Dynamic Analysis of Switching Converters (2)

8/18/2019 Dynamic Analysis of Switching Converters (2)

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Chapter 6

Dynamic analysis

of switching converters


2/134

Power switchin Dynamic analysis of s 2

Overview

Continuous-Time Linear Models Switching converter analysis using classical control

techniques

Averaged switching converter models Review of negative feedback using classical-control

techniques

eedback compensation

State-space representation of switching converters !nput "#! filters


3/134


Overview

Discrete-time models Continuous-time and discrete-time domains

Continuous-time state-space model

$iscrete-time model of the switchingconverter

$esign of a discrete control system withcomplete state feedback


4/134


$ynamic analysis

Dynamic or small-signal analysis of the switching

converter enables designers to predict the dynamic

performance of the switching converter to reduce

prototyping cost and design cycle time

Dynamic analysis can be either numerical or

analytical


5/134


$ynamic analysis

Switching converters are non-linear time-variant circuits

Nevertheless, it is possible to derive a continuous time-invariant linear model to represent a switching converter

Continuous-time models are easier to handle, but notvery accurate

Since a switching converter is a sampled system, adiscrete model gives a higher level of accuracy


6/134


%inear model of a switching

converter

Lo ref

o L

Z V kV

Z Z =

+


7/134


&'# modulator model

Sensitivity of the duty cycle with respect to vref

Voltagemode control

ref

p

V D

V =

$ !ref ref

ref p

Dd v v

V V = =$ $


8/134


9/134

Power switchin Dynamic analysis of s (

&'# modulator model

Variation of the duty cycle due to a pertur#ation in the output voltage

$urrent mode control

! % &d cr V V L= −

'

) * % &+cd cr V V v L= − +

!r I DT = ∆

'

) % &r I D d T = ∆ +

!

' '

% &

) % & * % &+

d c

cd c

V V Lr I DT

r I D d T V V v L

−∆= =

∆ + − +'

'

'

c

cd c

vd D

V V v

= − −

' '

c

d c

Dd v

V V

= −

$

d c

d D

V V v

∂= −∂ $


10/134

Power switchin Dynamic analysis of s !,

&'# modulator model

Variation of the duty cycle due to a pertur#ation on the pea- current

$urrent mode control

''

p I d T r =

d cV V

r L

−=

''

p

d c

I L

d T V V = −

'

'

!

d c p

d L

T V V I

∂=

−∂


11/134

Power switchin Dynamic analysis of s !!

Averaged switching converter

models

.hreeterminal averagedswitch model

/veragedswitch model for voltagemode control

$!

apV v d

D=

$! ci I d =


12/134

Power switchin Dynamic analysis of s !2


models

01amples of switching converters with an averaged switch


13/134



models

Smallsignal averagedswitch model for the discontinuous mode

$

$

! 2

2 2

3 2

ai

ac

a

pac

ac

p

p

o

cp

I g

V

I I d

D

I I v

V

I I d D

I g

V

=

=

=

=

=

$


14/134



models

Smallsignal model for currentmode control

2

2% & !

e

n z n

s s H s

Qω ω = + +

2 z Q

π −=

n

sT

π ω =

2

! 2

%! &

2

!

% &

s i f

s ir

m

n c s

DT R Dk

L

D T Rk

L

and

F S S T

− = − ÷

−=

=+


15/134


Output filter model

utput filter of a switching converter

o

o so

oo

o

1 R

s! V "s# $ "s# %V 1

& sL R s!

÷

o oo

' s

o oo o

1

"s#V ! L $ %

s 1"s#V & & s! ! R L

'o

' 'o o

H"s# $ ( & ' s & s

ω

ζ ω ω o

o

1 $

! 'R

L

ζ oo o

1 $ %

! Lω


16/134


Output filter model

agnitude response of the output filter for several values of the

output resistance o

22 2

2, log % & !, log ! 4o o

) ω ω

ω ζ ω ω

÷ ÷= − − + ÷ ÷ ÷ ÷ ÷


17/134


Output filter model

Phase response of the output filter for several values of the output

resistance o

!

2

2

% & tan

!

o

o

ω ζ

ω ω

ω

ω

−

÷ ÷Φ = − ÷

÷− ÷ ÷


18/134

Power switchin Dynamic analysis of s !"

Output filter model

utput filter with a capacitor esr

o esr esr o

o o esr oo'o o esr

o o esr o o esr o o

1 s&

R R ! R H"s# $ %&! L R R R" & # L R R

& s& s " & # " & #! ! L R R L R R

e

!

2 6 *SR

sr o

f ! π

=


19/134

Power switchin Dynamic analysis of s !(

Output filter model

agnitude response of

an output filter with a

capacitor having a Resr

for several values of

the output resistance Ro

22 2

2 22, log % & !, log%! & !, log ! 4o o

) ω ω

ω ω τ ζ ω ω

÷ ÷= + − − + ÷ ÷ ÷ ÷ ÷


20/134

Power switchin Dynamic analysis of s 2,

Output filter model

Phase response of an output filter with a capacitor having a Resr

for several values of the output resistance Ro

!2

2

tan tan

!

+1o L!

esr

o

f

f f $ + % f f

f

ζ

θ −

÷

÷ ÷ ÷ ÷− ÷ ÷

e% &

oo

o o o sr

R

L ! Rω =

+

e

2

o oo sr

o

L!

R

ω ζ

= + ÷

e

o sr ! τ =


21/134

Power switchin Dynamic analysis of s 2!

"(ample 6)*

The boost converter shown in igure !"#$ has the

following parameters% &in ' #$ &, &o ' !$ &, f s ' #

()*, L ' #$ m), C ' #$$ + and L ' !$ " The

reference voltage is . &" The converter operates inthe continuous-conduction mode under the voltage-

mode" /sing 0a1 the averaged-switch model,

calculate the output-to-control transfer function, and

0b1 Matlab to draw the 2ode plot of the transferfunction found in 0a1 "


22/134


"(ample 6)*

Smallsignal model of the #oost converter

0a1

The nominal duty cycle can be calculated as,

d

,

d

d

,

V

V!D

V

VD!

D!

!

V

V−=⇒=−⇒

−=

for the given input and output voltages, we have D'$"."

,ap

,

,

,$

VV

6

VD&%!D&%!

−=

−−=−−=

( )

( )( ) ( )

+

−+

−−

+

=∧

∧

!D!

8s

D!

8$sD!

s8!DV

d

v

22

22

$ap,

( )

8$

D!9

−=⇒

( )D!2 !

$

8:

−=⇒


23/134


"(ample 6)*

Pc

d !

Vv

∧

∧

=

D

V

!

V $P = ref PV

V !,D

V ⇒ = =


24/134


"(ample 6)*

;ode plot of the smallsignal transfer function of the #oost converter


25/134


Small-signal models of switching

converters


26/134



converters


27/134



converters


28/134

Power switchin Dynamic analysis of s 2"


converters


29/134

Power switchin Dynamic analysis of s 2(


converters


30/134



converters


31/134



converters


32/134



converters


33/134


Review of negative feedback

;loc- diagram representation for a closedloop system


34/134


Review of negative feedback

Closed-loop gain

Loop gain

or TL33#

Stability analysis

!

o

ref

V ,

V ,β =

+

LT ,β =

!o

ref

V

V β =

!!

% & % & !",

, , or

p-ase p-ase ,

β β β

== − + =

o


35/134


Relative stability

Definitions of gain and phase margins


36/134


Relative stability

8oop gain of a system with three poles


37/134


Closed-loop switching

converter


38/134


eedbac( networ(

2

! 2

)a o R

V V R R

=+


39/134


"rror amplifier compensation

networks

P $ompensation networ-

'

' 1

1 '

1 '

1 1 & R

s! s! H"s# $ %

1 1 & & R R

s! s!

÷ ÷

÷

' 1

1 1 ' ' 1 '

1 & sR ! H"s# $ %

" & & # sR ! ! sR ! !

p

' '

1 $ f

' ! Rπ z ' 1

1 $ f

' ! Rπ

tan+1 1

.ead

z

f $

f θ

÷

!tan

+1.ag

p

f $ %

f θ

÷ ÷

.he total phase lag!

tan tano +1 +11

z p

f f $ + & %'/0

f f θ

÷ ÷ ÷


40/134



networks


41/134



networks

Phase response of the P compensation networ-


42/134



networks

PD $ompensation networ-

' 1 1

'' 1 1 1 ' 1 '

"1& 2 # "1& 2 " & # #! ! R R R H"2 # $ %

+ & 2 " & # & 2! ! ! ! ! R R R R

ω ω ω

ω ω ω

! p

1 $ f

' ! Rπ 2

1 '

p

' 1 '

" & #! ! $ f

' ! ! Rπ

! z

' 1

1 $ f

' ! Rπ 2 z

1

1 $ f

' " & #! R Rπ

'1

1

R $ 3

R

' 1 '

1

" & # R R R $ 3

R R


43/134



networks

agnitude response of the PD compensation networ-


44/134



networks

agnitude response of the PD compensation networ-


45/134



networks

Phase response of the PD compensation networ-


46/134



networks

/symptotic appro1imated magnitude response of the PD

compensation networ-

' 1 1

1 1 $

' ' " & #! ! R R R

π π

' 1 1 $ " & # %! ! R R Rtan+1 1

zd

zd

f $ '

f θ

÷

1 '

' 1 '

1 " & #! ! $

' '! ! ! R Rπ π

1 '

' 1 '

! ! R $ %!

&! ! R

!tan

+1 pd

pd

f $ ' %

f θ ÷ ÷

!tan tan

o +1 +11

zd pd

f f $ + ' & ' %'/0

f f θ

÷ ÷ ÷


47/134


48/134


Compensation in a buc( converter

with output capacitor 4S

f #, is chosen to be one-fifth of the switching

fre5uency

4

4

R

& R R

log10

'0 5 " '%6 6# 7 $ +8 d9

log s p10'0 5 " # 7V V

6 6e

5

% & %!,, !, &%!,, !, &%5 ,5&!5!7

2 2

o

o o o sr

o

R

L ! R : : f kHz

π π

− −+ += = =

+8

1$ $ %1; kHz

' "0%6#100:10π


49/134


50/134



with output capacitor 4S

!

2

2

tan 7,!tan

!

+1o L!

esr

o

f

f f $ + %

f f f

ζ

θ −

÷

÷ = ÷ ÷

÷− ÷ ÷

o

3!5 7,! 244(ea $ $ %θ −o o o

64(tan tan+1 +1

z p

6 6

+ $ % f f

÷ ÷ ÷

o

64(tan tan+1 +1 1 f< + < $ %

f

÷

o

7741' z

1 $ $ pF !

' f Rπ

3"'

' p

1 $ $ pF %

! ' f Rπ


51/134



with no output capacitor 4S

!tan tan

+1 +1 o o o1

zd pd

f f ' + ' $ + $ %'/0 16 16

f f

÷ ÷ ÷

Gtan tan+1 +1 o1' f< + $ %16

f

÷

pd

1 $ $ 0%18 F %!

' f R µ

π

1 1

'

" & #! R R $ $ 0%1 F %!

R µ

1

''

1

'

R! ! R $ $ 6%'= nF %! R

5 + 7! ! R

!% & 36,H Hde.a> p-ase de.a> t f = −

C


52/134


Compensation in a buck converter

with no output capacitor "SR

agnitude response of the #uc- converter

openloop /;$

closedloop FBC8

error amplifier D0


53/134


%inear model of a voltage

regulator including e(ternal

perturbancesref D! o

o ref D! oV V iv ) v ) v ) i= + +$ $ $ $

, ,

,,

D! D!

ref

oo

v v

oo

V ref ref ii

v v)

v v

= =

==

∂= =

∂

$ $

$$

$

$

,,

, ,

ref ref

D!

o o

vv

ooV

D! D! i i

v v)

v v

==

= =

∂= =

∂

$$

$ $

$

$

audio suscepti#ility

output impedance,,

, ,

ref ref

o

D! D!

vv

ooi o

oo v v

V v) Z

I i

==

= =

∂= = =

∂

$$

$ $

$

$

6 t t i d d t bilit


54/134


6utput impedance and stability

utput impedance

!

oo

of

Z

Z ,β = +

! ! !

of o L Z Z Z = +

!

! !oof L

Z Z Z

−

= − ÷ ÷

!

! !o

of L

Z Z Z

−

= − ÷ ÷

! ,β ?of oo Z Z =

of L Z Z = of o Z Z ≈

St t t ti f


55/134


State-space representation of

switching converters

Review of Linear System Analysis/ simple secondorder lowpass circuit

!! 2? L : :•

= +

!1d: : $

dt

•

! 2' : : $ ! : &

R

• •

2'd: : $

dt

•

!' 1 : ? : $ + &

L L•

21 '+ : : : $ & %

! R!

• : $ , : & 9 ?• 1

'

: : $ (

:

0 1+ L

, $ (1 1

+! R!

1? $ 5 7 (?

1

9 $ % L

0

St t t ti f


56/134


State-space representation of

switching converters

Review of Linear System Analysis

/ simple secondorder lowpass circuit

s@"s# $ , @"s# & 9 A"s#

+1 @ "s# $ "s I + , 9 A"s# #

1

''

1 11 s& +

R! L A"s# L

1 s 0"s# @ !

$ % s 1"s# @ & & s

R! L!

1

'1

1 1"s & #"s# @ L R! $

s 1"s#A " & & # s R! L!

'

'1

1

"s# @ L! $ % s 1"s#A " & & # s

R! L!

St t S 7 i


57/134


State-Space 7veraging

approximates the switching converter as a

continuous linear system

re5uires that the effective output filter corner

fre5uency to be much smaller than the

switching fre5uency

St t S 7 i


58/134



Step #% 8dentify switched models over a switching cycle" Drawthe linear switched circuit model for each state of the switchingconverter 0e"g", currents through inductors and voltages acrosscapacitors1"

Step !% 8dentify state variables of the switching converter" 9ritestate e5uations for each switched circuit model using :irchoff;svoltage and current laws"

Step


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Step >% =erturb the averaged state e5uation toyield steady-state 0DC1 and dynamic 07C1terms and eliminate the product of any 7C

terms"

Step .% Draw the lineari*ed e5uivalent circuitmodel"

Step ?% =erform hybrid modeling using a DCtransformer, if desired"

St t S A d M d l f


60/134


State-Space Averaged Model for an

Ideal Buck Converter


61/134

State Space A eraged Model for an


62/134


State-Space Averaged Model for an

Ideal Buck Converter

0 1 0 1+ + L L

, $ d & "1+ d#1 1 1 1

+ +

! R! ! R!

0 1+ L

, $ %1 1

+! R!

1 d 0

9 $ d & "1+ d# $ % L L0

0 0

!

2

1

1

'

0 1d +

: : L $ & 5 7 % L ?

1 1 : : 0+! R!

•

•

A nonlinear continuous equivalent


63/134



circuit of the ideal buck converter

!'

1

d : : $ + & ?

L L

•

2 ! 2

1 1

: $ : + : %! R!

•

!! 2? d $ L : & :•

2!' : : $ ! : & %

R

•

A linear equivalent circuit of the


64/134



ideal buck converter

$!! !, : $ : & : (

$22 2, : $ : & : (

$!! !,? $ ? & ? (

Id $ D & d %

$ $( ) $( ) ( )! 22, !, !! !d : : : D d ? ?dt L L

= − + + + +

$$( ) $( )2 ! 2!, 2,

! !d : : : : :

dt ! R! = + − +

( ) $ $ $( )! 2 !2, !, !,! ! : : D? : D ? d ? L L• = − + + − + +

2I

I'0 '

10 1

1 1 : : : $ " + # & " + # % : :

! R ! R

•

'0 10

10 $ "+ & # : D?

L

'0

10

: $ D %

?

$ $ $2 ! !,

I % &1 1d: $ : D? d?dt L

− + + $2 III 11 10 d: : $ D & d + L %? ? dt

'010

1 :0 $ " + # :

! R

'010

: $ % :

R



65/134



ideal buck converter

$2 I

I'

1

d : 1 : $ " + # :

dt ! R

$2 I

I'

1

d : : $ ! & % :

dt R

$ $2 ! !2, !, !,

I% & : & : $ + L : & D ? &? & d ? %•

!2I

1 10 : $ & d + L : D? ?•

$ $

22,! 2!,

: & : : & : $ ! : &

R

•2

2!

: : $ ! : & %

R

•

A source reflected lineari+ed


66/134


A source-reflected lineari+ed

equivalent circuit of the ideal buck

converter

!! !,I '

'

d L :? & ? $ & : D D D D

•

( )2

22 2!

: : D D : D !

D D R

•

= +

A li i d i l i i f h


67/134


A lineari+ed equivalent circuit of the

ideal buck converter using a $C

transformer

State space averaged model for the


68/134


State-space averaged model for the

discontinuous-mode buck converter

( )! 2 3! 2 ! 2! , , d , d , d d = + + − −

( )! 2 3! 2 ! 2! 9 9 d 9 d 9 d d = + + − −

! : $ 0•

2' :! : & $ 0 %

R

•

3

0 0

, 10 +

R!

=

0 $ % 9

0

State space averaged model for the


69/134


State-space averaged model for the

discontinuous-mode buck converter

1 '

1 '

0 " & #d d +

L , $

& 1d d +

! R!

1d

9 $ % L

0

!

2

1 '1

11

1 ' '

0 " & #d d + d

: : L $ & L ? & 1d d :

: 0+! R!

•

•



70/134



circuit for the discontinuous-mode

buck converter

!1 ' 1

' 1

" & #d d d : $ + & : ?

L L

•

( )! ! ! 2 2? d d d := +

21 ' '

1

" & #d d : : $ + % :

! R!

•

! !%,& % & , : : T = =



71/134



circuit for the discontinuous-mode

buck converter

( )! ! ! 2 2? d d d := +

21 ' '

1" & #d d : : $ + % :

! R! •

11 1 '

s

d $ " + # : ? :

' Lf

I I II I1 1 '10 1 '0 '1 1 '" & #" & # $ " & & & #" & # (? ? : : D D Dd d d

$2 I I II1 ' '0 '1 '

10 1" & & & #d : " & # D D : :d d $ " & # + ( : :

dt ! R!

II I I

1 110 1 10 1 '0 '

s

" & # D d & $ " & + + # % : : ? ? : :' Lf

A lineari+ed equivalent circuit


72/134



for the discontinuous-mode

buck converter

State Space Averaged #odel for a


73/134


State-Space Averaged #odel for a

,uck Converter with a Capacitor

"SR

Switched models for the buck


74/134



converter with a R esr

! 2! 2 e sr ? $ L : : & ! :• •

+ 222!esr : & ! : R : $ ! : &

R

••

!

2

esr

1esr esr

1

'

esr esr

+ R +R R1

L" & R# L" & R# : R R : $ & 5 7 % L ?

R +1 : : 0

!" & R# !" & R# R R

•

•

! 22 esr 0 $ L : & : & ! : % R• •

222!

esr : & ! : R : $ ! : & R

••

e

ee!

2

e e

6

6 6

6 6

sr

1 sr sr

1

'

sr sr

+R +R

L" & R# L" & R# 0 : : $ & 5 7 %?

0 R +1 : :

!" & R# !" & R#

•

•



75/134




! 2 ! 2%! &

esr

esr esr

esr esr

+ R +R R

L" & R# L" & R# R R ,$ , d , d , , $ %

R +1

!" & R# !" & R# R R

+ − = =

! 2 %! &

d

9 $ 9 d 9 d $ % L

0

+ −

!

2

esr

1esr esr

1

'

esr esr

+ R +R Rd

L" & R# L" & R# : R R : $ & 5 7 L ? R +1 :

: 0!" & R# !" & R# R R

•

•

A nonlinear continuous


76/134


A nonlinear continuous

equivalent circuit for the buck


!esr

1 ' 1

esr esr

+ R R d R : $ + & : : ?

L" & R# L" & R# L R R

•

2'

1

esr esr

R : : $ + % :

!" & R# !" & R# R R

•

!esr

1 1 '

esr esr

R R Rd $ L : & &? : :

" & R# " & R# R R

•

!1 'd $ L : & >?•

esr 1 ' 'esr 1'

esr esr esr

R R R : : : R $ & $ " R# & > : R

" & R# " & R# " & R# R R R2

2esr '

1

" & ! : # : R $ ! : & % :

R

••

A lineari+ed continuous


77/134


A lineari+ed continuous

equivalent circuit for the buck

converter with a R esr $

!II I Iesr 10 1 '0 ' 10 1

esr esr

d : + R" & # R" & # "D&d#" & # : : : : ? ? R $ + &

dt L" & R# L" & R# L R R

$2 I I10 1 '0 '

esr esr

d : R" & # " & # : : : : $ + %dt !" & R# !" & R# R R

.he D$ terms are

esr 1010 '0

esr esr

+ R R D? R0 $ + & : :

L" & R# L" & R# L R R

'010

esr esr

R :0 $ + % :

!" & R# !" & R# R R

.he /$ terms are$

!II

I Iesr 1 10

&1 '

esr esr

d : + R R D d? ? R $ + & : :

dt L" & R# L" & R# L L R R

$2 I

I'

1

esr esr

d : R : $ + % :

dt !" & R# !" & R# R R



78/134



using $C transformer with a

turns-ratio of D

! !

I 'esr 1 10 1 '

esr

R :

& d $ L : & " R# & $ L : & > D? ? : R " & R# R

• •

2 1 '

esr esr

R 1! : $ + % : :

& R & R R R

•

State-Space Averaged #odel for


79/134



an !deal ,oost Converter

State-Space Averaged #odel


80/134

Power switchin Dynamic analysis of s ",

State Space Averaged #odel

for an !deal ,oost Converter

!1 $ L :?•

2'

'

: $ ! : &?

R

•

!

2

1 1

' '

1 00 0

: ? : L $ & %0 1

0 1+ : ? : R! !

•

•

!1 '$ L : &? :•

2'

1 '

: & $ ! : & : ?

R

•

!

2

1 1

' '

0 1 1 o+ : ? : L L $ & %

1 1 0 1 : ? : +

! R! !

•

•



81/134

Power switchin Dynamic analysis of s "!

State Space Averaged #odel for

an !deal ,oost Converter

! 2 %! &

0 10 0 +

L , $ , d , d $ d & "1+ d#0 1

1 1++ R!

! R!

+ −

0 +"1+ d#

L , $ %

"1+ d# 1+

! R!

! 2 %! &

1 0 1 0

L L 9 $ 9 d 9 d $ d & "1+ d#

0 1 0 1

! !

+ −

1 0

L 9 $ %

0 1

!

!

2

1 1

' '

0 +"1+ d# 1 0

: ? : L L $ &

"1+ d# 1 0 1 : ? : +! R! !

•

•

!

2

' 1

1 ' '

+"1+ d# : ? : L L

$ & %"1+d# : : ? : +

! R! !

•

•

onlinear continuous


82/134

Power switchin Dynamic analysis of s "2

onlinear continuous

equivalent circuit of the ideal

boost converter

!'

1

+"1+ d# 1 : : $ & ?

L L

•

!1

' L? $ : & :"1+ d# "1+ d#

•

21 ' '"1+d# : : ? : $ + &

! R! !

•

2' '1 ! ? : & $ : & % :"1+ d# "1+ d# R"1+ d#

•

%ineari+ed equivalent circuit of


83/134



the ideal boost converter

$!

II I'0 ' 10 1

d : +"1+ D + d# 1 $ " & # & " & # : : ? ?

dt L L

$2

II I I10 1 '0 ' '

d : "1+ D + d# 1 1 $ " & # + " & # & ( : : : : ?

dt ! R! !

$!

2, !,

I !II ' '0

d : +"1+ D# d 1 D 1 $ & & ? : ? : :

dt L L L L L

−− +

$2

II I I1 10 ' ' 10 '0

d : "1+ D# d 1 1 "1+ D# 1 $ + + & & + % : : : ? : :

dt ! ! R! ! ! R!



84/134




D$ solutions

10'0

"1+ D# ?0 $ + & :

L L

!10 '0

"1+ D#

0 $ + % : :! R!

'0

10

1 : $

"1+ D#?

'0

10

:

R $ : "1+ D#

1010 '

? $ ( :

R"1+ D #

%ineari+ed equivalent circuit


85/134


q

of the ideal boost converter

/$ solutions$

! II I' '0 1d :

L $ +"1+ D# & d & : : ?dt

$2 III I'1 10 'd : :! $ "1+ D# + d + & % : : ?

dt R

smallsignal averaged statespace e=uation

$ $ $

$

$

2,

!

!, 2

%! &!,

,%! & !

, !

: D

? L L : : d L D : ?

L R! !

•− −

= + + − − −



86/134


q


!

I'0

' 1

L d 1 : : $ + & & : ?

"1+ D# "1+ D# "1+ D#

•

2

I10

1 ' '

! d 1 1 : : $ + + & % : : ?

"1+ D# "1+ D# R"1+ D# "1+ D#

•

Source-reflected lineari+ed


87/134


Source-reflected lineari+ed

equivalent circuit for the ideal

boost converter

! I' '0 1 L : $ + "1+ D# & d & % : : ?

•

22

I10 '

1' '

: "1+ D#! d : ?5 : "1+ D#7 $ + + & ( :

"1+ D# "1+ D#"1+ D R"1+ D # #

•

%oad-reflected lineari+ed


88/134

Power switchin Dynamic analysis of s ""

%oad reflected lineari+ed

circuit for the ideal boost

converter

2 I'

1 10 '

:! : $ "1+ D# + + d & % : : ?

R

•

!

I1 '0

''

L d ? : : "1+ D# $ + & & % :

"1+ D# "1+ D#"1+D #

•

$C transformer equivalent


89/134

Power switchin Dynamic analysis of s "(

$C transformer equivalent

circuit for the ideal boost

converter

Switching Converter .ransfer


90/134

Power switchin Dynamic analysis of s (,

g

unctions

SourcetoState .ransfer


91/134

Power switchin Dynamic analysis of s (!

g

unctions

SourcetoState .ransfer


92/134

Power switchin Dynamic analysis of s (2

g

unctions

;K$C $V0.0

!

2

I10

1

1

'

0 1 D+ ?

: : L $ & 5 7 & d % L L?

1 1 : : 0 0+

! R!

•

•

0

0 1+ L

$ ,1 1

+! R!

0

D

$ L 90

10?

* $ % L

0

'e R

P P

"s# 51 & H"s#7 "s# + H"s# "s#V V @ d"s# $ $

V V

II '

P

H"s# "s# @ d"s# $ + %V

T

P

"s# $ 0 +H"s# F

V

[ ]T "s# $ 0 0 %Q



93/134


g

unctions

;K$C $V0.0

+1

10

1 P 1

'

s 1 H"s#? & D

"s# @ L LV $ 5 "s#7 L A

"s# 1 1 @ 0+ s&! R!

10

P 1

1

10''

P

1 H"s# 1? s& +" & # D

R! L LV 5 "s#7 L A

1 s 0"s# @ !

$ % s 1 H"s#"s# ? @ & & & s

R! L! L!V

1

10'1

P

D 15s& 7

"s# @ L R! $ s 1 H"s#?"s#

A & & &S R! L! L!V

'

10'1

P

D

"s# @ L! $ % s 1 H"s#?"s#A & & &S

R! L! L!V



94/134


g

unctions

;S. $V0.0

!

2

I

'0

1 1

10' '

0 +"1+ D# 1 0 :

: ? : L L L $ & & d %"1+ D# 1 0 1 : : ?

: + +! R! ! !

•

•

0

0 +"1+ D#

L $ , "1+ D# 1

+! R!

0

1 0

L $ 9 0 1

!

'0

10

:

L * $ % :

+!

+1

'0

1 1 P

' 10 '

P

s "1+ D# H"s# : 1 o&"s# "s# L A @ LV L

$ %o 1"s# +"1+ D# 1 H"s# "s# : A @

"s& #+! ! R! !V

+1

'0

1 1 P

' 10 '

P

s "1+ D# H"s# : 1 o&"s# "s# L A @ LV L

$ %o 1"s# +"1+ D# 1 H"s# "s# : A @

"s& #+! ! R! !V



95/134


g

unctions

;S. $V0.0

10

1 P

101'

'0' P

P

1 1 H"s# :5s& + 7

"s# L R! @ !V $ %

H"s#R :"s#A s51+ 7"1+ D H"S#"1+ D# # :V & & & s

R! L! L!V

10

'

1 P '

' 10 10'

' '

P P

+"1+ D# H"s#?51& 7

L! "s# "1+D # @ V $ %

"s# s H"s# "1+ D H"s# #A ? ?& 51+ 7 & 51& 7 s

R! L! "1+ D "1+ D # #V V

'

'

1 10 10'' '

P P

1+ D"s# @ L! $ %

"s# s H"s# "1+ D H"s# #A ? ?& 51+ 7 & 51& 7 s R! L! "1+ D "1+ D # #V V

'

'' 10 10'

' '

P P

s"s# @ !

$ %"s# s H"s# "1+ D H"s# #A ? ?& 51+ 7 & 51& 7 s

R! L! "1+ D "1+ D # #V V


96/134


Complete state feedback

This techni5ue allows us to calculate the

gains of the feedbac( vector re5uired to place

the closed-loop poles at a desired location

7ll the states of the converter are sensed and

multiplied by a feedbac( gain

$esign of a control system with


97/134


g y

complete state feedback

: , : 9 ?•

= × + ×

control strategy ? F := − × ,

% & : , 9 F :• = − ×

closedloop poles

det* + , s I , 9 F × − + =

.he closedloop poles can #e ar#itrarily placed #y choosing the

elements of F



98/134

Power switchin Dynamic analysis of s ("

g y


Pole selection


99/134

Power switchin Dynamic analysis of s ((

g y


7 buc( converter designed to operate in the

continuous conduction mode has the followingparameters% ' > @, L ' #"! &, and &a ' #! &"

Calculate 0a1 the open-loop poles, 0b1 the feedbac(gains to locate the closed loop poles at = ' #$$$ B -$"

01ample



100/134

Power switchin Dynamic analysis of s !,,


Solution

!, !! L !

! ! ,

L L , 9

! R!

− = = −

!, ,2 L 2

! ! ,

L , 9

! R!

− = = −

! 2 %! &L

! 2 %! &

, , D , D

9 9 D 9 D

= × + × −= × + × −

!,L

! ! ,

D L L , 9

! R!

− = = −



101/134

Power switchin Dynamic analysis of s !,!


poles8 ? eig%/&

poles8 ? !,,, H M !32(" N 24(6!i !32(" 24(6!iO' ' ' '

! 2

' ' '

2 ! 2

!% &

! !% &

: : D? d A L

: : :

! R

•

•

= − + +

= −

'

' ' '!

'

2

!,

! !, ,

D A : L

: ? d L L

:! R!

• − = + + −



102/134

Power switchin Dynamic analysis of s !,2


Step response of the lineariJed #uc- converter

sys8?ss%/;$,&

step%sys8&



103/134



design the control strategy

'

,? =' ' '

: , : * d

•

= + ,

A

* L

=

for voltagemode control' '

d ref ref

Dv

V =

f we apply complete state feed#ac-

' '

ref v F := −' ' '

% &ref

D : , : * F :

V

•

= + − ' '

% &ref

D : , * F :

V

•

= − !Lref

D , , * F

V = −



104/134



we calculate the feed#ac- gains asP?!,,, H*,32(" N ,!,i ,32(" ,!,i+)

% &ref

D F p.ace , * P

V =

.hen < ? M266,, ,32,2O

!L

ref

D , , * F

V = −

,2,,, ,,5!! !e4

!,63" ,266,!L ,

=

chec- the locations of the closed loop poleseig%/$8&L which gives

ans ? !eN2 H * 32(", N !,,,,i 32(", !,,,,i+

&Spice schematic


105/134


.ransient response of the


106/134


open-loop and closed-loop

converters


107/134

f


108/134

Power switchin Dynamic analysis of s !,"

!nput "#! filters

7n input 4M8 filter placed between the power sourceand the switching converter is often re5uired topreserve the integrity of the power source

The maHor purpose of the input 4M8 filter is toprevent the input current waveform of the switchingconverter from interfering with the power source

7s such, the maHor role of the input 4M8 filter is tooptimi*e the mismatch between the power sourceand switching converter impedances

! "#! fil


109/134

Power switchin Dynamic analysis of s !,(

!nput "#! filters

$ircuit model of a #uc- converter with an input 0 filter

! t "#! filt


110/134

Power switchin Dynamic analysis of s !!,

!nput "#! filters

The stability of a closed-loop switchingconverter with an input 4M8 filter can befound by comparing the output impedance ofthe input 4M8 filter to the input impedance ofthe switching converter

The closed-loop switching converter eGhibitsa negative input impedance

Sta#ility $onsiderations

!nput "#! filters


111/134

Power switchin Dynamic analysis of s !!!

nput impedance versus fre=uency for a #uc- converter

utput impedance of the 0 filter s I

*BI ' I s I I

& 2 R L $ % Z

"1 + # & 2! ! L R

ω

ω ω

'

L L oin e o' ''

L Lo o

1 ! R R$ " 5 & 7 & 2 5 + 7 # ( Z R L

1&" 1&" # # D ! ! R Rω

ω ω

/t the resonant fre=uency

ein '

R $ + % Z

D

/#ove the resonant fre=uency

in '

2 L $ + % Z

D

ω

! t "#! filt


112/134

Power switchin Dynamic analysis of s !!2

!nput "#! filters

The maGimum output impedance of the input 4M8 filter, I4M8,maG,must be less than the magnitude of the input impedance of the

switching converter to avoid instability

The switching converter negative input impedance incombination with the input 4M8 filter can under certain conditionsconstitute a negative resistance oscillator

To ensure stability, however, the poles ofshould lie in the left-hand plane


AAin *BI Z Z

ma1in *BI( % Z Z >>

! t "#! filt


113/134


!nput "#! filters

7 resistance in series with the input 4M8 filter inductor can beadded to improve stability

)owever, it is undesirable to increase the series resistance of theinput 4M8 filter to improve stability since it increases conductionlosses


! t "#! filt


114/134


!nput "#! filters

nput 0 filter with LR reactive damping

! t "#! filt


115/134


!nput "#! filters

nput 0 filter with R! reactive damping

! t "#! filt


116/134


!nput "#! filters

8t should be noted that high core losses in the input 4M8 filterinductor is desirable to dissipate the energy at the 4M8 fre5uency

so as to prevent it from being reflected bac( to the power source


! t "#! filt


117/134


!nput "#! filters

/ fourthorder input 0 filter with LR reactive damping

!nput "#! filters


118/134

Power switchin Dynamic analysis of s !!"

2 2

2'

L L oe o' ''

L Lo o

1 ! R R" # $ & & ( Z R L

1&" 1&" # # D ! ! R Rω ω

ω ω ∈

− ÷ ÷

nput impedance Z in"f# of the #uc- converter and output

impedance Z *BI "f# of the input 0 filter

( ) 2

'' s I

*BI ''

I s I I

&" # R L" # $ % Z

1+ &" #! ! L R

ω ω

ω ω


119/134

Power switchin Dynamic analysis of s !!(

&art 0

$iscrete-time models

Continuous-time and discrete-time

d i


120/134

Power switchin Dynamic analysis of s !2,

domains

continuoustime system % & % & : , : t 9 ? t = × + ×&

.he solution for the differential e=uation % &% & % & % &o

t

,t , t

o

t

: t e : t e 9 ? d τ τ τ × −= × + × × ×∫ 2 2 2 ,t e I , t , t = + × + × !+ ......123

% & !% &

o

t

, t , t

t

e 9 ? d e I , 9 ?τ τ τ × − × − × × × = − × × × ∫

[ ]% & !% &o

t

, t

t

e 9 ? d I , t I , 9 ?τ τ τ × − −× × × = + × − × × ×∫

% & % & % & ,t o o

: t e : t t 9 ? t ≈ × + × ×

Continuous-time and discrete-time

d i


121/134

Power switchin Dynamic analysis of s !2!

domains

the discretetime e1pression

% &o st n D T = + ×

% ! & st n D T = + + ×

*% ! & + *% & + *% & + s ,T

s s s s : n D T e : n D T T 9 ? n D T + + × = × + × + × × + ×

C ti ti t t d l


122/134



0=uivalent circuit during ton> /!

! ! : , : 9 ?•

= +

C ti ti t t d l


123/134



0=uivalent circuit during toff > /2

2 2 : , : 9 ?•

= +

Continuous time state space model


124/134



switching functions

! % &% &

, % & % !&

s n s

n s s

if nT t n d T d t

if n d T t n T

< < += + < < +

)% & ! % &d t d t = −

( ) ( )! 2% & )% & % & ! )% & 2 : d t , d t , : d t 9 d t 9 ?•

= + + +

nonlinear model



125/134



smallsignal model

$d d d = + ) !d d = −

! % &% &

, % & % !&

s s

s s

if nT t n D T d t

if n D T t n T

< < += + < < +

$ [ ]sgn% & % & % &% &,

n s n sd D if t n D T n d T d t

ot-erCise

− ∈ + +=

s s? V v= + $ $nnd D d = + $ : : := +



126/134



stead>+state e?ation

( ) ( )! 2 ! 2) ) s : d , d , : d 9 d 9 V •

= + + +

pert?rEation in t-e state vector

$ ( ) ( )! 2 ! 2 ! 2 ! 2II) ) s s : d , d , : d 9 d 9 v , , : 9 9 V d

•

= . + . + . + . + − + − .$

Discrete-time model of the switching

converter


127/134


converter

( ) ( ) ! s sn D T n T + +

$ $ $ ( )2 2 s n s s : , : 9 v 3 d T t n D T δ •

= + + − + $

( ) ( ) ( )! 2 ! 2 s s 3 , , : n D T 9 9 V − + + − B

$ ( ) $ ( ) $

( )

( )

( )

2 2 2

2

) ) )

!

!

2

! s s s

s

s

s

, D T , D T , D T n s s s

n T

, n T s

n D T

: n T e e : n D T e 3 T d

e 9 v d τ

τ

+ + −

+

+ = + + +

+ ∫ $

( ) ( )! ! s sn T n D T + + +

$ $! : , :

•=

$ ( ) $ ( ) $! 2 ! 2) )! s s s s , DT , D T , DT , D T n s s s : n D T e e : n D T e e 3 T d + + = + +

$esign of a discrete control

t ith l t t t


128/134

Power switchin Dynamic analysis of s !2"

system with complete state

feedback% !& % & % & : n , : n 9 ? n+ = × + ×

% & % &? n F : n= − × ,

% !& % & % & : n , F 9 : n+ = − × ×

det* + , z I , F 9× − + × =

.he closedloop poles can #e ar#itrarily placed #y choosing the

elements of F

$esign of a discrete control



129/134

Power switchin Dynamic analysis of s !2(


feedback

Pole selection

ne way of choosing the closedloop poles is to design a low

pass ;essel filter of the same order

.he step response of a ;essel filter has no overshoot thus it is

suita#le for a voltage regulator

.he desired filter can then #e selected for a step response that

meets a specified settling time


130/134

Power switchin Dynamic analysis of s !3,


feedbackVoltage mode control

( ) [ ]! 2 ! 2' ' '

) )! s s s s

, DT , D T , DT , D T n s s s : n T e e : nT e e 3 T d + = +

$ref

ref

D

d vV = $

' ' '

*% !& + * + n s : n T : nTs d + = Φ + Γ

! 2 )

! 2% &

s s , DT , D T

s

s

e e

3 T

3 9 9 V

Φ =Γ = Φ = −$ ( ) $[ ]! ref s s

ref

D : n T : nT v

V + = Φ + Γ

$

$[ ]ref sv F : nT = −$

$ ( ) $[ ]! s sref

D : n T F : nT

V

+ = Φ − Γ

!L

ref

D F

V

Φ = Φ − Γ

$ ( ) $[ ]! s !L s : n T : nT + = Φ

"(tended-state model for a

tracking regulator


131/134

Power switchin Dynamic analysis of s !3!

tracking regulator

Digital trac-ing system with fullstate feed#ac-

* +

* + * +* +

L

d c

a

i n

: n v n : n

=

,

L ,d d

a ac

Φ Γ Φ = Γ = Γ Φ

> c :=

[ ]! 2 L L L=

2a a > L :=

Current mode control


132/134


Sensitivities of the duty cycle

! 2

' ' ' '

! 2

n

n n n p

p

d d d d : : I

: : I

∂ ∂ ∂= + +

∂ ∂ ∂

!

2

L

!

: i

: v

=

=

' ' ' '

n

n n n p L c

L c p

d d d d i v I

i v I

∂ ∂ ∂= + +

∂ ∂ ∂

' '

% &n L

d c s

Ld i

V V T = −

−

' '

n c

d c

Dd v

V V =

−

' '

% & pn

d c s

Ld I

V V T =

−

! 2 3 ! 2

! 2

L L L * +n n n

p

d d d and

: : I ω ω ω ω ω

∂ ∂ ∂= = = Ω =

∂ ∂ ∂

[ ] {'

'

'' '

! 2 3'

F

!B

L

pn

c d

d

id I

v

ω ω ω = + 1 44 2 4 43

' ' '

n !B F d d d = +

Current mode control


133/134


' ' '

*% !& + * + n s : n T : nTs d + = Φ + Γ

' ' ' '

*% !& + * + % & F s !B

: n T : nTs d d

+ = Φ + Γ +' ' ' '

3*% !& + * + * + p s : n T : nTs : nTs I ω + = Φ + Γ Ω + Γ

' ' '

3*% !& + * + p s !B : n T : nTs I ω + = Φ + Γ

!B Φ = Φ + Γ Ω

II s p

F : nT I

= − ×

$ ( ) [ ] $[ ]3! s !B s : n T F : nT ω + = Φ − Γ

[ ]3!L !B F ω Φ = Φ − Γ

$

( )

$

[ ]!

s !L s : n T : nT

+ = Φ

Qith complete state feed#ac-

4Gtended-state model for a trac(ing

regulator


134/134

regulator

Digital trac-ing system with fullstate feed#ac-

* +

* + * +

L

d c

i n

: n v n

=

,

!B

dd Φ Γ

Φ = Γ = [ ]! 2 L L L=

Dynamic Analysis of Switching Converters (2)

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Transcript of Dynamic Analysis of Switching Converters (2)