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
-
-
@ 1976
IEEE.
* - 10, 1976,
CUK
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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ax+dv,-o - x - - ~ - h v , ;r-C?
dynom,c ioc ~ m o / / s 1 9 ~m/ 1o t / .
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 .
-
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 << 1. T h i s i s , however, shown t o be
e q u i v a l e n t t o t h e requirement f o r s m a l l o u t p u t
v o l t a g e 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 c o n v e r t e r s .
F i n a l l y , t h e s t a t e - s p a c e a v e r a g i n g approach
s e r v e s a s a b a s i s f o r d e r i v a t i o n of a u s e f u l
g e n e r a l c i r c u i t model t h a t d e s c r i b e s t h e 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-to-dc
converter.
1.3
New Canonical C i r c u i t Model
The c u l m i n a t i o n of any of t h e s e d e r i v a tions
along e i t h e r path a o r path b i n t h e
Flowchart of F i g . 1 i s an e q u i v a l e n t c i r c u i t
(block 5) , v a l i d f o r s m a l l - s i g n a 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 f u n c t i o n s
of i n t e r e s t f o r a s w i t c h i n g c o n v e r t e r . These
.re t h e l i n e v o l t a g e t o o u t p u t and d u t y r a t i o
to o u t p u t 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 c a n o n i c a l model
t h a t c o n t a i n s 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 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 t h e
A s seen i n b l o c k 5 f o r
detailed configuration.
the general c a s e , t h e model i n c l u d e s a n 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 r a n s f o r m a t i o n 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 v a l u e s depend upon
t h e dc d u t y r a t i o ; and a v o l t a g e and a c u r r e n t
g e n e r a t o r p r o p o r t i o n a l t o t h e d u t y r a t i o modulation input.
The c a n o n i c a l model i n b l o c k 5 of t h e Flowc h a r t can be o b t a i n e d f o l l o w i n g e i t h e r p a t h a o r
p a t h b , namely from b l o c k 4a o r 4b, a s w i l l be
&own l a t e r . However, f o l l o w i n g 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 b l o c k
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 c a n o n i c a l
model a r e made p o s s i b l e , which a r e o t h e r w i s e n o t
a c h i e v a b l e . Namely, even though f o r a l l c u r r e n t l y
known s w i t c h i n g dc-to-dc c o n v e r t e r s (such a s t h e
buck, b o o s 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 o n l y i n t h e d u t y - r a t i o dependent v o l t a g e
g e n e r a t o r b u t n o t i n t h e c u r r e n t g e n e r a t o r , and thein
~nbs,a@,a,fkaLs.f d e ~ L
. ~ i , r l ~ l e ~ z c r _ o ~ . . e o I ~ ionm i a l
p o l e s of t h e e f f e c t i v e f i l t e r network which
essentially c o n s t i t u t e t h e l i n e voltage t o output
t r a n s f e r f u n c t i o n . Moreover, i n g e n e r a l , both
d u t y - r a t i o dependent g e n e r a t o r s , v o l t a g e and c u r r e n t , a r e frequency dependent ( a d d i t i o n a l z e r o s
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
b o o s t o r buck-boost c o n v e r t e r s t h i s dependence
reduces t o a f i r s t o r d e r polynomial r e s u l t s from
t h e f a c t t h a t - t h e o r d e r of t h e system which i s
involved i n t h e s w i t c h i n g a c t i o n i s o n l y two.
Hence from t h e g e n e r a l r e s u l t , t h e o r d e r of t h e
Polynomial i s a t most o n e , though i t could r e d u c e
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 o f t h e new 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 be
reduced t o t h i s c a n o n i c a l 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
a;nt"y";;;f;
5-7 i;-g;;,;j,
;~;;er~f~~c:'iv:a
for
elements depend on d u t y r a t i o D ) , and t h e c o n f i -
g u r a t i o n chosen which o p t i m i z e s t h e s i z e and
weight. Also, comparison of t h e frequency dependence 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
p r o v i d e s i n s i g h t i n t o t h e q u e s t i o n 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 modelling
t h e c o n v e r t e 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 s w i t c h i n g a c t i o n ,
can e a s i l y be i n c o r p o r a t e d i n t o more complicated
systems c o n t a i n i n g dc-to-dc c o n v e r t e r s .
For
example, by m o d e l l i n g t h e modulator s t a g e a l o n g 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 s w i t c h i n g r e g u l a t o r .
Standard
l i n e a r feedback theory ran t h e n 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 p r o p e r d e s i g n of feedback compensating n e t works f o r m u l t i p l e loop a s w e l l a s s i n g l e - l o o p
regulator configurations.
2.
STATE-SPACE AVERAGING
In t h i s s e c t i o n t h e state-space averaging
method i s developed f i r s t i n g e n e r a l f o r any dcto-dc s w i t c h i n g c o n v e r t e r , and t h e n demonstrated
i n d e t a i l f o r t h e p a r t i c u l a r c a s e of t h e b o o s t
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 i n ductor) a r e included. General equations f o r
b o t h s t e a d y - s t a t e ( d c ) and dynamic performance
( a c ) a r e o b t a i n e d , from which i m p o r t a n t t r a n s f e r
f u n c t i o n s a r e d e r i v e d and a l s o a p p l i e d t o t h e
s p e c i a l c a s e of t h e b o o s t power s t a g e .
s w i t c h i n g 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 e l e m e n t s , i n d u c t a n c e s
and c a p a c i t a n c e s .
I n p r a c t i c e , t h i s f u n c t i o n may
be o b t a i n e d by use of t r a n s i s t o r s and d i o d e s
which o p e r a t e a s synchronous s w i t c h e s . 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 t h e soc a l l e d "continuous conduction" mode i n which t h e
i n s t a n t a n e o u s i n d u c t o r c u r r e n t does n o t f a l l t o
z e r o 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 o n l y
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 r e p r e s e n t e d by a l i n e a r c i r c u i t
model ( a s shown i n b l o c k l b of F i g . 1 ) o r by a
c o r r e s p o n d i n g s e t of s t a t e - s p a c e e q u a t i o n s ( b l o c k
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 i s customary and c o n v e n i e n t i n e l e c t r i c a l
networks t o a d o p t t h e i n d u c t o r c u r r e n t s and capac i t o r v o l t a g e s . The t o t a l number of s t o r a g e
elements t h u s d e t e r m i n e s t h e o r d e r of t h e system.
k e , E , , ~ s , " d , ~:~,~h,,B,,:h,~,i,c,~
e
:
f
LaLve_c,to,E
e q u a t i o n s f o r t h e two switched models:
(i)
i n t e r v a l Td :
(11) interval Td':
where Td denotes t h e i n t e r v a l when t h e switch i s
i n t h e on s t a t e a n d T ( l - d ) 5 ~ d i' s t h e i n t e r v a l
f o r which i t 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 f o r
t i e case when t h e output q u a n t i t y does not
s w ~ t c hdrive
k
1
Td
off
Td'
7
4r-
Fig. 2. D e f i n i t i o n o f t h e two switched i n t e r v a l s
Td and Td'
.
coincide w i t h any o f t h e s t a t e v a r i a b l e s , but
i s rather a c e r t a i n l i n e a r combination o f t h e
state variable$.
Our o b j e c t i v e now i s t o replace t h e s t a t e space d e s c r i p t i o n o f t h e two l i n e a r c i r c u i t s
emanating from t h e two successive phases o f t h e
switching c y c l e T by a s i n g l e state-space desc r i p t i o n which represents approximately t h e behaviour o f t h e c i r c u i t across t h e whole period T .
We t h e r e f o r e propose t h e following simple averaging s t e p : t a k e t h e average o f b o t h dynamic and
s t a t i c equations f o r t h e two switched i n t e r v a l s
( I ) , by summing t h e equations f o r i n t e r v a l Td
m u l t i p l i e d by d and t h e equations f o r i n t e r v a l
Td' m u l t i p l i e d by d ' . The following l i n e a r
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 i n e a r continuous system state-space d e s c r i p t i o n ,
we o b t a i n t h e b a s i c averaged state-space descript i o n (over a s i n g l e period T ) :
T h i s model i s t h e b a s i c averaged model which
i s the s t a r t i n g model f o r a l l o t h e r d e r i v a t i o n s
( b o t h state-space and c i r c u i t o r i e n t e d ) .
Note t h a t i n t h e above equations t h e d u t y
r a t i o d i s considered c o n s t a n t ; i t i s n o t a time
dependent v a r i a b l e ( y e t ) , and p a r t i c u l a r l y not a
switched discontinuous v a r i a b l e which changes
between 0 and 1 as i n [ l ]and [ 2 ] , but i s merely
a f i x e d number f o r each c y c l e . This i s e v i d e n t
from t h e model d e r i v a t i o n i n Appendix A. In
p a r t i c u l a r , when d = 1 ( s w i t c h
c o n s t a n t l y on)
t h e averaged model ( 3 ) reduces t o switched
and when d = 0 ( s w i t c h o f f ) i t
model ( 1 1 ) ,
reduces t o switched model ( 1 1 1 ) .
In e s s e n c e , comparison between ( 3 ) and ( 1 )
shows t h a t t h e system matrix o f t h e averaged
model i s obtained by taking t h e average o f two
switched model matrices A and A2, i t s c o n t r o l i s
t h e average o f two c o n t r o l v e c t o r s bl and b 2 , and
i t s output i s t h e 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 t h e nature o f the
approximation i n s u b s t i t u t i o n f o r 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 b a s i c approximation made, however, i s t h a t
o f approximation o f t h e fundamental matrix
eAt = I + At
by i t s f i r s t - o r d e r l i n e a r
term. T h i s i s , i n turn,shown i n Appendix B t o
be t h e same approximation necessary t o o b t a i n t h e
dc c o n d i t i o n independent o f t h e storage element
v a l u e s (L,C) and dependent on t h e dc d u t y r a t i o
o n l y . I t a l s o coincides w i t h t h e requirement f o r
low output v o l t a g e r i p p l e , which i s shown i n
Appendix C t o be e q u i v a l e n t t o f c / f << 1 ,
namely t h e e f f e c t i v e f i l t e r corner frequency
much lower than t h e switching frequency.
+
The model represented by ( 3 ) is an averaged
model over a s i n g l e period T . I f we now assume
t h a t t h e d u t y r a t i o d i s constant from c y c l e t o
c y c l e , namely, d = D ( s t e a d y s t a t e dc duty r a t i o ) ,
we g e t :
where
Since ( 4 ) i s a l i n e a r system, superposition
holds and i t can be p e r t ~ r b e d by i n t r o d u c t i o n o f
l i n e v o l t a g e v a r i a t i o n s v as v = V + C , where
V i s t h e dc l i n e i n p u t v 8 1 t a g e f cauging 8
c8rrespo:ding perturbation i n t h e s t a t e v e c t o r
x = X + X , where !gain X i s t h e dc value o f t h e
s t a t e v e c t o r and x t h e superFposed ac perturb a t i o n . S i m i l a r l y , y = Y + y , and
Separation o f the steady-state ( d c ) part
from the dynamic ( a c ) part then r e s u l t s i n t h e
steady s t a t e ( d c ) model
dc
term
ac
term
ac term
nonlinear term
The perturbed state-space d e s c r i p t i o n i s
nonlinear owing t o t h e presence o f the_ prodyct
o f the two time dependent q u a n t i t i e s x and d .
and t h e dynamic ( a c ) model
2.3
I t i s i n t e r e s t i n g t o note t h a t i n ( 7 ) t h e
steady s t a t e ( d c ) v e c t o r X w i l l i n general only
depend on the dc duty r a t i o D and r e s i s t a n c e s
i n t h e o r i g i n a l model, but not on t h e storage
element values ( L ' s and C ' s ) . T h i s i s so
because X i s t h e s o l u t i o n o f t h e l i n e a r system
o f equations
i n which L ' s and C's are proportionality cons t a n t s . T h i s i s i n complete agreement w i t h t h e
f i r s t - o r d e r approximation o f t h e exact dc
conditions shown i n Appendix B , which coincides
w i t h expression ( 7 ) .
Linearization and Final State-Space Averaged
Model
Let us now make t h e small-signal approximat i o n , namely t h a t departures from t h e steady s t a t e
values are n e g l i g i b l e compared t o t h e steady s t a t e
values themselves:
Then, using approximations ( 1 2 ) we n e g l e c t a l l
nonlinear terms such as t h e second-order terms i n
( 1 1 ) and obtain once again a line_ar system, but
including duty-ratio modulation d . A f t e r separ a t i n g steady-state ( d c ) and dynamic ( a c ) parts
o f t h i s linearized system we a r r i v e a t t h e following r e s u l t s for t h e f i n a l state-space averaged
model.
Steady-state ( d c ) model:
From t h e dynamic ( a c ) model, t h e l i n e
voltage t o state-vector t r a n s f e r functions can
be e a s i l y derived a s :
Dynamic (ac small-signal) model:
Hence a t t h i s stage both steady-state
( d c ) and l i n e t r a n s f e r functions are a v a i l a b l e ,
as shown by block 6a i n t h e Flowchart o f F i g . 1 .
We now undertake t o include t h e duty r a t i o
modulation e f f e c t i n t o t h e basic averaged
model ( 3 ) .
2.2
Perturbation
Suppose now t h a t the duty r a t i o changes from
where D
cycle t o c y c l e , t h a t i s , d ( t ) = D +
i_s the steady-state ( d c ) duty r a t i o as b e f o r e and
d i s a superimposed ( a c ) v a r i a t i o n . With t h e
c o r r e s ~ o n d i n aperturbatipn d e f i n i t i o n x = X + x ,
dc term
line
duty r a t i o v a r i a t i o n
variation
a
A
nonlinear second-order term
In these r e s u l t s , A, b and c
by ( 5 ) .
T
are given as b e f o r e
Equations ( 1 3 ) and ( 1 4 ) represent t h e smallsignal low-frequency model o f any N o - s t a t e
switching dc-to-dc converter working i n t h e continuous conduction mode.
I t i s important t o note t h a t by n e g l e c t o f
t h e nonlinear term i n ( 1 1 ) t h e source o f harmonics
i s e f f e c t i v e l y removed. T h e r e f o r e , t h e l i n e a r
description ( 1 4 ) i s a c t u a l l y a l i n e a r i z e d
describing function r e s u l t t h a t i s the l i m i t o f
t h e d e s ~ r i b i n g ~ f u n c t i o2s
n t h e amplitude o f t h e
input signals v e andlor d becomes vanishingly
explained i n [ I ] , 121, or la]' i n h i c h smailsignal assumption ( 1 2 ) i s preserved. Very good
agreement up t o close t o h a l f t h e switching
frequency has been demonstrated repeatedly
(111, [21, 131, 171).
2.4
Example:
i n which I i s t h e d c i n d u c t o r c u r r e n t , V i s
t h e d c c a p a c i t o r v o l t a g e , and Y i s t h e d c o u t p u t
voltage
Dynamic ( a c s m a l l s i g n a l ) model:
Boost Power S t a g e w i t h P a r a s i t i c s
.
We now i l l u s t r a t e t h e method f o r t h e b o o s t
power s t a g e shown i n F i g . 3.
F i g . 3.
Example f o r t h e s t a t e - s p a c e averaged
modelling: b o o s t power s t a g e w i t h paras i t i c ~i n c l u d e d .
i n which R'
2
(1-D) R
+
R~
+ D(~-D)(R~~~R;.
We now l o o k more c l o s e l y a t t h e d c v o l t a g e
transformation r a t i o i n (17):
Fig. 4.
Two switched c i r c u i t models of t h e
c i r c u i t i n P i g . 3 w i t h assumption of i d e a l
s w i t c h e s . A l l elements i n t h e f i n a l s t a t e s p a c e averaged mode4 (13) and (14) a r e
from a ) f o r i n t e r v a l
obtained: A b , c
Td, and ~ ~ , b i : ~ kb) f o r i n t e r v a l Td'
hom
With assumption of i d e a l s w i t c h e s , t h e two
switched models a r e a s shown i n Fig. 4. For c h o i c e
of s t a t e - s p a c e v e c t o r xT = ( 1 v ) , t h e s t a t e s p a c e
e q u a t i o n s become:
.--I
where
v
Y
V
V
_ r - = -
.
1
(1-D) 2R
1-D ( I - D ) ~ R + R p + D(1-D) (Rc
W L
T
ideal
correction factor
dc gain
/b)
(19)
T h i s shows t h a t t h e i d e a l d c v o l t a g e g a i n i s 1 / D '
when a l l p a r a s i t i c s a r e z e r o (R = 0 , Rc = 0) and
t h a t i n t h e i r p r e s e n c e i t i s s l f g h t l y reduced by
a c o r r e c t i o n f a c t o r l e s s t h a n 1. Also we o b s e r v e
..L-..
------- ---
.I
.
-
a..
----
/u
d i s c o n t i n u i t y of o u t p u t v o l t a g e was n o t in?iu$.&$+L
i n [Z], b u t was c o r r e c t l y accounted f o r i n 111.
From t h e dynamic model (18) one can f i n d t h e
d u t y r a t i o t o o u t p u t and l i n e v o l t a g 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 , which a g r e e e x a c t l y w i t h t h o s e
o b t a i n e d i n [ l ] by f o l l o w i n g a d i f f e r e n t method of
averaged model d e r i v a t i o n based on t h e e q u i v a l e n c e
of c i r c u i t t o p o l o g i e s of two s w i t c h e d networks.
Note t h a t (15) i s t h e s p e c i a l c a s e of (1) i n
which bl = b2 = b = [ I / L b]T.
Using (16) and (5) i n t h e g e n e r a l r e s u l t
(13) and ( 1 4 ) , we o b t a i n t h e f o l l o w i n g f i n a l
s t a t e - s p a c e averaged model.
S t e a d y - s t a t e (dc) model:
The fundamental r e s u l t o f t h i s s e c t i o n i s t h e
development of t h e g e n e r a l s t a t e - s p a c e averaged
model r e p r e s e n t e d by (13) and ( 1 4 ) , which can be
e a s i l y used t o f i n d t h e s m a l l - s i g n a l low-frequency
model of any s w i t c h i n g dc-to-dc c o n v e r t e r . T h i s
was demonstrated f o r a b o o s t power s t a g e w i t h
p a r a s i t i c s r e s u l t i n g i n t h e averaged model (17)
and ( 1 8 ) . I t i s i m p o r t a n t t o emphasize t h a t ,
unlike the t r a n s f e r function description, the
s t a t e - s p a c e d e s c r i p t i o n (13) and a 4 ) g i v e s t h e
complete system behaviour. T h i s i s v e r y u s e f u l
i n implementing two-loop and multi-loop feedback
when two o r more s t a t e s a r e used-in a feedback
p a t h t o modulate t h e d u t y r a t i o d. For example.
both o u t p u t v o l t a g e and i n d u c t o r c u r r e n t may
be r e t u r n e d i n a feedback l o o p .
3.
HYBRID MODELLING
In t h i s s e c t i o n it w i l l be shown t h a t f o r any
s p e c i f i c converter a u s e f u l c i r c u i t r e a l i z a t i o n
o f t h e b a s i c averaged model given by ( 3 ) can
always be found. Then, i n t h e following s e c t i o n ,
t h e perturbation and l i n e a r i z a t i o n s t e p s w i l l be
carried out on t h e c i r c u i t model f i n a l l y t o
a r r i v e a t t h e c i r c u i t model equivalent o f ( 1 3 ) and
(14)
+
Fig. 5.
The c i r c u i t r e a l i z a t i o n w i l l be demonstrated
f o r t h e same boost power stage example,for which
t h e b a s i c state-space averaged model ( 3 ) becomes:
Circuit realization o f the basic statespace averaged model ( 2 0 ) through hybrid
modelling.
d': I
Fig. 6 .
In order t o "connect" t h e c i r c u i t , we
express t h e capacitor v o l t a g e v i n terms o f t h e
desired output q u a n t i t y y a s :
R+Rc
v =
y
R
-
Basic c i r c u i t averaged model f o r t h e
boost c i r c u i t example i n F i g . 3 . Both dcto-dc conversion and l i n e v a r i a t i o n are
modelled when d(t)-D.
As b e f o r e , we f i n d t h a t t h e c i r c u i t model i n
Fig. 6 reduces f o r d = 1 t o switched model i n F i g .
4 a , and f o r d = 0 t o switched model i n F i g . 4b.
In both cases t h e a d d i t i o n a l r e s i s t a n c e R =
1
dd' ( R R) disappears, as it should.
II
(1-d)Rci
I f t h e duty r a t i o i s constant so d = D , t h e
dc regime can be found e a s i l y by considering
inductance L t o be short and capacitance C t o be
open f o r d c , and t h e transformer t o have a D ' : l
r a t i o . Hence t h e dc v o l t a g e gain ( 1 9 ) can be
d i r e c t l y seen from F i g . 6 . S i m i l a r l y , a l l l i n e
t r a n s f e r f u n c t i o n s corresponding t o ( 1 0 ) can be
e a s i l y found from Fig. 6 .
o r , i n m a t r i x form
From ( 2 2 ) one can e a s i l y reconstruct t h e c i r c u i t
r e p r e s e n t a t i o n shown i n F i g . 5 .
I t i s i n t e r e s t i n g now t o compare t h i s i d e a l
d ' : l transformer w i t h the usual ac transformer.
While i n t h e l a t t e r t h e t u r n s r a t i o i s f i x e d , t h e
one employed i n our model has a dynamic t u r n s r a t i o
d ' : l which changes when t h e d u t y r a t i o i s a funcI t i s through t h i s i d e a l
tion o f time, d ( t )
transformer t h a t t h e actual c o n t r o l l i n g f u n c t i o n i s
achieved when t h e feedback loop i s c l o s e d . I n
a d d i t i o n t h e i d e a l transformer has a dc t r a n s formation r a t i o d ' : l , while a r e a l transformer
works f o r ac s i g n a l s o n l y . N e v e r t h e l e s s , t h e
concept o f t h e i d e a l transformer i n Fig. 6 w i t h
such properties i s a very u s e f u l o n e , s i n c e a f t e r
a l l t h e switching converter has t h e o v e r a l l
property o f a dc-to-dc transformer whose t u r n s
r a t i o can be dynamically adjusted b y d u t y r a t i o
modulation t o achieve t h e c o n t r o l l i n g f u n c t i o n .
We w i l l , however, see i n t h e n e x t s e c t i o n how
t h i s can be more e x p l i c i t l y modelled i n terms o f
d u t y - r a t i o dependent generators o n l y .
The b a s i c model ( 2 2 ) i s v a l i d f o r t h e dc
regime, and t h e two dependent generators can be
modeled as an i d e a l d ' : l transformer whose range
extends down t o d c , as shown i n F i g . 6 .
Following t h e procedure o u t l i n e d i n t h i s
s e c t i o n one can e a s i l y o b t a i n t h e b a s i c averaged
c i r c u i t models o f t h r e e common converter power
stages,as shown i n t h e summary o f Fig. 7 .
S u b s t i t u t i o n o f ( 2 1 ) i n t o ( 2 0 ) gives
.
ideal
(a)
buck
4.1
Steqe :
power
Perturbation
--
-
I f t h e averaged model i n F i g . 7b i s perturbed
v +G iA-I+$, d m-a,
according t o v g
d' ,v
~
4
, the~
nonlinear
~ model
i n F i g . 8 results.
D'-a,
(b)
boost
Power
staar :
(c)
buck
boost
power
buck
boost
s
F i g . 8.
R
4.2
Fig. 7 .
Summary o f b a s i c c i r c u i t averaged models
f o r t h r e e common power stages: buck,
b o o s t , and buck-boost.
The two switched c i r c u i t state-space models
f o r t h e power stages i n Fig. 7 are such t h a t t h e
general equations ( 1 ) reduce t o t h e s p e c i a l cases
A1
A2
A, bl # b2 = 0 ( z e r o v e c t o r ) f o r t h e
b2 = b f o r t h e
buck parer s t a g e , and A1 # A2, b
boost power s t a g e , whereas f o r t h e buck-boost
power stage A
A2 and bl # b2 = 0 so t h a t t h e
general case 3s retained.
- -
+
4.
H'Y+~
-
CIRCUIT AVERAGING
Perturbation o f t h e b a s i c averaged c i r c u i t
model i n F i g . 6 includes t h e duty r a t i o
modulation e f f e c t 2, but r e s u l t s i n t h i s
nonlinear c i r c u i t model.
Linearization
Under t h e small-signal approximation ( 1 2 ) ,
t h e following l i n e a r approximations are obtained:
and t h e f i n a l averaged c i r c u i t model o f F i g . 9
results.
In t h i s c i r c u i t model we have f i n a l l y
obtained t h e c o n t r o l l i n g f u n c t i o n separated i n
terms o f duty r a t i o
dependent generators e,
and j , , while t h e transformer t u r n s r a t i o i s
dependent on t h e dc duty r a t i o D o n l y . The
c i r c u i t model obtained i n F i g . 9 i s equivalent t o
the state-space d e s c r i p t i o n given by ( 1 7 ) and ( 1 8 ) .
a
As indicated i n t h e Introduction,in t h i s
section t h e a l t e r n a t i v e path b i n t h e Flowchart
o f F i g . 1 w i l l be followed, and equivalence
w i t h t h e previously developed path a f i r m l y
established. The f i n a l c i r c u i t averaged model
f o r t h e same example o f t h e boost power stage
w i l l be arrived a t , which i s equivalent t o i t s
corresponding state-space deacription given by
(17) and (18)
.
The averaged c i r c u i t models shown i n ' F i g .
7 could have been obtained as i n [ 2 ] by d i r e c t l y
averaging t h e corresponding components o f t h e two
switched models. However, even f o r some simple
cases such as t h e buck-boost or tapped inductor
boost [ l ]t h i s presents some d i f f i c u l t y owing t o
t h e requirement o f having two switched c i r c u i t
models t o p o l o g i c a l l y e q u i v a l e n t , while t h e r e i s
no such requirement i n t h e outlined procedure.
In t h i s s e c t i o n we proceed with t h e perturbat i o n and l i n e a r i z a t i o n steps applied t o t h e c i r c u i t model, continuing w i t h t h e boost power stage
as an example i n order t o include e x p l i c i t l y t h e
duty r a t i o modulation e f f e c t .
Fig. 9.
Under
model
final
stage
5.
small-signal assumption (121, the
i n F i g . 8 i s l i n e a r i z e d and t h i s
averaged c i r c u i t model o f t h e boost
i n F i g . 3 i s obtained.
THE CANONICAL CIRCUIT MODEL
Even though t h e general f i n a l state-space
averaged model i n ( 1 3 ) and ( 1 4 ) gives t h e complete
d e s c r i p t i o n o f t h e system behaviour, one might s t i l l
wish t o derive a c i r c u i t model describing i t s
input-output and c o n t r o l properties as i l l u s t r a t e d
i n F i g . 10.
(a)
state
- space
i n p u t output
bas is
and a l s o a "modulation" resistance t h a t a r i s e s
from a modulation o f t h e svitching t r a n s i s t o r
storage time 111.
'
0
'5.1
-7G
control
F i g . 10.
control
D e f i n i t i o n o f the modelling o b j e c t i v e :
c i r c u i t averaged model describing inputoutput and control properties.
Derivation o f t h e Canonical Model through
State-Space
From t h e general state-space averaged model (13)
and ( 1 4 ) , we obtain d i r e c t l y using the Laplace
transform:
In going from t h e model o f F i g . 10a t o t h a t o f
F i g . l o b some information about the internal
behaviour o f some o f the s t a t e s w i l l c e r t a i n l y be
l o s b b u t , on t h e other hand, important advantages
w i l l be gained as were b r i e f l y outlined i n the
Introduction, and as t h i s section w i l l i l l u s t r a t e .
We propose t h e following fixed topology
c i r c u i t model, shown i n F i g . 11, as a r e a l i z a t i o n
control f u ~ t i o n basic dc-to-dc
transforrrration
via d
effective low-pass
f I lter network
Now, from t h e complete s e t o f t r a n s f e r functions
we single out those which describe t h e converter
input-output properties, namely
,.
A
i n which t h e G's are known e x p l i c i t l y i n terms o f
t h e matrix and vector elements i n ( 2 3 ) .
F i g . 11.
Canonical c i r c u i t model r e a l i z a t i o n o f t h e
"black box" i n F i g . l o b , modelling t h e
three e s s e n t i a l functions o f any dc-to-dc
converter: control, basic dc conversion,
and l w - p a s s f i l t e r i n g .
o f t h e "black box" i n F i g . l o b . We c a l l t h i s model
the canonical c i r c u i t model, because any switching
converter input-output model, regardless o f i t s
detailed configuration, could be represented i n
t h i s form. D i f f e r e n t converters are represented
simply by an appropriate s e t o f formulas for t h e
four elements e ( s ) , j ( s ) , p . He(s) i n t h e general
equivalent c i r c u i t . The polarity o f the i d e a l
p:1 transformer i s determined by.whether or not
t h e power stage i s polarity inverting. I t s t u r n s
r a t i o p i s dependent on t h e dc duty r a t i o D , and
since f o r modelling purposes the transformer i s
assumed t o operate down t o dc, it provides t h e
basic dc-to-dc l e v e l conversion. The single-sect i o n low-pass LeC f i l t e r i s shown i n F i g . 11
only f o r i l l u s t r a t i o n purposes, because t h e actual
number and configuration o f t h e L's and C's i n t h e
e f f e c t i v e f i l t e r t r a n s f e r function r e a l i z a t i o n
depends on t h e number o f storage elements i n t h e
original converter.
The resistance Re i s included i n the model
o f F i g . 11 t o represent t h e damping properties
o f t h e e f f e c t i v e low-pass f i l t e r . I t i s an
" e f f e c t i v e " resistance t h a t accounts for various
s e r i e s ohmic resistances i n t h e actual c i r c u i t
(such a s RE i n t h e boost c i r c u i t example), t h e
additional "switching" resistances due t o d i s continuity o f t h e output voltage (such as
Rl = DD'(R
R) i n t h e boost c i r c u i t example),
11
Equations (24) are analogous t o the two-port
network representation o f the terminal properties
o f the n:twork (output voltage y ( s ) and input
current i ( s ) ) . The subscripts designate t h e
corresponding t r a n s f e r funstions. For example ,
Gv i s t h e source voltage v t o output voltage y
t r % n s f e r funct;on, Gid i s tfFe duty r a t i o 8 t o
input current i ( s ) t r a n s f e r f u n c t i o n , and so on.
For the proposed canonical c i r c u i t model i n
F i g . 11, we d i r e c t l y g e t :
o r , a f t e r rearrangement i n t o t h e form o f (24) :
- .
Direct cornoarison o f (.2 4-) and (26)
~ r o v i d e sthe
solutions for H e ( s ) , e ( s ) , and j ( s ) i n terms o f
the known t r a n s f e r functions G v g , G v d , Gig and
Gid as:
Note t h a t i n ( 2 7 ) t h e parameter 1 / p represents
t h e i d e a l dc voltage gain when a l l t h e p a r a s i t i c s
are zero. For t h e previous boost power stage
example, from ( 1 9 ) we get u
1-D and t h e correct i o n factor i n ( 1 9 ) i s then associated w i t h t h e
e f f e c t i v e f i l t e r network H e ( s ) . However, l~
could be found from
-
?v
= -cT~-lb
8
-u
1
X (correction
factor)
,-.
i n which the output voltage y coinc!des w i t h the
state-variable capacitance voltage v .
.
From ( 2 8 ) and ( 2 9 ) one obtains p = D ' / D
W i t h use o f ( 2 9 ) t o derive t r a n s f e r f u n c t i o n s , and
upon s u b s t i t u t i o n i n t o ( 2 7 ) . there r e s u l t s
(28)
by s e t t i n g a l l p a r a s i t i c s t o zero and reducing
t h e correction f a c t o r t o 1.
The physical s i g n i f i c a n c e o f t h e i d e a l dc
gain p i s t h a t it a r i s e s as a consequence o f t h e
switching a c t i o n , so i t cannot be associated with
t h e e f f e c t i v e f i l t e r network which a t dc has a
gain ( a c t u a l l y attenuation) equal t o t h e correction factor.
The procedure f o r finding t h e four elements
i n the canonical model o f F i g . 11 i s now b r i e f l y
reviewed. F i r s t , from ( 2 8 ) t h e b a s i c dc-to-dc
conversion f a c t o r p i s found as a f u n c t i o n o f dc
duty r a t i o D . Next, from t h e s e t o f a l l t r a n s f e r
functions (23) only those defined by ( 2 4 ) are
actually calculated. Then, by use o f t h e s e
four t r a n s f e r functions G
vd' ' v " d' ' i f
(27) the frequency dependent geherators e s
and j ( s ) as, w e l l as t h e low-pass f i l t e r t r a n s f e r
function H ( s ) are obtained.
The two generators could be f u r t h e r put
i n t o t h e form
i n which V i s t h e dc output voltage.
The e f f e c t i v e f i l t e r t r a n s f e r f u n c t i o n i s
e a s i l y seen as a low-pass LC f i l t e r w i t h Le =
and w i t h load R . The two generators i n t h e
canonical model o f F i g . 11 are i d e n t i f i e d by
LID'^
We now derive the same model b u t t h i s time
using t h e equivalent c i r c u i t transformations and
path b i n t h e Flowchart o f F i g . 1 .
A f t e r perturbation and l i n e a r i z a t i o n o f the
c i r c u i t averaged model i n F i g . 7c ( w i t h
Re-0) the s e r i e s o f equivalent c i r c u i t s
12
i s obtained.
%-;ip.
-
where f l ( 0 )
f 2 ( 0 ) = 1 , such t h a t t h e parameters
E and J could be i d e n t i f i e d as dc gains o f t h e
frequency dependent functions e ( s ) and j( 6 ) .
F i n a l l y , a general s y n t h e s i s procedure 1101
f o r r e a l i z a t i o n o f L,C t r a n s f e r f u n c t i o n s
terminated i n a s i n g l e load R could be used t o
obtain a low-pass ladder-network c i r c u i t
r e a l i z a t i o n o f t h e e f f e c t i v e low-pass network
He(s). Though f o r t h e second-order example o f
He(s) t h i s step i s t r i v i a l and could be done by
i n s p e c t i o n , f o r higher-order t r a n s f e r functions
the orderly procedure o f t h e s y n t h e s i s [ l o ] i s
almost mandatory.
5.2
Example:
Ideal Buck-boost Power Stage
For t h e buck-boost c i r c u i t shorn i n Fin. 7c
p i g . 12.
Equivalent c i r c u i t transformations of t h e
f i n a l c i r c u i t averaged model ( a ) , leading
t o i t s canonical c i r c u i t r e a l i z a t i o n (c)
demonstrated on t h e buck-boost example o f
Pig. 7c ( w i t h Rf-0 , Rc-0 ).
The o b j e c t i v e of t h e t r a n s f ormations i s t o
reduce t h e o r i g i n a l f o u r d u t y - r a t i o dependent gene r a t o r s i n F i g . 1 2 a t o j u s t two g e n e r a t o r s ( v o l t age and c u r r e n t ) i n Fig. 1 2 which
~
a r e a t the inAs t h e s e c i r c u i t t r a n s put p o r t o f t h e model.
f o r m a t i o n s u n f o l d , one s e e s .how t h e frequency dependence i n t h e g e n e r a t o r s a r i e e s n a t u r a l l y , a s
i n ~ i g .12b. Also, by t r a n s f e r of t h e two gene r a t o r s i n F i g . 32b from t h e secondary t o t h e
primary of t h e l : D t r a n s f o r m e r , and t h e i n d u c t a n c e
L t o t h e secondary of t h e D ' : l transformer, the
cascade o f two i d e a l t r a n s f o r m e r s i s reduced t o
t h e s i n g l e transformer with equivalent turns
A t t h e same time t h e e f f e c t i v e f i l t e r
r a t i o D':D.
network Le, C, R i s g e n e r a t e d .
Expreseione f o r t h e elements i n t h e c a n o n i c a l
e q u i v a l e n t c i r c u i t can b e found i n a s i m i l a r way
f o r any c o n v e r t e r c o n f i g u r a t i o n . R e s u l t s f o r t h e
t h r e e f a m i l i a r c o n v e r t e r s , t h e buck, b o o s t , and
buck-boost power s t a g e s a r e summarized i n Table I.
Table
I
Definition of t h e elements i n t h e
c a n o n i c a l c i r c u i t model of Fig. 11
f o r t h e t h r e e common power s t a g e s
of F i g . 7.
It may be n o t e d i n T a b l e I t h a t , f o r t h e buckb o o s t power s t a g e , p a r a m e t e r s E and J have n e g a t i v e
s i g n s , namely E = - v / D ~ and J = - v / ( D ' ~ R ) .
However, as s e e n from t h e p o l a r i t y of t h e i d e a l
D':D t r a n s f o r m e r In F i g . 1 2 c t h i s s t a g e i s a n
i n v e r t i n g one. Hence, f o r p o s i t i v e i n p u t dc
v o l t a g e V , t h e o u t p u t d c v o l t a g e V is n e g a t i v e
(V < 0 ) s f n c e V/Vg
-D/D8. T h e r e f o r e E > 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.
-
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
=
.
- .-
--.
- - .
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.
--
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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