4.5 Cunrreiiit Programmed Dc-to-dc Converters
4.5.2 Inductor Carrenit to Output Voltage Transformation
I t was m e n tio n e d in S e c tio n 3.6 t h a t th e d e s ira b le w a y o f s e ttin g u p th e s lid in g s u rfa c e f o r s lid in g m o d e c o n t r o l is b y m e a n s o f o u t p u t e r r o r a n d its d e r iv a tiv e s . T he re a s o n f o r th is c h o ic e is t h a t th e a n a ly s is of s t a b ilit y is s im p le f o r s u c h a s lid in g s u rfa c e . I t was a ls o se e n in S e c tio n 4 .4 t h a t s u c h a s ta te a s s ig n m e n t is n o t p o s s ib le in case o f b u c k a n d b u c k - b o o s t c o n v e r te r s . I t b e co m e s th e n n e c e s s a ry to s e t u p th e s lid in g s u rfa c e in te r m s o f s u ita b le c o n tin u o u s p h y s ic a l v a r ia b le s . The in d u c t o r c u r r e n t i a n d th e o u tp u t v o lta g e a re c o n tin u o u s v a r ia b le s a n d q u a lify f o r s u ita b le s ta te s in te r m s o f w h ic h th e s lid in g s u rfa c e m a y be s e t up. In o r d e r to s tu d y th e s t a b ilit y o f th e t r a je c t o r ie s o n s u c h a s lid in g s u r fa c e , one h a s to t r a n s f o r m th e s lid in g s u rfa c e in t o o u tp u t e r r o r a n d it s d e riv a tiv e . T h e e q u iv a le n t s y s te m d e s c r ip tio n g iv e n b y Eqs. (4 .3 3 ) to (4 .3 5 ) m a y be u s e d to o b ta in th is t r a n s f o r m a t io n b e tw e e n in d u c t o r c u r r e n t a n d o u tp u t v o lta g e . The m e th o d is illu s t r a t e d f o r b o o s t c o n v e r t e r a n d th e r e s u lt s a re p re s e n te d f o r th e o th e r c o n v e r te r s .
F o r b o o s t c o n v e r te r , E q. (4 .3 4 ) m a y be m a n ip u la te d a n d r e w r it t e n as
v 02+JRCv0dv0/ dt = R i(v g—L d i / dt) (4 .3 9 )
In p r a c t ic a l c o n v e r te r s in o r d e r to o b ta in g o o d s m a ll s ig n a l as w e ll as p o w e r b a n d w id th , i t is n e c e s s a ry to c h o o s e L s u c h t h a t vg » L d i/d t . T h e n .E q . (4 .3 9 ) re d u c e s to
C l/o d l l n I/o2
£ = ---- ---- —-J--- —
vg dt Rvg
12 0
The above e q u a tio n g ive s th e r e la tio n s h ip b e tw e e n th e in d u c to r c u r r e n t a n d o u tp u t v o lta g e f o r b o o s t c o n v e r te r . T h is re la tio n s h ip is in g e n e ra l n o n lin e a r a n d is g iv e n h e re f o r th e o th e r c o n v e rte rs as w ell.
B u c k C o n v e rte r:
c£vq v 0
* = C~ d T + ~R (4 .4 0 )
B o o s t C o n v e rte r:
C vo d v 0 v 02
•i = --- -7T—+ —--- [i;„ » L d i / d t ] , . ... *
v g d t E v g 13 1 (4 .4 1 )
B u c k -b o o s t C o n v e rte r:
( v a — v 0) d v n v a —v n
4 = » L d i / d t ^ (4 .4 2)
In th e n e x t s e c tio n s lid in g m o d e v o lta g e c o n t r o l o f d c - to - d c c o n v e r te r s is d e s c rib e d w h e re th e s lid in g s u rfa c e is s e t up in te rm s o f in d u c to r c u r r e n t a n d o u tp u t v o lta g e . The r e la tio n s h ip b e tw e e n in d u c to r c u r r e n t a n d o u tp u t v o lta g e g iv e n b y Eqs. (4.40) to (4 .4 2 ) a re th e n u s e d to a n a ly z e th e s t a b ilit y o f s lid in g re g im e .
4 .6 V o lta g e C o n tr o l o f D c - to - d c C o n v e rte rs
In S e c tio n 4.4 th e p ro b le m s a s s o c ia te d w ith th e v o lta g e c o n t r o l o f b o o s t a n d b u c k - b o o s t c o n v e r te r s w e re h ig h lig h te d . One o f th e w ays o f o v e rc o m in g th e s e p ro b le m s is to e s ta b lis h an in n e r lo o p o f s lid in g m o d e c u r r e n t c o n tr o lle r , a n d th e n to d e sign a n o u te r
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s u p e rv is o ry lo o p o i v o lta g e c o n t r o lle r b a s e d o n th e s m a ll s ig n a l m o d e l o f th e c u r r e n t c o n tr o lle d c o n v e r te r s . The t r a n s f e r fu n c tio n s g iv e n by Eqs. (4.37) a n d (4.3 8 ) a re u s e fu l f o r th is a p p ro a c h .
An a lte r n a tiv e a p p ro a c h is to im p le m e n t a d ir e c t s lid in g m ode v o lta g e c o n tr o lle r . The c o n t r o l s tr a te g y d e v e lo p e d f o r th e b u c k c o n v e r te r is n o t a p p lic a b le in g e n e r a l s in c e th e d e riv a tiv e o f th e o u tp u t v o lta g e e x h ib its d is c o n tin u ity d u r in g s w itc h in g . The a lte rn a te a p p ro a c h c o n s is ts o f s e ttin g up th e s lid in g s u rfa c e in te rm s o f som e c o n tin u o u s s ta te s o i th e s y s te m , a n d th e n to r e la te th e r e s u lta n t d y n a m ic s to th e d e s ire d re s p o n s e . T h is m e th o d is a p p lic a b le fo r a ll th re e d c -to -d c c o n v e rte rs a n d is illu s t r a t e d h e re f o r th e b o o s t c o n v e rte r. The r e s u lts a re th e n p re s e n te d f o r th e o th e r c o n v e rte rs .
B o o s t C o n v e rte r: E xa m p le
F o r th e case o f b o o s t c o n v e r te r , th e s lid in g lin e m a y be s e t u p as u s u a l in te rm s o f som e c o n tin u o u s s ta te s o f th e syste m . The s ta te s c h o s e n fo r th is p u rp o s e a re th e e r r o r s in th e 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 . L e t
a = gx = KylRji +£'0] = 0
£ and € 0 a re th e e r r o r s in th e in d u c to r c u r r e n t a n d th e o u tp u t v o lta g e re s p e c tiv e ly , a is th e w e ig h te d s u m o f th e s ta te s o f th e s y s te m a n d a = 0 re p re s e n ts a s t r a ig h t lin e in th e phase p la n e . The in d u c to r c u r r e n t e r r o r z a n d th e o u t p u t v o lta g e e r r o r v 0 a re th e axes o f th e phase p la n e . The g a in p a r a m e te r s u se d in s e ttin g up th e s lid in g lin e a re a n d K^R^. The g a in p a r a m e te r Rs in p r a c tic e is th e c u r r e n t sen sin g re s is ta n c e . The c u r r e n t s e n sin g re s is ta n c e Rs is
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u s u a lly o rd e rs o f m a g n itu d e s m a lle r th a n th e lo a d re s is ta n c e R. The c o n d itio n s fo r th e e x is te n c e o f s lid in g m ode a re d e riv e d in a s u b s e q u e n t s e c tio n . S uppose t h a t s lid in g m ode e x is ts a n d th e s y s te m RP is c o n fin e d to m ove a lo n g th e s lid in g lin e a = 0. Kv does n o t in flu e n c e th e re sp o n se . I t is s ig n ific a n t o n ly as a s c a le f a c t o r o f th e p h a se p la n e . In id e a l s lid in g m ode c o n tr o l, s w itc h in g ta k e s p la c e in a n in fin ite s im a l v ic in it y a ro u n d th e s lid in g lin e a = 0 a n d Kv h a s no s ig n ific a n c e w h a ts o e v e r. In r e a l s lid in g m o d e c o n tr o l, s w itc h in g ta k e s p la c e in a fin ite A v ic in it y a ro u n d th e s lid in g lin e a n d , Ky d e te rm in e s th e n o m in a l s w itc h in g fr e q u e n c y f o r a g iv e n rip p le a n d v ic e v e rs a .
4 .6 .1 S ta b ility o f T r a je c to r ie s o n th e S lid in g L in e
“We now go o n to e x a m in e w h e th e r th e tr a je c to r ie s a lo n g th e s lid in g lin e lea d to a u n iq u e s te a d y s ta te o p e ra tin g p o in t. D efin e th e c u r r e n t a n d v o lta g e e r r o r s b y th e fo llo w in g r e la tio n s h ip .
i = r + i v 0 = V0°+V0
In th e b o o s t c o n v e rte r, th e r e la tio n s h ip b e tw e e n in d u c to r c u r r e n t i a n d o u tp u t v o lta g e v D was d e riv e d in S e c tio n 4 .5, Eq. (4 .4 1 ). This r e la tio n s h ip m a y be w r it te n in te r m s o f th e v o lta g e a n d c u r r e n t e r r o r s as follo w s.
„ . c- C (V 0* + v 0) d( V + ^ o ) . ( V + f ? o ) 2 I +z --- —--- + --- ---
vg dt Rvg
S e p a ra tin g th e dc a n d ac c o m p o n e n ts , we g e t
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123
CVD° d v 0 2 V 0
n + C Vr
v g d t E v 3 " u ' 2 v g d t - +‘ R u g
U s in g E q .(4 .4 4 ), we m a y t r a n s f o r m th e s lid in g lin e in to
a = CFo f?3 e«70 Rs ^
+ —— y n +
c££ ' 0
CRS d v 0z
2 V g d t
E q u a tio n (4 .4 5 ) m a y be r e a r r a n g e d a n d w r it t e n as a = K j [ a { t } v a + d v Q/ dt~\
w h e re , a(£)
b ( t )
i + y ° - [a+-g0/Rug CRSV0*
(4.44)
(4 .4 5 )
(4 .4 6 )
E q u a tio n (4 .4 6 ) is a f i r s t o r d e r d if f e r e n t ia l e q u a tio n . T he c o n d itio n f o r g u a r a n te e d s t a b ilit y o f re s p o n s e a lo n g th e s lid in g lin e is t h a t b o th a ( t) a n d &(£) a re p o s itiv e . O r, in o t h e r w o rd s th e r e la t iv e e r r o r Vo* is le s s t h a n 1. A lte r n a tiv e ly , E q. (4 .4 6 ) m a y be in t e r p r e t e d d if f e r e n t ly as fo llo w s . F o r la rg e e r r o r s , th e v0 2 te r m s d o m in a te a n d th e s lid in g lin e is d e te r m in e d m a in ly b y v 02 te r m s ; f o r s m a ll e r r o r s , te r m s d o m in a te . The re s p o n s e m a y be th o u g h t o f as h a v in g tw o p a r ts .
L a rg e s ig n a l re s p o n s e : _ „ R C <*®p2
° « * T ~ d T = a
S m a ll s ig n a l re s p o n s e :
(4 .4 7 )
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