Analysis off Duty Ratio Controlled Converters

c o n tr o lle d d c -to -d c c o n v e rte rs . T he d e s ig n m e th o d s f o r th e a p p lic a tio n o f s lid in g m ode c o n t r o l fo r th e d iffe r e n t ty p e s o f c o n v e r te r s a re th e n o u tlin e d .

1

So— c*

9 I

____ I

—o

F i g . 4 . 4 a D u t y r a t i o c o n t r o l o f s w i t c h i n g c o n v e r t e r s . T h e c o n t r o l v o l t a g e v c i s c o m p a r e d t o a c o n s t a n t f r e q u e n c y t r i a n g u l a r w a v e t o g e n e r a t e t h e d u t y r a t i o d .

c—1

1

—

u=0

(v,

A = f

(b )

F i g . 4 . 4 b D u t y r a t i o c o n t r o l l e d c o n v e r t e r s a s a v a r i a b l e s t r u c t u r e s y s t e m (V S S ) . The c o n s t a n t f r e q u e n c y t r i a n g u l a r w a v e i s r e p l a c e d b y a n i n t e g r a t o r , a n d c o m p a r a t o r w i t h a p p r o p r i a t e f e e d b a c k . T h e h y s t e r e s i s i n t h e c o m p a r a t o r i s a d j u s t e d t o o b t a i n c o n s t a n t s w i t c h i n g f r e q u e n c y .

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

th e d u ty r a t io d. T h e sam e c o n t r o l m e th o d is s h o w n b y a n e q u iv a le n t s c h e m e in Fig. 4 .4 b . T h is e q u iv a le n t s c h e m e is a m a th e m a tic a l a r t if ic e u sed, in o r d e r to be ab le to a p p ly th e p r in c ip le s o f s lid in g m o d e c o n t r o l f o r th e a n a ly s is o f d u ty r a t io c o n t r o lle d c o n v e r te r s . T he e q u iv a le n t s c h e m e in c o r p o r a te s a n in t e g r a t o r a n d a c o m p a r a to r w it h h y s te re s is . T he c o n t r o l in p u t is n u m e r ic a lly e q u a l to th e d u ty r a t io (O ^d = v cs£ 1). S u ppose t h a t th e h y s te r e s is A in th e c o m p a r a to r is a f u n c t io n o f vc, s u c h t h a t th e s w itc h in g fr e q u e n c y is m a in ta in e d c o n s ta n t a t a ll d u ty r a tio s . I t is to be s tre s s e d a g a in t h a t th e e q u iv a le n t s c h e m e in F ig. 4 .4 b is an a r t i f i c ia l r e p r e s e n ta tio n o f th e c o n s ta n t fr e q u e n c y d u ty r a t io c o n t r o lle d c o n v e r t e r s , in o r d e r to a p p ly th e s lid in g c o n t r o l p r in c ip le s . T he d y n a m ic e q u a tio n s o f th e c o n t r o lle r a n d th e c o n v e r t e r m a y n o w be w r it t e n as

y = d - u (4 .1 0 )

x = Ax + bu + c (4 .1 1 )

E q u a tio n (4 .1 0 ) is a s c a la r e q u a tio n . E q u a tio n (4 .1 1 ) d e s c rib e s th e c o n v e r t e r a n d is o f o r d e r tw o . E q u a tio n s (4 .1 0 ) a n d (4 .1 1 ) m a y be c o m b in e d in to a s in g le s e t o f a t h i r d o r d e r s y s te m as fo llo w s .

(4 .1 2 )

V ⁰ ⁰ y ^{- 1} d

X 0 A ^X +

b u +

The s t r u c t u r e - c o n t r o l la w is u = ¹ fo r c - y > h

0 fo r a = y < A (4 .1 3 )

U n d e r id e a l s lid in g m o d e c o n tr o l, a = a = ⁰ a n d u gg = d. T he o v e r a ll

s y s te m is o f o r d e r 3 a n d th e n u m b e r o f c o n t r o l in p u t is 1. As e x p la in e d in C h a p te r 3, th e e q u iv a le n t d e s c r ip tio n th e n is o f o r d e r 2.

B y s u b s titu tin g ^d = ^{u 3q} in Eq. (4 .1 2 ) we g e t a n e q u iv a le n t s y s te m d e s c r ip tio n o f o r d e r 2.

z = A x + b d + c (4 .1 4 )

T he above d e s c r ip tio n is e x a c t w h e n th e s y s te m m o tio n is a lo n g th e id e a l s lid in g s u rfa c e (A = 0). F in ite c o n s ta n t s w itc h in g fre q u e n c y im p lie s t h a t A^O. As a r e s u lt th e a c t u a l s y s te m m o tio n c o n s is ts o f n o n id e a litie s ow in g to fin it e A (s w itc h in g r ip p le ) , s u p e rim p o s e d on th e id e a l m o tio n g iv e n b y Eq. (4 .1 4 ). W hen th e s w itc h in g r ip p le is s m a ll, th e a c tu a l s y s te m m a y be r e p r e s e n te d b y th e id e a l s y s te m w ith in a fin it e e r r o r . T his sam e c o n d itio n m a y a ls o be s ta te d as fo llo w s : th e s w itc h in g fr e q u e n c y m u s t be s u ffic ie n tly la rg e c o m p a re d to th e n a t u r a l fr e q u e n c y o f th e c o n v e r te r .

E q u a tio n (4 .1 4 ) is in g e n e r a l n o n lin e a r . The o n ly c o n d itio n im p o s e d so f a r is t h a t th e s w itc h in g fr e q u e n c y be s u ffic ie n tly la rg e . H en ce Eq. (4 .1 4 ) m a y be u s e d to s tu d y b o th th e s m a ll s ig n a l as w e ll as la rg e s ig n a l b e h a v io u r o f th e c o n v e r te r s . A lte r n a tiv e ly Eq. (4 .1 4 ) m a y be lin e a r iz e d a ro u n d th e o p e r a tin g p o in t to o b ta in th e t r a n s fe r fu n c tio n d e s c r ip tio n o f th e c o n v e r te r s . In o r d e r t o r e la te th e re s u lts e a s ily to th o s e o b ta in e d b y th e s ta te sp a ce a v e ra g in g m e th o d , Eq.

(4 .1 1 ) m a y be r e w r itt e n as fo llo w s .

X - [ A x X + b l V g ] u + [ A z x + b z V g \ { l —u ) (4 .1 5 )

w h e re A1, A z . b l , b z a re as d e fin e d in S e c tio n 4 .1 .2 . U n d e r s lid in g m ode c o n tr o l u eq - d a n d so Eq. (4 .1 5 ) re d u c e s to

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f n BUCK CONVERTER BOOST CONVERTER BUCK-BOOST CONVERTER

V0 DVg Vg/ ( 3 - D ) - V gD / (3-O J

I Vq/ R Vq/ R ( 3 - D ) - ' / 0/ R (3-D J

3 ^D ^{i + S C R} ³ ^1+SCR D i + S C R

Vg R G ( S ) R ( l - D ) * G ( S ) R ( l - D ) k G(S)

D 1 ¹ ³ ^D ³

Vg ^{G( S )} ( 3 - D ) G ( S ) ( 3 - D ) G (SJ

2 V0 i + S C R 2 V0 i + S C R / 2 V0 ( 1 + 0 ) l + S C R / l l + D )

d RD G ( S) R ( 3 - D ) 2 G (S ) RD ( 3 - D ) 2 G (S)

Vo Vq i Vo 1 - S L / R ( 3 - D ) 2 Vo 3 - S L D /R ( 3 - D ) 2

d 0 G (S) ( 3 - D ) G (S) D ( 3 - D ) G (SJ

G (S ) i + S ~ + S sLC

1 + S _ _ . L +„a LC

1 WR ( i - D ) 2 ^ ( 3 - D ) 2 R (1- D ) 2 w ( 3 - D ) 2

T A B L E 4 .1 The. s t e a d y s t a t e , a n d t r a n s f e r f u n c t i o n s o f c o n s t a n t f r e q u e n c y d u t y r a t i o c o n t r o l l e d d c - t o - d c c o n v e r t e r s .

z = [dA1 + ( 1 — d)A 2]x + [db x + ( 1 — d)b2]vg = Az + bVg (4.16)

Eq. (4 .1 6 ) is id e n tic a l to Eq. (4 .1 ) o b ta in e d fo llo w in g s ta te space a v e ra g in g m e th o d . E q u a tio n (4 .1 6 ) m a y be lin e a r iz e d to o b ta in th e s m a ll s ig n a l tr a n s fe r fu n c tio n s id e n tic a l to th o s e o b ta in e d (Eq. (4.5) a n d (4 .8 )) fo llo w in g th e s ta te spa ce a v e ra g in g m e th o d a n d g iv e n in T a b le 4.1.

4 .3 .2 D is c o n tin u o u s In d u c t o r C u r r e n t M ode (BCM)

D c -to -d c c o n v e r t e r s m a y a ls o be o p e ra te d in th e DCM ra n g e o f o p e r a tio n . In DCM o p e ra tio n , th e in d u c to r c u r r e n t s ta r ts fr o m z e ro in e v e ry c y c le , goes to z e ro b e fo re th e end o f th e s w itc h in g p e rio d , a n d r e s ta r ts f r o m z e ro a t th e b e g in n in g o f th e n e x t cyc le . S u c h a m ode o f o p e r a tio n a ris e s o u t o f th e f a c t t h a t th e m o s t in e x p e n s iv e r e a liz a tio n o f s w itc h e s in d c -to -d c c o n v e rte rs is b y m e a n s o f u n id ir e c t io n a l s w itc h e s , w h ic h a re c a p a b le o f c a r r y in g c u r r e n t in o n ly one d ir e c tio n . As a r e s u lt w h e n th e in d u c to r c u r r e n t re a c h e s z e ro d u rin g a p a r t o f th e s w itc h in g c y c le , th e u n id ir e c tio n a l s w itc h in h ib its r e v e r s a l o f c u r r e n t a n d b lo c k s . F ig u re 4.5 shows th e b a s ic c o n v e r te r s in th e DCM ra n g e a n d th e c o rre s p o n d in g in d u c to r c u r r e n t s . The u n id ir e c t io n a l n a tu r e o f th e s w itc h is show n b y a diode o n th e th r o w a r m o f th e SPDT s w itc h e s . The d c -to -d c c o n v e r te is in DCM th e n h a ve th re e c h a r a c t e r is tic p e rio d s d u rin g a c y c le - a c tiv e (at - ¹), n o n a c tiv e (u z ~ ¹), a n d b lo c k in g (u' = l ) —. S uppose we define th e s w itc h in g v a ria b le s as

Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission.

BUCK

CONVERTER:

(Ul=l --- &» 2 V O

■v

BOOST CONVERTER:

Vo

BUCK-BOOST CONVERTER:

u '^ i Vo

i

■*> t i

t

P i g . 4 . 5 Th e S a s ic d c - t o - d c c o n v e r t e r s i n t h e D i s c o n t i n u o u s C o n d u c t io n M o d e (D C M ). T h e s c h e m a t i c d io d e i n th e t h r o w o f t h e S P D T s w i t c h i n d i c a t e s t h e u n i d i r e c t i o n a l n a t u r e o f t h e s w i t c h . T h e c u r r e n t i n t h e i n d u c t o r f a l l s to z e r o b e f o r e t h e e n d o f e v e r y c y c l e .

u =

u '

1 for kTs ^ t ^ (k-bd')Ts 0 fo r (A + ii)T s s t == (7c + l) T s

0 fo r k T s < £ < (7c+d)2^

1 fo r ( k + d ) T s «= t < ( j f c + d + d 2) 7 ;

0 fo r ( & + d + d 2)7^. ^ (Jc + l ) T s 0 fo r kTs < f ( / c + d + d 2) r s 1 fo r { k + d + d ^ T s < £ < (& + l)!T s

i i + i i2+ ti' = 1 fo r a l l t

The d y n a m ic e q u a tio n s o f th e s y s te m m a y be w r it t e n as x - Axz -kby'dg fo r u = 1

x = Azx + bz'ug fo r i i2 = 1

x = 4 sz + &3i/ff fo r iz' = 1

The o v e r a ll s y s te m e q u a tio n s a re th e n g iv e n b y

x = [ A i X + b i V g A z x + b zv g A 3x + & 3-uff];u ( 4 . 1 7 )

w h e re u = [u u z i i ' ] r . T he o v e r a ll s y s te m g iv e n b y Eq. (4 .1 7 ) is lin e a r w ith r e s p e c t to c o n tr o l. T he e q u iv a le n t d e s c r ip tio n is fo u n d b y r e p la c in g th e d is c o n tin u o u s s w itc h in g v a r ia b le s b y t h e ir a ve ra g e v a lu e s o v e r a c y c le .

x — [ y l i Z + b i i / g A 22 + f e 27jg ,A3z + &3u g] [ d d z d ' ] T (4 .1 8 ) The o v e r a ll d y n a m ic s y s te m d e s c rib e d b y Eq. (4 .1 8 ) is id e n tic a l to th e s y s te m d e s c r ip tio n o b ta in e d b y th e s ta te s p a c e a v e ra g in g m e th o d [8].

A lth o u g h th e d y n a m ic e q u a tio n s s h o w u p th r e e s w itc h in g v a ria b le s d , d z ,d', th e r e is o n ly one in d e p e n d e n t c o n t r o l in p u t d . The o th e r tw o in p u t v a ria b le s d z a n d d ' a re fu n c tio n s o f d , th e c ir c u it e le m e n ts o f

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th e c o n v e r te r a n d th e s w itc h in g p e r io d Ts. The m e th o d o f o b ta in in g th e t r a n s fe r fu n c tio n s is g iv e n in d e ta il in r e fe r e n c e [8].

4.3.3 CCM Operation under Variable Switching Frequency

I t was seen in S e c tio n 4.3 .1 t h a t in th e case o f d u ty r a t io c o n tr o l, th e o v e ra ll s y s te m d e p e n d s o n th e c o n v e r te r e le m e n ts a n d th e d u ty r a tio , a n d in d e p e n d e n t o f th e s w itc h in g fre q u e n c y ; th e o n ly r e s t r ic t io n o n th e s w itc h in g fr e q u e n c y is t h a t i t be s u ffic ie n tly la rg e c o m p a re d to th e n a t u r a l fr e q u e n c y o f th e c o n v e r te r . W hen th e s w itc h in g fre q u e n c y is c o n s ta n t, s ta te sp a ce a v e ra g in g o r th e e q u iv a le n t c o n t r o l m e th o d m a y be u s e d to o b ta in th e tr a n s fe r f u n c tio n d e s c rip tio n o f th e s y s te m . T h is p r o p e r t y h a s b e e n u s e d in th e p a s t to in d e p e n d e n tly c o n tr o l tw o d if fe r e n t o u tp u ts o f a m u ltip le o u t p u t d c -to -d c c o n v e r te r [9 ]. W hen th e s w itc h in g fr e q u e n c y is n o t c o n s ta n t, d u ty r a t io is n o t d e fin e d a n d i t is n o t p o s s ib le to tim e a v e ra g e th e syste m . H ow e ve r th e e q u iv a le n t c o n t r o l m e th o d c a n v e r y c o n v e n ie n tly be u se d to a r r iv e a t th e tr a n s f e r fu n c tio n d e s c rip tio n s . W ith re fe re n c e to Fig. 4.4,

y = ve—il

x = Ax + bu + c

y fo o' y ^{- 1} K

I I

“ [o A ^X + b ^u + c

T he s tr u c t u r e - c o n t r o l la w is th e sa m e as b e fo re .

_ 1 fo r ct = y > A (4 .2 0 )

u ~ 0 fo r a - y < A

100

fn BUCK CONVERTER BO O ST C O N VE R TE R B U C K -B O OS T CONVERTER

Vo

Vc Vg V9 / f l - V cJ - V g Vc/ l l - V c)

I V g / R V0 / R ( l - V c ) - V 0 / R f l - V cJ

1 Vo

^{1+S C R} ^Vo ^{1 + S C R} (Vg-VgJ V0 1+S C R

v9

RVg G IS) RVg2 G fS J RVg G (SJ

Vo 1 Vo 1 V 0 1

Vg ^{Vg G}^(S) ^Vg ^{G (SJ} ^{Vg G fS J}

i ^{Vg 1+SC R} ^{2 V 03} l + S C R / 2 (Vg-VoJ 3 1 + S C R -V 0/ (Vg-VoJ

Vc ^R ^G^(S) ^{R V |} ^{G IS )} ^RVg2 ^{G (SJ}

v0 \f

¹ ^Vo2 ¹ (Vg-VoJ S1 + S L V 0 (V g -V g J /R V g 2

3 G fSJ Vg G fS J Vg G (SJ

G fSJ 1 + S

^

^{+ s 2l c}

L V 02 V i

4 sO 1 r»c. p L f Vg - V 0 J2 L C f V g -V o J 2

4 I r* f i 1 o _ “ LL* _

RVg2 Vg2 1 I J

RVg2 Vg2

T A B L E 4 . 2 The s t e a d y s t a t e , an d . t r a n s f e r f u n c t i o n s o f v a r i a b l e f r e q u e n c y d u t y r a t i o c o n t r o l l e d d c - t o - d c c o n v e r t e r s . Vc is th e n o r m a l i z e d , n o n d i m e n s i o n a l c o n t r o l i n p u t .

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

101

J u s t as b e fo re , u n d e r s lid in g m o d e c o n t r o l, a - b = 0 a n d u 3q = v c.

The e q u iv a le n t s y s te m r e p r e s e n ta tio n is

z = Ax + bvc + c (4 .2 1 )

F o r c o n v e n ie n c e , Eq. (4 .2 1 ) m a y be r e s t r u c t u r e d as fo llo w s w h e re AitAg,bi, ^ 2 a re a ll as d e fin e d in S e c tio n 4 .1 .2 .

x = [v cA 1 + { l - v c)A z \x + [v cb 1 + { l - ' i j c) b z\v g (4 .2 2 )

E q u a tio n (4 .2 2 ) is in g e n e ra l n o n lin e a r . x , v c,vg m a y be p e r tu r b e d to g e t th e s te a d y s ta te s o lu tio n s a n d th e v a r io u s t r a n s f e r fu n c tio n s . The v a r io u s tr a n s f e r fu n c tio n s a re g iv e n in T a b le 4 .2 . T hese a re s im ila r to th o s e o b ta in e d f o r c o n s ta n t s w itc h in g fr e q u e n c y c o n v e r te r s a n d g iv e n in T a b le 4.1. T he o n ly d iffe re n c e is t h a t th e t r a n s f e r fu n c tio n s a re e x p re s s e d in te r m s o f th e o p e r a tin g v o lta g e s a n d c u r r e n t s in s te a d o f th e d u ty r a t io . The c o n d itio n t h a t th e s w itc h in g fr e q u e n c y be s u ffic ie n tly la rg e s t i l l h o ld s .

4 .4 S lid in g M ode C o n tr o l o f B u c k C o n v e r te r

In S e c tio n 4 .3 , we d ig re s s e d to a p p ly th e e q u iv a le n t c o n t r o l m e th o d to o b ta in s m a ll s ig n a l t r a n s f e r fu n c tio n s o f th e d u ty r a t io c o n tr o lle d d c - to - d c c o n v e r te r s . T h e p u r p o s e o f t h a t e x e rc is e w a s to show th e u s e fu ln e s s o f e q u iv a le n t c o n t r o l m e th o d as a n a n a ly s is m e th o d f o r p ro g ra m m e d s w itc h in g s t r u c tu r e s . A m o re fu n d a m e n ta l a s p e c t o f VSS is to be ab le to s y n th e s iz e th e d y n a m ic s t r u c t u r e - c o n tr o l la w (as a g a in s t th e p r o g ra m m e d s t r u c t u r e - c o n t r o l la w o f d u ty r a tio c o n tr o l) to a c h ie v e th e d e s ire d s te a d y s ta te a n d d y n a m ic

102

p e r fo r m a n c e o f th e c o n v e r te r . The s te p s in v o lv e d in s u c h a d e s ig n p ro c e s s a re as fo llo w s .

i) to r e la te th e s te a d y s ta te a n d d y n a m ic r e q u ir e m e n ts in t o a n a p p r o p r ia te s lid in g b o u n d a ry .

ii) to e s ta b lis h a s t r u c t u r e - c o n t r o l la w w ith r e fe r e n c e to th e s e le c te d s lid in g b o u n d a ry , s u c h t h a t th e c o n d itio n s o f re a c h in g a n d e x is te n c e o f s lid in g re g im e a re s a tis fie d .

I t w as m e n tio n e d in C h a p te r 3 t h a t th e above d e s ig n p ro c e s s is s im p le w h e n th e c o n tr o lla b le s ta te s o f th e s y s te m ( o u tp u t a n d its d e r iv a tiv e s ) a re c o n tin u o u s a n d a c c e s s ib le . D c -to -d c b u c k c o n v e r te r s a tis fie s th e s e r e q u ir e m e n ts a n d is th e s im p le s t o f a ll th r e e d c - to - d c c o n v e r t e r s f o r th e a p p lic a tio n o f s lid in g m o d e c o n tr o l. T h is e x a m p le w as s h o w n b r ie fly in C h a p te r 3. In th e fo llo w in g s e c tio n , th e d c - to - d c b u c k c o n v e r t e r is f i r s t ta k e n u p f o r th e a p p lic a tio n o f s lid in g m ode v o lta g e c o n tr o l.

4 .4 .1 B u c k C o n v e r te r in P h a s e V a ria b le C a n o n ic a l F o r m

F ig u re 4.6 show s th e b u c k c o n v e r te r , its in d u c t o r c u r r e n t , a n d th e o u tp u t v o lta g e u n d e r s te a d y s ta te . The VSS d e s c r ip tio n o f th e b u c k c o n v e r te r is

x = Ax + bu (4 .2 3 )

x = A = [ 0 - 1 / L

l l / C - 1 / RC o =

Vg/ L

T he in d u c t o r c u r r e n t % a n d th e o u tp u t v o lta g e v 0 a re b o th c o n tin u o u s fu n c tio n s . T he d e r iv a tiv e o f th e o u tp u t v o lta g e (dv0/ d t) is

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103

F ig . 4

U =1

o t

00 0

88 a 0

=u

.6 The b u c k d c - t o - d c c o n v e r t e r a n d th e v a r i o u s s t e a d y s t a t e w a v e f o r m s . The p o l a r i t y o f th e s w i t c h i n g f u n c t i o n o d e t e r m i n e s th e s t a t e o f th e s w i t c h .

104

p r o p o r t io n a l to th e c a p a c it o r c u r r e n t , w h ic h is a ls o a c o n tin u o u s fu n c tio n . L e t th e d e s ir e d o u t p u t v o lta g e b e V0*. T h e n th e o u t p u t e r r o r is d e fin e d as v 0 — V0 *. F o r d c - t o - d c b u c k c o n v e r te r s V0* is c o n s ta n t, a n d V0* < v g . E q u a t io n 4 .2 3 m a y t h e n b e t r a n s f o r m e d as

y = A "y + b *it + c * ( 4 .2 4 )

0 1 0

1 1 ; b * =

L C RC.

y

⁼ ^{v 0- V}

rf(^⁰- V ) dt

;

c*

⁼

0 - V

T h e d c -to - d c b u c k c o n v e r t e r is a s e c o n d o r d e r s y s te m w ith o n e c o n t r o l in p u t. T h e r e f o r e th e d y n a m ic re s p o n s e o b ta in a b le u n d e r s lid in g m o d e c o n t r o l is o f o r d e r 1. T h e d e s ir e d s te a d y s ta te a n d d y n a m ic re s p o n s e m a y th e n b e e x p r e s s e d as a o n e d im e n s io n a l s lid in g s u r fa c e (s lid in g lin e ) in th e p h a s e p la n e . C o n s id e r

d(^o -V)

a = ( v 0 - V 0* ) + r -

dt ⁼ 0 = [1 r ] Ty = g y ( 4 .2 5 )

E q u a tio n ( 4 .2 5 ) d e s c r ib e s a s ta b le t r a j e c t o r y in t h e p h a s e p la n e w ith s te a d y s ta te o p e r a t in g p o in t u 0 = Vi,*, a n d f i r s t o r d e r t r a n s ie n t r e c o v e r y w ith a tim e c o n s t a n t r .

T h e n e x t s te p is to s e le c t t h e s t r u c t u r e - c o n t r o l la w s u c h t h a t t h e lin e a = 0 is a s lid in g b o u n d a r y . T h e c o n d itio n f o r th e e x is te n c e o f s lid in g r e g im e is

l i m a < 0 ; l i m a > 0

a - o + <7 = 0 ~

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

105

L e t

f o r a > 0

i l~ f o r o < 0

The c o n d itio n f o r th e e x is te n c e o f s lid in g re g im e is gA°y +gb °u*+gc * < 0 < gA*y +gb *u~-t-gc *

E x p a n d in g a n d a p p ly in g th e c o n d itio n t h a t dv0/ d t = 0, we g e t

^d v ^/ d t \ = u * + Y q ( v 9u + - v o) < 0

0 < tdvo/ d t ~\u=u- + -£c(vgv - ~ - v 0) (4 .2 6 )

F ro m Eq. (4.26) u + a n d u~ a re s e le c te d as 0 a n d 1 r e s p e c tiv e ly . The c o n d itio n f o r th e e x is te n c e o f s lid in g m o d e is th e n r > RC. The sam e c o n d itio n m a y also be a r r iv e d a t f r o m th e s te a d y s ta te w a v e fo rm s show n in Fig. 4.6.

s ta r tin g fr o m a n y w h e re in th e p h ase p la n e , th e s y s te m RP w ill r e a c h th e s lid in g lin e a = 0. R e c a llin g th e th e o r e m in S e c tio n 3.4 o n th e s u ffic ie n t re a c h in g c o n d itio n , i t m a y be v e r if ie d fr o m Fig. 4.7 t h a t th e s lid in g lin e p a r titio n s th e p h a se p la n e in t o tw o re g io n s a n d t h a t th e s te a d y s ta te o p e ra tin g p o in t f o r th e c o n t r o l in p u t in e a c h o f th e s e re g io n s lie s in th e o p p o s ite re g io n . The s w itc h in g b o u n d a r y a - ⁰ th e n q u a lifie s as a s lid in g lin e .

i) a = 0 is a s ta b le t r a je c t o r y s a tis fy in g th e s te a d y s ta te ( ^ 0 = V0") (4 .2 7 ) The n e x t s te p in th e d e s ig n p ro c e s s is to e n s u re t h a t

106

F i g . 4 . 7 Th e s w i t c h i n g f u n c t i o n a = 0 i s s e e n i n th e p h a s e p l a n e . a = 0 p a r t i t i o n s t h e p h a s e p l a n e i n to t w o r e g i o n s (a < 0.

a n d a > 0 ). The c o n t r o l i n p u t i s r e s p e c t i v e l y u ~ a n d u + i n th e s e t w o r e g i o n s .

Vo

F i g . 4 . 8 The s t a r t i n g t r a n s i e n t i n th e i n d u c t o r c u r r e n t i a n d th e o u t p u t v o l t a g e v 0 u n d e r s l i d i n g m o d e c o n t r o l . T h e d u r a t i o n f r o m t = 0 to t = t l i s t h e i n i t i a l t r a n s i e n t i n r e a c h i n g t h e s l i d i n g L in e a = 0. Th e s e c o n d p a r t o f th e t r a n s i e n t f r o m t = to t = t 2 i s th e s y s t e m m o t i o n a lo n g t h e s l i d i n g l i n e to r e a c h th e s t e a d y s t a t e o p e r a t i n g p o i n t (v 0 = Y0*).

Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission.

107

a n d d y n a m ic (e x p o n e n tia l e r r o r r e c o v e r y w ith a tim e c o n s ta n t r ) re q u ir e m e n ts .

ii) The b o u n d a ry a = 0 is a s lid in g re g im e w h e n u = 1f o r a < A a n d u = 0 f o r a > A. F u r th e r , c o n d itio n s f o r th e e x is te n c e o f th e s lid in g m o d e a re r > E C a n d vg > V0*.

iii) S ta r tin g fr o m a n y a r b it r a r y i n i t i a l c o n d itio n o n th e p h ase p la n e , th e s y s te m e v e n tu a lly re a c h e s th e s lid in g lin e .

4 .4 .2 O v e rc u rre m t P r o te c tio n

In Fig 4.8 a t y p ic a l s ta r tin g t r a n s ie n t is sh o w n . I t c o n s is ts o f tw o p a rts . F ro m tim e t = 0 to t - ty is th e tim e ta k e n f o r th e s y s te m RP to re a c h th e s lid in g lin e . F r o m tim e t - t x to t - t z is th e tim e ta k e n to r e a c h th e s te a d y s ta te o p e r a tin g p o in t (u0 = F0’ ) a lo n g th e s lid in g lin e . The in it ia l t r a n s ie n t tim e t x d e p e n d s o n th e s y s te m p a r a m e te r s vg,L ,R , a n d C. The s lid in g m o d e t r a n s ie n t to t z is e x p o n e n tia l w ith tim e c o n s ta n t r a n d is in d e p e n d e n t o f th e s y s te m p a r a m e te r s . D u rin g th e tr a n s ie n t, th e in d u c t o r c u r r e n t i is seen to r e a c h le v e ls m u c h h ig h e r th a n th e s te a d y s ta te le v e l. T h is la rg e t r a n s ie n t in r u s h c u r r e n t is o b je c tio n a b le f o r tw o re a s o n s . F ir s t ly i t w o u ld s a tu r a te th e in d u c to r m a g n e tic c i r c u i t le a d in g to s t i l l la r g e r in r u s h c u r r e n ts . S e c o n d ly th e e le c tr o n ic SPDT s w itc h in th e c o n v e r t e r m a y n o t be c a p a b le o f w ith s ta n d in g th is la r g e in r u s h c u r r e n t . I t is th e r e fo r e a h e a lth y p r a c tic e to l im i t th is in r u s h c u r r e n t . T h is fe a tu r e is e a s ily in c o r p o r a t e d u n d e r s lid in g c o n t r o l by th e fo llo w in g s im p le m o d ific a tio n o f th e s lid in g lin e .

Fig. 4 .9 The m o dified s lid in g lin e in c o rp o ra tin g ov e rcu rre n t protection. The slid in g lin e is n o w lim it e d to I *.

Fig. 4.1 0 A p i c t o r i a l re p re s e n ta tio n o f the control la w f o r the buck dc-to-dc c o n v e rter shown on the phase plane.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

109

F ig u re 4.9 in d ic a te s th e m o d ifie d s lid in g lin e in c o r p o r a tin g th e o v e r c u r r e n t p r o te c tio n . The m o d ifie d s lid in g lin e a" = ⁰ sh o w n in Fig. 4.9 c o n s is ts o f th re e p ie ce s o f s tr a ig h t lin e s .

a =

fo r

= 0 fo r d f o o - V )

dt d (u0- V )

d i fo r < *fro-F o*)

dt < ( 4 .2 8 )

The m o d ifie d s lid in g lin e also s a tis fie s th e r e a c h in g c o n d itio n . T his m a y be seen f r o m th e s te a d y s ta te o p e ra tin g p o in ts f o r th e tw o c o n tr o l in p u ts u = 1 a n d u = 0. I t was a lre a d y seen t h a t th e e x is te n c e c o n d itio n s o f s lid in g re g im e is s a tis fie d in th e m id d le p o r tio n o f th e s lid in g lin e a* = 0. In th e c u r r e n t lim ite d r e g io n th e s lid in g lin e m a y be w r it t e n as

** ~ [0 l] ? / — Imas 3 1J Arms

o * —g " y

EC LC

Vn -fmas Vg

' EC + T c 1 Vg

LC (4 .2 9 )

When \Imax\ is s u ffic ie n tly low , n + = 0 a n d u~ = 1 s a tis fy Eq. (4 .2 9 ). I t m a y also be seen f r o m th e o r ie n ta tio n o f a 9 = 0 th a t th e o v e r a ll t r a je c t o r y a lo n g cr* = 0 is s ta b le le a d in g to th e s te a d y s ta te o p e ra tin g p o in t v 0 = V0\

no

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Dalam dokumen SLIDING MODE CONTROL OF PO"WER CONVERTERS (Halaman 97-116)