A
Disc e e-Time Equi alen Sys em App oach o
he Pe iodic Response
o
Nonlinea
Au onomous Ci cui s
P.
Pala
Schondde and
J.
M. Mi o Sans
=-Depa men o Signal Theo y and Communica ions
PO
Box 30.002
08080
Ba celona
-
Spain
(+34)
3
401 70 72
[email p o ec ed]
ABSTRACT
The p oblem o compu ing he s eady s a e esponse o
nonlinea au onomous ci cui s is sol ed making use o a
disc e e- ime equi alen sys em app oach. Wi h he
applica ion o an s-plane o z-plane mapping, he ci cui
equa ions a e disc e ized and w i en in ec o o m. Using
hus
echnique, i is no necessa y o epea edly compu e
ans o ms be ween he ime and he equency domain.
An
e icien scheme o build he jacobian ma ix wi h exac
pa ial de i a i es wi h espec o he oscilla ion pe iod and
wi h espec o he samples o he unknown a iables
is
desc ibed. Applica ion examples on
m
widely s udied
ci cui s a e p o ided o alida e he p oposed echnique.
INTRODUCTION
Se e al echniques ha e been de eloped o sol e he
p oblem o de e mining he s eady-s a e esponse o
nonlinea au onomous ci cui s.
An
impo an class
o
ime-
domain me hods o he s eady-s a e analysis o o ced
nonlinea ci cui s a e he so-called shoo ing me hods which
y o ind he ini ial condi ions ha make ze o he ansien
esponse. Al hough hls echmque has been op imized
successi ely, i s applica ion is expensi e in e ms o
compu ing ime. Addi ional d icul ies a ise in he case o
au onomous ci cui s whe e he e is no a-p io i knowledge o
he oscilla ion pe iod.
F equency-domain me hods di ide he ci cui in o a linea
and a nonlinea pa , aking ad an age o he
high
e iciency han may be achie ed sol ing he linea
pa .
Howe e ,
as
he nonlinea i ies a e bes e alua ed in he ime
domain, i is necessa y o ake successi e ans o ma ions
be ween bo h domains, consuming a subs an ial amoun
o
ime while limi ing accu acy. Ex ension o au onomous
ci cui s equi es he compu a ion o sensi i i ies wi h
espec o he oscilla ion pe iod in o de o be able o imbed
his added unknown in o an i e a i e sol ing scheme. I
should
be
poin ed ou ha when he ci cui a iables ha e a
b oad spec um -i.e., elaxa ion oscilla o s- he numbe
o ha monics equi ed o sa is ac o ily app oxima e he
wa e o ms may
be
e y high, equi ing excessi e compu e
ime.
In his pape
we
desc ibe an e icien echnique o sol ing
he s eady-s a e esponse o nonlinea au onomous ci cui s
based on he o mula ion o he ci cui equa ions in he ime
domain, making i unnecessa y o ake any u he
ans o ma ion. A e dlsc e izing he equa ions wi h he use
o an s-plane o z-plane mapping and he applica ion o he
in e se-z ans o m, he p oblem is w i en in ec o o m,
making i possible o ake ad an age o pa allel p ocessing
echniques.
An
ex emely e icien me hod o he
compu a ion o he jacobian ma ix, wi h exac pa ial
de i a i es wi h espec o he samples o he ci cui
a iables and he oscilla ion pe iod, has been implemen ed.
EQUATION FORMULATION
Conside
a
nonlinea au onomous ci cui whe e all bias
sou ces and all nonlinea elemen s ha e been ex ac ed.
E e y nonlinea elemen may be modeled
by
he pa allel
combina ion o
a
linea elemen and
a
nonlinea con olled
sou ce [l]. Figu e
1
depic s an idealized example whe e
only one bias sou ce
Ub
and one nonlinea elemen
desc ibed
by
he nonlinea unc ion
i=F(x)
a e included o
he sake o simplici y. The ex ension o an a bi a y numbe
o bias sou ces and nonlinea elemen s is s aigh o wa d.
By applica ion o supe posi ion, he con ol a iable o he
nonlinea i y
x
and he desi ed ou pu a iable
y
may be
exp essed
as
U
Figu e
1
whe e
A,
B, C
and
D
ep esen linea ope a o s. In he case
o dynamic ci cui s, he e ms
A
{
ub},
B{F(x)}
co espond o
con olu ions in he ime domain. The e alua ion o
=A{ub}
is simple, since i is a linea p oblem. The
esul ing p oblem is hen o sol e
x= +B{F(x)}
o he
a iable
x,
assumed o be pe iodic wi h
unknown
pe iod
T.
This equa ion may be exp essed in he ime domain
as
whe e
h( )
is he impulse esponse
o
he linea ci cui
L
o
he exci a ion o he nonlinea sou ce.
An
e icien way o sol e his in eg al equa ion consis s in
disc e izing he analog con olu ion.
To
ob ain he
equi alen disc e e in insic impulse esponse
b(n),
F ey and
No man [l] p opose o i s apply an s-plane o z-plane
mapping o he Laplace ans o m o
b(0.
The esul ing
unc ion o
z
is hen in e se-z ans o med o he sequence
domain. In his p ocess, i is impo an ha he numbe o
samples pe pe iod
N
sa ides he Nyquis c i e ion o
p e en aliasing. The applica ion o his s a egy allows o
exp ess
(1)
as
Since he a iables a e pe iodic wi h pe iod
T,
i is possible
o ew i e
(2)
in ec o o m
as
whe e x=[xl.
x2,
...
,
x$,
=[ l.
2.
...
,
N]‘, (x)=[ (xl),
(x2),
._.
,
(xN)l and
B
is
a ci culan ma ix ha ope a es on
a
ec o o samples
x
in he equi alen o m
as
he analog
ope a o
B(s)
ope a es on he signal
x(Q.
The ma ix
B
may
be
compu ed
as
ollows.
In gene al,
B(s)-ls)/d(s).
Then, a e applying he map
ST(+),
a
a ional ans e unc ion
B(z)=q(z’)+(i+)
is
ob ained. The e ec o
B(z)
on a ec o o inpu samples
may
be
desc ibed
by
he di e ence equa ion
which, in he case o pe idc sequences, may be exp essed
in ec o o m
poy
+
p1Sy+.
. .
+PAS ky
=
q,x
+
q,sx+...
+q,S’x
whe e
S
is he ma ix ci cula
shi
ope a o , S=ci c[O 1
0
...
01.
Finally,
WQx,
and he ou pu sequence may be
ob ained
as
y=Bx,
wi h
B=PIQ.
To ob ain he s eady-s a e esponse, i is necessa y o ind
he solu ion o he nonlinea se o equa ions
(3),
whe e he
unknowns
a e he
N
samples o he con ol a iable
x
and
he pe iod
o
oscilla ion
T.
The ac ha he oscilla ion
pe iod
T
is
an
unknown
is
an
added di icul y wi h espec
o he analysis o o ced ci cui s whe e, in gene al, he
pe iod
o
he a iables is ixed
by
he ex e nal signal
sou ces. Ha ing
N+I
unknowns
and only
N
es ic ions
indica es he p esence o an in ini e numbe o solu ions. In
pa icula , no e ha i a ec o
5
is
a
solu ion o
(3),
Sq
also a solu ion o any in ege
q.
The exis ence
o
1
is
a
con inuum o solu ions makes i possible o ix he alue o
one
o
he samples o
x
[2].
In he ollowing, and wi hou
loss o gene ali y, he alue o he i s sample
x,
will be
assumed ixed.
In he case o o ced ci cui s, i is possible o implemen
me hods o sol ing
(3)
based on elaxa ion s a eges,
aking ad an age
o
he esul ing pa allelism. Howe e , hs
app oach is no
so
appealing in he au onomous case
because
o
he di icul y
o
adequa ely embedding he pe iod
T
in he elaxa ion scheme. The e icien me hod o
compu ing he pa ial de i a i es wi h espec o he
samples and wi h espec o he pe iod o be desc ibed nex
makes i pa icula ly in e es ing o make use o lughe o de
solu ion me hods ha make use
o
he jacobian ma ix.
COMPUTATION
OF
PARTIAL DERIVATIVES
To simpll y he compu a ion o he pa ial de i a i es wi h
espec o he oscilla ion pe iod
T,
i is con enien o
e o mula e he ini ial equa ion
(3).
An
equi alen
o mula ion is
(4)
P(7‘)x-
P(T)
-
Q(7‘)
(x)
=
0
which may be exp essed
as
The dependence o
(4)
wi h he pe iod appea s implici ly
when he s-plane o z-plane mapping is pe o med. In e e y
mapping
s=To(z)/A[2],
he dependence wi h he oscilla ion
pe iod
T
appea s in he e m
A=T/N.
In ou
wo k,
he
second o de
Gea
map
s=(3-4z-'+ 2)/(2A)
has been used.
To compu e he pa ial de i a i e o he k- h equa ion wi h
espec o he pe iod, some p e ious s eps a e necessa y.
Le P,,(s) be a gene ic polynomial in he a iable
s
Pn(s)
=
pnsn
+
pn-ls"-l+.*'+pls+ Po
A e applyng he mapping
s=T,(z)lA,
he ollowing
exp ession is ob ained.
Pn
A"
A
=
i0
+ i,Z-l+...+ imZ-m
whe e, o he Gea -2 map,
m=2n.
The associa ed ma ix is
hen ob ained
as
Now,
=
$A
+jjiZ-l+...+$Z-m
and he associa ed ma ix may be compu ed
as
m
P
=
C&Sk
k=O
Using his no a ion, he pa ial de i a i e o
(5)
wi h espec
o he pe iod o oscilla ion is ob ained
as
k=l
k=l
k=l
Finally, he jacobian ma ix
J
is ob ained
as
Fo
implemen a ion pu poses, i is possible o sa e a
sign ican amoun o compu e memo y alung ad an age
o he
ac
ha he ci culan s P and
Q
only ha e
N
di e en
en ies. I should be
also
poin ed ou ha he special
s uc u e o
J
allows o e icien LU decomposi ion codmg.
Also,
since in he au onomous case he e is no ex e nal
signal inpu , he ec o
is cons an
as
i is he esponse
due o he bias sou ces. The esul o applying
T$(z-')
on a
ec o
o
cons an samples is
a
ze o ec o , excep o
k=O.
The inexpensi e a ailabili y o he exac jacobian ma ix
allows o implemen e icien algo i hms
o
sol ing he
nonlinea sys em o equa ions
(3).
I is wll-known ha
New on's me hod con e ges quad a ically when he s a ing
poin is su icien ly close o he solu ion bu may e en no
con e ge on ce ain si ua ions. To o e come his acul y,
globally con e gen modi ica ions
o
New on's me hod
[3]
ha e been implemen ed.
APPLICATION EXAMPLES
To alida e he desc ibed echnique, i s applica ion o wo
kinds o widely s uQed classic oscilla o ci cui s is
p esen ed.
1.
Van de
Pol
oscilla o
The an de Pol equa ion
x
+
p(x2
-
1)X
+
x
=
0
may be modeled
as
depic ed in igu e
2
++~~=-p(+)
Dependmg on he alue o
pa ame e
p,
he ci cui
beha io o ixed alues
o
L
and
C
(L=l,
C=l)
Fig.
2
anges om a nea ly
sinusoidal oscilla o o
small alues o
p
b4.1)
o
a
elaxa ion oscilla o
@-
4..10),
wi h an impo an numbe o signi ican ha monics.
Applying ha monic balance echmques equi es o conside
an impo an numbe o ha monics -wi h he
co esponding compu a ional cos - in o de o achie e
su icien accu acy.
"
(~
The wa e o ms esul ing o Me en alues o
p
along wi h
he co espondmg oscilla ion pe iod ha e been ob ained
wi h he desc ibed echnique. The esul ing wa e o ms and
pe iods ha e been compa ed wi h he esul s ob ained wi h
36
Veb,
Vcb
(V)
161
I
0
10
20
ime
(s)
Figu e 3
he use o ime-domain in eg a ion me hods. The ob ained
pe iod has also been compa ed wi h he alues gi en in
e e ence
[4].
Good
ag eemen has been obse ed in bo h
cases. The esul s ob ained in
[5]
using
a
ha monic balance
app oach show signi ican ly mo e dis o ion wi h
a
compa able compu a ional cos . Fo illus a ion pu poses,
he ob ained wa e o m o
p=5
is depic ed in igu e 3.
2.
Colpi s oscilla o
IlOV
The Colpi s oscilla o in
'
I
,
igu e
4
has been analyzed
/
using he desc ibed echnique,
~~j~
~-
/,>
wi h no malized elemen
2
,
alues Rc=lO, L=O.1,
C1=2,
C2=0.8
and Re=20. Fo he
bipola ansis o he Ebe s-
T
Moll model wi h pa ame e s
R~'.;
~2
a,=0.99,
aR=0.015
and
VT=0.025 has been used. This
LL
ci cui p esen s a ela i ely
high
Q
ac o . Fo his eason
Fig.
4
classical ime domain
in eg a ion me hods a e less
e icien . Howe e , he wa e o ms ob ained wi h hese
me hods ag ee wi h he esul s ob ained om he desc ibed
echnique. In his example
N=30
samples pe pe iod ha e
been used. The implemen ed solu ion algo i hm s a ed wi h
1
V
ampli ude sinusoidal wa e o ms wi h
T=ls
o bo h
unknowns V, and V, wi h
a
dc componen o 1OV o
V,
ob ained om ci cui inspec ion. The inal oscilla ion
pe iod
was
T=1.52
s
and he esul ing wa e o ms a e shown in
igu e
5.
~~~ ~~
--
A
~L,
~
lc~462E-
1
2
,
/es=7E-
12
,
~
-2OV
CONCLUSIONS
0
1
2
3
ime(s>
Figu e
4
me hod is based on he o mula ion and subsequen
disc e iza ion o he ci cui equa ions in he ime domain,
so
ha i is no necessa y o
pe o m
any addi ional ans o m
be ween he ime and equency domains. The Qsc e ized
equa ions a e hen w i en in ec o o m, allowing o ake
ad an age
o
pa allel p ocessing. In he esul ing
o mula ion, he unknowns o be sol ed o simul aneously
a e he samples o he con ol a iables o he nonlinea
elemen s and he oscilla ion
pe iod.
An
e icien me hod o
cons uc ing he jacobian ma ix wi h exac compu a ion o
he sensi i i ies wi h espec o he samples and he pe iod
is desc ibed.
REFERENCES
[l]
D.
F ey and
0.
No man,
"An
in eg al equa ion
app oach o he pe iodic s eady-s a e p oblem in
nonlinea ci cui s", IEEE
T ans. Ci cui s and Sys ems,
ol. 39,
pp.
744-755,
Sep embe 1992.
[2]
L.
Chua,
P.
Lin,
Compu e -Aided Analysis
o
Elec onic Ci cui s,
Engled Cli s, NJ: P en ice
Hall, 1975.
[3]
J.
E.
Dennis and
R.
B.
Schnabel,
Nume ical Me hods
o Uncons ained Op imiza ion and Nonlinea
Equa ions,
Engled Cli s, NJ, P en ice Hall, 1983.
[4]
D.
Hen e and
R. H.
Jansen, "F equency domain
con inua ion me hod o he analysis and s abili y
in es iga ion o nonlinea mic owa e ci cui s",
P oc.
IEE, ol. 133,
P .
H.
p~.
351-362, Oc obe
1986.
[5]
E. Ma in,
AnaIysis
o
Nonlinea Au onomous Ci cui s
wi h Ha monic Balance Techniques,
Ph.
D.
Thesis,
Mad id, Sep embe 1990.
In his pape ,
an
e icien echnique o he analysis o
nonlinea au onomous ci cui s
has
been desc ibed. The