Odd/Even Scroll Generation with Inductorless Chua s and Wien Bridge Oscillator Circuits

Transcription

1 Watcharn Jantanate, Peter A. Chayasena, Sarawut Sutorn Odd/Even Scroll Generaton wth Inductorless Chua s and Wen Brdge Oscllator Crcuts Watcharn Jantanate, Peter A. Chayasena, and Sarawut Sutorn * School of Electrcal Engneerng, Insttute of Engneerng School of Mathematcs, Insttute of Scence Suranaree Unversty of Technology Nakhon Ratchasma, Thaland, *correspondng author: sarawut@sut.ac.th Abstract: - An nductorless Chua s crcut, and a scroll generator are presented n ths paper. The scheme employs only RCs and op-amps, and provdes a very relable generaton of multple odd and even chaotc scrolls. The desgn formulas and examples ncludng smulaton and practcal results are llustrated. The prncple has been appled successfully to generate some odd/even scrolls wth the Wen brdge oscllator as llustrated by the smulaton and the practcal results. Key-Words: - Chaos, Chua s crcut, nductorless, Wen brde oscllator, scroll generaton. Introducton CHAOTIC phenomenon has receved much nterest for a few decades. Such behavour has been successfully appled to sgnal transmsson and cryptography []-[3]. Several types of oscllators have been studed and appled for generatng chaos, e.g. Collpts, Wen brdge, Chua, Lorenz, etc.. 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 opamps [4]-[6], whose structures are compact and open for smple adustments. More complcated dynamcs of the Chua s crcut have been nvestgated. These nclude antmonotoncty, and bubble formaton [7]-[8]. It s also possble to replace the pecewse lnear characterstc of the Chua s dode wth a smooth cubc functon [9]. After successful applcatons of the double scroll oscllators, n-scroll chaotc crcuts producng a more complcated behavour have been developed. The key concept of the scroll generator s to add multple breakponts nto the characterstcs of the Chua s dode [0]-[]. The generator has not been appled wth the nductorless Chua s crcut, and the Wen brdge oscllator before. In ths paper, the backgrounds of the Chua s, Wen brdge, and scroll generatng crcuts are revewed n the next secton. Secton 3 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 4. Backgrounds. 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 dv C = ( v v ) + () 3 dt R d3 L dt = v (3) f ( v ) = Gv + ( G G0 ){ v + E v E} (4) Fg. Chua s crcut. ISSN: Issue 7, Volume 9, July 00

2 Watcharn Jantanate, Peter A. Chayasena, Sarawut Sutorn Fg. V-I characterstc of Chua s dode. Fg. 3 Vector feld regons of double scroll. The dynamcs of the Chua s crcut are descrbed 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 obtaned from PSIM.. The chaotc Wen brdge oscllator Wen brdge s a very well-known oscllator. When t s connected to the Chua s dode, the crcut s represented by the dagram n Fg. 5, and produces a double scroll chaotc behavour. The crcut s very senstve to ntal condtons whle the nterconnected resstor, R, plays a key role for the adustment of the ntal condtons. The dynamc of the crcut can be represented by Eqs. (5)-(7). The prevously descrbed pecewse lnear characterstc of the Chua s dode s stll appled. The smulated responses of the crcut are shown n Fg. 6. (a) crcut dagram Fg. 4 An nductorless Chua s crcut. (b) smulated phase portrat Fg. 5 A chaotc Wen brdge oscllator. ISSN: Issue 7, Volume 9, July 00

3 Watcharn Jantanate, Peter A. Chayasena, Sarawut Sutorn dv C = ( v v ) f ( v ) (5) dt R dv C = ( v v ) + v (6) 3 dt R R8 dv R C = v v (7) dt RR 4 R.3 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 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 ( G G ) v + E v E n = (a) tme-doman G0v = Gv + ( G G ) E = G v + ( G G ) E n = (8) (b) phase portrat Fg. 6 Dynamc responses of the Wen brdge oscllator. As an example, an -scroll attractor requres segments of the pecewse-lnear functon as shown n Fgs. 7(a)-(b). Referrng to Fg. 7(b), the center manfold contans the attractor D wth the equlbrum pont 0 P =. For a gven breakpont E, the equlbrum 0 0 ponts, P + and P, n dfferent regons can be calculated accordng to Eq. (9), (a) pecewse lnear functon ISSN: Issue 7, Volume 9, July 00

4 Watcharn Jantanate, Peter A. Chayasena, Sarawut Sutorn (b) vector feld Fg. 7 The vector manfold of -scroll attractor. [ ] P + = v+ v+ 3+ ( G G ) E G ( G G ) E = = = 0 ( G + G ) ( G + G ) P = [ v v 3 ] ( G G ) E G ( G G ) E = = = 0 ( G + G ) ( G + G ) (9) accordng to Eq. (0). The crcut uses op-amps, and Rs as the buldng block shown n Fg. 8. Fg. 8 Buldng block of the scroll generator. where =,, 3,...; and v +, v +, 3+, v, v, 3 are the proectons of the ponts P + and P to the correspondng v, v and 3 axes, respectvely. The condton E < v+ < E must hold for symmetrcal + scrolls, and the breakpont voltages can be determned from (0) where G = /R = /.65k = S. ( ) G G E = E+ E ( G + G) = ( =,, 3,...) (0) 3 Realzaton, Smulatons, and Experments Realzaton of the scroll generatng crcut follows the procedures descrbed n [0]. After specfyng the conductance Gs, and the breakpont E, the other breakponts can be calculated As an example, the components of the Chua s crcut are C = 5.06µ F, C = 47.9µ F, R =.65kΩ, and L = 9.3mH. The nductor s realzed by an opamp wth C 3 = 0.0µ F, R 4 = Ω, R5 = 980kΩ, and R6 = 0kΩ. To acheve the odd numbers of scroll of 3, 5, 7 and 9, based on the double scroll Chua s, and the crcut, the followng breakpont ( E ) conductances are assgned: E 0.V =, G0 = G = G4 =... = G8 = 0.3mS, and G = G3 =... G7 = 0.85 ms. Hence, the obtaned breakponts are E = [ E E E3... E8 ] = [ ]. ISSN: Issue 7, Volume 9, July 00

5 Watcharn Jantanate, Peter A. Chayasena, Sarawut Sutorn r ( ) R = = = (), E + sat R, E (,,3,...,0) R 0 G8 = ( v E8) R3 R0 ( ) + ( ) G8 = G9 + + r E8 v < E9 = R3 r + + (,,,3,...,0) () R0 r0 = = R3G8 R0 + ( ) ( ) + r R +, r + = = = R +, R3 G8 G9 ( ) ( ),,3,...,0 (3) The + and nput pns of the op-amp used as the buldng block are alternately ted to ground. The normalzed resstances r and r + can be calculated usng the formulas n Eqs. ()-(3), respectvely. Consequently, the followng normalzed resstances of the scroll generator can be obtaned. [ ] [ ] r = r0 r r4 r6 r8 r0 r r4 r6 = r + = r3 r5 r7 r9 r r3 r5 r 7 = [ ] [ ] Fg. 9 llustrates the scroll generatng crcut to acheve 3, 5,7 and 9 scrolls based on the nductorless Chua s crcut. The statuses 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. 0, both smulaton and expermental results confrm the successful applcatons. The crcut structure as well as the desgn formulas can be readly appled to generate even numbers of scroll. Consder the cases of 4 and 6 scrolls, the followng conductances are assgned: G0 = G = G4 = 0.85 ms, and G = G3 = G5 = 0.3 ms. Eqs. (0)-(3) are ppled to calculate the breakponts, and the normalzed resstances obtaned as: E E E E3 E ], and = [... ] = [ [ ] [ r = r0 r r4 r6 r8 r0 = ] r + = [ r3 r5 r7 r9 r r3 r5 r7] = [ ]. Usng the same buldng block shown n Fg. 8, one can realze the even scroll generatng crcut as llustrated n Fg.. The crcut dagrams n the nsets of Fg. (a) represent the Chua s dode, the nductorless Chua s crcut, and the Wen brdge oscllator. The frst chaotc core s the nductorless Chua s crcut of whch smulated responses are depcted n Fg. (b). When the Wen brdge oscllator s n use, the nductorless Chua s crcut s dsconnected at the pont a at whch the Wen brdge oscllator s connected nstead. There are man swtches for scroll generaton. To generate 4 and 6 scrolls, the swtches (S,S) must be (on,off), and (on,on), respectvely. Fg. llustrates the expermental results of even scroll generaton wth the nductorless Chua s crcut, and the Wen brdge oscllator used as the chaotc core crcuts. Both ISSN: Issue 7, Volume 9, July 00

6 Watcharn Jantanate, Peter A. Chayasena, Sarawut Sutorn smulated and expermental results show very nce agreements, and confrm the successful applcatons of the scroll generatng technque. Durng the experments, settng up the ntal condtons through adustng the correspondng components was not dffcult provdng that good qualty components were used. (a) scroll generatng crcut and smulaton results ISSN: Issue 7, Volume 9, July 00

7 Watcharn Jantanate, Peter A. Chayasena, Sarawut Sutorn (b) expermental results Fg. 9 Odd-scroll generatng crcut based on the nductorless Chua s crcut. (a) scroll generatng crcut and smulaton results ISSN: Issue 7, Volume 9, July 00

8 Watcharn Jantanate, Peter A. Chayasena, Sarawut Sutorn (b) expermental results Fg. 0 Odd-scroll generatng crcut based on the Wen brdge oscllator. (a) scroll generatng crcut ISSN: Issue 7, Volume 9, July 00

9 Watcharn Jantanate, Peter A. Chayasena, Sarawut Sutorn (b) smulaton results Fg. Even-scroll generatng crcut based on the nductorless Chua s crcut and the Wen brdge oscllator. (a) 4 and 6 scrolls generated from the nductorless Chua s crcut. (b) 4 and 6 scrolls generated from the Wen brdge oscllator. Fg. Expermental results. 4 Concluson The prncples of the scroll generatng crcut, the Chua s crcuts, and the Wen brdge oscllator have been explaned n the paper. The paper presents the development of scroll generatng crcuts capable of generatng both odd and even numbers of scroll. Both smulaton and practcal results are llustrated for successful scroll generatons wth the nductorless Chua s crcut, and the Wen brdge oscllator. Settng up the ntal condtons for the crcuts can be smply done through adustng some nterconnected resstances. 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 40, 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 , 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 , 995. ISSN: Issue 7, Volume 9, July 00

10 Watcharn Jantanate, Peter A. Chayasena, Sarawut Sutorn [4] M. P. Kennedy, Robust op amp realzaton of Chua s crcut, FREQUENZ, vol. 46, pp , Apr. 99. [5] O. Morgul, Inductorless realzaton of Chua oscllator, Electroncs Letters, vol. 3, pp , Aug [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, 008, pp [7] I. M. Kyprands, New chaotc dynamcs n Chua s canoncal crcut, WSEAS Trans. Crcuts and Systems, vol. 5, no., pp , 006. [8] I. N. Stouboulos, I. M. Kyprands and M. S. Papadopoulou, Chaotc dynamcs n a modfed Chua s crcuts, bd, pp [9] I. M. Kyprands and M. E. Fotadou, Complex dynamcs n Chua s canoncal crcut wth a cubc nonlnearty, WSEAS Trans. Crcuts and Systems, vol. 5, no. 7, pp , 006. [0] 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 , Mar [] Y. Smn, Q. Shusheng and L. Qnghua, New results of study on generatng multplescroll chaotc attractors, Scence n Chna (Seres F), vol. 46, no., pp. 04 5, Apr [] Y. Smn, M. Zaguang and Q. Shusheng, Generaton and synchronzaton of N-scroll chaotc and hyperchaotc attractors n fourthorder systems, Chnese Physcs, vol. 3, no. 3, pp , Mar ISSN: Issue 7, Volume 9, July 00