IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL.
CAS-32,
NO. 10, OCTOBER
1985 1083
whe e
P=blockdiagonal(Z,,Z,;~~,~~,,Pi,Zi+,,~~~,ZP)
(1.4
whe e Zk is k
X
k iden i y ma ix, and k is he dimension o
componen 2,.
Using he componen connec ion app oach, a model o he
in e connec ed sys em Z is desc ibed wi h
i=(F+GLH)z+GMu Wa)
y=Hz W)
whe e
F=blockdiagonal(A,,A,;..,A,_,,~,A,+,,...,A,)
(I3c)
G=blockdiagonal(B,,B,,...,Bi_,,Gi,Bi+,,...,B,)
(134
H=blockdiagonal(C,,C,;..,C,-,,H,,C,+,,...,C,).
Equa ions (3c), (lOc), (12), and (13~) imply
FP= PA.
Equa ions (3d), (3e), (lOd), (lOe), (12), and (13d) imply
( GLH) P =
PBLC.
The e o e, (14a) and (14b) imply
(F+GLH)P= P(A+BLC).
Simila ly
G=PB
HP=C.
(13e)
(144
WI
(154
(15b)
(154
b) Necessi y: The condi ion s a ed in he heo em esul s
om he de ini ion o he componen connec ion model. In his
amewo k he s a e ec o o he composi e sys em is ob ained
by s acking he s a e ec o s o he indi idual componen s.
Rema k: I (lb) is modi ied o
y, = C,x, + D, ;
hen, (8b) needs o be modi ied o
y = Hz + Ju
and condi ions (9) will include
J= D. (94
Consequen ly, he heo em s ill holds and he composi e sys em
model is desc ibed by [6]
i=(A+BQLC)x+BQMu
y=(Z+DQL)Cx+DQMu
whe e
Q=(Z-LD)-‘.
The agg ega e model o he composi e sys em is desc ibed by
i=(F+GQLH)z+GQMu
y=(Z+DQL)Hz+DQMu
whe e he agg ega e model s a e a iables z a e as de ined abo e.
V.
CONCLUSION
Whene e dealing wi h in e connec ed la ge-scale sys ems he
size o he sys em becomes an issue. Model educ ion echniques
a e used o alle ia e hese di icul ies. Mos o he echniques a e
applied o he linea ized model o he composi e sys em model. ’
In his pape , i is shown ha in he componen connec ion
model amewo k he composi e sys em is agg egable i and only
i a leas one o i s componen s is agg egable. Because o he
ela i ely small size o he indi idual componen s, i is easie o
agg ega e he componen model a he han he composi e sys em
model. The impo an consequence o his esul is ha when
dealing wi h in e connec ed la ge scale sys ems in he componen
connec ion model amewo k one does no need o agg ega e he
la ge-scale composi e sys em bu a he can conside he agg ega-
ion o each, ela i ely smalle scale componen , independen ly.
111
I21
[31
[41
PI
I61
PI
REFERENCES
H. H. Happ,
Diakop ics
und
Ne wo ks.
New Yo k: Academic, 1971.
E. J. Da ison, “Decen alized s abiliza ion and egula ion in la ge mul i-
a iable sys ems,” in
Di ec ions in La ge SC& Sys ems,
H. and Mi e ,
Eds., pp. 303-323, 1975.
A. Feliachi and A. P. Meliopoulos, “Modeling and Sim&a ion o La ge
Scale In e connec ed Sys ems,”
Ad ances in Modeling and Simulu ion,
AMSE P ess, F ance, ol. 1, no. 1, pp. 33-49, 1984.
N. R. Sandell, P. Va aiya, M. A hens, and M. G. Sa ono , “Su ey o
decen alized con ol me hods o
la ge scale
sys ems,”
IEEE T ans.
Au oma . Con .,
ol. AC-23, pp. 108-125, Ap . 1978.
0. Wasynczuk and R. A. DeCa lo, “The componen connec ion model
and s uc u e p ese ing model o de educ ion,”
Au omu ica, ol.
17, no.
4, pp. 619-626,198l.
R. A. DeCa lo and R. Saeks,
In e connec ed Dynomicc l Sys em. New
Yo k: Ma cel Dekke , 1981.
M. A&i, “Con ol o la ge scale dynamic sys ems by agg ega ion,”
IEEE
T uns. Au oma . Con ., ol.
AC-13, pp. 246-253, 1968.
Chaos in a Swi ched-Capaci o Ci cui
ANGEL B. RODRIGUEZ-VAZQUEZ, JOSE L. HUERTAS,
AND
LEON 0. CHUA
Abs ac
-We epo chao ic phenomena obse ed om a simple nonlin-
ea swi ched-capaci o ci cui . The expe imen ally measu ed
bi u ca ion
ee
diag am e eals a
pe iod-doubling ou e
o chaos. This ci cui is
desc ibed by a i s -o de disc e e equa ion which can be ans o med in o
he
logis ic
map whose chao ic dynamics is well known.
Se e al nonlinea ci cui s which exhibi a ious ypes o chao ic
phenomena ha e been epo ed ecen ly [l]--[5]. Ou objec i e in
his le e is o epo an expe imen al esul showing he
ubiqui ous chao ic phenomena can also occu in a swi ched-
capaci o ci cui . Since swi ched-capaci o ci cui s a e impo an
in VLSI echnology, any po en ial anomaly o ailu e mechanisms
Manusc ip ecei ed May 28,1985. This pape was suppo ed in pa by he
O ice o Na al Resea ch unde Con ac NOOO14-76-C-0572, by he Na ional
Science Founda ion unde G an ECS-8542885, and by he Spain CAICYT
unde Con ac 0245/81.
A B. Rod igua-Vazquez and J. J. Hue as a e wi h he Depa men o
Elec ical and Elec onic Enginee ing, Uni e si y o Secille, Se ille, Spain.
L. 0. Chua is wi h he Depa men o Elec ical Enginee ing and Compu e
Sciences, Uni e si y o Cali o nia, Be keley, CA 94720.
0098-4094/85/1000-1083$01.00 01985 IEEE
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1084
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL.
CAS-32,
NO. 10, OCTOBER
1985
+T,Y
(b)
Fig. 1. A nonlinea swi ched-capaci o ci cui and i s associa ed iming di-
ag am.
due o he onse o chaos should be ully analyzed. This chao ic
ci cui is also o ci cui - heo e ic in e es because i s dynamic
equa ion is equi alen o he well-known logis ic map [6] whose
chao ic dynamics ha e been ex ensi ely s udied and is now well
unde s ood. Since he logis ic map is he simples chao ic poly-
nomial disc e e map, he chao ic ci cui o be desc ibed below is
he simples chao ic ci cui desc ibed by a i s -o de disc e e
map.
Conside he swi ched-capaci o ci cui in Fig. l(a): i is made
o a ba e y I’,, a linea capaci o C,, a nonlinea swi ched-capa-
ci o componen [7], [8], and h ee analog swi ches. The s a e (on
o o ) o he swi ches is con olled by a s anda d wo-phase
clock de ined by he iming diag am shown in Fig. l(b). The
swi ches labeled S’ ( esp., So) u n on in synch oniza ion wi h
he ising edge o he clock signal @’ ( esp., +‘).
The nonlinea swi ched-capaci o componen -hence o h
called an FESC ( o wa d. Eule swi ched capaci o ) esis o -is
de ined by
Q, -Q,-, = kV,2_, A AQ, (1)
whe e AQ, is he ne cha ge lowing in o he FESC esis o
du ing he n h clock pe iod, V,-, is he ol age sampled ac oss
he FESC esis o du ing he (n - 1) h pe iod, and
K is an
a bi a y posi i e cons an .
We ha e buil he ci cui in Fig. l(a) wi h C, =l nF and
k = 0.5 nF/V using o - he-shel componen s and obse ed he
s eady-s a e ol age wa e o m samples V o di e en alues o
he ba e y ol age ,. Con a y o ou in ui i e expec a ion o a
single- alued ela ionship be ween V and V,, we ound he ela-
ionship o be mul iple- alued o e some anges o he “pa ame-
e ” y , and unde ined i.e., chao ic, o o he anges. This
obse a ion is summa ized by he bi u ca ion ee measu ed
expe imen ally om his ci cui . The amilia cascades p eceding
he chao ic egion implies a pe iod-doubling ou e o chaos [6].
To de i e a ecu si e ela ionship o V,, we no e ha he ne
cha ge AQ, + i lowing in o he FESC esis o du ing he (n + 1) h
clock pe iod mus be equal o he ne cha ge lowing ou o he
linea capaci o C, (cha ge conse a ion p inciple), and hence:
V
n+l = , - g ,‘.
(4
s
We can ans o m (2) in o se e al mo e amilia equi alen o ms
by de ining
X
n+l
= uV,+~ + b.
I we choose a = l/y and b = 0, we would ob ain
(3)
X n+I =l- xx,
Fig. 2. Bi u ca ion ee.
+
V
4
Fig. 3. O - he-shel ealiza ion o he FESC esis o in Fig. l(a).
whe e
X=k /;/C,.
I we choose
and b = i, we would ob ain he well-known logis ic map
X n+1=4Gl(1- 41)
whe e
(5)
k
XP- 4aC, .
Bo h equa ions, (3) and (5) ha e been in ensi ely s udied [6]
and hei global quali a i e beha io s a e now well classi ied and
unde s ood. Consequen ly, Fig. l(a) ep esen s he i s eal
physical ci cui whose chao ic dynamics can be compIe ely
analyzed.
Fo eade s in e es ed in epea ing ou expe imen s, he FESC
esis o in Fig. l(a) can be ealized by he ci cui shown in Fig. 3.
ACKNOWLEDGMENT
The au ho s would like o hank G eg Be ns ein o s imula ing
discussions which led o his pape .
REFERENCES
[l] Y. S. Tang, A. I. Mees, and L. 0. Chua, “Synch oniza ion and chaos,”
IEEE T ans. Ci cui s Sys .,
ol. CAS-30, pp. 620-626, Sep . 1983.
[2]
A. Azzouz, R. Duh , and M. Hasle , “T ansi ion o chaos in a simple
nonlinea ci cui d i en by a sinusoidal ol age sou ce,”
IEEE T ans.
Ci cui s Sys .,
ol. CAS-30, pp. 913-914, Dec. 1983.
[3] T. Ma sumo o, L. 0. Chua, and S. Tanaka, “Simples chao ic nonau ono-
mous ci cui ,”
Phys. Reu.
A, ol. 30, pp. 1155-1157, Aug. 1984.
[4] T. Ma sumo o, “A chao ic a ac o om Chua’s ci cui ,”
IEEE T uns.
Ci cui s Sys .,
ol. CAS-31, pp. 1055-1058, Dec. 1984.
[5] G-Q. Zhong and F. Ay om, “Expe imen al con i ma ion o chaos om
Chua’s ci cui ,”
In . J. Ci cui Theo y App ., ol.
13, pp. 93-98, 1985.
[6] P. C i ano ich
(Ed.), Uni e si y in Chaos: A Repo Selec ion.
B is ol,
England: Adam Hilge , 1984.
[7] J. L. Hue as, L. 0. Chua, A. B. Rod iguez-Vazquez, and A. Rueda,
“Nonlinea swi ched-capaci o ne wo ks: Basic p inciples and piecewise-
linea design,”
IEEE T ans. Ci cui s Sys .,
ol. CAS-32, pp. 305-319, Ap .
(4)
1985
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[8] A. Rod iguez-Vazquez, J. L. Hue as and L. 0. Chua, “On a class o SC
esis o s and i s applica ion o he syn hesis o nonlinea d i ing-poin
and ans e -cha ac e is ic plo s,”
In . J. Ci cui Theo y Appl.,
o be
published.
[9] A. Rod iguez-Vazquez, J. L. Hue as, and L. 0. Chua: “Ra ional SC
ne wo ks,” in p epa a ion.
Nodal Vol age Simula ion o Ac i e RC Ne wo ks
P. V. ANANDA MOHAN, V. RAMACHANDRAN,
AND
M. N. S. SWAMY
Abs ac --I is shown ha he loa ing nodes p esen in ac i e
RC
ne wo ks can he elimina ed by nodal ol age simula ion, leading o new
ac i e
RC
ne wo ks. The new ac i e
RC
ne wo ks hus gene a ed can be
easily used o ealize s ay-insensi i e SC ne wo ks. Fqi alence o ce ain
ac i e
RC
opologies, esul ing om applica ion o his me hod, is also
demons a ed.
I.
INTRODUCTION
The heo y and design echniques o ac i e RC il e s a e
conside ably ma u e a p esen . In eg a ed ci cui implemen a ion
necessi a ed ci cui s using g ounded capaci o s, o elimina e he
e ec o bo om-pla e pa asi ic capaci ances [l]. Wi h he ad en
o swi ched-capaci o echnique, i has become necessa y o
elimina e he e ec o pa asi ic capaci ances al oge he , since
p edis o ion as well as imming a e una ac i e [2]. I is na u-
al, he e o e, o a emp o de i e s ay-insensi i e SC opologies
om ac i e RC il e s using g ounded capaci o s. Th ee ech-
niques iz., pa asi ic compensa ion [3], [4], nodal ol age simula-
ion [5], [6], and s ay-capaci ance elimina ing ans o ma ions [7]
ha e been epo ed in he li e a u e. In his le e , we examine he
second
app oach in de ail. I may be no ed ha his app oach has
been used in de i ing bilinea SC ladde il e s om doubly
e mina ed LC il e s [8]. O he a ia ions such as nodal ans e
unc ion simula ion [9], [lo] a e also a ailable in he li e a u e.
II. NODALVOLTAGESIMULATION
METHOD
We s udy he nodal ol age simula ion echnique wi h e e -
ence o speci ic ac i e RC ne wo ks, in wha ollows, in o de o
demons a e i s u ili y.
A. Applica ion o Mul iple Feedback Type Single -Ampli ie Low -
Pass Fil e s
Conside he mul iple- eedback ac i e RC low-pass il e o
Fig. l(a). The in e nal node x in his ne wo k is desc ibed by
KCL as
Y , [
1 1 1
F+F= ” x+R+R+SC,. I (1)
1 3 1 2 3
I is easy o ealize V, using OA’s wi h g ounded nonin e ing
inpu as shown in Fig. l(b), by he ci cui wi hin do ed lines.
Augmen ing his ne wo k by he emaining ci cui consis ing o
Manusc ip ecei ed Ap il 10, 1984.
P. V. Ananda Mohan was wi h he Depa men o Elec ical Enginee ing,
Conco dia Uni e si y, Mon eal, Canada. He is now wi h T ansmission R&D,
Indian Telephone Indus ies, Bangalo e, India.
V. Ramachand an and M. N. S. Swamy a e wi h he Depa men o Elec i-
cal Enginee ing, Conco dia Uni e si y, Mon eal, Canada.
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL.
CAS-32,
NO. 10, OCTOBER
1985
1085
L-------------l (b)
Fig. 1. (a) Mul iple eedback low-pass il e . (b) Ac i e
RC
ne wo k equi -
alen o Fig. l(a).
R,, C, and OA A, o ealize V, om V,, we ob ain he ne wo k
o Fig. l(b), which in e es ingly, is he well-known Tow-Thomas
biquad [ll]. The a ailabili y o band-pass ou pu a node x and
low-pass ou pu a V0 in he ci cui o Fig. l(a) is ans e ed o
he ne wo k in Fig. l(b). No e ha , bo h hese ci cui s a e based
on induc ance simula ion [12], [13]. Fu he , we obse e ha he
ci cui o Fig. l(a) is ha ing damping due o esis o s R,, R,,
and R,. This damping can be educed by mul iplying he e ec
o R, and R, h ough a nega i e esis ance o g ound a node x
[14]. The esul ing ac i e RC il e is shown in Fig. 2(a). P oceed-
ing in he same manne as in he case o Fig. l(a), we ob ain he
equi alen RC ac i e il e o Fig. 2(b). The ans e unc ion
ealized is gi en by
Kl - l/R,
-=
K
(4
s2ClC2 R, - SC, R,
Thus bo h low-pass and band-pass ans e unc ions a e s ill
a ailable, while la ge Q’s can be ob ained h ough he use o
nea ly equal R, and R, alues. The Q-sensi i i ies a e hus
la ge:
and $--=1-F.
5
No e ha Fische and Moschy z [15] used such a modi ica ion o
Fleische -Lake ’s biquad [16] o educing he capaci ance sp ead.
In applica ions whe e equal capaci o alues a e p e e ed in
he ci cui o Fig. l(a), posi i e eedback can be applied [17]. This
leads o he modi ica ion o he Tow-Thomas biquad, as shown
in Fig. 3(a). The ans e unc ion ealized is
, l/R,
-=
Y
s2ClC2 R, s s!p$)+(&AL)
(3)
No e, howe e , ha he band-pass ans e unc ion is no ealiz-
able a he ou pu s o he OA’s A, and A,. The design equa ions
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