Latest Trends on Crcuts, Systems and Sgnals Scroll Generaton wth Inductorless Chua s Crcut and Wen Brdge Oscllator Watcharn Jantanate, Peter A. Chayasena, and Sarawut Sutorn * Abstract An nductorless Chua s crcut, and a scroll generator are presented n ths paper. The scheme employs only RCs and opamps, and provdes a very relable generaton of multple chaotc scrolls. The desgn formulas and examples ncludng smulaton and practcal results are llustrated. The prncple s appled successfully to generate some odd scrolls wth the Wen brdge oscllator. Keywords Chaos, Chua s crcut, nductorless, Wen brde oscllator, scroll generaton. C I. INTRODUCTION HAOTIC phenomenon has receved much nterest for a few decades. Such behavour has been successfully appled to sgnal transmsson and cryptography []-[3]. Several type of oscllators, for example Collpts, Wen brdge, Chua, Lorenz, etc., have been studed and used for generatng chaotc attractors. Among those, Chua s crcut stands out and always provdes a relable result. The orgnal Chua s crcut has been modfed to have only RCs and op-amps [4]-[6], whose structures are compact and open for smple adustments. After successful applcatons of the double scroll oscllators, n- scroll chaotc crcuts producng a more complcated behavour have been developed by ntroducng addtonal breakponts nto the characterstcs of the Chua s crcut [7]-[9]. In ths paper, the backgrounds of the Chua s and scroll generatng crcuts are revewed n the next secton. Secton III presents the desgn examples, smulaton and practcal results based on the nductorless Chua s crcut as well as the Wen brdge oscllator. Concluson follows n secton IV. II. BACKGROUNDS dv C = ( v v + ( 3 dt R d3 L = v (3 dt f ( v = Gv + G { v + E v E } (4 Fg. Chua s crcut. Fg. V-I characterstc of Chua s dode. A. The Chua s crcut The Chua s crcut as shown n Fg. s a rch 3 rd -oder dynamcal system. The NR stands for the nonlnear resstor whch s commonly mplemented va op-amps and 6 resstors known as the Chua s dode. dv C = ( v v f ( v ( dt R The authors are wth the School of Electrcal Engneerng, and the School of Mathematcs, Suranaree Unversty of Technology (SUT, Thaland. Fnancal supports from SUT and Raamangala Unversty of Technology Isan (Khonkaen Campus, Thaland, are greatly acknowledged. * Correspondng author: sarawut@sut.ac.th. Fg. 3 Vector feld regons of double scroll. ISSN: 79-434 48 ISBN: 978-96-474-8-
Latest Trends on Crcuts, Systems and Sgnals The dynamcs of the Chua s crcut are expressed by Eqs. (-(3. Eq.(4 expresses the nonlnear functon f ( v whose characterstc curve s depcted n Fg.. Fg. 3 represents the vector regons of a double scroll case. The attractors D and D slde from one plane to another alternately through the bond orbt. The nductor n the orgnal Chua s crcut can be replaced by a smulated nductor. Fg. 4 llustrates the nductorless Chua s crcut wth ts smulated behavour usng PSIM. Gv = Gv + G = G v + G n = As an example, an -scroll attractor requres segments of the pecewse-lnear functon as shown n Fgs. 5 (a-(b. (5 (a Chua crcut smulaton (nductor smulated.8.6.4 (a. VC(v -. -.4 -.6 -.8 - -5-4 -3 - - 3 4 5 VC(v (b Fg. 4 An nductorless Chua s crcut: (a the crcut dagram, (b the smulaton result. B. The scroll generators In the paper, t s shown that the nductorless Chua s crcut can be used as the core to generate the chaotc behavour for mult-scroll generaton. Referrng to Fg. 3, the left and rght boundares of the bond orbt are separated by the breakpont voltages, E and E respectvely. More breakponts can be nserted to generate multple attractors and bond orbts [8]-[9] based on the pecewse-lnear functon concept. Accordng to ths prncple, the nonlnear functon, f ( v, of the n-scroll chaotc attractor can be expressed by n { } f ( v = G v +.5 G v + E v E n = (b Fg. 5 An example of -scroll attractor: (a the pecewselnear functon, (b the vector feld. Referrng to Fg. 5(b, the center manfold contans the attractor D wth the equlbrum pont, P =. For a gven breakpont E, the equlbrum ponts, P + and P, n dfferent regons can be calculated accordng to Eq. (6, P + = [ v + v+ 3 + ] G G G = = = + G + G (6 P = [ v v 3 ] G G G = = = + G + G ISSN: 79-434 49 ISBN: 978-96-474-8-
Latest Trends on Crcuts, Systems and Sgnals where =,, 3,...; and v +, v +, 3+, v, v, 3 are the proectons of the ponts P + and P to the correspondng v, v and axes, respectvely. 3 E < v + < E must hold for + symmetrcal scrolls, and the breakpont voltages can be determned from (7 where G=/R = /.65k =.66 S. ( G G E = ( =,, 3,... = E+ E + G (7 Fg. 6 Buldng block of the scroll generator. III. REALIZATION, SIMULATIONS, AND EXPERIMENTS Realzaton of the scroll generatng crcut follows the procedures descrbed n [7]. After specfyng the conductance Gs, and the breakpont E,the other breakponts can be calculated accordng to Eq.(7. The crcut uses op-amps, and Rs as shown n Fg. 6. The + and nput pns of the op-amp used as the buldng block are alternately ted to ground. The normalzed resstor r and r + can be calculated usng the formulas n Eq. (8- (, respectvely. Consequently, the followng normalzed resstors of the scroll generator can be obtaned. r ( R = = = (8, E + sat R, E (,,3,..., R G8 = ( v E8 R3 R ( + ( G8 = G9 + + r E8 v < E9 = R3 r + + R r = = R3G8 R + ( ( + r R +, r + = = = R +, R3 G8 G9 ( (,,,3,..., (,,3,..., (9 ( As an example, the components of the Chua s crcut are C = 5.6µ F, C = 47.9µ F, R =.65kΩ,and L = 9.3mH. The nductor s realzed by an op-amp wth C 3 =.µ F, R = Ω, R5 = 98kΩ, and R6 = kω. To acheve the odd 4 numbers of scroll of 3, 5, 7 and 9, based on the double scroll E, and the Chua s crcut, the followng breakpont ( conductances are assgned: E =.V, G = G = G4 =... = G8 =.3mS, and G = G3 =... G7 =.85mS. Hence, the obtaned breakponts are E = [ E E E3... E8 ] = [..6654.654.538.938.396.796 3.66]. [ ] [ 9.346.43.49 7.5] = [ ] = [.649 9.984 66.99] r = r r r4 r6 r8 r r r4 r6 =.64 4.3844 4.4 5.9678 6.463 r + r3 r5 r7 r9 r r3 r5 r7 3.7 3.865 4.689 5.968 7.7797 Fg. 7 llustrates the scroll generatng crcut to acheve 3, 5,7 and 9 scrolls based on the nductorless Chua s crcut. The status of the swtches (S, S, S 3 are as follows: (off, off, off for 3, (on, off, off for 5, (on, on, off for 7, and (on, on, on for 9 scrolls, respectvely. Smulaton and expermental results are also depcted n the same fgure. The same prncple and the crcut are appled to the case of Wen brdge oscllator. As shown n Fg. 8, both smulaton and expermental results confrm the successful applcatons of ths scroll generator. ISSN: 79-434 5 ISBN: 978-96-474-8-
Latest Trends on Crcuts, Systems and Sgnals (a scroll generatng crcut and smulaton results (b expermental results Fg. 7 Odd-scroll generatng crcut based on the nductorless Chua s crcut. ISSN: 79-434 5 ISBN: 978-96-474-8-
Latest Trends on Crcuts, Systems and Sgnals (a scroll generatng crcut and smulaton results (b expermental results Fg. 8 Odd-scroll generatng crcut based on the Wen brdge oscllator. ISSN: 79-434 5 ISBN: 978-96-474-8-
Latest Trends on Crcuts, Systems and Sgnals IV. CONCLUSION The prncples of the scroll generatng crcut, and the Chua s crcuts have been explaned n the paper. The desgn of an odd-scroll generatng crcut wth smulaton and practcal results are presented. The cases of 3, 5, 7, and 9 scrolls based on the nductorless Chua s crcuts, and the Wen brdge oscllator have been llustrated. The prncples descrbed heren can be readly appled for generatng evenscrolls. REFERENCES [] G. Kolumban, M. P. Kennedy, and L. O. Chua, The role of synchronzaton n dgtal communcatons usng chaos-part II: Chaotc modulaton and chaotc synchronzaton, IEEE Trans. Crcuts Syst. (part-i, vol. 45, no., pp. 9 4, 998. [] T. Yang, C. W. Wu, and L. O. Chua, Cryptography based on chaotc systems, IEEE Trans. Crcuts Syst. (part-i, vol. 44, no.5, pp. 469-47, 997. [3] A. S. Dmtrev, A. I. Panas, S. O. Starkov, Experments on speech and musc sgnals transmsson usng chaos, Int. J. Bfurc. Chaos, vol. 5, no. 4, pp. 49-54, 995. [4] M. P. Kennedy, Robust op amp realzaton of Chua s crcut, FREQUENZ, vol. 46, pp. 66-8, Apr. 99. [5] O. Morgul, Inductorless realzaton of Chua oscllator, Electroncs Letters, vol. 3, pp.43-44, Aug. 995. [6] C. Ass and D. Kazakos, An mproved realzaton of the Chua s crcut usng RC-op amps, n Proc. 7 th WSEAS Int. Conf. on Sgnal Processng, 8, pp. 5-8. [7] M. E. Yalcn, J. A. K. Suykens and J. Vandewalle, Expermental confrmaton of 3- and 5-scroll attractors from a generalzed Chua s crcut, IEEE Trans. Crcuts Syst. (part-i, vol. 47, pp. 45-49, Mar.. [8] Y. Smn, Q. Shusheng and L. Qnghua, New results of study on generatng multple-scroll chaotc attractors, Scence n Chna (Seres F, vol. 46, no., pp. 4 5, Apr. 3. [9] Y. Smn, M. Zaguang and Q. Shusheng, Generaton and synchronzaton of N-scroll chaotc and hyperchaotc attractors n fourth-order systems, Chnese Physcs, vol. 3, no. 3, pp. 37-38, Mar. 4. ISSN: 79-434 53 ISBN: 978-96-474-8-