Stability of Driving Systems with Induction Motors. A New Method of Analysis

Transcription

1 Stability of Diving Syte with nduction Moto. A New Method of Analyi Electoechanical aculty niveity of aiova 7 Decebal Steet, aiova, 44 OMANA enache@e.ucv.o Abtact: - hi pape analyze the tability of the diving yte with induction oto by two ethod.he fit one conit in uing the geoetical locu of the oot and the econd one i an oiginal ethod developed by autho. he pape detail the atheatical odel which ae ued, the eult obtained by copute iulation and the concluion obtained by caying out the analyi. Key-Wod: - Diving yte, tability, tatic convete, induction oto, iulation. ntoduction he diving yte with induction oto involve, obviouly, oe dynaic egie. hee one, at thei tun, ipoe thoough analye of the tability which confi the poibility of thei adequate opeation. n ode to cay out an adequate tudy in thi field, it i ipoed to ue an adequate atheatical odel. he odel i optiu fo the dive point of view when it geneate an a iple a poible tuctue of the contol yte. n thi ituation, becaue of the geat nube of equation and paaete occuing in the cae of the induction oto peed adjutent by odifying the upply fequency, oe iplification have been neceay. But thi fact lead to a odel which doe not eflect coectly the dynaic behaviou of the yte. So, it i ipoed to ue an a coplete a poible odel, ateialized in a geat nube of diffeential equation. o witing thi atheatical odel, which i to facilitate the tudy of the influence of the filtation cicuit paaete of a voltage and fequency tatic convete on the tability of the VS-induction oto aebly, the tuctue of uch a yte will be taen into account; thi tuctue i depicted in figue. he notation have the following eaning: - - ectifie; - - filtation cicuit; - - invete; - - indutial obot. ig.. Stuctue of diving yte. Opeation equation of yte he atheatical odel of the convete-induction oto aebly will be etablihed futhe on. hu, in accodance with efeence, the two axe voltage equation of a quiel cage induction achine, witten in opeational, have the following atix fo: h h h h d q h h h h q d d q whee i the aplace opeato. SSN: ue, Volue 7, Octobe 8

2 he equation get the following fo, by poceing it adequately and by conideing in addition that the tato voltage phao i oiented afte the eal axi : u q q h q d h d q d h h h h h h d h q h q d h d q d he following notation will be ued futhe on: - tie contant of the tato winding; - tie contant of the oto winding; - leaage coefficient of the achine winding; d h d q h q 3 he elation becoe with thee notation: q d d q d 4 he otion equation i added to thi yte: M M dt d p J 5 whee 3 q d d q h i i i i p M 6 hee eult fo 3: [ ] d h d [ ] q h q 7 he following elation eult by applying aplace tanfoation to the equation 5 and by eplacing then and given by 7: [ ] 3 p M p J d q 8 he filtation cicuit will be iulated futhe on. hi one i decibed by the following equation in opeational: 9 he peviou yte can be alo witten in the fo: SSN: ue, Volue 7, Octobe 8

3 t will be alo conideed that the invete behave lie a coon aplifie without delay tie, chaacteized by the odulation facto : d he elation can be witten by conideing that i alway equal to zeo and by eplacing adequately in accodance with the elation : d n concluion, the atheatical odel of the convete-induction achine aebly ha the following fo, by taing into account the elation 4, 8 and : q d q c c c d c q d dd D D D dd D qd 3 whee J p c. he yte i theefoe decibed by even equation which can be witten again in the nown copact fo: B Y A Y. 4 whee the input quantitie ae, and. 3 Siulation eult A Matlab poga ha been conceived with the help of the atheatical odel detailed befoe; it eult ae peented futhe on. n ode to ephaize the filtation capacito influence on tability, a nueical analyi of the poble ha been pefoed fo eveal value of the capacito. igue ha been obtained by unning the coputation poga. A few value of the even oot coeponding to fou value of the filtation capacito ae peented futhe on in ode to ephaize the eeging concluion. SSN: ue, Volue 7, Octobe 8

4 ig.. Geoetic locu of oot fo 47 μ -,88 tep 5 μ. able. oot coeponding to the filtation capacito. 47 μ 94μ,4 μ,88-499, 94,7i -3465, -444,4-4394,9-499, -94,7i -53,9-853,6-63, 3-55,5-55,5-55,6-55,5 8,4i 8,5i 8,6i 8,8i 4-55,5-55,5-55,6-55,5-8,4i -8,5i -8,6i -8,8i 5-8,4-8,4-8,4-8,5 9,i 9,i 9,i 9,i 6-8,4-8,4-8,4-8,5-9,i -9,i -9,i -9,i 7-3, -3, -3, -3, hu, while the oot 3, 4, 5, 6 and 7 ae alot "iobile", the oot and diplace at the ae tie with the inceae of the capacito value in accodance with the aow. f it value doe not exceed 85 μ the oot and have contant eal pat, the yte tability eaining pactically unchanged. Ove thi value, the tability begin to deceae becaue of the fact that coe neae the iaginay axi. An inceaed value i ipoed fo fo having an a good a poible filtation of the ectified voltage. Howeve, in the cae of the tudied convete, although thee ae eeved on the bae boad fou eat fo fixing the filtation capacito, till only two capacito of 47 μ have been connected in paallel 94 μ. One of the eaon fo pefeing thi cobination i in accodance with the one entioned befoe even the inceaed tability of the yte fo that cae. o tudying the filtation inductivity influence the following eult have been obtained fig. 3 and table. ig. 3. Geoetic locu of oot fo,-,6 H tep H. he tudied yte i alway table becaue all oot have negative eal pat. n the peviou epeentation the aow how the ene fo the filtation inductivity inceae. n ode to jutify thei ene futhe on we peent the oot value fo fou paticula value of the inductivity. able. oot coeponding to the inductivity., H, H,3 H,4 H -8789,7-3465, -665,8-49, 876,i 47,3i -8,4-53,9-665,8 49, - 876,i -47,3i 3-55,5-55,5-55,5-55,5 8,5i 8,5i 8,5i 8,5i 4-55,5-55,5-55,5-55,5-8,5i - 8,5i - 8,5i - 8,5i 5-8,4-8,4-8,4-8,4 9,i 9,i 9,i 9,i 6-8,4-8,4-8,4-8,4-9,i - 9,i - 9,i - 9,i 7-3, -3, -3, -3, A one can notice fo the peviou table, the chaacteitic equation have five iobile oot in the coplex plan 3, 4, 5, 6 and 7. he othe two, depending on the filtation inductivity value, diplace in accodance with the aow indication they coe neae the iaginay axi at the ae tie with it inceae. he eeging concluion i the ae a that one deduced in the cae, theefoe when the filte inductivity value inceae the yte tability deceae. he odulation facto influence i analyzed in the ae way. he following eult have been obtained by doing a in the peviou cae fig. 4 and table 3. SSN: ue, Volue 7, Octobe 8

5 ig. 4. Geoetic locu of oot fo -,5 tep,. able 3. oot coeponding to.,,4,5-3464,8-3465, -3465,5-3465,9-535, -534,3-53,4-59,3 3-54,6-55, -56, -57, 8,4i 8,4i 8,5i 8,6i 4-54,6-55, -56, -57, -8,4i -8,4i - 8,5i - 8,6i 5-8,5-8,5-8,4-8,4 9,5i 9,4i 9,i 8,9i 6-8,5-8,5-8,4-8,4-9,5i -9,4i -9,i -8,9i 7-3, -3, -3, -3,3 A it can be noticed, the odulation facto influence on tability i vey all. Howeve, owing to the fact that at the ae tie with it inceae the oot coe neae the iaginay axi while the othe oot ae alot fixed, we can ay that the tudied yte tability, in the entioned ituation, tend to deceae. ig. 5. Geoetic locu of oot fo 4- Ω. he aow how the ene of the oot diplaceent in the cae of the oto winding inceae, fact confied by data fo table 4, too. able 4. oot coeponding to. 3,5 Ω 5,5 Ω 7,5 Ω 8,5 Ω -3464,8-3464,8-3464,8-3464,8-535, -535, -535, -535, 3-45,9-54,6-64, -79,9 38,i 8,4i 9,i 5,i 4-45,9-54,6-64, -79,9-38,i -8,4i -9,i -5,i 5-9,3-8,5-9, -38,7 4,9i 9,5i 7,5i 4,4i 6-9,3-8,5-9, -38,7-4,9i -9,5i -7,5i -4,4i 7-66,5-3, -35, -56,4 he induction achine paaete influence on tability when conideing the convete paaete too, can alo be tudied with the help of the elation 3. he concluion ae identical with the one obtained by uing the ethod hown in efeence. he oot geoetic locu when odifying the oto winding eitance i depicted futhe on fo exeplification. he concluion eeging fo the analyi of thee eult i identical with the one fo efeence, theefoe at the ae tie with the inceae of the oto winding eitance the tudied yte tability alo inceae. he chaacteitic depicted in the peviou figue have been obtained fo the cae when: 94 μ,,3h, f5 Hz, 5 V, d,84 A, q A value eaued on the convete EGEE VNV 44. t ha alo been conideed that the achine opeate without load. hi ean that it can be conideed that by neglecting the oto cuent coponent ove the tato cuent coponent,84 A, A. 4 A new analyi ethod of tability he equation yte that i ued have the following fo [5]: d Ψ Ψ Ψ j Ψ 5 dt d Ψ Ψ Ψ j Ψ dt d h [ Ψ ] Ψ dt xt hee equation ae lineaized futhe on. SSN: ue, Volue 7, Octobe 8

6 n ode to do thi thing it i conideed that the pulation odifie in altu with a vey low value. hi vaiation will lead iplicitly to a voltage odification, in altu too, with the ae value, o that the two quantitie atio to eain contant. n thi hypothei the yte will odify a follow. Ψ Ψ Ψ Ψ d Ψ Ψ dt. [ ] j Ψ [ Ψ Ψ Ψ Ψ ] Ψ d Ψ dt j Ψ d h Ψ dt xt {[ Ψ Ψ ] Ψ Ψ }. Ψ Ψ 6 By applying aplace tanfoation to the fit two equation of the yte and, by ubtacting ebe by ebe and by neglecting the poduct of the fo, one obtain: 7 j Ψ Ψ j Ψ Ψ Ψ j Ψ d Ψ Ψ Ψ Ψ h dt x t whee the opeational vaiable i noted with. t ut alo be noticed that fo iplifying the witing and fo not poducing confuion, both in the peviou elation and in the following one, it ha been given up both to indicate the quantitie depending on, etc. and to note the with capital. f it i conideed that i not le than, in the peviou elation the following appoxiation ay be ade: j Ψ and jψ. 8 hi way, the fit two elation fo 3 becoe: j Ψ Ψ Ψ Ψ 9 he analyi of thee elation can be iplified if it i conideed. But thi iplifying hypothei lead to atifactoy eult only in the field,5. So it i ipoed to analyze the ituation when, but conideing that the tudied phenoenon i lineaized. o thi, it i conideed that the oto opeated no-load befoe odifying the fequency. n thi ituation, owing to the low fequency of the oto cuent, it active coponent ay be neglected. hu, one can wite: Ψ Ψ i id jiq id. dx By olving the yte 5 elatively to Ψ and Ψ by eplacing thee elation in 6, afte coputation, it i obtained: i d j ε j ε whee the following notation ha been ued: j ε. x x When, it eult that it can be conideed with appoxiation: Ψ and Ψ j 3 n thee condition, by applying aplace tanfoation to the elation 9, it i obtained: h e Ψ Ψ, 4 xt o, equivalently h e Ψ d Ψ d, 5 xt epectively i d h. 6 SSN: ue, Volue 7, Octobe 8

7 5 Quantitative eult uthe on, fo the tudy of the induction oto tability, the elation 7 and etablihed befoe ae ued. he fit elation can be witten in the fo: id G i h d, 7 with G 8 h he econd elation i poceed analogouly: b Aplitude phae chaacteitic. i d G, 9 whee j G ε j ε j ε j 3 By uing 3 and 5 the following configuation can be dawn. c Pulation-aplitude chaacteitic. ig. 6. Bloc chee of the achine in the entioned ituation. uthe on it i poible to pa to the tability tudy in ou concete cae by uing all thee intoductive notion. hi analyi will be ade with the help of a Matlab poga achieved on the bai of the chee depicted in the figue and of the elation 3, 4, 5 and 6. By unning thi poga the following gaphic have been obtained. d Phae-pulation chaacteitic. ig. 7. Gaphic dependence coeponding to the cae 7,5 Ω continuou line and,5 Ω dot line. a anfe locu. a anfe locu. SSN: ue, Volue 7, Octobe 8

8 Obevation With the help of a pecial achieved Matlab poga and of the chaacteitic coeponding to the cae when a paaete fo the one depicted in the econd colun of the able 5 i ucceively odified ove the initial cae, the agin of phae depicted in the thid colun of the ae table ae obtained. b Aplitude phae chaacteitic. able 5 Pa. Ab. value Pe unit Pe unit Phae [ Ω ], [H], pa. value agin [ g [ degee] ] 7,5,988 75,54,5,33 74, c Pulation-aplitude chaacteitic. d Phae-pulation chaacteitic. h J 5,5,75 75,54 4,5,593 53,7,59,97 75,54 x,549,735 69,3,58,865 75,54 x,548,694 67,3,498,63 75,54 x,438,838 75,76,4 3,4 75,54 h,3 4,3 47,65 ig. 8. Gaphic dependence coeponding to the cae ' 5,5 Ω continuou line and ' 4,5 Ω dot line. Obevation n ode to deteine the chaacteitic depicted in the peviou figue it ha been conideed that the induction oto ha the following paaete: 7,5 Ω; 5,5 Ω;,59 H;,58 H; h,498 H; J,4 g, that lead to the following pe unit value:,989;,75; x,97; x,865; x,63 x t,456 ; ; x t,45 ;,46;,958;,944; h3,4; ε,458. n the able 6 i peented the pe cent vaiation of the paaete and the pe cent vaiation of the phae agin. able 6 Paaete Pe cent vaiation of the paaete Pe cent vaiation of the phae agin 66, 6,4 8, 8,89 3,64 8,48 3,93,89 h,4,9 J 5 36,9 SSN: ue, Volue 7, Octobe 8

9 6 oncluion he following concluion can be ephaized, by analyzing the peviou eult: - the deceae of the tato winding eitance lead to the tability deceae; - the oto eitance deceae ha alo a an effect, the deceae of the achine tability and conveely; - the inceae of the tato winding inductivity lead to the tability deceae; - at the ae tie with the oto inductivity inceae the yte tability deceae; - the ain inductivity inceae ha a nontabilizing effect; - the inetia oent inceae contibute to the tability inceae. n ode to catch quantitatively thee intedependence, the following table can be filled. efeence: [] A. apeanu, S. Enache,. Vlad, oncluion egading plication of the nduction Moto Paaete on Stability in the ae of Static onvete Supply, Poceeding of EEOMOON '99, Pata, Gecia, 999, [] [3] [4] [5] [6] pp hi, Dynaic Stability Analyi Baed on Enegy-Paivity onideation, WSEAS anaction on icuit and Syte, ue 3, Volue 7, Mach 8, pp B. Deleontey, B. Jacquot,. ung, B. De onel, J. Bavad, Stability Analyi and Stabiliation of an nduction Moto Dive with nput ilte, 6 th Euopean onfeence on Powe Electonic and Application, Sevilla, 995. M.A. Enache, M. Dobiceanu, About Stability of Mechatonic Syte Diven by Aynchonou Moto, WSEAS anaction on icuit and Syte, SSN: 9-734, ue, Volue 7, Januay 8, pp S. Enache, A. apeanu,. Vlad, M. Enache, A New Method fo nduction Moto Stability Analyi when Supplying at Vaiable ecquency, Poceeding of WSEAS SAS 7, Athen, 7, SSN: 79-55, SBN: , pp. 7-. S. Enache, A. Bitoleanu, M. Enache, M. Dobiceanu, Stability of Soe Diving Syte Analyzed by Mean of oot Geoetic ocu, Poceeding of WSEAS SAS 8, odo, 8, SSN: 79-55, SBN: , pp. -5. [7] S. Enache, A. apeanu,. Vlad, M.A. Enache, Deteination of Aynchonou Machine Paaete Mean of Exhautive Exploing Method, hallenge in Powe High Voltage and Machine, Poceeding of the WSEAS 7th ntenational onfeence on Electic Powe Syte, High Voltage, Electic Machine POWE 7, Venetia, talia, Novebe -3, 7, SBN: , SSN: 79-57, pp. -6. [8] S. Enache,. Vlad, M. Enache, onideation egading nfluence of nduction Moto eitance upon Stability in cae of Opeation at Vaiable equency. hiinau, SEMEN 5, Vol., SBN GEN , 5, pp [9] S. Enache.,. Vlad, M.A. Enache, Apect egading Dynaic egie of nduction Moto ontol by the Stato lux, EEE Powe Engineeing, Enegy and Dive, Setubal, Potugal, ntenational Volue, ue, Apil 7, SBN , pp [] S. Enache,. Pejbeanu, A. apeanu,. Vlad, Apect egading Siulation of the Satuated nduction Moto ontol by the Voltage nvete oanded in uent, EEE egion 8, EOON7 - he ntenational onfeence on opute a a ool, Waaw, Poland, Septebe 7, SBN: X, p. 86. [] A. Gace, A.J. aub, J.N. ittle,.m. hopon, e' Guide to the ontol Syte oolbox, he MathWo, nc.,. [] P. Kundu, Powe Syte Stability and ontol, McGaw-Hill, nc.,. [3] J.. in,. G. Shiau, On Stability and Pefoance of nduction Moto Speed Dive with Sliping Mode uent ontol, Aian Jounal of ontol, Vol., No.,, p.. [4]. ipo, Stability Analyi of a ectifieinvete nduction Moto Dive, EEE anaction on nduty Application, 88,, 993, pp [5] H. Moull, Stabilization of an nduction th Moto Dive with eonant nput ilte, Euopean onfeence on Powe Electonic and Application, Deden, Geany, 5, pp [6] Y.V. Maaov, A New igidity Method fo Stabilizing anient Pocee By oodinated ontol Action, Poceeding of the ut ntitute of ailway Enginee, Vol., ut,, pp SSN: ue, Volue 7, Octobe 8

10 [7] M.J. Mwandoya, V. Subahanya, M.. Mohaed, Stability Analyi od a Vaiable equency nduction Moto ing D - opoition Method, Achiv fu Electotechni 68,, pp [8] Ju. H. Pa, S. Won, Stability Analyi fo Neutal Delay-diffeential Syte, Jounal of he anlin ntitute Engineeing and Applied Matheatic, Vol. 337,, No., Philadelphia SA. [9] W. Sangtungtong, S. Sujitjon, Adaptive Sliding - Mode oad - oque Obeve : t Stability Apect, WSEAS anaction on Syte, ue 7, Volue 7, July 8, pp [] B. Shahian, M. Haul, ontol Syte Deign uing MAAB, Pentice Hall,. [] V. Subahanya, K. Suenda, Stability Analyi of Vaiable equency nduction Moto ing D - Decopoition Method, Electic Machine and Powe Syte,, 996, pp. 5-. [] N. Vinatou, eoia iteelo, Editua niveitaia, aiova, 996. SSN: ue, Volue 7, Octobe 8