A discrete-time equivalent system approach to the periodic response of nonlinear autonomous circuits

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