Nonlinear Resonances Analysis of a RLC Series Circuit Modeled by a Modified Van der Pol Oscillator

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1 Nonliner Resonnces Anlysis of RLC Series Circuit Modeled by Modified Vn der Pol Oscilltor Y. F.Kpomhou * C. Midiwnou, R. G. Agbokpnzo, L. A. Hinvi nd.k.k. Adjï eprtment of Industril nd Technicl Sciences, ENSET-Lokoss, University of Abomey, Abomey, Bénin eprtment of Physics, ENS-Ntitingou, University of Abomey, Abomey, Bénin eprtment of industril Computer nd Electricl Engineering, UIT-Lokoss, University of Abomey, Abomey, Bénin eprtment of Physics, University of Abomey-Clvi, Abomey-Clvi, Bénin Abstrct In this pper, the nonliner resonnces nlysis of RLC series circuit modeled by modified Vn der Pol oscilltor is investigted. After estblishing of new generl clss of nonliner ordinry differentil eqution, forced Vn der Pol oscilltor subjected to n inertil nonlinerity is derived. From this eqution the multiple scles method is used to find the vrious resonnt sttes. As nlyticl results primry resonnce, sub-hrmonic resonnce of order / nd super-hrmonic resonnce of order re obtined. The stedy-stte solutions nd theirs stbilities re determined. Numericl simultions disply bistbility, hysteresis, jump nd bifurction phenomen. The effects of different prmeters on the system behvior re investigted nd results re presented grphiclly nd discussed. Keywords: Nonliner RLC circuit, Modified Vn der Pol oscilltor, Nonliner resonnces, Bistbility, Hysteresis, jump nd bifurction.. Introduction It is well known tht the resonnt electricl circuit composing most of the electric or electronic devices disply rich vriety of nonliner dynmicl behvior [-]. Therefore, such electricl circuit is governed by nonliner ordinry differentil equtions. The inherent nonlinerity source in resonnt electricl systems rises from resistive, inductive nd cpcitive elements []. Given the importnce of RLC circuit systems in mny res of physics nd modern engineering pplictions, the understnding of nonliner electricl properties becomes necessity from theoreticl nd prcticl point of view. Therefore this requires of developing mthemticl model tking into ccount the nonliner chrcter of the three fundmentl elements of the electricl circuits nd proceeding to its numericl simultion before ll optimiztion * E-mil : fkpomhou@gmil.com (Corresponding uthor)

2 process nd design [5-6]. In this perspective, mthemticl modeling of nonliner electricl systems ws become chllenge for reserchers in these three lst decdes. For instnce, Kufmn nd Roberts [7] hve derived from RLC series circuit consisting of nonliner resistor nd nonliner cpcitor clss of nonliner differentil equtions contining the Riccti's eqution nd Abel's eqution of the first kind s specil cse. Recently vrious nonliner mthemticl models of different compleities hve been proposed in open literture for nlyzing the nonliner oscilltions generted by RLC series circuit [-]. But it hs been remrked tht no simplest nonliner oscilltory circuit tking simultneously into considertion nonliner resistor nd nonliner inductor hs been developed for generting clss of nonliner differentil equtions. Therefore this problemtic is tken into ccount in this work for investigting new clss of nonliner differentil equtions governing the dynmic of the nonliner RLC series circuits nd nlyzing the nonliner oscilltions produced by such electricl circuit system. The min objectives of this work consist firstly to derive new generl clss of nonliner differentil equtions governing the nonliner dynmicl behvior of RLC series circuit with nonliner resistor nd nonliner inductnce nd secondly to nlyze the nonliner oscilltions tht cn rise in such electricl circuit through nonliner oscilltor belonging to this generl clss. In order to ttin the fied objectives in this pper, we generte t first sight, clss of nonliner differentil equtions (Section ) nd we investigte the vrious resonnt sttes of nonliner RLC series circuit by mens of the multiple scles method (Section ). We present fterwrd the numericl results nd discussions (Section ). Finlly the conclusion of this reserch work will be drwn (Section 5).. Mthemticl modeling.. Formultion of problem We consider n electricl circuit composed of nonliner resistor R (i), nonliner inductnce L(i) nd liner cpcitnce C connected in series nd driven by voltge source E(t) s shown in Fig. From this figure, it concerns to build new generl clss of nonliner equtions describing the dynmicl behviors of the nonliner resonnt circuits using Kirchoff's voltge nd current lws.

3 Fig.. Nonliner RLC series Circuit.. New generl clss of mied Liénrd type equtions Applying Kirchoff's voltge nd current lws t Fig., one obtin the following eqution: d dt i where U R i q C i Et d U L L( i) i, U R i) R( i) i dt ( nd cross the inductor, resistor nd cpcitor respectively. Et () q U C represent the voltge drop C denotes the pplied voltge, q mens the chrge t the cpcitnce, nd i (t) designte the flu nd current respectively. The differentition with respect to time of the eqution () yields fter some lgebric mnipultions to the following eqution: d i di di d E t i i U R i i () with dt dq i dt dt dt C dt The so obtined eqution () with nonliner functions describes the dynmic behvior of the nonliner RLC series circuits. This eqution is known s nonutonomous mied Liénrd-type equtions [] which will hve mny pplictions in modern electricl engineering, since it is possible to chieve n electronic circuit from nonliner differentil eqution []. Before ll nlysis of nonliner RLC series circuit under considertion, it is needed to mke cler the nonliner functions, U R (i) nd the pplied voltge E(t) epressions. At this stge, it is very importnt to point out tht vrious epressions of i i nd U R (i) re used in open literture for nlyzing the dynmicl response of nonliner electricl circuits [5-7]. For this, we choose the epressions of these functions (section ) in order to investigte the dynmic responses of the considered nonliner resonnt circuit. i

4 . Nonliner resonnces nlysis.. Mthemticl problem In this subsection one purpose to investigte the dynmic responses of nonliner RLC series circuit subjected to hrmonic voltge source of the form: Et E sin t. For this purpose, we choose the following epressions for i nd U R (i) : U R ( i) R i R i () nd ( i) L i L i () where R, R, L nd L re constnt prmeters. Substituting Eqs.() nd () into Eq.(), we obtin fter few mthemticl opertions the following eqution d i di di L L i 6L i ( R R i ) i E cos t (5) dt dt dt Now, using i i nd t, where i, nd denote the normliztion L C current nd the dimensionless vribles respectively, Eq.(5) becomes Q F cos C (6) L with i, Q L R C R i, F, L RQ ELC nd L C. i It is very importnt to show tht when the dimensionless prmeter, Eq. Q F cos, which represents the fmous Vn (6) becomes der Pol oscilltor intensively studied in the open literture in contet of vrious problems. Therefore Eq.(6) represents the generlized Vn der Pol oscilltor. Now, in view of importnce of the mied Liénrd-type equtions in electronic res nd other brnches of sciences, the problem here is to investigte the vrious resonnt sttes of the eqution (6) by pplying the multiple scles method.

5 .. Primry resonnce stte In this cse of oscilltion, the detuning prmeter nd the ecittion frequency re relted ccording to. In order to pply the multiple scles method, it is necessry to introduce into Eq.(6) the smll perturbtion prmeter. Tht mking, Eq.(6) becomes Q F cos (7) with, Q Q, nd F F Eq.(7) is known s wekly nonliner eqution. From this eqution, we define the fst time sclet, which ssocites with chnges occurring t the frequencies nd nd the slow time scles T which ssocites with modultions in the mplitude nd phse cused by nonlinerity. In terms of the new time scles, the first nd second time derivtive become d d d d () where n T n Now, we begin to ssume tht the pproimte solution of Eq.(7) cn be written in the following form:, T, T T T (9), Substituting Eqs.() nd (9) into Eq.(7) nd equting coefficients of like powers of, we get () Q F cos T () The solution of Eq.() cn be epressed in the comple form: it it T T AT e AT e, () 5

6 where AT is the comple conjugte of A T, nd A T is the comple mplitude function which cn be determined by eliminting the seculr terms t the higher levels of pproimtion eqution. Inserting Eq.() into Eq.(), yields to F it it i A A A iq A ia A e e c. c NST () where NST denotes the terms does not produce seculr terms nd c.c designtes the comple conjugte terms. Now eliminting the seculr terms from Eq.() nd introducing the mplitude A by the following polr form: T T i T e A T () we obtin fter seprting rel nd imginry prts the following modultion equtions: Q sin F (5) F cos (6) where T To determine the stedy-stte solution, we put into Eqs.(5) nd (6). Thus we obtin F sin Q (7) F cos () Eqs.(7) nd () show tht there no trivil solution t =. For non-trivil solution ( ), eliminting from these equtions yields to F Q (9) 6

7 7 Eq.(9) represents the frequency-response eqution for primry resonnce. In order to nlyze the stbility of the non-trivil fied points of modultion equtions (5) nd (6), we let:, () where nd represent the non-trivil solutions nd nd re ssumed to be infinitesiml. Substituting Eqs.() into Eqs.(5) nd (6) nd keeping only liner terms in the perturbtion quntities, we get the following vritionl equtions describing the stbility of the stedy-stte solution: ' cos F Q () ' sin cos F F () Eqs.()nd () dmit solutions of the form,, T e c c provided tht: p p p () with sin F Q p nd sin cos cos Q F F F p Therefore the stedy stte solution is unstble when p is greter thn zero... Super-hrmonic nd sub-hrmonic resonnces In this cse, we consider hrd resonnt ecittion, tht is to sy F =(). In this sitution, Eqs.() nd () re modified to cos T F () Q (5) The solution of Eq.() is cc e e A T T T T i it, (6)

8 where F i A [ [ [ with. Substituting Eq.(6) into Eq.(5) we get: A AAA iq A AAA it AA iq i AAe it i ] e i ] A e i( ) T i() T i ] Ae cc NST ia A e it (7) Eq.(7) shows two possibles cses of resonnces such s nd. We tret in following section these cses of resonnces.... Super-hrmonic resonnce ( ) In this cse of oscilltion, the detuning prmeter nd the dimensionless ecittion frequency cn be written ccording to the following reltionship () Inserting Eq.() into Eq.(7) we obtin it W ( T ) e cc NST (9) where it i A A AA iq A A AA ia A i e W T Eliminting the seculr terms from Eq.(9) nd introducing the epression of A given by Eq.() into W T, we obtin fter some mthemticl T opertions the following modultion equtions: Q 5 cos sin () cos sin () 7 9 where T. Thus we get 9F. To determine the stedy-stte solution, we let 6

9 9 cos sin 9 5 Q () 9 7 cos 9 5 sin () Eqs.() nd Eq.() show tht there re no trivil solution t =. Eliminting from these two equtions, the solution of super-hrmonic resonnce of order is given by the following eqution: Q () To nlyze the stbility of the non-trivil fied points of modultion equtions () nd () we substitute Eqs.() into () nd () nd keeping only liner terms in the perturbtion quntities, we get: ' (5) ' (6) with Q, cos 9 5 sin, sin cos 9 5 nd sin 9 5 cos Eqs.(5) nd (6) dmit solution of the form,, T e where nd re constnts, provided tht (7) Consequently, the stedy stte solution is unstble when >.... Sub-hrmonic resonnce of order To tret this cse of nonliner resonnce, we introduce the detuning prmeter ccording to () into Eq.(7), we get

10 it Z( T ) e cc NST (9) where Z 6 9 it T i A A AA iq A A AA ia A A 5 i e Replcing the epression of mplitude T Z T nd eliminting the seculr terms from Eq.(9), we obtin fter seprting rel nd imginry prts, the following modultion equtions: A given by Eq.() into Q 5 cos sin () 5 cos sin () F where T. In order to determine the stedy-stte solution, it is 6 enough to set. Thus, we obtin 5 Q cos sin 5 cos sin () () From Eqs.() nd () we observe tht there re no trivil solution t =. Then eliminting the oscilltion phse from these two equtions we get fter some lgebric mnipultions the following frequency response eqution for subhrmonic resonnce of order /: Q () In order to nlysis the stbility of the non-trivil solution, we follow the steps used in the preceding subsection. Thus, we obtin the following vritionl equtions describing the stbility of the stedy-stte solutions: ' M M (5) (6) ' M M

11 with M M Q 5 sin cos, 5 M sin cos, 5 cos sin nd M 5 cos sin Eqs.(5) nd (6) dmit solutions of the form,, T e where nd re constnts, provided tht M (7) M M M M M M M Therefore, the stedy stte solution is unstble if nd only if M M MM >.. Numericl Results nd discussions This section presents numericl solutions of the frequency-response equtions. Frequency response equtions (9), () nd () re nonliner lgebric equtions in the mplitude ( ). The stedy-sttee equtions nd stbility conditions re solved numericlly. Solid/dottedd lines designte the stble/unstble solution brnches respectively. The numericl solutions re plotted in group of Figs.-, which represent the vrition of the mplitude ( ) versus the detuning prmeter ( ). Figs..- present the frequency-response curves of the primry resonnce. Figure. The frequency response curves of the primry resonnce solution for the prmeters:,., Q. nd F.

12 In Fig.. we observe tht the response mplitude hs two brnches which bent to the left showing the softening behvior. The upper brnch hs stble solution while the lower brnch presents unstble nd stble solutions. Therefore, we conclude tht there ppers in this system, the jump nd bifurction phenomen. When the liner dmping fctor (the inverse of qulity fctor) increses, the mimum mplitude of the response decreses considerbly nd the instbility region disppers s shown in Fig.(). In the sme wy, incresing of nonliner dmping coefficient (vn der Pol coefficient) decreses considerb bly the instbility domin nd the mimumm mplitude of the response (see Fig.(b)). In Fig.(c), when the inertil nonlinerity decreses nd tkes the vlues, we lwys note the softening behvior. Moreover the mimum mplitude of the response stys unchnged but its loction vlue increses. However the hrdening behvior is obtined when tkes the vlue -. In such sitution, we cn conclude tht there pper in the system under considertion the jump nd bifurction phenomen. In Fig.(d) we note thtt when F decreses the mimum mplitude of the response decreses highly s well s the instbility region. The system displys lwys the softening behvior nd therefore it ppers in such system the jump nd bifurction phenomen.

13 Figure. Frequency response curves ehibiting the effect of: () Q ; (b) ; (c) nd F on the primry resonnce solution Figures -6 represent the frequency-response curves for super-hrmonic resonnce of order three. In Fig. we observe tht the response mplitude hs multi-vlued curves which bent to the right showing tht there ppers in the system the hrdening behvior. The multi-vlued curves consists of two stbles brnches. Then bistbility phenomenon is obtined. Figure.Typicl frequency response curves of the super-hrmonic resonnce solution for the prmeter rs:,., Q. nd F. As in Fig. (), the sme vritions of liner dmping fctor Q is obtined in Fig.5(). In Fig.5(b), when increses we note n importnt displcement of the curves towrds the right with respect to typicl frequency response curves shown in Fig... Moreover the mimum mplitude of the response of these curves decresess widely. For certin negtivee vlues of, the softening behvior

14 is observed in the system. In Fig5(c) we notice tht the system ehibits the softening behvior when becomes negtive. The upper nd lower brnches hve stble solutions. When increses the mimum mplitude decreses highly nd the correspon nding curves move to the right with respect to typicl frequency response curves with. For positive vlues of hrdening behvior is obtined nd the mimum mplitude of the response decreses with decresing of (Fig.5(d)). Fig.6 shows the vrition of ecittion mplitude F. In this figure, we note eqully the sme effects produced by in Fig.5(d) ).

15 Figure 5. Frequency response curves illustrting the effects of: () Q ; (b) nd (c,d) on the :super-hrmonic resonnce solution Figure 6. Frequency response curves showing the effects of F on the :super- resonnce of order /. In Fig.7 we note thtt the response mplitude is obtined for positive vlues of detuning prmeter. This solution is ovl nd it is hrmonic resonnce solution Figures7-9 represent the frequency-response curves of the sub-hrmonic composed of two brnches whichh bent to the right showing the hrdening behvior. The upper nd lower brnches hve stble nd unstble solutions nd there eist two sddle node bifurctions. 5

16 Figure 7. Typicl frequency response curves of the / sub-hrmonic resonnce solution for the prmeter rs:,, Q. nd F When the liner dmping fctor Q increses, the mimum mplitude s well s instbility domin increse with ppernce of two sddle node bifurctions (Fig.()). In Fig.(b), when decreses nd tkes the vlue.55 the response mplitude hs ovl nd it is symmetric bout.75. For this response mplitude the upper nd lower brnches hve stble nd unstble solutions. Therefore, theree eist two sddle node bifurctions in the system. When, softening behvior ppers in the system under study. The upper brnch is stble nd unstble solutions while the lower brnch hs unstble solution. Then it eists in the system the bifurction phenomenon. When increses positively, the minimum nd mimum mplitudes increse. With., the response mplitude is symmetric bout.75 nd it presents two sddle node bifurctions. When tkes the vlues.5 the two brnches of the mplitude response bent to the left showing the softening behvior. The upper nd lower brnches hve stble nd unstble solutions with ppernce of two sddle node bifurctions (Fig.(c)). For negtivee vlues of hrdening behvior is observed with high vlues of detuning prmeter.we lso note the presence of two sddle node bifurctions in the system (Fig(d)). We conclude thtt some positive vlues of, the system my switch from the softening behvior to the hrdening behvior nd vice vers. But for negtive vlues softening behvior is only obtined. In Fig.9, we observe tht when F increses the mimum mplitude increse with incresing of the lso eist two sddle node bifurctions. minimumm mplitude nd instbility domin. There 6

17 5. Conclusion In this work the nonliner resonnces nlysis of nonliner RLC seriess circuit modeled by modified Vn der Pol oscilltor is investigted. The multiple scles method is used to investigte the vrious resonnt sttes of the system. It is found tht the system modeled by modified Vn der Pol oscilltor presents three resonnt sttes such s primry resonnce, sub-hrmonic resonnce of order / nd super-hrmonic resonnce of order. The stedy-stte solutions nd their stbilities in ech resonnce re solved numericlly. The obtined numericl results disply tht the system presents hrdening or softening behvior, bistbility, hysteresis, jump nd bifurction phenomen. The effects of different prmeters of the system on the ech stedy-stte response re studied. It is found tht the prmeters of the system cn be used to control the mplitude of the nonliner response of the system. Finlly, we cn point out tht the coefficients of inertil non-linerity nd vn der pol respectively re responsible for the hrdening or softening behvior of the system in the three resonnce sttes nd the sub-hrmonic nd super-hrmonic resonnces obtined. 7

18 Figure. Frequency response curves showing the vritions of: nd (c, d) on the sub-hrmonic resonnce of order / solution () Q ; (b) Figure 9. Frequency response curves ehibiting the effect of F on the sub- hrmonic resonnce of order / solution

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