EEKTRYKA 06 ezyt (38) Ro XII Agez JAKUBOWSKA-CISEK, Juz WACAK Sle Uverty of Techology ANAYSIS OF RESONANCE PHENOMENA IN COMPEX FRACTIONA ORDER CIRCUITS Sury. Aly of the reoce pheoe coplex crcut of the cl R β C α,.e. cotg t let three frctol-order eleet, h ee preeted the pper. Fro the geerl reoce codto relto decrg reoce frequece hve ee derved. The uer of reoce frequece deped o the vlue of: α, β, β d C α. The ly h ee llutrted y ulto exple. Keyword: phe d pltude reoce, coplex crcut of the cl R β C α, frctol order cpctce d ductce. ANAIA JAWISK REONANSOWYCH W ŁOŻONYCH OBWODACH UŁAMKOWEGO RĘDU Strezczee. W rtyule przedtwoo lzę zw rezoowych w złożoych owodch ly R β C α, tz. zwerących co e trzy eleety ułowego rzędu. ogólych wruów rezou wyprowdzoo zleżośc oreślące czętotlwośc rezoowe. Ozue ę, że lcz czętotlwośc rezoowych oże zleżeć od wrtośc pretrów: α, β, β C α. Uzye wy lzy zlutrowo przyłde yulcyy. Słow luczowe: rezo fzy pltudy, złożoy owód ly R β C α, poeość ducyość ułowego rzędu.. INTRODUCTION Theoretcl reerch to frctol-order crcut focue otly o two ue: ly of reoce properte [] d tret tte [] thee crcut. Prole of cotructg ew frctol-order eleet [3] cottute eprte feld of reerch. Aly of reoce effect tg plce frctol order crcut cotg rectce eleet β, C α d tdrd (clcl) eleet h ee the uect of y pulcto [4-8].
8 A. Juow-Cze, J. Wlcz The curret pper devoted to ly of phe d pltude reoce pheoeo hgher-order crcut cotg rectce eleet β, C α. The ter "hgher order" relte to crcut cotg t let three rectce eleet of frctol order.. FORMAIATION OF THE PROBEM Ay two-terl pve crcut (Fg.) cotg frctol order eleet wth pedce defed : ~ ~, () C ~, ~ ~,, ~, C ~ W, () C ~ y e decred y pedce () ee fro put terl. Th pedce defed y rtol fucto wth o-tegrl expoet:,, W. (3) Fg.. Model of the pve two-terl crcut cotg frctol order cpctce d ductce Ry.. Model pywego dwó zwerącego poeośc ducyośc ułowego rzędu I frequecy do pedce () rtol fucto of frctol order defed y forul: Y. (4) The codto for phe d pltude reoce geerto uch crcut hve ee lyzed uequet ecto of the pper.
Aly of reoce pheoe coplex 9 3. PHASE AND AMPITUDE RESONANCE Bg o phe reoce codto: 0 I, (5) 0 I Y, (6) we c coclude tht reot frequece for crcut how Fg. y e foud y olvg oler equto: 0, (7) for vrle ω. Slrly, for pltude reoce codto d voltge upply: 0 I, (8) d for curret upply: 0 U, (9) the followg reltohp y e derved:, (0) () Solvg thee equto for vrle ω llow u to detere reoce frequece for phe reoce (Eq.7) d pltude reoce (Eq.9 d 0). Solvg oler equto (7), (0) d () pole y uercl ethod. The etto of uer of potve oluto y qulttve ethod very dffcult [9]. 4. EXAMPE OF FRACTIONA-ORDER CIRCUIT Exple of reoce crcut cotg three frctol-order eleet how Fg.. It copoed of two del frctol-order ductce β d β coected prllel d frctol-order cpctce C α coected ere.
0 A. Juow-Cze, J. Wlcz Fg. Model of the exeplry coplex frctol-order crcut of the cl R β C α Ry. Model przyłdowego złożoego owodu ułowego rzędu ly R β C α Ipedce (ω) of th crcut defed : C, () Tg to ccout phe reoce forul (Eq.5), the followg oler equto derved: C co co -. (3) Soluto of Eq.3 y e oted y fdg terecto pot of rght-de d left-de equto chrctertc,.e. fucto defed :, C f, (4) co co, f -. (5) Th ethod llutrted Fg.3. Crcut preter hve ee elected : C α =0.045 F/ (- α), β = 0.4 H (-β), β = 0.05 H (-β), β = 0.4, β = 0.86. The clculto hve ee crred out for chge of preter α ϵ <0, >. Sulto hve ee ru wth d of Mthetc oftwre.
Aly of reoce pheoe coplex Ry. 3 Wyzcze czętotlwośc rezou fzowego z wruu (7) dl ułdu z ry.. Fg. 3 Deterto of phe reoce fro codto defed y Eq.(7) for crcut how Fg.. The ulto how tht depedg o the electo of crcut preter vlue, th crcut chrcterzed y oe, two or ore phe reoce frequece. 5. CONCUSION Reult of ly of reoce pheoe coplex frctol-order crcut,.e. crcut cotg t let three frctol-order rectce eleet re preeted the pper. Bg o phe d pltude reoce codto, geerl reltohp decrg reoce frequece hve ee derved. Thee reltohp re defed y trogly oler trcedetl equto; olvg uch equto effectvely pole y uercl ethod. Aly of reoce effect h ee llutrted wth the help of ulto exple. If crcut preter le elected pproprtely, the t pole to detfy two reoce frequece for ller vlue of frctol-order cpctor' α preter. I lyzed ce, f α creed, the gle reoce frequecy y pper. REFERENCES. Rdw A.G.: Reoce d qulty fctor of the R α C α frctol crcut. IEEE J. Eerg. Sel. Top. CAS 3, 03, p. 377 385.. Juow A., Wlcz J.: Aly of the tret tte ere crcut of the cl R β C α, Crcut. Syte d Sgl Proceg, Specl Iue: Frctol-Order Crcut d Syte: Theory, Deg d Applcto 06, Vol. 35, No. 6, p. 83-853. 3. Trpthy M.Ch., Modl D., Bw K., Se S.: Experetl tude o relzto of frctol ductor d frctol-order dp flter. Itertol Jourl o Crcut Theory d Applcto 05, o. 43, pp. 83-96.
A. Juow-Cze, J. Wlcz 4. Wlcz J., Juow A.: Aly of the reoce pheoe ere RC crcut wth upercpctor, Meureet Autoto d Cotrol, vol.0, pp. 05 08, 03. 5. Wlcz J., Juow A.: Reoce ere frctol order R β C α crcut, Przegląd Eletrotechczy, r. 90, vol. 4, 04, pp. 0-3. 6. Wlcz J., Juow A., Aly of the prllel reoce crcut wth upercpctor. Poz. Uv. Tech. Acd. Jour. Electr. Eg. 04, o. 77, pp. 93-99. 7. Serocu D., Srw, M. Twrdy: Reoce pheoe crcut wth ultrcpctor, th It. Cof. o Evroet d Electrcl Egeerg (EEEIC), 03, pp. 97-0. 8. Rdw A.G., Reoce d qulty fctor of the R α C α frctol crcut. IEEE Jourl o Eergg d Selected Topc CAS 03, Vol. 3, pp. 377-385. 9. Turowcz A.: Geoetr zer weloów. PWN, Wrzw, 967. Mgr ż. Agez JAKUBOWSKA-CISEK, Prof. dr h. ż. Juz WACAK Sle Uverty of Techology Fculty of Electrcl Egeerg, Ittute of Electrcl Egeerg d Coputer Scece ul. Adec 0 44-00, Glwce Tel. (3) 37-0-44; e-l: gez.uow@poll.pl Tel. (3) 37--90; e-l: uz.wlcz@poll.pl