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A SIMPLIFIED PRESENTATION OF UNCUSTOMARY BEHAVIOR OF NONLINEAR CIRCUITS

Article · January 1992

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2 authors: Some of the authors of this publication are also working on these related projects: MOBILIS FP6 (2006-2009) View project Florin Constantinescu Polytechnic University of Bucharest

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Miruna Nitescu Polytechnic University of Bucharest

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iI SItrTIII-[FIEIi I}tIEST,]\I.TTI05 (}Ir 0F- \()It,t.\IiAIt I.i\(.t. STOlt.\ltY BIIIt.tI I 0I'Il {IIt(.t.II',S

\ I ' l ' I r S ( . i . *

O s c rir. clc lt' rtrtn re p rir-in tl co rn q ,o rt.,a a a

• bisnr r itii si neoltisir uitli a cir enitt:lr lr

t f i n a n t i c t . n c lillirii' c itrclttz.itrtl trp a rilia strb a rrrror r icclor - si conr por tar ea lr acticr 'r sur iL i l t r s t r a t e p r i r t sirrtrrla rc p e ca lcu l.ltrlr. Sotrtc ihetlrcl.tts o11 cttstonrart' :ttrrl rrncustorrrrtrv behar.'iou5of nonlinear- < l t 'i l a t t r i c c i r crtits ittclu tijrrg tlrr a p ;.it' a ra n ce

• f s ulthanr r onics itr r r i tir e dhaotic l:r ir a-

v i u u r a r c i l l t rsir itte rl J-l1 ' ln e zrn s

• f cririrp u Ie r sirrrulatior r .
• l. Introtluction

An Iiie't:tricai trjt;girrcerilg liundiirntrrtals co[rr,st' gir-cn jn tire .qc(j,Jii(l all(l 111iI'tl setticstet's t() litt' stu(i('n1.s

• f lilectrorritrs

a]ld 'l'clecot]]nrurirrations Flitrultr- ltlrs it-i ll(,.ut's tietlic'iitt'ti 1g r.it'c.it;i theOi,l'; 27 ir'rit.s:ti,e (le\-Oteri fo t11'rt_zrrriit'cit'c'uits alr(l llte'irist tr-,o t-rf lltertr <leal t-i11r tlrl pres(rnt?rtiotr 9t tltl?ilitati\'€ pLol-)('rties. ,lrt tltt,st'tu'o t.gttr'se li131ll'S 1t'e l)r,eSrrltett theor.erls op cristen{'{, an(1 rtr}i(lucn(,ss r;f soliltioiis. iilr(l customar.l- nrrri noncustomar.) cir"ctrit}-rehrrvjtiui'. ,\il tlrcs('rlir-'ot,t,nr:r ilr,a'given rvithoirt llroof but iil,e accolri- panied b1- exanlplcs. ftr ot'tler to ilhistrlte sttl.rltirnnonic responses atld chaotic behaliour colnptlt('r'sirlltllatiott iriis bet'tr use(I.

• tl1'this sa1- the studr.nts can luiorv

the 'utrcustomart- l:t'jravioru' of uorrliltear circuits \\,ithout, using higir ler el math.enratical tools ,rs Poincare nrilpping or L)'apuno\- function t1].

• 2. Theorv

'Iht' c'nstomal')-

• l' utlclltiturri.irrv

liehaliorit of a cilcuit is riefineri in the following \\-a]': a cittrtiii hus ir t'ristoruar'1'beirirvioul jf it has ther pr.oper.- titrs : f ', - ttrtitlue :L - periotlic I'esllolrscr to a T - per.ioilie ercitatiorr Pz - spectnrnt cons(.r\'atic:i .Ps __ e\pon(-)rltial trirrr-rient tit,i.r,r]-. Othelri'ist', the cirt.uit ]rii.q :1ir uncustoniat.)- jrt,haliour..

* t.<lrrg.

• dr. ing: Conf.

BucharesL elr. ing., Elecl.rical Engineering Dept., poll-technic Inslitute of

l B r t

• frl. (,o n st.a tttitre sctr. ]lir utta

Nitcsctt

In rlrtler 1lt shou. Ihi' t.orrrlitions fclr lt t'it'crtit to ltavt lirt' 1)l'(tl)el'- lics P. -- It" solll{' ttcx' tit'rittitiotts irl'r.' liecessar'}'. Tire -stroirgi.r- klcal lrir.sile cirtrracter of a (n --{- 1)-leltrrirral ('n-1lott) r-esist{)r u'iili tlic t'orrslitutivt' i't,liltiori 17 : l1(p\h: Iin-- Jl" ltt'ittg :'t ctrttti- liotls funt:tiorr) is rir.fiirt'rl lrv tirc e-ristet:r:e

• f tlte const.ili)ts

:)' ^,' so flrttt fol attv e', :i'" tlit' t't'lrttion : ^ i l . r '

• - . 1 " / ' < ( ; i ' - ; t " ' ' , 1 '

U r ( r ' ) - l t ( , r ' " ) ] < i ; l l { 1 , '

• - i " i l l

is yalitl, q'hr,r'e li 'il is t|<. t'1t.1ir1carr rrt;r'ni. It lrillrri's tl)ttt lt,t'lt si,lrtttglr'

fi;:t Jl;:';,: 'itilt;';X-l;t,, ;,,T,..-u/il ';1,, ,,,;;.r.ii,i. is trt,{ii*,ri iir :, ,si.r;1,ir

pliii1'l(.i' lr)' c,6rrsirlcr,ilg t[at j/ -. /r(,r') is tlr. : (rr(rr) (i" -= ,'r1.1,r1)

• t' q. ==

: 4"(ri1 (11..

• - i?c,(r1r))

u'ht're stantlartl riotatioti i+l ltits J)i't-'tt ttsetl. A cirt,irit has att uttitlttt. ?'- irelittriic i'cspollsc'e(t) =, \qcft),, ()r.(1)) ttr tl ?-iteriodic t5tcit:rtion if rt'spotrst's c'(/) nttil :"(.t) (ctilrcsllttrrtlirrg,t.o tht irri'Lial cortrtitiotrs (q'c(g), q\'L(O)) lrrtl 1q""{{)\, (r"r(rp; satisfr- tlrt' t't'latiott:

l:t:r 11 3'(l)

• 2"(t)11

:0

:iri(f , oirviriusl,v, (qc,(l), <Dr(1)) alt' T - peliotlir'. 'l'lle sour,(,(] sltectrtrnr cotnllinatiott "t" asor'iittt't1 u itiL tl.c. siltll'{rei: i11tri rr..(,. sr)tu'1(,s

• 1t1't'('rlrtettcics

c,;x is flt6: s('t of ft'ctlttenci('s o,: F tt*.t[r, *.itt,t.C ir,- e }-. I f i l r t c i r c L r i t h a s : i l ) t . l ' i o r i i r ' l ( , s i ) o t ] - { t ' \ \ ' r ' { ' i I 1 ' } A s s o ( ' i i t t t ' t i ' i i l t i { t , i l t ' i ' t s - ponst| spcttlurtr r.orrrltirrzit ioii .\r. ,\ t'it't'uit irits l irc spet'ttrtlit t'rtttset'r.:ti itttt pr'opelt)' if Nr - N.r.

i f l i l ' ( s i s l t i i . : i l r t l l t ' > o t i t ' < . r ' \ ( , ( , l r r l 'I'ltr',t)'t.tn '! .: I

• l l i t ' t ' e i s t : o 1 , , , , 1 ,

, , r o l l a g c ( { . 1 1 t , r , ( , i l l , - r r r . . t . i l ' t . t t i l i t : l - l r ' : . , : , l r l l ( , l l p : i ( . i l r i . , r ,

• l i l l t ' c s i s t o r , : .

t l t i , r t l t c s 0 u l ( , ( , \ ( , ( . l r ) , l ll'hrrt,.r ttr )l .: ] , : O l l t ' t ' l i t , s i r , 1 r , , 1 i l , l 1 I ' t ' t t , i s t ^ o 1 , 1 1 . ; , , l : r r r ' 1 , , r , , ' i ' { ' n i ) . i i r u , . i r i l t , : r 1 ; i i r . l 1 { 1 : . ; , , , , : i t j i t ' t ' s i s l o t ' . i i t ( . . ; , , t l r t ' l i t t y ' \ l r l ' . \ ; l , l j l i i , ) ( i i t . a r r t l i i . ( / , i : j i l r : 7 ' i s : l t , t , l i l i , l , r r . l l t t ' t , t n : r i i 1 1 ; 1 1 , , . r , l

. :

[ : { ) l ; ' : .

I 1 i s r r o l ; l i l ! i r . r i i : I i i ; r ' 1 ) t . i s s i l r i l i t f - r n l r : r , , , I r r i l r u o t i i t . s o l i i l i i ; r r . I . ' , , , r i L l r i ' i t ( . l i ' t ' i s l i r . i l r 1 ' : . I

( L

'\ circrtit is saitl 1o ltitvc the if an,v pair of soltt.iititrs :'(l) lirrtl .3"(l) (q'c(O) {D' r.(g) arrtt ( Q" r'',0). (r"r(O)

t'rpont'ttt ial t llilts!t'ttt titc:tr' ltroltel'l1' (t:ol'r't'slltltl(liltg io 1 ltt' irritial t'outlil itltts satisties t6e t.cltliorr :

fun,i,, l] z'(A'l

• 3"(O),1

,'-

tr-1tc;'t' lr,ri': i,,r,r.: /,',u;,r. /,'n,*-* il;'t'lttlsitit't'('tlltstlttrtS.

• [.'l,olr!

1it+-'1rr.t.r'icr:s lessot:s it is ol.lr.ir>r:s thal ?. litli'ltt'll.r'ttililt!t'ctit't'tiit i',r1s iil,,.' i,'toilt'l'tit:r It, --'1'r. :ig1i1 r.i2;:r:-r{'s gi tir;iiLit}cat' t,it't'uii,r I iilt t'lrsir,iltlir'1' ll,.'l'ii uilrtll' ('lil} irc rir- i'irrt.<i. f 'll<' l'<;llgri'irrg' thtt>r't'uts gilt :-,t,if it'it'ltt t'i;tttiiiirltrs (iili\-. '!'irer;ie*i I [2 i). -\ r,it't.riit rvit j: l!rrelr' il<ltlt'1t;t s ttllll t'ii1';li'!tc::r iii16 the pl'opcr'tits lt, - /',, it : llrel'c j.r r:r; ir,t;ll (r'11-sct) 1oi'rilerl

• ttl;,'11.v

t'trlracitlli's {irttll:'.'tt)i'.') i{il(i,ol' vcltagi' (titi'i'.';-it) si;i;t'r'l'rl * 111 r,ill::,ir:itoi.s lrrrtl in<juttcx's ai't' litt,-riif . l)it.rsrjl-e ititti f i'6'i1;l'iltai ai1 calrac,itors a,ntl irttluctoi's are liilcili't passiYe il.li(l i'e(1iix'tlt'tll I t i . 1 , 1 1 ' 1 q 1 , t h a t t l r l l r , , i . r

. . . : i t , . , t . i t . , t , i l t . r l l i l o l i l ( . : l ; l r l i i ( i i r : . i . : r l r

r r { i i " . l i , r i l ) , { . ( f

t , ; 1 , 1 i i . . , l t . {

• i - _ , - 1 l i . i r ,

. r , !

i l ; l r l i t ' i l t l i I , r , < . t , ; i . i l i : i ( ) i i s r , i t ' < , t i i t t , : i i t . 1 i , . . , i i . { , l x l t p u l c r , 5 i r r r r i l l . L ' l r e r r i r r : r r i I i r r t i ! . _ l , i i ' t t r i t l , l ' t i t t , r - i t i . i : t . i r , , , r i li1)r.lrr' (.:*lja{.itoi, i\.tlii ir (

. \ S i r n p l i f i t d I ' r t s e r t t a t i o t r

• f i' ttcttslo trta l' \' IJtltaviottl' tif Noltlitleal

( 'ir cttit't

1 8 1

rr11 l'('sisloi's :11'(' st,t'tttt1l1r' ltlt'lt1 1)ii;{sivc' tht, sour(,r. \'(.ctor rr,,(1) is {'1 itiiil f

• -"1:eritltlit'.

'f 1111t.(:lp 2 [2l. A It('r,irt,uil ha.s the Lllol)cl'1i('s l,tt.. .,P, if : tlrt't,e'is r:e 1t,ir1l (r.trt-st,t) folurt'(i otrh- J,i-t':rp:rt'i1t;t's (irrihrt'1ut's) atttt,'oi' r.oltagc (t'rtri'c'ilt) sotrt't'ts ciicuit iuls tto itt<lttt'tot's (callilt'itols) 111 r,21:r-tt'i1ot's (ilrtltrclols) ?).1'(' stt'otrgl.r' Lrtrltl ltiissiVtr all r'csistrtt's itt't' ltttt'rlt'. ltassiVc tl1l(l t't"'i1lt'tlt'a1 tirc sour('c \.e('t{)i'ir"(l) is (-'r itiitl ?' - li('lilrtiir'. '-l'ltc1;'r,ttt J [2l. ,\ ]i i'{' cit'crrit \\.itiI "stnA1l-sigttill" il,irnl s hits ttritl l);'r)ll('r'1ics l" ir tttl l" is ; tlrt.rr.is 16 1r6ir it.rit-s1t) tr;r.rrrt.ti llr-r.alr:rr.ittils (iiirhictrlt's) ailtljot'r01* tir ge (cut'i't'ttt ) sttttt'tt's iril r,ir1;ar.it3r,s :rrirl intltrt.l or s ir r t' r'i'ci1lr'oca1 it I irl st I tlrrg^11' l0t':r1 Fassi r-r.r zrll t'csislol's iIl'(' sttrlrlglr' lrlt'ill ltas-iivt' 15i' titttg r':tt'r'itrg {'t}l}Ip{)ii('1lt tlt 'it'*(ir'"(1) : tr''(l) -- ti"'') is ('(x11illlt(}1is' q:r.i,i,ltiit.2prl li i,.r(i)lll.-D f6r,ii11 t> al \\,il(.t'r' ..i is u stttf ic'ierrtll'snre11 tltttttllt't iiil{i 7' is :l t'cit1 iititttllet'. tht t.lrrrsiitrrl iyl r'r,littir;rr r;i' tiit' ('it jlli('i1{)t's ittltl itrtlttt't{]r's itl'(' ('n f tttic- t i o r l s . It is rr6t rliftitrrrlt tri illusti,atc lri'titp1lls ol'ir.tr tr.itirl1-tii:irl t'ontprltillit.ttt iitc pgs:.ilrili1]- io hitvt,nli,11i1llc stc:trir'-str1c 1tt'r'ir,riic solutitltts ct'ott sttll-

• lriilrrl6riir. sillrrl
• igir. Iigl, (...iill'rU)lc

litc t'ilttrit itt f

• ig. 1.;r. hlis tlttt i'csisttli'

chat'ut'tet'istic itt fig'. 1.1r.

n r l l

L

\

l t L

*

• )

It i-q. il r it

I1 i" r,i,,:ai.th:rt titc r,t,sisioi'is irot stt'otiglr.-1rlr:itl 1;lissivt'. Tiit'sltttt't.tlrittlitltis . . - ; , . ' ; 1 . , i L

• 1

; 1 i , . " - i r . , , r t ; s i i l . T l i e i ' o j l o u ' i i r g s l r ' i l t l r - - s i : t t l " s t t J r - I s r i l r r i i r ; i r s { , : l n l l t ' t . l i s i i . r ' r ' < ' r ' i i ! t ' ' i l : t r " ( 1 ;

• . . s ! t i ( 1
• 2 l ; r , i } i , ' i r . l . l ) : i : '

t ' i t ' < ' t - i i t t ' i i i t t t l ) i ' t ' : ' i r i i t i t i l t l r i i i l r : l i i i t i l ' t j t l i i t l l i ' l : i i , ( i : r

l i . { , s * l l ; t t t l t ' l ' S i r r r t l l l r i i l i ; t

'lllre t:ir,r:llit irr

• tig. l.a. iii I is siittuiirtr,:rl

rr'jtll i.'Si'I('li. 'i'l:i't''1t,ii'aierit r.ir"r uit r;f tlrt' I'ai.acirirj rlirir-ir.' (ifg. 2.i;). lirrl, lt rtorili!l('?Ll' t't'sistttt' ili.(i a i1()li- litii,ar, call:rt.itci' l'r1]i it (l' ('i)rii{iit rrtir',i' Lt'littioil iitr(i it iiortliilt'ii}' (ri'},];il(litttt'

L , I l 1 C : 1 A l t l

I+.trr-l l---r

c >"il

I_-*_-_-T

. r i

• Iil. (,o n sta lltirresctl' Jiir r tilil \itcscil

! i r r r p l i { i e r l

I U')

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d

t / ,

/:'

i : r .

a.,\..),:

Pi,r."bo.F"qtl: J **rr.>*..# \[

?{ t'}itY

t :

.._::.. ... . . . :

ril

{ , 1 i i ' \ . ( ' . 1 ( i ( i ) , r . ( / ) ) , i1 i)i)('ii I'2Il lCi' 01' stt I rir

• ,.','\'('{i

(fiU. ;ti). ? l i c c . o r i < l i l i r , r ( ' 1 , l I \ t i t i l t i r - r ' t ' ( ' l i i 1 i . Thc ilt'ltiivi(;u {':r I I ('()lisi<lt't' :i r r:lnt r ' i r t i i r g l x ) i l i t t l l o \ . r ' > 'l'itc sirrrrtiitlir : , t r c l - i t i g t h e J ' S I '

• \tr erluililri';r

r { | )il1's{\ ri0t sn't ttll0,

, ) l ( i \ l l : l t t t i t . r , i t ' ( , u i l r i ) o s ( , l t 1 l ) ( ' \ \ ' r I ) - o 1 ' i r l t , s . \ \ - t , t h i r r l i 1 l r r r i r l l r ( ' l t r i l l l l l ( , 1 ' 1 o i l i l

. . - . + ';:cl':

• 4'

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• ' i 1 . 1 1 i r )

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• 1lt'r itr dic r i'sponse oi {r er Jttcncr I
• ptr jotlic

r csllollst

• ! l r ctilttttc)

l 2 .- cltaotic beir al'ior tr I' r ' o r ' c c tl i r r .q s l I..l .- 1 ... r I . ( t . ( . l t t t t i . I t .

• \ ' .

1 i r , s l c i t t l r

• s l

l ( i . \ s - 2 i i I . 1 . - \ : : o r i : . l l . I ) t t l t r . i t s i r t tr s o i r l i I ) e c . 1 1 r E : j . 1 . . ( ) . t l l t t t u . O . - \ . L

'\ s i r r r l r l i l i c(l .l )t' cst' llti.ltio ll

• i l' n tirslo r:rlr1 1 ' Ilchar .ior r r of \or r lir r c:- r r .

lir ctr it.s

1 {i:i

\\ iilr tor' 1

t .

I r { t l ( ,

• fig. ;1.

(llcri- It the

,4ti,.,1.iA,'ti

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/r. (.orrelusiorrs

; r .

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( F : i . { . * . .

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a)' ;i;t*.r,.- ,.... ... .:.'!i.i.i:.. . .

il.;:

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tris 2 t r l r r V c . l ( i ( i ) , ( ' ( / ) )

• - 0 i s ( i ] ' ? i \ 1 ' t t .

l l t c 1 ; c ' r ' i o r l i c t . l r a ; ' l i t . t t ' i . c l 1 h c r . { , s J X ) . s e , l i r t illii)('lit'?ll:ci'ol stilrititttIi{)l}it'S:rrxl tlit't,li;vrtir. lx'irlrvi,,"i, ,,:iii fn, "n*ilrl ,ril_ si'i'\'e{i (fis'. jl). Tlte r'otitiitioti:r tif iitt'0tt'irr ,'i al{' rr01 frrltiii,:rl (a cal)rrtitol has ,a Co {'(rtis1 itul i vi' 1,(,1ilt i(}ll ). Tht' ilt'ltat'itiltt' itt 1]rt' ('lls(' ir is r'ustorliilr'\' lrecaust' 1,r;r. sf iiil e(l) \r.o {'illI colisi<lt't'tt ,,sttt:t1i-sigrialt' t:cit-iilt iir' (._(lrriralc,t r,ircuii-;^(ttr,, iii,rCli',rt;;l l':rtillg lroiril lll{)\'('s in a" srnall r:ciglrltrnrr,lir;ort tLt ilx,^,lrlioii,;.

'lirt' sirttultrlit,ji jl- ltei'foilre(i i, :i, api,lic,til,;,' ir,;,;;: *,ii5 strirle*ts

i,trr;l-jirg 1lit. lrSI'f ('li t.r.r ilolntctit.

• \tr etluilibr':t1't'tl

sti'itt:titt'c rtf urr lilct.tlirlrl iirrgirrcering l.rrirria*rcrrtals {'rt111's1' tltlt'sn't ail0rv l}l(}t'e than tso r'<lri'st'houls 1or t1'rrlitittir.c lii.operties r.rf tlr-tl:illli('t'i|r'trits. I)tre to ihe srnall aru,ii,r.,1

• f ar.iiilaSlir

ti'rt, rvei havc t'ltosen tilt' tt':tt' rlf illustt'at,irrg tluaiitalit',. lx.rtlt'r,t!es lrr- llerrns ol. exam- l l l t ' s ' \ \ ' e t h i l r l i l h a t i r i i r r l i i i t r r t ' n ' o i ' l i i i r c c o n u l i l c r . s i r r - r u l a t i . * i s t h e . r r i - tlllc l}l2lItttt't'1o iliuslr':rtiVt'1lte e.lt:rortic llcira.,'ior

• f an clcct'iciil crir.cuit.

I , t E t r E t i i r N C i i s l' l)roct'cdirtgs 1I',I',I'-, r-tl . 75,'o. 8 (arrg-. 1glJ7). Spccial issrre

• n cliaotic

sr.stc.rs. :1. L.. ().

• Lhuq. '|).

l-. (irt,t,rr,.\ rlrralitaiir.el analviis ,,t tt," ltt,lrar iol of clr.nanric nonlinear slt':irlr'-slatc silltrlitttl Of ttottatrt0rr()unlous rrct$.oi'lis, Ii.,Ilij.l.rarrs. t;rr ( . . \ S _ 2 3 p p . J i ) 0 _ t i 0 ( s e p t . 1 g 7 t j ) . l'' -l' 'l::rrrr:' I?. I)urtr, rI. IItr.rrcr, frarisition to eira.s irr a sirriyrlc '''lirrcar circuit a s i t t r r s o i t l a l Y o l t a g t ' r i o t l r c c , I l i l i I l 1 ' r a n s . u r r ( , i l ' c u i t s u r r r l i i r s t c r r r s . . r o l . l ) c c . 1 f ) f i 3 , n o . . 1 2 . p 1 , . . t ) 1 3 _ 9 1 : ) . l l" ()' (lhtr.! (,'' 'l ' 1)estttr. rr' ^s.

• trrrlr. Linear anrr lrorrlirrrur

t.irciril-s. ]lc-(irau,-l

n ctrl orlis, ( l- \ S vr ii. tir ivcl Jr r ' CA S

• :i0,
• Iiil. I {',!7.

V i e w p u b l i c a t i o n s t a t s V i e w p u b l i c a t i o n s t a t s