power stages is developed, whose starting point is the unified ... modelling power stages by introduction of state- ...... Power Electronics Specialists Conference,.
A GENERAL UNIFIED APPROACH TO MODELLING SWITCHING-CONVERTER POWER STAGES RD.MIDDLEBROOK AND SLOBODAN
ABSTRACT A method f o r modelling switching-converter power s t a g e s i s developed, whose s t a r t i n g p o i n t i s t h e u n i f i e d s t a t e - s p a c e r e p r e s e n t a t i o n of t h e switched networks and whose end r e s u l t i s e i t h e r a complete s t a t e - s p a c e d e s c r i p t i o n o r i t s e q u i v a l e n t small-signal low-frequency l i n e a r c i r c u i t model.
A new c a n o n i c a l c i r c u i t model i s proposed, whose f i x e d topology c o n t a i n s a l l t h e e s s e n t i a l i n p u t - o u t p u t and c o n t r o l p r o p e r t i e s of any dc-todc s w i t c h i n g c o n v e r t e r , r e g a r d l e s s of i t s d e t a i l e d c o n f i g u r a t i o n , and by which d i f f e r e n t c o n v e r t e r s can be c h a r a c t e r i z e d i n t h e form of a t a b l e conv e n i e n t l y s t o r e d i n a computer d a t a bank t o prov i d e a u s e f u l t o o l f o r computer a i d e d d e s i g n and o p t i m i z a t i o n . The new c a n o n i c a l c i r c u i t model p r e d i c t s t h a t , i n genera1,switching a c t i o n i n t r o duces both z e r o s and p o l e s i n t o t h e d u t y r a t i o t o o u t p u t t r a n s f e r f u n c t i o n i n a d d i t i o n t o t h o s e from t h e e f f e c t i v e f i l t e r network.
1. INTRODUCTION 1.1 B r i e f Review of E x i s t i n g Modelling Techniques I n modelling of s w i t c h i n g c o n v e r t e r s i n g e n e r a l , and power s t a g e s i n p a r t i c u l a r , two main approaches one based on s t a t e - s p a c e modelling and t h e o t h e r u s i n g an a v e r a g i n g technique have been developed e x t e n s i v e l y , but t h e r e h a s been l i t t l e c o r r e l a t i o n between them. The f i r s t approach remains s t r i c t l y i n t h e domain of e q u a t i o n m a n i p u l a t i o n s , and hence r e l i e s h e a v i l y on numerical methods and computerized Implementationa. Its primary advantage i s i n t h e u n i f i e d d e s c r i p t i o n of a l l Pover s t a g e s r e g a r d l e s s of t h e t y p e (buck. b o o s t . buck-boost o r any o t h e r v a r i a t i o n ) through u t i l i z a t i o n of t h e e x a c t s t a t e - s p a c e e q u a t i o n s of t h e two switched models. -On t h e o t h e r hand, P r o c e s e i n g Systems .'I
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@ 1976
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* - 10, 1976,
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based on e q u i v a l e n t c i r c u i t m a n i p u l a t i o n s , resulting i n a single equivalent l i n e a r c i r c u i t model of t h e p a r e r s t a g e . T h i s h a s t h e d i s t i n c t advantage of p r o v i d i n g t h e c i r c u i t d e s i g n e r w i t h p h y s i c a l i n s i g h t i n t o t h e behaviour of t h e o r i g i n a l switched c i r c u i t , and of a l l o w i n g t h e powerful t o o l s of l i n e a r c i r c u i t a n a l y s i s and s y n t h e s i s t o be used t o t h e f u l l e s t e x t e n t i n design of r e g u l a t o r s i n c o r p o r a t i n g s w i t c h i n g converters. 1.2
Proposed New State-Space Averaging Approach
The method proposed i n t h i s paper bri-dges t h e gap e a r l i e r c o n s i d e r e d t o e x i s t between t h e s t a t e space t e c h n i q u e and t h e a v e r a g i n g t e c h n i q u e of modelling power s t a g e s by i n t r o d u c t i o n of s t a t e space averaged modelling. At t h e same time i t o f f e r s t h e advantages of both e x i s t i n g methods t h e g e n e r a l u n i f i e d t r e a t m e n t of t h e s t a t e - s p a c e approach, a s w e l l a s an e q u i v a l e n t l i n e a r c i r c u i t model a s i t s f i n a l r e s u l t . Furthermore, it makes c e r t a i n g e n e r a l i z a t i o n s p o s s i b l e , which o t h e r w i s e could n o t be achieved. The proposed s t a t e - s p a c e a v e r a g i n g method, o u t l i n e d i n t h e Flowchart of F i g . 1, a l l o w s a u n i f i e d t r e a t m e n t of a l a r g e v a r i e t y of power s t a g e s c u r r e n t l y used, s i n c e t h e a v e r a g i n g s t e p i n t h e s t a t e - s p a c e domain i s v e r y simple and c l e a r l y d e f i n e d (compare b l o c k s l a and 2 a ) . I t merely c o n s i s t s of a v e r a g i n g t h e two e x a c t s t a t e - s p a c e d e s c r i p t i o n s of t h e switched models over a s i n g l e c y c l e T, where f s = 1 / T i s t h e s w i t c h i n g frequency ( b l o c k 2 a ) . Hence t h e r e i s no need f o r s p e c i a l "know-howf' i n massaging t h e two switched c i r c u i t models i n t o t o p o l o g i c a l l y e q u i v a l e n t forms i n o r d e r t o a p p l y c i r c u i t - o r i e n t e d procedure d i r e c t l y , a s r e q u i r e d i n [ l ] (block l c ) . N e v e r t h e l e s s , through a hybrid modelling technique (block 2c), t h e c i r c u i t s t r u c t u r e of t h e averaged c i r c u i t model ( b l o c k 2b) can be r e a d i l y recognized from t h e averaged s t a t e - s p a c e model ( b l o c k 2 a ) . Hence a l l t h e b e n e f i t s of t h e p r e v i o u s a v e r a g i n g t e c h n i q u e a r e r e t a i n e d . Even though t h i s outI n e i t h e r c a s e , a p e r t u r b a t i o n and l i n e a r i z a t i o n
Reprinted, vith permission, f r w Proceedings of the IEEE Pwer E l e c t r ~ n i c sSpecialists Conference, J~~~ Cleveland, OH.
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F i g . 1, Flowchart of averaged modelling approaches
process required t o include the duty r a t i o modulation e f f e c t proceeds i n a v e r y s t r a i g h t f o r ward and formal manner, t h u s emphasizing t h e corner-stone c h a r a c t e r of b l o c k s 2a and 2b. At t h i s s t a g e (block 2a o r 2b) t h e s t e a d y - s t a t e (dc) and l i n e t o o u t p u t t r a n s f e r f u n c t i o n s a r e a l r e a d y a v a i l a b l e , a s i n d i c a t e d by b l o c k s 6a and 6b respectively, while the duty r a t i o t o output transfer function is available a t the final-stage model (4a o r 4b) a s i n d i c a t e d by b l o c k s 7a and 7b. The two f i n a l s t a g e models (4a and 4b) then g i v e t h e complete d e s c r i p t i o n of t h e s w i t c h i n g c o n v e r t e r by i n c l u s i o n of b o t h independent cont r o l s , t h e l i n e v o l t a g e v a r i a t i o n and t h e d u t y r a t i o modulation. Even though t h e c i r c u i t t r a n s f o r m a t i o n p a t h
b might be p r e f e r r e d from t h e p r a c t i c a l d e s i g n standpoint, the state-space averaging path a is i n v a l u a b l e i n r e a c h i n g some g e n e r a l c o n c l u s i o n s about t h e s m a l l - s i g n a l low-frequency models of any dc-to-dc s w i t c h i n g c o n v e r t e r (even t h o s e y e t t o be i n v e n t e d ) . Whereas, f o r p a t h b , one has t o be prekented w i t h t h e p a r t i c u l a r c i r c u i t i n o r d e r t o proceed w i t h modelling, f o r p a t h a t h e f i n a l s t a t e - s p a c e averaged e q u a t i o n s (block 48) g i v e t h e complete model d e s c r i p t i o n through
g e n e r a l m a t r i c e s A1, A2 and v e c t o r s bl, b2' c T, and c2T of t h e two s t a r t i n g switched models ( h o c k l a ) . T h i s i s a l s o why a l o n g p a t h b i n t h e Flowchart a p a r t i c u l a r example of a boost power s t a g e w i t h p a r a s i t i c e f f e c t s was chosen, w h i l e a l o n g p a t h a g e n e r a l e q u a t i o n s have been retained. S p e c i f i c a l l y , f o r t h e b o o s t power b. T h i s example w i l l be l a t e r s t a g e bl = b2 pursued i n d e t a i l a l o n g both p a t h s .
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I n addition t h e state-space averaging approach o f f e r s a c l e a r i n s i g h t i n t o t h e q u a n t i t a t i v e n a t u r e of t h e b a s i c a v e r a g i n g approximation, which becomes b e t t e r t h e f u r t h e r t h e e f f e c t i v e low-pass f i l t e r c o r n e r frequency f is below t h e s w i t c h i n g frequency f,, t h a t i s , f z / f s 0, J > 0 and c o n s e q u e n t l y t h e p o l a r i t y of t h e v o l t a g e and c u r r e n t d u t y - r a t i o dependent g e n e r a t o r s i s n o t changed b u t i s as shown i n F i g . 12c. Horeo v e r , t h i s i s t r u e i n g e n e r a l : r e g a r d l e s s of any i n v e r s i o n p r o p e r t y of t h e power s t a g e , t h e p o l a r i t y of two g e n e r a t o r s s t a y s t h e same a s In Fig. 11.
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5.3
S i g n i f i c a n c e of t h e Canonical C i r c u i t Model and R e l a t e d G e n e r a l i z a t i o n s =
.
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--.
- - .
p a s s f i l t e r i n g ( r e p r e s e n t e d by t h e e f f e c t i v e lowp a s s f i l t e r n e t v o r k H,(s)). Note a l s o t h a t t h e current generator j ( s ) a i n the canonical c i r c u i t model, eyen though e u p e r f l u o u e when t h e s o u r c e v o l t a g e v (a) i e i d e a l , i s n e c e s s a r y t o r e f l e c t t h e i n f l u k c e of a m n i d e a l s o u r c e g e n e r a t o r (with some i n t e r n a l impedance) o r of a n i n p u t f i l t e r [ 7 ]
upon t h e b e h a v i o u r of t h e c o n v e r t e r . I t s p r e s e n c e e n a b l e s one e a s i l y t o i n c l u d e t h e l i n e a r i z e d c i r c u i t model of a s w i t c h i n g c o n v e r t e r power s t a g e i n o t h e r l i n e a r c i r c u i t s , aa t h e n e x t s e c t i o n w i l l illustrate. Another s i g n i f i c a n t f e a t u r e o f t h e canoni c a l c i r c u i t model i s t h a t any s w i t c h i n g dc-to-dc c o n v e r t e r can b e reduced by u s e of ( 2 3 ) , (24). (27) and (28) t o t h i s f i x e d topology form, a t l e a s t a s f a r a s i t s i n p u t - o u t p u t and c o n t r o l prope r t i e s a r e concerned. Hence t h e p o s s i b i l i t y a r i s e s f o r u s e of t h i s model t o compare i n an e a s y and unique way v a r i o u s performance c h a r a c t e r i s t i c s of d i f f e r e n t c o n v e r t e r s . Some examples of such comparisons a r e given below.
1. The f i l t e r networks can b e compared w i t h respect t o t h e i r e f f e c t i v e n e s s throughout t h e dynamic d u t y c y c l e D r a n g e , b e c a u s e i n g e n e r a l t h e e f f e c t i v e f i l t e r elements depend on t h e s t e a d y s t a t e d u t y r a t i o D. Thus, one h a s t h e o p p o r t u n i t y t o choose t h e c o n f i g u r a t i o n and t o o p t i m i z e t h e s i z e and weight. 2. B a s i c dc-to-dc c o n v e r s i o n f a c t o r s pl(D) and p2(D) can b e compared as t o t h e i r e f f e c t i v e range. For some c o n v e r t e r s , t r a v e r s a l of t h e range of d u t y r a t i o D from 0 t o 1 g e n e r a t e s any c o n v e r s i o n r a t i o ( a s i n t h e i d e a l buckb o o s t c o n v e r t e r ) , w h i l e i n o t h e r s t h e convers i o n r a t i o might b e r e s t r i c t e d ( a s i n t h e Weinberg c o n v e r t e r [ 4 ] , f o r which k p - 3 ) . 2 3. I n t h e c o n t r o l s e c t i o n of t h e c a n o n i c a l model one can compare t h e f r e q u e n c y dependences of t h e g e n e r a t o r s e ( s ) and j ( s ) f o r d i f f e r e n t c o n v e r t e r s and s e l e c t t h e c o n f i g u r a t i o n t h a t b e s t f a c i l i t a t e s s t a b i l i z a t i o n of a feedback r e g u l a t o r . For example, i n t h e buck-boost conv e r t e r e ( ~ )i s a polynomial, c o n t a i n i n g a c t u a l l y a r e a l zero i n the r i g h t half-plane, which undoubtedly c a u s e s some s t a b i l i t y problems and need f o r p r o p e r compensation.
4. F i n a l l y , t h e c a n o n i c a l model a f f o r d s a v e r y convenient means t o s t o r e and f i l e i n f o r mation on v a r i o u s dc-to-dc c o n v e r t e r s i n a comp u t e r memory i n a form comparable t o T a b l e I. Then, thanks t o t h e f i x e d topology o f t h e c a n o n i c a l c i r c u i t model, a s i n g l e computer program can b e used t o c a l c u l a t e and p l o t v a r i o u s q u a n t i t i e s as f u n c t i o n s of frequency ( i n p u t and o u t p u t impedance, a u d i o s u s c e p t i b i l i t y , d u t y r a t i o t o o u t p u t t r a n s f e r r e s p o n s e , and s o o n ) . Also, v a r i o u s i n p u t f i l t e r s a n d / o r a d d i t i o n a l o u t p u t f i l t e r networks can e a s i l y be added i f desired. f u n c t i o n s o f complex frequency s. Hence, g e n e r a l b o t h some new z e r o s and p o l e s a r e i n t r o duced i n t o t h e d u t y r a t i o t o o u t p u t t r a n s f e r f u n c t i o n owing t o t h e s w i t c h i n g a c t i o n , i n a d d i t i o n t o t h e p o l e s and z e r o s of t h e e f f e c t i v e f i l t e r network ( o r l i n e t o o u t p u t t r a n s f e r func t i o n ) . However, i n s p e c i a l c a s e s , a s i n a l l
t h o s e shown i n Table I, t h e frequency dependence might r e d u c e simply t o polynomials, and even f u r t h e r i t might show up o n l y i n t h e v o l t a g e dependent g e n e r a t o r s ( a s i n t h e b o o s t , o r buckb o o s t ) and r e d u c e t o a c o n s t a n t ( f ( s ) i 1 ) f o r t h e c u r r e n t g e n e r a t o r . ~ e v e r t g e l e s s ,t h i s does n o t p r e v e n t u s from modifying any of t h e s e C i r c u i t s i n a vay t h a t would e x h i b i t t h e g e n e r a l result i n t r o d u c t i o n of b o t h a d d i t i o n a l z e r o s a s v e l l as poles.
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L e t ue now i l l u s t r a t e t h i s g e n e r a l r e s u l t on a climple m o d i f i c a t i o n o f t h e f a m i l i a r b o o s t c i r c u i t , w i t h a r e s o n a n t L1,C1 c i r c u i t i n s e r i e s w i t h t h e i n p u t i n d u c t a n c e L, a s shown i n Fig. 13.
F i g . 13.
Modified b o o s t c i r c u i t a s a n i l l u s t r a t i o n of g e n e r a l f r e q u e n c y b e h a v i o u r of t h e g e n e r a t o r s i n t h e c a n o n i c a l c i r c u i t model of F i g . 11.
By i n t r o d u c t i o n of t h e c a n o n i c a l c i r c u i t model f o r t h e b o o s t power s t a g e ( f o r t h e c i r c u i t t o t h e r i g h t o f c r o s s s e c t i o n AA') and u s e of d a t a from Table I , t h e e q u i v a l e n t averaged c i r c u i t model of Fig. 14a i s o b t a i n e d . Then, by a p p l i c a t i o n of t h e e q u i v a l e n t c i r c u i t t r a n s f o r m a t i o n a s o u t l i n e d p r e v i o u s l y , t h e averaged model i n t h e c a n o n i c a l c i r c u i t form i s o b t a i n e d i n F i g . 14b. A s can be s e e n from Fig. 14b, t h e v o l t a g e generator h a s . a double pole a t t h e resonant f r e quency U r = of t h e p a r a l l e l L1,C, n e t work. However, t h e e f f e c t i v e f i l t e r t r a k f e r f u n c t i o n h a s a double z e r o ( n u l l i n magnitude) a t P r e c i s e l y t h e same l o c a t i o n s u c h t h a t t h e two
1/m
p a i r s e f f e c t i v e l y c a n c e l . Hence, t h e r e s o n a n t n u l l i n t h e magnitude r e s p o n s e , w h i l e p r e s e n t i n the l i n e voltage t o output t r a n s f e r function, is n o t s e e n i n t h e d u t y r a t i o - t o o u t p u t t r a n s f e r funct i o n . T h e r e f o r e , t h e p o s i t i v e e f f e c t of r e j e c t i o n of c e r t a i n i n p u t f r e q u e n c i e s around t h e r e s o n a n t frequency w i s n o t accompanied by a d e t r i m e n t a l e f f e c t on t g e l o o p g a i n , which w i l l n o t cont a i n a n u l l i n t h e magnitude r e s p o n s e . T h i s example d e m o n s t r a t e s y e t a n o t h e r import a n t a s p e c t o f m o d e l l i n g w i t h u s e of t h e a v e r a g i n g t e c h n i q u e . I n s t e a d of a p p l y i n g i t d i r e c t l y t o t h e whole c i r c u i t i n F i g . 1 3 , we have i n s t e a d implemented i t o n l y w i t h r e s p e c t t o t h e s t o r a g e element network which e f f e c t i v e l y t a k e s p a r t i n t h e s w i t c h i n g a c t i o n , namely L, C, and R. Upon s u b s t i t u t i o n of t h e switched p a r t of t h e network by t h e averaged c i r c u i t model, a l l o t h e r l i n e a r c i r c u i t s of t h e complete model a r e r e t a i n e d a s t h e y a p p e a r i n t h e o r i g i n a l c i r c u i t (such as L1,C1 i n Fig. 1 4 a ) . Again, t h e c u r r e n t g e n e r a t o r i n F i g . 14a i s t h e one which r e f l e c t s t h e e f f e c t of t h e i n p u t r e s o n a n t circuit. I n t h e n e x t s e c t i o n , t h e same p r o p e r t y i s c l e a r l y displayed f o r a closed-loop regulatorconverter with o r without the input f i l t e r .
6.
SWITCHING MODE REGULATOR MODELLING
This s e c t i o n demonstrates t h e ease with which t h e d i f f e r e n t c o n v e r t e r c i r c u i t models developed i n p r e v i o u s s e c t i o n s c a n be i n c o r p o r a t e d i n t o more complicated systems s u c h a s a s w i t c h i n g mode r e g u l a t o r . I n a d d i t i o n , a b r i e f d i s c u s s i o n of m o d e l l i n g of modulator s t a g e s in g e n e r a l i s i n c l u d e d , and a complete g e n e r a l switching-mode r e g u l a t o r c i r c u i t model i s g i v e n . A g e n e r a l r e p r e s e n t a t i o n .of a switching-mode r e g u l a t o r i s shown i n F i g . 1 5 . For c o n c r e t e n e s s , t h e switching-mode c o n v e r t e r i s r e p r e s e n t e d by a buck-boost power s t a g e , and t h e i n p u t and p o s s i b l e a d d i t i o n a l o u t p u t f i l t e r a r e r e p r e s e n t e d by a ,unrroulated
Fig. 1 4 .
Equivalent c i r c u i t transformation leading t o t h e c a n o n i c a l c i r c u i t model (b) of t h e c i r c u i t i n F i g . 13.
nnout
rrqufated output 7
i n p u t and o u t p u t f i l t e r s . The b l o c k diaLC gram i s g e n e r a l , and s i n g l e - s e c t i o n f i l t e r s and a buck-boost c o n v e r t e r a r e shown a s t y p i c a l r e a l i z a t i o n s .
s i n g l e - s e c t i o n low-pass LC c o n f i g u r a t i o n , b u t t h e d i s c u s s i o n a p p l i e s t o any c o n v e r t e r and any f i l t e r configuration. The main d i f f i c u l t y i n a n a l y s i n g t h e switchr e g u l a t o r l i e s . i n t h e modelling of i t s nonl i n e a r P a r t , t h e switching-mode c o n v e r t e r . HOWever, we have succeeded i n p r e v i o u s s e c t i o n s i n o b t a i n i n g t h e s m a l l - s i g n a l low-frequency c i r c u i t model of any "two-state" s w i t c h i n g dc-to-dc conv e r t e r , o p e r a t i n g i n t h e continuous conduction mode, i n t h e Canonical C i r c u i t form. The o u t p u t f i l t e r i s shown s e p a r a t e l y , t o emphasize t h e f a c t t h a t i n averaged modelling of t h e switching-mode c o n v e r t e r only t h e s t o r a g e elements which a r e a c t u a l l y involved i n t h e s w i t c h i n g a c t i o n need be t a k e n i n t o account, t h u s minimizing t h e e f f o r t i n i t s modelling.
V, +
cg
converter and modulator model
ing
The n e x t s t e p i n development o f t h e r e g u l a -
tor equivalent c i r c u i t i s t o o b t a i n a model f o r t h e modulator. T h i s i s e a s i l y done by w r i t i n g an e x p r e s s i o n f o r t h e e s s e n t i a l f u n c t i o n of t h e modul a t o r , which i s t o Convert an (analog) c o n t r o l v o l t a g e Vc t o t h e s w i t c h duty r a t i o D. T h i s exp r e s s i o n can b e w r i t t e n D = V /Vm i n which, by d e f i n i t i o n , Vm i s t h e range oh c o n t r o l s i g n a l r e q u i r e d t o sweep t h e d u t y r a t i o o v e r i t s f u l l range from 0 t o 1. A s m a l l v a r i a t i o n vc superimposed upon Vc t h e r e f o r e produces a correspond i n g v a r i a t i o n a = Gc/vm i n D, which can be g e n e r a l i z e d t o account f o r a nonuniform frequency response a s
'm
C
i n which fm(0) 1. Thus, t h e c o n t r o l v o l t a g e t o d u t y r a t i o s m a l l - s i g n a l transmission c h a r a c t e r i s t i c of t h e modulator can be r e p r e s e n t e d i n gene r a l by t h e two parameters Vm and f m ( s ) , regardl e s s of t h e d e t a i l e d mechanism by which t h e modul a t i o n i s achieved. Hence, by s u b s t i t u t i o n f o r a from (32) t h e two g e n e r a t o r s i n t h e c a n o n i c a l c i r c u i t model of t h e s w i t c h i n g c o n v e r t e r can b e expressed i n terms of t h e a c c o n t r o l v o l t a g e sad t h e r e s u l t i n g model i s t h e n a l i n e a r a c e q u i valent c i r c u i t t h a t represents the small-signal t r a n s f e r p r o p e r t i e s of t h e n o n l i n e a r p r o c e s s e s i n t h e modulator and c o n v e r t e r .
cc,
It remains simply t o add t h e l i n e a r amplif i e r and t h e i n p u t and o u t p u t f i l t e r s t o o b t a i n t h e a c e q u i v a l e n t c i r c u i t o f t h e complete closedloop r e g u l a t o r as shown i n Fig. 16. The modulator t r a n s f e r f u n c t i o n h a s been i n c o r p o r a t e d i n t h e g e n e r a t o r d e s i g n a t i o n s , and t h e g e n e r a t o r symbol h a s been changed from a c i r c l e t o a s q u a r e t o emphasize t h e f a c t t h a t , i n t h e closed-loop r e g u l a t o r , t h e g e n e r a t o r s no l o n g e r a r e independent b u t a r e dependent on a n o t h e r s i g n a l i n t h e same system. The connection from point Y t o the e r r o r amplifier, v i a the reference v o l t a g e summing node, r e p r e s e n t s t h e b a s i c v01t a g e feedback n e c e s s a r y t o e s t a b l i s h t h e system a s a v o l t a g e r e g u l a t o r . The dashed connection from P o i n t Z i n d i c a t e s a p o e s i b l e a d d i t i o n a l feedback s e n s i n g ; t h i s second feedback s i g n a l may
E
e,(s)= - f , ( ~ ) f ~ ( s ) ~ ~ Vm
Pig. 16.
------dc ref
General s m a l l - s i g n a l a c e q u i v a l e n t c i r c u i t f o r t h e switching-mode r e g u l a t o r of F i g . 15.
be d e r i v e d , f o r example, from t h e i n d u c t o r f l u x , inductor current, o r capacitor current, a s i n v a r i o u s "two-loop" c o n f i g u r a t i o n s t h a t a r e i n use t91. Once a g a i n t h e c u r r e n t g e n e r a t o r i n Fig. 16 i s r e s p o n s i b l e f o r t h e i n t e r a c t i o n between t h e switching-mode r e g u l a t o r - c o n v e r t e r and t h e i n p u t f i l t e r , t h u s c a u s i n g performance d e g r a d a t i o n and/ o r s t a b i l i t y problems when an a r b i t r a r y i n p u t f i l t e r i s added. The problem of how p r o p e r l y t o d e s i g n t h e i n p u t f i l t e r is t r e a t e d i n d e t a i l i n (71 As shown i n Fig. 1 6 we h a v e succeeded i n obt a i n i n g t h e l i n e a r c i r c u i t model o f t h e complete s w i t c h i n g mode-regulator. Hence t h e w e l l - k n m body of l i n e a r feedback t h e o r y can b e w e d f o r both a n a l y s i s and d e s i g n of t h i s t y p e of r e g u l a tor.
7.
CONCLUSIONS
A g e n e r a l method f o r m o d e l l i n g power s t a g e s o f any s w i t c h i n g dc-to-dc c o n v e r t e r h a s been developed through t h e s t a t e - s p a c e approach. The fundamental s t e p i s i n replacement of t h e s t a t e space d e s c r i p t i o n s of t h e two s w i t c h e d networks by t h e i r average over t h e s i n g l e s w i t c h i n g p e r i o d T, which r e s u l t s i n a s i n g l e continuous s t a t e s p a c e e q u a t i o n d e s c r i p t i o n (3) d e s i g n a t e d t h e b a s i c averaged s t a t e - s p a c e model. The e s s e n t i a l approximations made a r e i n d i c a t e d i n t h e Append i c e s , and a r e shown t o b e j u s t i f i e d f o r any p r a c t i c a l dc-to-dc s w i t c h i n g c o n v e r t e r .
The subsequen t p e r t u r b a t i o n and l i n e a r i z a t i o n step under t h e s m a l l - s i g n a l aesumption (12) l e a d s t o t h e f i n a l s t a t e - s p a c e averaged model given by (13) and (14). These e q u a t i o n s then s e r v e as t h e b a s i s f o r development of t h e most i m p o r t a n t q u a l i t a t i v e r e s u l t of t h i s v o r k , t h e c a n o n i c a l c i r c u i t model o f F i g . 11. D i f f e r e n t c o n v e r t e r s a r e r e p r e s e n t e d simply by a n appropria t e s e t of formulas ((27) and (28)) f o r f o u r elements i n t h i s g e n e r a l e q u i v a l e n t c i r c u i t . Bes i d e s i t s u n i f i e d d e s c r i p t i o n , o f which s e v e r a l
examples a r e given i n Table I , one of t h e sdvant a g e s of t h e c a n o n i c a l c i r c u i t model is t h a t v a r i o u s performance c h a r a c t e r i s t i c s of d i f f e r e n t s w i t c h i n g c o n v e r t e r s can b e compared i n a quick and e a s y manner. Although t h e s t a t e - s p a c e modelling approach has been developed i n t h i s paper f o r two-state s w i t c h i n g c o n v e r t e r s , t h e method can be extended t o m u l t i p l e - s t a t e c o n v e r t e r s . Examples of t h r e e s t a t e c o n v e r t e r s a r e t h e f a m i l i a r buck, b o o s t , and buck-boost power s t a g e s o p e r a t e d i n t h e d l s continuous conduction mode, and dc-to-ac switchi n g i n v e r t e r s i n which a s p e c i f i c o u t p u t waveform i s "assembled" from d i s c r e t e segments a r e examples of m u l t i p l e - s t a t e c o n v e r t e r s . I n c o n t r a s t w i t h t h e s t a t e - s p a c e modelling approach, f o r any p a r t i c u l a r c o n v e r t e r an a l t e r n a t i v e p a t h v i a h y b r i d modelling and c i r c u i t t r a n s f o r m a t i o n could be followed, which a l s o a r r i v e s f i r s t a t t h e f i n a l c i r c u i t averaged model e q u i v a l e n t of (13) and (14) and f i n a l l y , a f t e r equivalent c i r c u i t transformations, again a r r i v e s a t t h e c a n o n i c a l c i r c u i t model. Regardless of t h e d e r i v a t i o n p a t h , t h e c a n o n i c a l c i r c u i t model can e a s i l y be incorporat e d i n t o an e q u i v a l e n t c i r c u i t model of a comp l e t e s w i t c h i n g r e g u l a t o r , a s i l l u s t r a t e d i n Fig. 16. Perhaps t h e most i m p o r t a n t consequence of t h e c a n o n i c a l c i r c u i t model d e r i v a t i o n v i a t h e g e n e r a l s t a t e y a p a c e averaged model (13). ( 1 4 ) , (23) and (24) i s i t s p r e d i c t i o n through (27) of a d d i t i o n a l z e r o s a s w e l l a s p o l e s i n t h e duty r a t i o to output t r a n s f e r function. I n addition frequency dependence i s a n t i c i p a t e d i n t h e duty r a t i o dependent c u r r e n t g e n e r a t o r of Fig. 11, even though f o r p a r t i c u l a r c o n v e r t e r s considered i n Table I, i t reduces merely t o a c o n s t a n t . Furthermore f o r some a w i t c h i n g networks v h i c h would e f f e c t i v e l y i n v o l v e more than two s t o r a g e elements, h i g h e r o r d e r polynomials should b e expected i n f l ( s ) a n d / o r f 2 ( s ) of Fig. 11. The i n s i g h t s t h a t have emerged from t h e g e n e r a l s t a t e - s p a c e modelling approach s u g g e s t t h a t t h e r e i s a whole f i e l d of new s w i t c h i n g dcto-dc c o n v e r t e r pover s t a g e s y e t t o be conceived. This encourages a renewed s e a r c h f o r i n n o v a t i v e c i r c u i t d e s i g n s i n a f i e l d which is y e t young, and promises t o y i e l d a s i g n i f i c a n t number of i n v e n t i o n s i n t h e s t r e a m of i t s f u l l development. This p r o g r e s s w i l l n a t u r a l l y be f u l l y s u p p o r t e d by new t e c h n o l o g i e s coming a t an e v e r i n c r e a s i n g pace. However, even though t h e e f f i c i e n c y and performance of c u r r e n t l y e x i s t i n g c o n v e r t e r s w i l l i n c r e a s e through b e t t e r , , f a s t e r t r a n s i s t o r s , more i d e a l c a p a c i t o r s ( w i t h lower e a r ) and s o on, it w i l l be p r i m a r i l y t h e r e s p o n s i b i l i t y of t h e c i r c u i t d e s i g n e r and i n v e n t o r t o p u t t h e s e components t o b e s t use i n an o p t i ~ ltopology. Search f o r new c i r c u i t c o n f i g u r a t i o n s , and h w b e s t t o use p r e s e n t and f u t u r e t e c h n o l o g i e s , w i l l b e of prime importance i n a c h i e v i n g t h e u l t i m a t e g o a l of neari d e a l g e n e r a l s w i t c h i n g dc-to-dc c o n v e r t e r s .
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