Frequency scaling simulation of Chua s circuit by automatic determination and control of step-size

Transcription

1 Avlle onlne t Appled Mthemtcs nd Computton 94 (7) Frequency sclng smulton of Chu s crcut y utomtc determnton nd control of step-sze E. Tlelo-Cuutle *, J.M. Muñoz-Pcheco, J. Mrtínez-Crlldo INAOE Lus Enrque Erro No., Tonntzntl, Puel 784, Mexco Astrct A procedure s ntroduced to speed-up tme smulton of Chu s crcut y mens of determnng the egenvlues of ts stte mtrxes. A multstep method s ppled wth utomtc control of step-sze (h), where h s determned y the computton of the egenvlues of the stte mtrxes whch chnge ccordng to the slopes of Chu s dode. It s shown tht the proposed smulton procedure enhnces the estmton of the vlues of the crcut elements n Chu s crcut nd the explorton on ts frequency sclng ehvor. Besdes, t s not necessry to gve n ntl step-sze for ntlzng the smulton. Ó 7 Elsever Inc. All rghts reserved. Keywords: Numercl smulton; Stte vrles pproch; Chu s crcut; Step-sze control; Egenvlues. Introducton The tendency of electronc desgn utomton (EDA) s focused on the development of fst, optml nd sutle computer-ded desgn (CAD) tools. An EDA tool s sed on the use of nlog ehvorl models (ABMs) [], where the ABM s smlr to su-crcut n SPICE, ut the dfference s tht ts nteror s mplemented n terms of lgerc or dfferentl equtons rther thn y netlst, nd therefore, t cn e used to effcently descre rod rnge of models. In ths mnner, s lredy demonstrted n [], n ABM s well suted to smulte Chu s crcut t the electronc system level (ESL). On the other hnd, n [3] ws shown tht tme smulton cn e mproved y utomtc control of step-sze (h). Addtonlly, the ABM to descre Chu s dode hs een lredy desgned wth novel electronc devces, s shown n [4], where t offers wde rnge of vlues to propose synthess process from hgh-level smulton of Chu s crcut []. Henceforth, ths pper s focused to enhnce tme smulton of Chu s crcut y utomtc determnton of the egenvlues to select nd control h. On the other hnd, snce Chu s crcut s smulted t the ESL heren, the frequency sclng ehvor s explored usng the proposed smulton procedure. * Correspondng uthor. E-ml ddresses: e.tlelo@eee.org (E. Tlelo-Cuutle), mpcheco@noep.mx (J.M. Muñoz-Pcheco) /$ - see front mtter Ó 7 Elsever Inc. All rghts reserved. do:.6/j.mc.7.4.5

2 . Hgh-level modellng of Chu s crcut Bfurcton nd chotc phenomen cn potentlly pper n mny nonlner systems, nd most prctcl electronc crcuts re nonlner to some degree. Besdes, Chu s crcut s the only chotc system whch cn e esly ult, smulted, nd trctle mthemtclly [5]. As shown n Fg., t conssts of fve crcut elements: one lner resstor (R), one nductor, two cpctors, nd one nonlner resstor known s Chu s dode. The ehvor of Chu s dode s depcted n Fg., whose current ( NR ) s descred y (), nd ts pecewse-lner I V chrcterstc s well descred y (), whch s ts correspondng ABM. Furthermore, Chu s crcut cn e modeled y pplyng stte vrles pproch s shown y (3), whch hs the form _x ¼ Ax þ Bu. NR ¼ mv C þ Ix; ðþ 8 >< g V C þðg g ÞBP V C < BP; NR ¼ g V C BP 6 V C 6 BP; ðþ >: g V C þðg g ÞBP V C > BP; 6 4 _V C _V C _I L ¼ 4 E. Tlelo-Cuutle et l. / Appled Mthemtcs nd Computton 94 (7) RC þ m C RC RC RC C L V C V C I L 3 3 Ix C þ 4 7 5: ð3þ 3. Smulton of Chu s crcut wth utomtc control of h The numercl smulton of (3) y pplyng the thrd-order Adms Moulton (AM) mplct lgorthm gven y (4), hs een lredy shown n []. x nþ ¼ x n þ hf5=ða^x nþ þ Þþ8=ðAx n þ Þ =ðax n þ Þg: ð4þ In [3] ws shown tht f the order of the numercl method does not chnge durng the ntegrton process, then h cn e optmzed y selectng the hghest possle vlue for whch the locl error s lmted elow mxmum-error gven y h Æ e mx, to gurntee the stlty of the numercl method. The formuls used for the locl error truncton cn e found n [,3]. Fg.. Chu s crcut. I NR g g3 g BP BP -BP -BP g g3 V C Fg.. I V pecewse-lner chrcterstc of Chu s dode.

3 488 E. Tlelo-Cuutle et l. / Appled Mthemtcs nd Computton 94 (7) AM wth utomtc control of step sze (h). R=65..6 AM, R=65, h=.e Vc Vc Fg. 3. Smulton of Chu s crcut usng (4) to generte the doule-scroll ehvor [5]: () wth utomtc control of h (55 tertons), nd () wthout control of h (, tertons)..5 x -3 AM wth utomtc control of steo sze (h), R=65.5 x -3 AM, R=65, h=.e-6, E Fg. 4. Error generted from: () Fg. 3, nd () Fg. 3. For nstnce, y settng: R = 65, C = 45 pf, C =.5 nf, L = mh, g = /358, g = /464, g 3 =/ 6, BP =.4 V, BP =.4 V, h =.e 6, V C () =. V, V C () = V nd I L () = A, Fg. 3 shows the smulton of Chu s crcut y utomtc control of h nd wthout control of h. The error generted y Fg. 3 s shown n Fg. 4, where t cn e pprected tht e T 6 e mx, nd where e mx =.5e Automtc determnton of egenvlues for utomtc control of the ntl h The determnton of the egenvlues of system cn e computed y evlutng (5), where A s the stte mtrx, I s the dentty mtrx nd D express the determnnt of ja kij. Eq. (5) s known s the chrcterstc equton of A, nd ts evluton genertes chrcterstc polynoml, from whch re clculted the egenvlues DðA kiþ ¼: ð5þ In [,3], the ntl h to the thrd-order AM lgorthm ws gven y the user nd f the vlues of the components chnge, the user needs to estmte new ntl step-sze. Furthermore, the necessry vlues, such s: nferorerror, superor-error nd mxmum-error gven n [3], to ntlze the smulton wth utomtc control of h

4 were lso gven y the user. Therefore, n ths secton procedure s ntroduced to compute the egenvlues n Chu s crcut to utomtclly determne the ntl h long wth the necessry vlues to ntlze tme smulton. As result, the numer of opertons s reduced respect to [3], ledng to more gn to speed-up tme smulton. Also, the proposed procedure enles the user to chnge the vlues of components (R, C, C nd L) to explore on the frequency sclng ehvor. The evluton of (5) results n the chrcterstc equtons gven y (6) nd (7). Agn, y settng: C = 45 pf, C =.5 nf, L = mh, g = /358, g = /464, g 3 = /6 nd R = 65, the evluton of (6) nd (7) genertes the egenvlues gven y (8) nd (9), respectvely. k 3 A þ ðlc g RLC þ LC Þ RLC C k 3 B þ ðlc g RLC þ LC Þ RLC C E. Tlelo-Cuutle et l. / Appled Mthemtcs nd Computton 94 (7) k A þ ðrc g LÞ k A þ ð g RÞ ¼ ; ð6þ RLC C RLC C k B þ ðrc g LÞ k B þ ð g RÞ ¼ ; ð7þ RLC C RLC C k A ¼ 596:88; k A;3A ¼ 3373: :83; ð8þ k B ¼ 94:; k B;3B ¼ 9: :45: ð9þ In ths mnner, the ntl h s computed from (), where k mn s the mnmum solute vlue of ll the egenvlues expressed y (). W n () s estmted y pplyng the smple theorem, nd therefore, ts mnmum vlue s. h ¼ ð=k mnþ W ; ðþ k mn ¼ mnfjk A j; jk A j; jk 3A j; jk B j; jk B j; jk 3B jg: ðþ The vlues of e, e s nd e mx re gven y (), (3) nd (4), respectvely. e ¼ 4:ð: Kþ Þðh Kþ Þ; e s ¼ 4:5ð: Kþ Þðh Kþ Þ; e mx ¼ 94:6ð: Kþ Þðh Kþ Þ: ðþ ð3þ ð4þ The evluton of () results n k mn = , nd therefore, () results n h e 6 wth W = 9, for nstnce. Furthermore, y solvng (), (3) nd (4), one gets e.e 3, e s.e 3 nd e mx.5e Frequency sclng smulton y utomtc selecton nd control of h In ths secton, the smulton of Chu s crcut s shown usng the thrd-order AM lgorthm descred n [], usng utomtc control of h s shown n [3], nd y ncludng the utomtc determnton of h, e, e s nd e mx, s descred n the prevous secton. In Fg. 5, the smulton of Chu s crcut s shown y utomtc determnton of ntl h, e, e s nd e mx ; nd re shown the vlues of h t ech terton. In Fg. 6 the smulton of Chu s crcut s shown wth sclng frequency of 4 over the vlues of C, C nd L. By settng: R = 65, C =.5 pf, C = 375 pf, L = 5 lh, g = /358, g = /464, g 3 = /6, BP =.4 V, BP =.4 V, V C () =. V, V C () = V,I L () = A, the proposed procedure computes h = e 7, e = e 33, e s = e 33 nd e mx = e 33.InFg. 7 the sgnl n V C s shown to pprecte the sclng frequency. Fnlly, n Tle the pproprteness of ddng utomtc determnton nd control of step-sze to speed-up tme smulton s demonstrted. The proposed method s gong to e mplemented to reproduce n expermentl relzton of nry sgnls trnsmsson usng chos [6], ut t the ESL.

5 49 E. Tlelo-Cuutle et l. / Appled Mthemtcs nd Computton 94 (7) Vc h sze, step 3.5 x Fg. 5. Smulton of Chu s crcut: () wth utomtc determnton nd control of h (349 tertons), nd () vlues of h x -6 Vc step sze, h Fg. 6. Smulton of the frequency sclng of Chu s crcut y 4: () wth utomtc determnton nd control of h (333 tertons) nd () vlues of h tme x -4 tme x -4 Fg. 7. Sgnls n V C : () from Fg. 5, nd () from Fg. 6. Tle Smultons comprsons usng utomtc determnton nd control of h Algorthm Itertons Tme Mytes Fnl tme Thrd-order AM wthout utomtc control of h, e 3 Thrd-order AM wth utomtc control of h e 3 Thrd-order AM wth utomtc determnton nd control of h (egenvlues) e 3

6 E. Tlelo-Cuutle et l. / Appled Mthemtcs nd Computton 94 (7) Concluson The pproprteness of mplementng utomtc determnton nd control of h to speed-up tme smulton of chotc osclltors, e.g. Chu s crcut were shown. The smultons were crred-out y pplyng the thrd-order AM lgorthm, nd t ws shown the frequency sclng smulton for whch the gn n speedng tme smulton s mntned. As result, the proposed smulton procedure cn e very useful to dopt top down desgn pproch for chotc osclltors, nd to smulte synchronzton nd trnsmsson systems t the ESL. Acknowledgement Ths work hs een prtlly supported y CONACyT/MEXICO under the project numer Y. References [] Kenneth S. Kundert, Olf Znke, The desgner s gude to Verlog AMS, Kluwer Acdemc Pulshers, 4. [] E. Tlelo-Cuutle, J.M. Muñoz-Pcheco, Numercl smulton of Chu s crcut orented to crcut synthess, Interntonl Journl of Nonlner Scences nd Numercl Smulton 8 () (7) [3] E. Tlelo-Cuutle, J.M. Muñoz-Pcheco, Smulton of Chu s crcut y utomtc control of step-sze, Appled Mthemtcs nd Computton, n press, do:.6/j.mc [4] E. Tlelo-Cuutle, A. Gon-Hernández, J. Grcí-Delgdo, Implementton of chotc osclltor y desgnng Chu s dode wth CMOS CFOAs, Anlog Integrted Crcuts nd Sgnl Processng 48 () (6) [5] L.O. Chu, M. Komuro, T. Mtsumoto, The doule scroll fmly, IEEE Trnsctons on Crcuts nd Systems 33 () (986) [6] C. Cruz-Hernndez, D. López, V. Grcí, H. Serrno, R. Núñez, Expermentl relzton of nry sgnls trnsmsson usng chos, Journl of Crcuts, Systems nd Computers 4 (3) (5)