英文 状态空间平均法

A GENERAL UNIFIED APPROACH TO MODELLING SWITCHING-CONVERTER POWER STAGES RD.MIDDLEBROOK AND SLOBODAN CUK ABSTRACT A method f o r modelling switching-converter power s t a g e s is developed, whose s t a r t i n g po in t is the u n i f i e d s ta te - space 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 tate-space d e s c r i p t i o n o r i t s equiva len t small-signal low-frequency l i n e a r c i r c u i t model. A new canonica l c i r c u i t model is proposed, whose f ixed topology conta ins a l l t h e e s s e n t i a l input-output and c o n t r o l p r o p e r t i e s of any dc-to- dc switching conver te r , r e g a r d l e s s of i t s d e t a i l e d conf igura t ion , and by which d i f f e r e n t conver te r s can be charac te r ized i n t h e form of a t a b l e con- ven ien t ly s t o r e d i n a computer d a t a bank t o pro- v i d e a u s e f u l t o o l f o r computer a ided design and opt imizat ion. The new canonical 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 zeros and p o l e s i n t o t h e duty r a t i o t o output t r a n s f e r func t ion i n add i t ion t o those from the e f f e c t i v e f i l t e r network. 1. INTRODUCTION 1.1 Brief Review of Exis t ing Modelling Techniques I n modelling of switching conver te r s i n genera 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 ta te - space modelling and t h e o t h e r us ing an averaging technique - have been developed ex tens ive ly , but t h e r e has 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 equa t ion manipulat ions, and hence r e l i e s heav i ly on numerical methods and computerized Implementationa. Its primary advantage is 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 type (buck. boost . 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 exac t s ta te-space equa t ions of t h e two switched models. -On t h e o t h e r hand, Proceseing Systems .'I based on equiva len t c i r c u i t manipulat ions, r e s u l t i n g i n a s i n g l e equ iva len t l i n e a r c i r c u i t model of t h e p a r e r s tage . This has t h e d i s t i n c t advantage of providing t h e c i r c u i t designer with phys ica l i n s i g h t i n t o t h e behaviour of the o r i g i n a l switched c i r c u i t , and of al lowing 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 ex ten t i n design of r e g u l a t o r s incorpora t ing switching conver te r s . 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 considered t o e x i s t between t h e s t a t e - space technique and t h e averaging technique 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 genera l u n i f i e d t reatment of t h e s ta te - space approach, a s w e l l a s an equiva len 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 otherwise could no t be achieved. The proposed s ta te - space averaging method, o u t l i n e d i n t h e Flowchart of Fig. 1, al lows a u n i f i e d t rea tment 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 averaging s t e p i n t h e s ta te - space domain is very simple and c l e a r l y defined (compare blocks l a and 2a) . I t merely c o n s i s t s of averaging t h e two exac t s ta te - space 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 is t h e switching frequency (block 2a) . 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 equ iva len t forms i n o r d e r t o apply c i rcu i t -o r ien ted procedure d i r e c t l y , a s requ i red i n [ l ] (block l c ) . Nevertheless , through a hybrid model l ing 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 (block 2b) can be r e a d i l y recognized from t h e averaged s tate-space model (block 2a) . Hence a l l t h e b e n e f i t s of t h e previous averaging technique a r e r e t a i n e d . Even though t h i s out- I n e i t h e r case , 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 @ 1976 IEEE. Reprinted, vith permission, f r w Proceedings of the IEEE Pwer Electr~nics Specialists Conference, J~~~ * - 10, 1976, Cleveland, OH. stolc - spore C ~ Y O ~ I O ~ S rlrody slotc ldct modd d - o:d, d '~ ' -2 , x - x + ; , ax+dv , -o - x - - ~ - h v , ; r-C? y . ~ + i , v,-Lj.i dynom,c ioc ~ m o / / s 1 9 ~ / 1 m o t / . Fig. 1, Flowchart of averaged modelling approaches process requi red t o inc lude the duty r a t i o modulation e f f e c t proceeds i n a very s t r a i g h t f o r - ward and formal manner, thus emphasizing the corner-stone charac ter of blocks 2a and 2b. At t h i s s t age (block 2a o r 2b) t h e s teady-s ta te (dc) and l i n e t o output t r a n s f e r funct ions a r e a l ready ava i l ab l e , a s indica ted by blocks 6a and 6b r e spec t ive ly , while t he duty r a t i o t o output t r a n s f e r funct ion is a v a i l a b l e a t t h e f ina l -s tage model (4a o r 4b) a s i nd i ca t ed by blocks 7a and 7b. The two f i n a l s t age models (4a and 4b) then g ive t h e complete desc r ip t ion of t he switching converter by inc lus ion of both independent con- t r o l s , t he l i n e vo l t age v a r i a t i o n and t h e duty r a t i o modulation. Even though t h e c i r c u i t t ransformation pa th b might be p re fe r r ed from the p r a c t i c a l design s tandpoint , t he s ta te -space averaging path a is invaluable i n reaching some genera l conclusions about the small-signal low-frequency models of any dc-to-dc switching conver ter (even those y e t t o be invented) . Whereas, f o r path b , one has t o be prekented with t he p a r t i c u l a r c i r c u i t i n order t o proceed with modelling, f o r path a t h e f i n a l s ta te -space averaged equat ions (block 48) give t h e complete model desc r ip t ion through genera l mat r ices A1, A2 and vec to 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 ) . This i s a l s o why a long path 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 wi th p a r a s i t i c e f f e c t s was chosen, while along pa th a genera l equat ions have been re ta ined . S p e c i f i c a l l y , f o r t h e boost power s t age bl = b2 - b. This example w i l l be l a t e r pursued i n d e t a i l along both paths. I n add i t i on t h e s ta te -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 averaging approximation, which becomes b e t t e r t he 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 corner frequency f is below the switching frequency f,, t h a t is , f z / f s << 1. This is , however, shown t o be equiva lent t o t h e requirement f o r smal l output vol tage r i p p l e , and hence does n o t pose any s e r i o u s r e s t r i c t i o n o r l i m i t a t i o n on modelling of prac r i c a l dc-to-dc conver ters . F i n a l l y , t h e s ta te -space averaging approach se rves a s a b a s i s f o r de r iva t ion of a u s e f u l genera l c i r c u i t model t h a t desc r ibes t h e input- output and c o n t r o l p rope r t i e s of any dc-to-dc conver ter . 1 .3 New Canonical C i r c u i t Model The culminat ion of any of t h e s e der iva- tions along e i t h e r pa th a o r pa th b i n t h e Flowchart of Fig. 1 is an equiva len t c i r c u i t (block 5) , v a l i d f o r smal l - s igna l low-f requency v a r i a t i o n s superimposed upon a dc o p e r a t i n g p o i n t , t h a t r e p r e s e n t s t h e two t r a n s f e r func t ions of i n t e r e s t f o r a swi tch ing conver te r . These .re the l i n e vo l tage t o ou tpu t and duty r a t i o to output t r a n s f e r f u n c t i o n s . The e q u i v a l e n t c i r c u i t i s a canonica l model t h a t con ta ins t h e e s s e n t i a l p r o p e r t i e s of dc-to-dc swi tch ing c o n v e r t e r , r e g a r d l e s s of t h e de ta i led conf igura t ion . A s seen i n block 5 f o r the general case, t h e model inc ludes an i d e a l transformer t h a t d e s c r i b e s t h e b a s i c d c - t e d c t ransformation r a t i o from l i n e t o o u t p u t ; a low-pass f i l t e r whose element va lues depend upon the dc duty r a t i o ; and a v o l t a g e and a c u r r e n t generator p r o p o r t i o n a l t o t h e duty r a t i o modula- t i o n input . The canonica l model i n block 5 of t h e Flow- c h a r t can be obtained fol lowing e i t h e r pa th a o r path b, namely from b lock 4a o r 4b, a s w i l l be &own l a t e r . However, fol lowing t h e g e n e r a l d e s c r i p t i o n of t h e f i n a l averaged model i n block ha, c e r t a i n g e n e r a l i z a t i o n s about t h e canonica l model a r e made p o s s i b l e , which a r e o therwise no t achievable. Namely, even though f o r a l l c u r r e n t l y known switching dc-to-dc conver te r s (such a s t h e buck, boos t , buck-boost, Venable 1 3 1 , Weinberg [ 4 ] and a number of o t h e r s ) t h e frequency dependence appears on ly i n t h e du ty- ra t io dependent v o l t a g e generator bu t n o t i n t h e c u r r e n t genera tor , and the ~n bs,a@,a,fkaLs.f d e ~ . L ~ i , r l ~ l e ~ z c r _ o ~ . . e o I ~ o m i a l i n po les of t h e e f f e c t i v e f i l t e r network which e s s e n t i a l l y c o n s t i t u t e t h e l i n e v o l t a g e t o ou tpu t t r a n s f e r func t ion . Moreover, i n g e n e r a l , both duty- ra t io dependent g e n e r a t o r s , v o l t a g e and cur- r e n t , a r e frequency dependent ( a d d i t i o n a l ze ros and p o l e s ) . That i n t h e p a r t i c u l a r c a s e s of t h e boost o r buck-boost conver te r s t h i s dependence reduces t o a f i r s t o rder polynomial r e s u l t s from the f a c t t h a t - t h e order of t h e system which i s involved i n t h e swi tch ing a c t i o n i s only two. Hence from t h e g e n e r a l r e s u l t , t h e o rder of t h e Polynomial i s a t most one, though i t could reduce t o a pure c o n s t a n t , a s i n t h e buck o r t h e Venable converter [ 3 ] . The s i g n i f i c a n c e of t h e new c i r c u i t model i s t h a t any swi tch ing dc-to-dc conver te r can be reduced t o t h i s canonica l f ixed topology form, a t l e a s t a s f a r a s i t s input-output and c o n t r o l a;nt"y";;;f; 5-7 i;-g;;,;j, ;~;;er~f~~c:'iv:a for elements depend on duty r a t i o D ) , and t h e conf i - gura t ion chosen which opt imizes t h e s i z e and weight. Also, comparison of t h e frequency depen- dence of t h e two d u t y - r a t i o dependent g e n e r a t o r s provides i n s i g h t i n t o t h e ques t ion of s t a b i l i t y once a r e g u l a t o r feedback loop i s c l o s e d . 1.4 Extension t o Complete Regulator Treatment F i n a l l y , a l l t h e r e s u l t s obtained i n model l ing t h e conver te r o r , more a c c u r a t e l y , t h e network which e f f e c t i v e l y t a k e s p a r t i n swi tch ing a c t i o n , can e a s i l y be incorpora ted i n t o more complicated systems conta in ing dc-to-dc conver te r s . For example, by model l ing t h e modulator s t a g e along t h e same l i n e s , one can o b t a i n a l i n e a r c i r c u i t model of a closed-loop swi tch ing r e g u l a t o r . Standard l i n e a r feedback theory ran then be used f o r both a n a l y s i s and s y n t h e s i s , s t a b i l i t y c o n s i d e r a t i o n s , and proper des ign of feedback compensating ne t - works f o r m u l t i p l e loop a s w e l l a s s ingle- loop r e g u l a t o r c o n f i g u r a t i o n s . 2. STATE-SPACE AVERAGING In t h i s s e c t i o n t h e s ta te - space averaging method i s developed f i r s t i n g e n e r a l f o r any dc- to-dc switching c o n v e r t e r , and then demonstrated i n d e t a i l f o r the p a r t i c u l a r c a s e of t h e boost power s t a g e i n which p a r a s i t i c e f f e c t s ( e s r of t h e c a p a c i t o r and s e r i e s r e s i s t a n c e of t h e in - duc tor ) a r e included. General equa t ions f o r both s teady-s ta te (dc) and dynamic performance (ac) a r e obtained, from which important t r a n s f e r in func t ions a r e der ived and a l s o a p p l i e d t o the s p e c i a l c a s e of t h e boost power s t a g e . swi tch ing between two l i n e a r networks c o n s i s t i n g of i d e a l l y l o s s l e s s s t o r a g e elements , inductances and capaci tances. I n p r a c t i c e , t h i s func t ion may be obtained by use of t r a n s i s t o r s and d iodes which o p e r a t e a s synchronous swi tches . On t h e assumption t h a t t h e c i r c u i t o p e r a t e s i n the so- c a l l e d "continuous conduction" mode i n which t h e ins tan taneous induc tor c u r r e n t does no t f a l l t o ze ro a t any p o i n t i n t h e c y c l e , t h e r e a r e only two d i f f e r e n t " s t a t e s " of t h e c i r c u i t . Each s t a t e , however, can be represen ted by a l i n e a r c i r c u i t model ( a s shown i n block l b of Fig. 1 ) o r by a corresponding s e t of s t a t e - s p a c e equa t ions (block l a ) . Even though any s e t of l i n e a r l y independent v a r i a b l e s can be chosen a s t h e s t a t e v a r i a b l e s , i t is customary and convenient i n e l e c t r i c a l networks t o adopt t h e i n d u c t o r c u r r e n t s and capa- c i t o r vo l tages . The t o t a l number of s t o r a g e elements t h u s determines t h e order of t h e system. k e , E , , ~ s , " d , ~ :~,~h,,B,,:h,~,i,c,~ e:f LaLve_c,to,E equa t ions f o r t h e two switched models: (i) i n t e rva l Td : (11) in t e rva l Td ' : where Td denotes the in t e rva l when the switch i s i n the on s t a t e a n d T ( l -d ) 5 ~ d ' i s the in terval for which it i s i n t h e o f f s t a t e , as shown i n Fig. 2 . The s t a t i c equations y l = clTx and y = c2Tx are necessary i n order t o account for t i e case when t h e output quanti ty does not s w ~ t c h d r i v e o f f 7 k r- 1 Td Td' 4 Fig. 2. De f in i t i on o f t h e two switched in t e rva l s Td and Td' . coincide with any o f t h e s t a t e var iab le s , but i s rather a cer ta in l inear combination o f the s ta t e variable$. Our objec t ive now i s t o replace the s ta te- space description o f t h e two l inear c i r c u i t s emanating from the two successive phases o f the switching cyc le T by a s ingle state-space des- cr ip t ion which represents approximately the beha- viour o f the c i r c u i t across the whole period T . We there fore propose the following simple avera- ging s t ep : take t h e average o f both dynamic and s t a t i c equations for the two switched in t e rva l s ( I ) , by summing t h e equations for in t e rva l Td mult ipl ied by d and the equations for in t e rva l Td' mult ipl ied by d ' . The following l inear continuous system r e s u l t s : A f t e r rearranging ( 2 ) i n t o t h e standard l inear continuous system state-space descr ip t ion , we obtain the basic averaged state-space descrip- t i o n (over a s ingle period T ) : This model i s the bas ic averaged model which i s the s ta r t ing model f o r a l l other derivations (both state-space and c i rcu i t o r i en ted ) . Note tha t i n the above equations t h e duty r a t i o d i s considered constant; it i s not a time dependent var iable ( y e t ) , and particularly not a switched discontinuous variable which changes between 0 and 1 as i n [ l ] and [ 2 ] , but i s merely a f ixed number for each cycle . This i s evident from the model der ivat ion i n Appendix A. In par t icular , when d = 1 (swi tch constantly on) the averaged model ( 3 ) reduces t o switched model (11) , and when d = 0 (swi tch o f f ) i t reduces t o switched model (111) . In essence, comparison between ( 3 ) and ( 1 ) shows tha t the system matrix o f the averaged model i s obtained by taking t h e average o f two switched model matrices A and A2, i t s control i s the average o f two control vec tors bl and b 2 , and i t s output i s t he average o f two outputs yl and y over a period T . 2 The j u s t i f i c a t i o n and the nature o f the approximation i n subs t i t u t ion for t h e two switched models o f (1) by averaged model (3) i s indicated i n Appendix A and given i n more d e t a i l i n [b]. The bas ic approximation made, however, i s that o f approximation o f t h e fundamental matrix eAt = I + At + by i t s f i rs t -order l inear term. This i s , i n turn,shown i n Appendix B t o be the same approximation necessary t o obtain the dc condit ion independent o f t h e storage element values (L,C) and dependent on the dc duty r a t i o only. I t a l so coincides w i th the requirement for low output voltage r i p p l e , which i s shown i n Appendix C t o be equivalent t o f c / f << 1 , namely the e f f e c t i v e f i l t e r corner frequency much lower than the switching frequency. The model represented by ( 3 ) is an averaged model over a s ingle period T . I f we now assume tha t the duty r a t i o d i s constant from cycle t o c y c l e , namely, d = D (steady s t a t e dc duty r a t i o ) , we get : where Since ( 4 ) i s a l i near system, superposition holds and it can be per t~rbed by introduction o f l i n e voltage var ia t ions v as v = V + C , where V i s the dc l i n e input v81tagef cauging 8 c8rrespo:ding perturbation i n the s t a t e vector x = X + X , where !gain X i s t he dc value o f the s t a t e vector and x the superFposed ac pertur- bation. S imi lar l y , y = Y + y , and Separation o f the steady-state (dc) part from the dynamic (ac ) part then resu l t s i n the steady s ta te (dc ) model and the dynamic (ac ) model I t i s