See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/280155434 A SIMPLIFIED PRESENTATION OF UNCUSTOMARY BEHAVIOR OF NONLINEAR CIRCUITS Article · January 1992 CITATIONS READS 0 18 2 authors: Florin Constantinescu Miruna Nitescu Polytechnic University of Bucharest Polytechnic University of Bucharest 158 PUBLICATIONS 341 CITATIONS 122 PUBLICATIONS 298 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: MOBILIS FP6 (2006-2009) View project All content following this page was uploaded by Florin Constantinescu on 23 July 2015. The user has requested enhancement of the downloaded file.
-+ iI SItrTIII-[FIEIi I}tIEST,]\I.TTI05 (}Ir I.i\(.t. STOlt.\ltY BIIIt.tI I 0I'Il 0F- \()It,t.\IiAIt {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 obisnr 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 o 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 o f cririrp u Ie r sirrrulatior r . l. Introtluction An Iiie't:tricai trjt;girrcerilg gir-cn jn tire .qc(j,Jii(l liundiirntrrtals co[rr,st' a]ld 'l'clecot]]nrurirrations setticstet's all(l 111iI'tl t() litt' stu(i('n1.s of lilectrorritrs Flitrultr- ltlrs it-i ll(,.ut's tietlic'iitt'ti 1g r.it'c.it;i (le\-Oteri theOi,l'; 27 ir'rit.s:ti,e fo t11'rt_zrrriit'cit'c'uits alr(l llte'irist tr-,o t-rf lltertr <leal tlrl pres(rnt?rtiotr t-i11r 9t tltl?ilitati\'€ pLol-)('rties. ,lrt tltt,st'tu'o t.gttr'se li131ll'S 1t'e theor.erls op l)r,eSrrltett 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')- ol' 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]-. :1ir uncustoniat.)- jrt,haliour.. Othelri'ist', the cirt.uit ]rii.q * t.<lrrg. dr. ing: Conf. ing., Elecl.rical Engineering Dept., poll-technic Inslitute of elr. BucharesL
frl. (,o n st.a tttitre sctr. ]lir utta Nitcsctt l B r t fclr lt t'it'crtit to ltavt lirt' 1)l'(tl)el'- In rlrtler 1lt shou. Ihi' t.orrrlitions i f l i l ' ( s i s l t i i . : i l r lics P. -- It" solll{' ttcx' tit'rittitiotts irl'r.' liecessar'}'. l l t ' > o t i t ' < . r ' t \ ( , ( , l r r l klcal lrir.sile cirtrracter of a (n --{- 'I'ltr',t)'t.tn '! 1)-leltrrirral ('n-1lott) Tire -stroirgi.r- .: I : Iin-- Jl" ltt'ittg :'t ctrttti- i't,liltiori 17 l1(p\h: - r-esist{)r u'iili tlic t'orrslitutivt' l l i t ' t ' e i s t : o 1 , , , , 1 , , , :)' ^,' lrv tirc e-ristet:r:e so flrttt liotls funt:tiorr) is rir.fiirt'rl of tlte const.ili)ts r o l l a g c ( { . 1 1 t , r , ( , i l l , - r r r . . fol attv e', :i'" tlit' t't'lrttion : t . i l ' t . t t i l i t : l - l r ' : . , : , l l ( , l l l r p : i ( . i l r i . , r , U r ( r ' ) - l t ( , r ' " ) ] < - - i " i l l ^ i - - . 1 " / ' < ( ; i ' - ; t " ' ' , 1 ' - i ; l l { 1 , ' l i l l t ' c s i s t o r , : . l . r ' t l t i , r t l t c s 0 u l ( , ( , \ ( , ( . l r ) , l is yalitl, q'hr,r'e li 'il is t|<. t'1t.1ir1carr tl)ttt lt,t'lt si,lrtttglr' .: rrt;r'ni. It lrillrri's ll'hrrt,.r ttr )l ';1,, fi;:t Jl;:';,: 'itilt;';X-l;t,, O l l t ' t ' l i t , s , , 1 i l , i r , 1 r ] , : l is trt,{ii*,ri iir :, ,si.r;1,ir ;,,T,..-u/il ,,,;;.r.ii,i. 1 I ' t ' t t , i s t ^ o 1 , 1 1 . ; , , (rr(rr) (i" -= ,'r1.1,r1) l : r r r ' 1 , , r , , ' i ' { ' n i ot' q. == ) . i i r u , . is tlr. : pliii1'l(.i' lr)' c,6rrsirlcr,ilg j/ -. /r(,r') t[at -- i?c,(r1r)) ; , , , , : ttsetl. i r i l t , : r 1 ; i i r . l 1 { 1 : . : u'ht're stantlartl riotatioti i+l ltits J)i't-'tt 4"(ri1 (11.. ()r.(1)) =, \qcft),, ttr i t j i t ' t ' s i s l o t ' . ?'- i'cspollsc'e(t) i i t ( . . ; , , A cirt,irit has att uttitlttt. irelittriic c'(/) nttil :"(.t) (ctilrcsllttrrtlirrg,t.o tht ; l , l tl ?-iteriodic t5tcit:rtion if rt'spotrst's t l r t ' l i t t y ' \ l r l ' . \ (q'c(g), q\'L(O)) lrrtl 1q""{{)\, (r"r(rp; satisfr- tlrt' t't'latiott: j l i i , ) ( i i t . irri'Lial cortrtitiotrs 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 . - 2"(t)11 :0 3'(l) l l t t ' t , t n : r i 11 i 1 1 ; 1 , . r 1 , , l . : l:t:r [ : { ) l ; ' : . I 1 i s r r o l ; l i l ! i r . r i i : I alt' T - <Dr(1)) peliotlir'. , oirviriusl,v, (qc,(l), :iri(f i i ; r ' 1 ) t . i s s i l r i l i t f - r n l r : r , , , 'l'lle sour,(,(] I r r i l r u o t i i t . "t" asor'iittt't1 u itiL tl.c. siltll'{rei: s o l i i l sltectrtrnr i i ; r r . I . ' , , , cotnllinatiott tt*.t[r, r i L l r i ' i t ( . l ' : . I is flt6: s('t of ft'ctlttenci('s o,: i ' t ' i s l sr)tu'1(,s o1t1't'('rlrtettcics c,;x i r . i l r 1 i11tri rr..(,. F *.itt,t.C ir,- e }-. ( L \ \ ' r ' { ' i I 1 ' } i { t , i l t ' i ' t s - h a s : i l ) t . l ' i o r i i r ' l ( , s i ) o t ] - { t ' A s s o ( ' i i t t t ' t i ' i i l t I f i l r t c i r c L r i t itttt ioii .\r. ,\ t'it't'uit irits l irc spet'ttrtlit t'rtttset'r.:ti ponst| spcttlurtr r.orrrltirrzit pr'opelt)' if Nr - N.r. llilts!t'ttt titc:tr' ltroltel'l1' '\ circrtit is saitl 1o ltitvc the t'rpont'ttt ial t ltt' irritial t'outlil itltts pair of soltt.iititrs :'(l) lirrtl .3"(l) (t:ol'r't'slltltl(liltg io 1 if an,v (r"r(O) {D' r.(g) arrtt ( satisties t6e t.cltliorr : (q'c(O) r'',0). Q" - 3"(O),1 fun,i,, z'(A'l ,'- l] 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 /,',u;,r. /,'n,*-* il;'t'lttlsitit't'('tlltstlttrtS. lr,ri': i,,r,r.: tr-1tc;'t' l t . r l l i l o l i l ( . : l ; l r l i i ( i i r : . i . : r -[.'l,olr! l r thal ?. litli'ltt'll.r'ttililt!t'ctit't'tiit lessot:s it is ol.lr.ir>r:s " . l i , r i l ) , { . 1it+-'1rr.t.r'icr:s r r { i i ( f iil,,.' i,'toilt'l'tit:r It, --'1'r. - i - _ , - 1 l i . i r , i',r1s t , ; 1 , 1 i i . . , l t . { . r , ! uilrtll' ('lil} gi tir;iiLit}cat' t,it't'uii,r I iilt t'lrsir,iltlir'1' ll,.'l'ii :ig1i1 r.i2;:r:-r{'s i l ; l r l i t ' ; i l t l i I , r , < . t , f 'll<' (iili\-. gilt :-,t,if it'it'ltt t'i;tttiiiirltrs irc rir- i'irrt.<i. l'<;llgri'irrg' thtt>r't'uts i . i l i : i ( ) i i s r , i t ' < , t i i t t , : i i t . 1 i , . . , '!'irer;ie*i I [2 i). -\ r,it't.riit rvit j: l!rrelr' il<ltlt'1t;t iii16 s ttllll t'ii1';li'!tc::r lt, - /',, it : the pl'opcr'tits i i . { , l x l t p u l c r , 5 i r r r r i l llrel'c j.r r:r; ir,t;ll (r'11-sct) 1oi'rilerl i{il(i,ol' ottl;,'11.v t'trlracitlli's {irttll:'.'tt)i'.') * (titi'i'.';-it) si;i;t'r'l'rl vcltagi' l . L ' l r e r r i r r : r r i I i r r t i ! . _ lrrrtl in<juttcx's ai't' litt,-riif ititti f i'6'i1;l'iltai . l)it.rsrjl-e 111 r,ill::,ir:itoi.s l , i i ' t t r i t l , l ' t i t t , r - i t i . i : t . i r , , , r i ai1 calrac,itors a,ntl irttluctoi's are liilcili't passiYe il.li(l i'e(1iix'tlt'tll li1)r.lrr' (.:*lja{.itoi, i\.tlii ir (
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