First-order piecewise-linear dynamic circuits
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1 Frs-order pecewse-lnear dynamc crcus. Fndng he soluon We wll sudy rs-order dynamc crcus composed o a nonlnear resse one-por, ermnaed eher by a lnear capacor or a lnear nducor (see Fg.. Nonlnear resse one-por C c C Nonlnear resse one-por (a (b Fg. The one-por consss o lnear and nonlnear ressors and DC sources. The olage curren characersc o each nonlnear ressor nsde he one por wll be approxmaed by a pecewse lnear uncon, conssng o seeral lnear segmens as shown n Fg.. Consequenly, he drng-pon characersc o he one-por s also a pecewse lnear. Fg. e us concenrae on he crcu shown n Fg..a and nd he soluon ( ( C, assumng he nal capacor olage C ( VO. The por olage ( C ( a an arbrary nsan o me and he correspondng por curren ( can be consdered as he coordnaes o a pon on he drng-pon characersc. As me ncreases he pon moes along he characersc sarng rom he gen nal pon. Thus, he pon races some dreced roue, called a dynamc roue. Snce he dynamc roue consss o lnear segmens we can oban he soluon along each o he segmens separaely, usng eece echnques o he lnear
2 analyss. In hs way he analyss o he nonlnear (pecewse lnear dynamc crcus s ransormed o a sequence o analyses o lnear dynamc crcus. The descrbed approach wll be llusraed usng a numercal example. Example e us consder he crcu shown n Fg. 3.a, where he pecewse lnear approxmaon o he drng-pon characersc o he nonlnear one-por s depced n Fg. 3.b. [V] 3 P Nonlnear resse one-por 5 P P P 3 - [A] (a (b Daa:.mH, ( A. Fg. 3 To deermne he dynamc roue we wre he equaon Subsung and yelds d. ( d d d. ( The equaon ( saes ha or >, d /d < and he curren s a decreasng uncon o me. For <, d /d > and he curren s an ncreasng uncon o me. Snce ( ( A, he dynamc roue consss o he lnear segmens P P, P P, and P P 3. A he pon P 3, hence, d /d (see equaon (. The pon where d /d s called an equlbrum pon. Thus, A speces he equlbrum pon correspondng o he seady sae soluon. Below we sudy separaely all here segmens o he dynamc roue. P P The segmen P P les on he sragh lne descrbed by he equaon 5, (3 whch can be consdered as a descrpon o a lnear ressor hang he ressance R 5Ω. Thus, as long as he pon (, ( remans on he segmen P P he dynamc crcu can be represened by he model shown n Fg. 4. R 5Ω Fg. 4
3 Snce he crcu depced n Fg. 4 s lnear, s soluon s gen by he ormula where, ( ( + ( ( ( e τ ( A τ ms, (. Hence, we hae, (4 5 R 5 ( e, where s n mllseconds. The equaon (5 s ald as long as ( (, holds hence e 5 3 ln 4.6 ms. 5 Thus, or 4.6 ms he curren ( s speced by he equaon (5. P P The segmen P P les on he sragh lne descrbed by he equaon, 3 (5. For or (, + 5, (7 whch can be consdered as a descrpon o seres connecon o ressor R 5Ω and olage source V V. Hence, we oban he crcu shown n Fg. 5. (6 R 5Ω The soluon where ( V V s gen by he equaon Fg. 5 ( ( + ( ( ( ( e, (8 τ V ( ( A, ( A, τ ms. R R 5 Subsung no (8 yelds ( ( 5 3e. (9, remans on he segmen P P. The equaon (9 s ald as long as he pon ( ( Denoe he me a whch by. Then, ( and usng he equaon (9 we hae ( e. ( We sole he equaon ( or ln.73 ms. 5
4 4 P P 3 In hs case we hae + ( and he crcu s modeled as shown n Fg. 6. R Ω V V Fg. 6 The soluon s gen by he equaon where Hence, we oban ( ( + ( ( ( ( e, ( τ (, ( A, τ. ms. (. 73 ( e. (3 As he curren ends o he equlbrum ( A. To plo ( we use he equaons (5, (9 and (3 and he correspondng me nerals (see Fg. 7. ( ( e (. 73 e [ms] 3e 5 ( 4. 6 Fg. 7
5 5 Relaxaon oscllaor. Consder he crcu shown n Fg. 8 conssng o a resse pecewse lnear one por, ermnaed by a capacor. R C C C R R Fg. 8 The drng pon characersc o he resse one-por s shown n Fg. 9 where R β. R + R E + R R sa P Q B ( o Vo C βe sa R R R βe sa Q A P P R E R sa Fg. 9 To deermne he dynamc roue we wre he equaon dc C C. (4 d Snce, we oban C C
6 6 d. (5 d C The equaon (5 saes ha < hen d /d > and ncreases as ncreases. I >, hen d /d < and decreases as ncreases. Thus, he dynamc roue mus moe oward he le n he upper hal plane and oward he rgh n he lower hal plane, as ndcaed by he arrow heads n Fg. 9. Suppose ha he nal olage C ( ( V, hen he nal pon on he drng pon characersc s P (see Fg.9. The pon ((, ( moes along he loer segmen o he characersc and reaches, a some me, he breakpon Q A. Snce he curren ( s negae, Q A s no equlbrum pon. Snce boh segmens meeng a he pon Q A are oposly dreced s mpossble o connue he dynamc roue along hese segmens. Thereore we assume ha he soluon jumps rom he pon Q A o he pon P as shown n Fg. 9. Nex he soluon moes oward he le, reachng aer some me he breakpon Q B and he dynamc roue s connued by jumpng o he pon P. Thus, aer some ransen me rom P o P he dynamc roue P -Q A -P -Q B -P s closed and he soluon waeorm s perodc. The crcu operaes as an oscllaor. The waeorms o ( and ( are skeched n gures and. ( Q A, P Q A, P P P ( Fg. Q B, P P P Q B Q A Q A P P P Fg.
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