!



Circuit SimulationTime Domain

(e.g. Transient Response)Frequency Domain

(e.g. Bode Plots)

LinearizationSmall Signal

Analytical Models Transfer Functions


Buck Converter Model:Define Switch Network And Ports

Port 1: v1, i1Port 2: v2, i2


Buck Converter Model:Sketch Waveforms

v1(t)

i1(t)

v2(t)

i2(t)

0

vg

vg

iL

iL

<v1(t)>

<v2(t)>

<i1(t)>

<i2(t)>

0 d·TS TS

0

0

0


Buck Converter Model:Average Switch Network Variables

v1(t)

i1(t)

v2(t)

i2(t)

0

vg

vg

iL

iL

<v1(t)>

<v2(t)>

<i1(t)>

<i2(t)>

0 d·TS TS

0

0

0

1

1

2

2

'

' 0

' 0

'

g g

g

L

L

g

g

L L

L

v t d v t d v t

v t

i t d i t d

d i t

v t d v t d

d v t

i t d i t d i t

i t


Buck Converter Model:Eliminate Non-Switch Variables

1

1

2

2

'

' 0

' 0

'

g g

g

L

L

g

g

L L

L

v t d v t d v t

v t

i t d i t d

d i t

v t d v t d

d v t

i t d i t d i t

i t

2 1gv t d v t d v t


Buck Converter Model:Eliminate Non-Switch Variables

2 1gv t d v t d v t

1 2Li t d i t d i t

1

1

2

2

'

' 0

' 0

'

g g

g

L

L

g

g

L L

L

v t d v t d v t

v t

i t d i t d

d i t

v t d v t d

d v t

i t d i t d i t

i t


Buck Converter Model:Create Switch Network Model 1 2i t d t i t 2 1v t d t v t

1v t 2v t

1i t 2i t

d t

1d t v t 2d t i t

!


Buck Converter Model:Complete The Converter Model

!


Buck Converter Model:Complete The Converter Model

!


Another View Of TheAverage Switch Model

+

A

P

C



A

PC

+



AP

Cg

g Q

L

D

D

CR

CL

+



A

P

C

1v t 2v t

1i t 2i t

d t

1d t v t 2d t i t

+



1v t 2v t

1i t 2i t

d t

1d t v t 2d t i t

A

P

C

+



1: d tA Ai t

P

C Ci t

APv t CPv t

+


Simulating Buck Converter WithAveraged Switch Model

• FSWITCH = 500 kHz• Vg = 12 Vdc• VO = 3 Vdc• D = 0.25• IO = 3 A• RO = 1 Ω

• ∆IL = 20% IO = 0.6 A L = 7.5 µH

• C = 33 µF– ESR = 50 mΩ– FZERO = 96.5 kHz– ∆VC= 9.1 mV

(capacitor only)

• F0 = 10.1 kHz• Q = 2.1 = 6.4 dB


LTspice Model: DC Sweep

Vg

12

L1

7.5µ

C1

33µ

ESR0.05 RO

Ro

Vdhat

0

B2

V=V(d)*Vg

B1

I=V(d)*I(L1)

VD_nom

Dnom

dd

v2 vov1

.dc VD_nom 0 0.95 0.01

.param Vg = 12

.param Vo_nom = 3

.param Ro = 1

.param Dnom = (Vo_nom/Vg)

.param Io_nom = Vo_nom/Ro

Input Parameters (Vg, Vo_nom, Ro) Calculated Parameters (Io_nom, Dnom)

VD_nom sets the dc duty cycle operating point

Vdhat sets the variation around the dc operating pointSet to zero for dc sweep analysis

+


LTspice Model: DC Sweep

0mV 100mV 200mV 300mV 400mV 500mV 600mV 700mV 800mV 900mV0V

1V

2V

3V

4V

5V

6V

7V

8V

9V

10V

11V

12VV(vo)

+


Vg

12

L1

7.5µ

C1

33µ

ESR0.05 RO

Ro

Vdhat

PULSE(0.025 -0.025 0 1u 1u 1m 2m)

B2

V=V(d)*Vg

B1

I=V(d)*I(L1)

VD_nom

Dnom

dd

v2 vov1

.tran 5m

.param Vg = 12

.param Vo_nom = 3

.param Ro = 1



Input Parameters (Vg, Vo_nom, Ro)Calculated Parameters (Io_nom, Dnom)


Vdhat sets the variation around the dc operating point

LTspice Model: Transient Response


0.0ms 0.5ms 1.0ms 1.5ms 2.0ms 2.5ms 3.0ms 3.5ms 4.0ms 4.5ms 5.0ms0.4A

0.8A

1.2A

1.6A

2.0A

2.4A

2.8A

3.2A

3.6A

4.0A

4.4A2.4V

2.5V

2.6V

2.7V

2.8V

2.9V

3.0V

3.1V

3.2V

3.3V

3.4V

3.5V

220mV

225mV

230mV

235mV

240mV

245mV

250mV

255mV

260mV

265mV

270mV

275mV

280mV

I(L1) -I(Vg)

V(vo)

V(d)

LTspice Transient Simulation


LTspice Model: Plotting Gvd

Vg

12

L1

7.5µ

C1

33µ

ESR0.05 RO

Ro

AC 1

Vac

0

B2

V=V(d)*Vg

B1

I=V(d)*I(L1)

VD_nom

Dnom

dd

v2 vov1

.param Vg = 12

.param Vo_nom = 3

.param Ro = 1



.ac dec 50 25 250k

Input Parameters (Vg, Vo_nom, Ro)Calculated Parameters (Io_nom, Dnom)


Vac is the sweep source


LTspice Model

100Hz 1KHz 10KHz 100KHz-30dB

-25dB

-20dB

-15dB

-10dB

-5dB

0dB

5dB

10dB

15dB

20dB

25dB

30dB

-200°

-180°

-160°

-140°

-120°

-100°

-80°

-60°

-40°

-20°

0°

20°

40° V(vo)


Flyback Simulation WithAverage Switch Model

+



+



+



+

AP

C

1: d tA Ai t

P

C Ci t

APv t CPv t



+

AP

C

1:dt

A

Ait

P

C

Cit

AP

vt

C

Pv

t



+



+

Vg

380

B1

I=-V(d)*I(B2)

B2

V=V(d)*(V(Vg)-V(Va))

Lpri12m

B3

V=-V(Va)/TR

B4

I=-I(VD)/TR

C0

47µ

R03

Vdhat

0

VD_nom

D0

VD

0.4

Va

dd

Vg Vo

VL

.op.IC V(Vo) = 12.IC I(Lpri) = 0.494

.param D0 0.2646

.param TR 11


Vdhat sets the variation around the dc operating



+

Vg

380

B1

I=-V(d)*I(B2)

B2


Lpri12m

B3

V=-V(Va)/TR

B4

I=-I(VD)/TR

C0

47µ

R03

Vdhat

0

VD_nom

D0

VD

0.4

Va

dd

Vg Vo

VL

.op.IC V(Vo) = 12.IC I(Lpri) = 0.494

.param D0 0.2646

.param TR 11


Vdhat sets the variation around the dc operating



+

Vg

380

B1

I=-V(d)*I(B2)

B2


Lpri12m

B3

V=-V(Va)/TR

B4

I=-I(VD)/TR

C0

47µ

R03

0

Vdhat

VD_nom

1

VD

0.4

Va

dd

Vg Vo

VL

.dc VD_nom 0.0 0.95 0.025*.IC V(Vo) = 12*.IC I(Lpri) = 0.494

*.param D0 0.param TR 11





+

0mV 100mV 200mV 300mV 400mV 500mV 600mV 700mV 800mV 900mV-70V

0V

70V

140V

210V

280V

350V

420V

490V

560V

630V

700VV(vo)

D =0 D =0.95

V = 0

V = 630



+

Vg

380

B1

I=-V(d)*I(B2)

B2


Lpri12m

B3

V=-V(Va)/TR

B4

I=-I(VD)/TR

C0

47µ

R03

Vdhat1

PULSE(-0.05 0.05 2m 10n 10n 2m 5m)VD_nom

D0

VD

0.4

Va

dd

Vg Vo

VL

.tran 20m.IC V(Vo) = 12.IC I(Lpri) = 0.494

.param D0 0.2646

.param TR 11


Vdhat sets the variation around the dc operating p



+

0ms 2ms 4ms 6ms 8ms 10ms 12ms 14ms 16ms 18ms 20ms80mA

160mA

240mA

320mA

400mA

480mA

560mA

640mA

720mA

800mA

880mA6V

7V

8V

9V

10V

11V

12V

13V

14V

15V

16V

17V

18V

I(Lpri)

V(vo)

Vout

IL_PRI



+

Vg

380

B1

I=-V(d)*I(B2)

B2


Lpri12m

B3

V=-V(Va)/TR

B4

I=-I(VD)/TR

C0

47µ

R03

AC 1

Vdhat1

VD_nom

D0

VD

0.4

Va

dd

Vg Vo

VL

.ac dec 100 10 100k.IC V(Vo) = 12.IC I(Lpri) = 0.494

.param D0 0.2646

.param TR 11





+

10Hz 100Hz 1KHz 10KHz 100KHz-15dB

-10dB

-5dB

0dB

5dB

10dB

15dB

20dB

25dB

30dB

35dB

40dB

45dB

-300°

-270°

-240°

-210°

-180°

-150°

-120°

-90°

-60°

-30°

0°

30°

60° V(vo)



+

Vg

380

B1

I=-V(d)*I(B2)

B2


Lpri12m

B3

V=-V(Va)/TR

B4

I=-I(VD)/TR

C0

47µ

R03

0

Vdhat

VD_nom

1

VD

0.4

RQ

1

Va

dd

Vg Vo

VL

.dc VD_nom 0.0 0.99 0.005*.IC V(Vo) = 12*.IC I(Lpri) = 0.494

*.param D0 0.param TR 11


Vdhat sets the variation around the dc operating po


0mV 100mV 200mV 300mV 400mV 500mV 600mV 700mV 800mV 900mV-30V

0V

30V

60V

90V

120V

150V

180V

210V

240V

270V

300V

330VV(vo)


+

D =0 D =0.99


Averaged Switch Small Signal Model

1 1 1

2 2 2

1 1 1

2 2 2

ˆ

ˆ

ˆ

ˆ

ˆ

v t V v t

v t V v t

i t I i t

i t I i t

d t D d t

1v t 2v t

1i t 2i t

d t

1d t v t 2d t i t

!

2 1

1 2

v t d t v t

i t d t i t


Discard Higher Order Nonlinear

Terms


DC And Small Signal Terms!

2 1

2 2 1 1

1 1 1 1

1

2 2 1 1 1

1 1 1

ˆˆ ˆ

ˆ ˆˆ ˆˆ ˆ 0

ˆˆ ˆˆˆ

v t d t v t

V v t D d t V v t

D V D v t V d t d t v t

d t v t

V v t D V D v t V d t

D V v t V d t



1 2

1 1 2 2

2 2 2 2

2

1 1 2 2 2

2 2 2

ˆˆ ˆ

ˆ ˆˆ ˆ

ˆ ˆ 0ˆˆ ˆ

ˆˆ

i t d t i t

I i t D d t I i t

D I D i t I d t d t i t

d t i t

I i t D I D i t I d t

D I i t I d t


Construct The Switch Model

2 2 1 1 1ˆˆ ˆV v t D V v t V d t 1 1 2 2 2

ˆˆ Î i t D I i t I d t

1 1I i t

1 1V v t 2Î d t 2 2D I i t 1 1D V v t

1ˆV d t

2 2ˆV v t

ˆD d t

2 2I i t


Construct The Switch Model

2 2 1 1 1ˆˆ ˆV v t D V v t V d t 1 1 2 2 2

ˆˆ Î i t D I i t I d t

1 1I i t

1 1V v t 2Î d t

1ˆV d t

2 2ˆV v t

ˆD d t

2 2I i t1: D


Averaged Switch Small SignalBuck Converter Model

ˆg gV v t

ˆg gI i t

2Î d t

1: D ˆ

gV d t L

C ˆC CV v t R


Boost Switch Model


1 1V v t

1 1I i t 2

ˆV d t' :1D

1Î d t 2 2ˆV v t

2 2I i t

ˆD d t


General Two-Switch Network


1v t

1i t

2v t

2i t

1 1V v t

1 1I i t

ˆD d t

2 2ˆV v t

2 2I i t' :D D

1 ˆ'

V d tD D

2 ˆ'

I d tD D


Averaged Switch Small Signal Model: Flyback

1 1V v t

1 1I i t

ˆD d t

2 2ˆV v t

2 2I i t' :D D

1 ˆ'

V d tD D

2 ˆ'

I d tD D



C R

:1N

1 1V v t

1 1I i t

ˆD d t

2 2ˆV v t

2 2I i t' :D D

1 ˆ'

V d tD D

2 ˆ'

I d tD D

L

ˆgv t

gi t

ˆLv t Li t

ˆCv t


C R

:1N

1 1V v t

1 1I i t

ˆD d t

2 2ˆV v t

2 2I i t' :D D

1 ˆ'

V d tD D

2 ˆ'

I d tD D

L

ˆgv t

gi t

ˆLv t Li t

ˆCv t


NowWhat?


C R

:1N

1 1V v t

1 1I i t

ˆD d t

2 2ˆV v t

2 2I i t' :D D

1 ˆ'

V d tD D

2 ˆ'

I d tD D

L

ˆgv t

gi t

ˆLv t Li t

ˆCv t


Simulation: Maybe

1 'g D CV D V D N V V

21

CI VR



Re-drawingMay Help

This Arrangement Based On Work Of Christophe Basso

L

C R

:1N

1 ˆ'

V d tD D

2 ˆ'

I d tD D

1 1I i t

1 1V v t 2 2ˆV v t

2 2I i t

' :D D

ˆgv t

gi t

ˆLv t Li t

ˆCv t



Q1

D1

C R vC(t)

vg(t)

ig(t) N:1

LvL(t)

iL(t) iC(t)

+

Re-drawAs For Averaged

Model



Q1 D1

C R vC

(t)

ig(t) N:1

LvL(t)iL(t)

iC(t)

VD

+



C R vC(t)

vg(t)

ig(t) N:1

LvL(t)iL(t)

iC(t)

1 ˆ'

V d tD D

2 ˆ

'I d t

D D

' :D D

+



+

This LooksMore

Promising

1 ˆ'

V d tD D

2 ˆ

'I d t

D D

' :D D

L

C R

DV:1N

ˆgv t

gi t

ˆLv t

Li t

Ci t

ˆCv t



Derive Transfer Functions?Still Very Tedious

!

1 ˆ'

V d tD D

2 ˆ

'I d t

D D

' :D D

L

C R

DV:1N

ˆgv t

gi t

ˆLv t

Li t

Ci t

ˆCv t


1 ˆ'

V d tD D

2 ˆ

'I d t

D D

' :D D

L

C R

DV:1N

ˆgv t

gi t

ˆLv t

Li t

Ci t

ˆCv t


The Point Of This Example?Averaged Switch

Small Signal ModelingHas Limitations

!


ErrorAmp/

Compensator

Small Signal Model OfPulse-Width Modulator

Vref Vout+ –PowerStagePWM

VoltageDividerNot LTI

We Also Need A Small Signal Model For The Pulse Width Modulator



0

VRAMP_MIN

VRAMP_MAX

0

vC(t)

PWM(t)

0 d·TS TS

VRAMP

vRAMP(t)



_

_

C RAMP MIN

RAMP

RAMP MINC

RAMP RAMP

v t Vd t

VVv t

V V

ˆ

ˆC C C

d t D d t

v t V v t

_

_

_

ˆˆ

ˆ

ˆˆ

RAMP MINC C

RAMP RAMP

C RAMP MINC

RAMP RAMP

C RAMP MIN

RAMP

C

RAMP

VV v tD d t

V VV Vv t

V VV V

DV

v td t

V


Pulse WidthModulator


1

RAMPV

PWM Has A Fixed Small Signal Gain

ˆCv t d t

!


More On Small Signal Modeling

• “Fundamentals of Power Electronics”, 2nd ed., Erickson and Maksimovic, Chapter 7

• V. Vorperian, "Simplified analysis of PWM converters using model of PWM switch. Continuous conduction mode," Aerospace and Electronic Systems, IEEE Transactions on, vol. 26, pp. 490-496, 1990.

• V. Vorperian, "Simplified analysis of PWM converters using model of PWM switch. II. Discontinuous conduction mode," Aerospace and Electronic Systems, IEEE Transactions on, vol. 26, pp. 497-505, 1990.

• Papers, seminars, books by Ray Ridley and Christophe Basso


Summary

• We Need Small Signal Models To Use Standard Control Tools To Design The Loop

• Two Averaging Methods Shown To Derive A Small Signal Model Of A Switching Converter:– Average Inductor Voltages And Capacitor Currents– Average Switch Network Port Voltages And

Currents

• Transfer Functions Of Interest Can Be Found From The Small Signal Models


You Can Download The Latest Version Of The SeminarIn Adobe Acrobat (.pdf)

And Microsoft PowerPoint Show Format (.ppsx)From The Embedded Power Labs website:

http://www.embeddedpowerlabs.com/publications.html


About The PresenterBob White has over 30 years experience in power electronics. He has held managerial and individual contributor positions in product development, technology development, applications and systems engineering, and technical marketing. His areas of expertise include power systems for computing and telecommunications systems, digital power, and applications of wide bandgap power semiconductor devices. Bob is currently the president and chief engineer of Embedded Power Labs, a power electronics consulting company.

Bob has been very active in the IEEE Power Electronics Society and the APEC committees, including twice serving as the APEC General Chair.

He is a Fellow of the IEEE, has a BSEE from MIT, a MSEE from Worcester Polytechnic Institute and is currently pursuing a Ph.D. in power electronics at the University of Colorado-Boulder. He is also an Honorably Discharged veteran of service in the United States Air Force.

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