Positive unstable electrical circuits

Transcription

1 Taz KZOEK alyto Uvrty of Tchology Faclty of Elctrcal Egrg Potv tabl lctrcal crct btract: Th tablty for th potv lar lctrcal crct compo of rtor col coator a voltag crrt orc ar ar Thr ffrt cla of th potv tabl lar lctrcal crct ar propo a aalyz It how that potv lctrcal crct ar tabl for all val of thr paramtr f th lctrcal crct ha at lat o mh cotag oly ctac a orc voltag Strzcz W artyl rozpatrywa ą tabl oat low obwoy ltrycz złożo z rzytorów cw oatorów źrół apęca lb prą alzowa ą trzy róż lay oatch obwoów ltryczych tór ą tabl la wzytch wartośc woch paramtrów tz rzytacj cyjośc pojmośc Wyazao ż oat obwoy ltrycz ą tabl la wzytch wartośc woch paramtrów jżl zawrają o przyajmj jo oczo złożo tylo z cw źrół apęca Doat tabl obwoy ltrycz Kywor: potvty lctrcal lar crct tablty Słowa lczow: oatość lowy obwó ltryczy tablość Itrocto yamcal ytm call potv f t trajctory tartg from ay ogatv tal tat rma forvr th potv orthat for all ogatv pt ovrvw of tat of th art potv thory gv th moograph [ ] Varty of mol havg potv bhavor ca b fo grg coomc ocal cc bology a mc tc Potvty of lar lctrcal crct ha b ar [ ] Th fractoal lctrcal crct hav b vtgat [5 6] ymptotc tablty of potv coto-tm lar ytm wth mtal tat-fbac ha b aalyz [7] a of lctrcal crct wth tatfbac [8] Th robt tablty of potv crttm lar ytm of fractoal orr ha b ar [9] Potv lar ytm wth ffrt fractoal orr a lctrcal crct hav b cor [] a potv lar ytm cotg of bytm wth ffrt fractoal orr [] I th papr th tablty of potv lar lctrcal crct compo of rtor col coator a voltag crrt orc wll b aalyz Thr ffrt cla of th potv tabl lar lctrcal crct wll b propo Sffct coo wll b tablh r whch th potv lctrcal crct ar tabl for all val of thr paramtr Th papr orgaz a follow I cto th prlmar a problm formlato ar gv Th potv tabl lar lctrcal crct of typ ar ar cto Th tablty of potv lar ytm of typ ar aalyz cto a for potv lctrcal crct of typ cto 5 oclg rmar ar gv cto 6 Th followg otato wll b : - th t of ral m m mbr - th t of m ral matrc - th t of m matrc wth ogatv tr a matrc wth ogatv off-agoal tr I tty matrx - th t of tzlr matrc ral - th Prlmar a problm formlato or a lar lctrcal crct compo of rtor col coator a voltag crrt orc Ug th Krchhoff law w may crb th trat tat th lctrcal crct by tat qato [-7] x t x t t y t x t D t m whr x t t y t ar th tat m pt a otpt vctor a p pm D th tat varabl x t x th compot of t xt th crrt th col a voltag o th coator ar cho th compot of th pt vctor t ar orc voltag or orc crrt a th compot of th otpt vctor yt ar crrt a voltag of th lctrcal crct It wll-ow [5 -] that ay lar lctrcal crct compo of rtor col coator a voltag crrt orc ca b crb by th tat qato Dfto Th lctrcal crct crb by th qato hortly lctrcal crct call trally potv f for ay x x a vry m t t w hav x t a y t t Thorm [ 5] Th lctrcal crct potv f a oly f m p p p pm D Dfto Th potv lctrcal crct call aymptotcally tabl f lm x t for ay x t Th potv lctrcal crct wll b call tabl f t ot aymptotcally tabl Thorm [ 7] Th potv lctrcal crct aymptotcally tabl f a oly f for = whr = ar th gval ot carly tct of th tzlr matrx [ th zro of th polyomal 5 t[ I a ] trac a a a a j ] j a a t[ ] mma Th potv lctrcal crct tabl f 6 t PZEĄD EEKTOTEHNIZNY Elctrcal vw ISSN N 5a/ 87

2 88 PZEĄD EEKTOTEHNIZNY Elctrcal vw ISSN N 5a/ Proof From 5 a t t follow that f 6 hol th at lat o gval of th matrx zro y Thorm th potv lctrcal crct tabl I th followg cto thr cla of lctrcal crct wll b prt whch ar potv a tabl for all val of thr rtac = q ; ctac j j = q a capactac = q hortly for thr paramtr Potv tabl lctrcal crct I th cto a cla of lctrcal crct compo of rtor wth rtac = q ; col wth ctac j j = q a orc voltag = m whch ar potv a tabl for all val of j wll b propo Exampl or th lctrcal crct how o Fgr wth gv rtac ctac a orc voltag Fg Elctrcal crct Ug th Krchhoff law w obta th followg qato 7 Th qato 7 ca b wrtt th form 8a whr 8b Th lctrcal crct potv for all val of a ozro c a Not that 9 t From 8 t follow that t f at lat o of zro Thrfor th lctrcal crct how o Fg potv a tabl for = a all val of a ozro or for = a all val of Not that th potv crct tabl f t ha at lat o mh cotag oly ctac a orc voltag Exampl or th lctrcal crct how o Fgr wth gv rtac ctac a orc voltag Fg Elctrcal crct Ug th Krchhoff law w obta th followg qato Th qato ca b wrtt th form a whr b Th lctrcal crct potv for all val of a ozro c a Not that t From t follow that t f at lat o of zro

3 PZEĄD EEKTOTEHNIZNY Elctrcal vw ISSN N 5a/ 89 Thrfor th lctrcal crct how o Fg potv a tabl for = a all val of a ozro or for = a all val of or = a all val of Not that th potv crct tabl f t ha at lt o mh cotag oly ctac a orc voltag It ay to chc that f two of th rtac ar zro th th matrx ha a obl zro gval = = Exampl or th lctrcal crct how o Fgr wth gv rtac ctac a orc voltag Fg Elctrcal crct for xampl Ug th Krchhoff law w obta th followg qato 5 Th qato 5 ca b wrtt th form 6a whr 6b Th lctrcal crct potv for all val of a ozro c a Not that 7 t From 7 t follow that 8 t f at lat o of zro Thrfor th lctrcal crct how o Fg potv a tabl for o zro rtac a for all val of th rmag rtac a all val of ozro ctac Th potv crct tabl f t ha at lat o mh cotag oly ctac a orc voltag Th gral ca w hav th followg thorm Thorm Th potv lctrcal crct of typ tabl f t ha at lat o mh cotag oly col a voltag orc Proof To mplfy th otato th proof wll b accomplh for th potv crct how o Fg = Th tablty pt of th pt a w may am = = If = th for th mh cotag ctac w hav 8 a th mpl lar pc of row a colm of th matrx a t = y mma th potv lctrcal crct tabl Potv tabl lctrcal crct I th cto a cla of lctrcal crct compo of rtor wth coctac = q ; coator wth capactac j j = q a orc crrt = m whch ar potv a tabl for all val of j wll b propo Th corato ar mlar al to th corato cto Exampl or th lctrcal crct how o Fgr wth gv coctac capactac a orc crrt Fg Elctrcal crct

4 9 PZEĄD EEKTOTEHNIZNY Elctrcal vw ISSN N 5a/ Ug th Krchhoff law w obta th followg qato 9 Th qato 9 ca b wrtt th form a whr b Th lctrcal crct potv for all val of a ozro c a Not that t From t follow that t f at lat o of zro Thrfor th lctrcal crct how o Fg potv a tabl for = a all val of a ozro or for = a all val of If = th potv tabl crct ha o o wth brach cotag oly coator a crrt orc Exampl 5 or th lctrcal crct how o Fgr 5 wth gv rtac capactac a orc crrt Fg 5 Elctrcal crct to xampl 5 Ug th Krchhoff law w obta th followg qato Th qato ca b wrtt th form a whr b Th lctrcal crct potv for all val of a ozro c a Not that 5 t From 5 t follow that 6 t f at lat o of zro Thrfor th lctrcal crct how o Fg 5 potv a tabl for = a all val of a ozro or for = a all val of or = a all val of Th potv crct tabl f = I th ca crct ha o o wth brach cotag oly coator a crrt orc I gral ca w hav th followg thorm Thorm Th potv lctrcal crct of typ tabl f t ha at lat o o wth brach cotag oly coator a crrt orc Proof mlar to th proof of Thorm Potv tabl typ lctrcal crct I th cto a cla of lctrcal crct compo of rtor wth rtac = q ; col wth ctac j j = q ; coator wth coctac = q a orc voltag l l = m whch ar potv a tabl for all val of paramtr wll b propo Exampl 6 or th lctrcal crct how o Fgr 6 wth gv rtac octac ctac capactac a orc voltag Fg 6 Elctrcal crct Ug th Krchhoff law w obta th followg qato

5 7 Th qato 7 ca b wrtt th form 8a whr 8b Th lctrcal crct potv for all val of a ozro c a From 9 t t follow that t f at lat o of zro Thrfor th lctrcal crct how o Fg 6 potv a tabl for = a all val of a ozro or for = a all val of a ozro Not that for = th crct ha o mh cotag oly th ctac a orc voltag Exampl 7 or th lctrcal crct how o Fgr 7 wth gv rtac coctac ctac capactac a orc voltag Fg 7 Elctrcal crct Ug th Krchhoff law w obta th followg qato Th qato 55 ca b wrtt th form a whr b Th lctrcal crct potv for all val of a ozro c a From t t follow that t f at lat o of zro Thrfor th lctrcal crct how o Fg 7 potv a tabl for at lat o of rtac or o of th coctac zro for all val of ozro ctac a capactac If = or = th th potv tabl crct ha o mh cotg of brach wth oly ctac a orc voltag I gral ca cor th lctrcal crct how o Fgr 8 wth gv rtac coctac ctac coctac a orc voltag Fg 8 Elctrcal crct Ug th Krchhoff law w ca wrt th qato PZEĄD EEKTOTEHNIZNY Elctrcal vw ISSN N 5a/ 9

6 5 j j j j j ; j whch ca b wrtt th form 6a whr 6b ag From 6 t follow that th lctrcal crct potv for all val of th rtac a coctac a ozro ctac a capactac Not that 7 t a th potv lctrcal crct tabl f at lat o of or o of zro for ay ozro val of a From th agoal form of th matrx t follow that th mltplcty of t zro gval qal to th mbr of zro coctac a rtac If at lat o of zro th th potv lctrcal crct tabl a t ha at lat o mh cotg of brach wth oly ctac a orc voltag Thrfor w hav th followg thorm Thorm 5 Th potv lctrcal crct of typ tabl f t ha at lat o mh cotag oly th ctac a orc voltag mlar al thorm ca b formlat for potv lctrcal crct havg o o wth brach cotg oly of coator a crrt orc Thorm oclg rmar Th tablty for th potv lar lctrcal crct compo of rtor col coator a voltag crrt orc ha b ar Thr ffrt cla: typ typ a typ of th potv tabl lar lctrcal crct hav b propo a aalyz It ha b how that vry potv lctrcal crct of th thr cla tabl for all val of thr paramtr Thorm a 5 f t ha at lat o mh wth brach cotag oly col a voltag orc Th corato ar lltrat by xampl of lctrcal crct blogg to thr cla of potv a tabl crct Th corato ca b aly xt to potv fractoal lar crct [] Not that th papr oly ffct coo for tablty of th potv lar lctrcal crct hav b tablh op problm to tablh th cary a ffct coo for th tablty of th potv lctrcal crct cowlgmt Th wor wa pport by Natoal Scc tr Pola r wor S/WE// EFEENES [] Fara al S Potv ar Sytm; Thory a pplcato J Wly Nw Yor [] Kaczor T Potv D a D ytm Sprgr Vrlag oo [] Kaczor T Potvty a rachablty of fractoal lctrcal crct cta chaca t tomatca Vol No - [] Kaczor T Potv lctrcal crct a thr rachablty Proc of omptr pplcato Elctrcal Egrg prl - Poza Pola [5] Kaczor T aly of fractoal lctrcal crct trat tat OITNS Szczyr Pola [6] Kaczor T Sglar fractoal lar ytm a lctrcal crct It J ppl ath ompt Sc Vol No 79-8 [7] Kaczor T ymptotc tablty of potv coto-tm lar ytm wth mtal tat-fbac Elctrcal vw Vol 87 No 6-66 [8] Kaczor T Elctrcal crct wth tat-fbac a r yamcal proprt armt tomato a otorg Vol 56 No [9] łowcz obt tablty of potv crt-tm lar ytm of fractoal orr ll Pol ca Sc Tch Vol 58 No [] Kaczor T Potv lar ytm wth ffrt fractoal orr ll Pol ca Sc Tch Vol 58 No 5-58 [] Kaczor T Potv lar ytm cotg of bytm wth ffrt fractoal orr IEEE Tra rc a Syt Vol 58 No 6 - [] tal PJ chl N ar Sytm rhar oto 6 [] Kaczor T ar otrol Sytm Vol J Wly Nw Yor 99 [] Kaczor T Slct Problm of Fractoal Sytm Thory Sprgr-Vrlag rl [5] Kalath T ar Sytm Prtc-Hall Eglwoo lff Nw Yor 98 [6] obroc HH Stat-Spac a ltvarabl Thory J Wly Nw Yor 97 [7] Wolovch W ar ltvarabl Sytm Sprgr-Vrlag Nw Yor 97 thor: prof r hab ż Taz Kaczor alyto Uvrty of Tchology Faclty of Elctrcal Egrg Wja 5D 5-5 alyto Pola E-mal: aczor@ppwpl 9 PZEĄD EEKTOTEHNIZNY Elctrcal vw ISSN N 5a/