Original Method for Stability Analysis of Mechatronic Systems with Asynchronous Machines

Transcription

1 Soin Enache, Auel Campeanu, Oiginal Method fo Stability Analyi of Mechatonic Sytem with Aynchonou Machine SORIN ENACHE, AUREL CAMPEANU, ION VLAD, MONICA ADELA ENACHE Electomechanical Faculty Univeity of Caiova Decebal Steet, Caiova, ROMANIA Abtact: - Thi pape detail an oiginal method fo the tability analyi of mechatonic ytem diven by aynchonou machine. The mathematical bai of the method and a few chaacteitic obtained with an adequate Matlab pogam ae peented. The window of the pogam fo monitoing of the expeimental tet ae alo depicted. Finally a eie of expeimental tet and the concluion confiming the validity of the popoed method ae peented. Key-Wod: - tability, mechatonic ytem, aynchonou machine, vaiable fequency, paamete, oftwae. 1 Intoduction Nowaday, the utilization on an eve lage cale of the aynchonou machine a an execution element in mechatonic ytem, ha impoed an eve ample appoach, in peciality pape, of poblem concening the dynamic egime of it and, implicitly, of tability poblem. The peciality liteatue [1], [12], [13], [16], [17] etc. peent a eie of method fo the tability analyi of uch ytem. Unfotunately, thee method eithe ae vey difficult to be implemented numeically o they have the dawback that they do not allow to tudy the inetia moment influence on tability, a vey impotant thing epecially in the cae of the low powe machine. In ode to eliminate thee dawback, a new method fo the tability tudy ha been conceived, with the help of the equation with epeentative phao witten in pe unit. 2 The mathematical model The equation ytem that i ued ha the following fom [8]: d k ( k j d 0 k ( k j( d h k x t Im [( ] m (1 Thee equation ae lineaized futhe on. In ode to do thi thing, it i conideed that the pulation modifie in altu with a vey low value. Thi vaiation will lead implicitly to a voltage modification, in altu too, with the ame value, o that the two quantitie atio to emain contant. In thi hypothei the ytem (1 will modify a follow. k 0 k j( [ k( ] j( ( [ k( ] k h. xt.im ( {[( ( ] ( } m (2 The following elation i obtained by applying Laplace tanfomation to the fit two equation of the ytem (1 and (2, by ubtacting membe by membe and by neglecting the poduct of the fom : 19 Iue 1, Volume 3, Januay 2008

2 Soin Enache, Auel Campeanu, ( k j k k j 0 k k ( k j( k h Im( (, x t (3 whee i the opeational vaiable. It mut alo be noticed that fo implifying 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 them with capital. If it i conideed that i not le than 0,1 in the peviou elation, the following appoximation may be made: j 1 and j k. (4 Thi way, the fit two elation fom (3 become: 0 ( k j k k k ( k k ( k (5 ( The analyi of thee elation can be implified if it i conideed that R 0. But thi implifying hypothei lead to atifactoy eult only inide the inteval (0,5 1. So it i impoed to analyze the ituation when R 0, but conideing that the tudied phenomenon i lineaized. In thi pupoe, it i conideed that the moto opeated without load befoe modifying the fequency. In thi ituation, owing to the low fequency of the oto cuent, it active component may be neglected. Thu, one can wite: i k id j iq id. (6 dx The following elation i obtained by computation, by olving the ytem (5 elatively to and, by eplacing thee elation in (6: id 2 ( k k k( j j k ( whee the following notation ha been ued: 2 ( 1 k k x x j (7. (8 When 0, 1 it eult that it can be conideed (with appoximation: ( 1 and jk (9 In thee condition, the following elation i obtained y applying Laplace tanfomation to the elation (9: k h Re( k, (10 xt o, equivalently k h Re( d k d, (11 xt epectively h k i d. (12 3 Simulation. Quantitative Reult Futhe on, fo the tudy of the induction moto tability, the elation (7 and (12 etablihed befoe ae ued. The fit elation can be witten in the fom [8]: k id G1 ( i h d, (13 with k G1 ( (14 h The econd elation i poceed analogouly: d 2 i G ( (, (15 whee j G 2( k 2 ( k k j k ( j k ( j (16 The following configuation can be dawn by uing (13 and ( Iue 1, Volume 3, Januay 2008

3 Soin Enache, Auel Campeanu, Fig.1. Machine block cheme in the mentioned ituation. Futhe on it i poible to pa to the tability tudy in ou concete cae by uing all thee intoductive notion. Thi analyi will be made with the help of a Matlab pogam conceived on the bai of the cheme depicted in figue 1 and of the elation (13, (14, (15 and (16. The following gaphic have been obtained by unning thi pogam. b Fig.3. Tanfe locu (a and amplitude-phae chaacteitic (b obtained in the cae of the inductance modification: L 0, 528 H (1 and L 0,548 H (2. a a b Fig.2. Tanfe locu (a and amplitude-phae chaacteitic (b obtained in the cae of the inductance modification: L 0, 529 H (1 and L 0,549 H (2. b Fig.4. Tanfe locu (a and amplitude-phae chaacteitic (b obtained in the cae of the inductance modification: Lh 0, 498 H (1 and L 0,558 H (2. h Obevation 1 In ode to etablih the chaacteitic depicted in the peviou figue it ha been conideed that the induction moto ha the following paamete [6]: 1m 0,0989; 0,0725; x 2,1907; x 2,1865; x 2,0623; x t 0,2456; x t 0,2451; k 0,4026; k 0,2958; k0,9414; h32,4; 0,0458. a Obevation 2 With the help of a pecially conceived Matlab 21 Iue 1, Volume 3, Januay 2008

4 Soin Enache, Auel Campeanu, pogam and of the chaacteitic coeponding to the cae when a paamete fom the one depicted in the econd column of the table 1 i ucceively modified (ove the initial cae, the magin of phae depicted in the thid column of the ame table ae obtained. Table 1 Pa. Ab. value [ ], [H], [kgm 2 ] Pe unit pa. Pe unit value Phae magin [degee] 7,5 0, ,54 R 2,5 0, ,20 5,5 0, ,54 4,5 0, ,71 0,529 2, ,54 L x 0,549 2, ,13 0,528 2, ,54 x 0,548 2, ,31 0,498 2, ,54 L h x 0,438 1m 1, ,76 0,004 32,4 75,54 J h 0,003 24,3 47,65 R L The following concluion can be emphaized, 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 machine tability and conveely; - the inceae of the tato winding inductivity lead to the tability deceae; - at the ame time with the oto inductivity inceae, the ytem tability deceae; - the main inductivity inceae ha a nontabilizing effect; - the inetia moment inceae contibute to the tability inceae. In ode to catch quantitatively thee intedependence, the following table can be filled. Table 2 Paamete Pe cent vaiation of the paamete Pe cent vaiation of the phae magin R 66,6 2,04 R 18,2 28,89 L 3,64 8,48 L 3,93 10,89 L h 12,04 0,29 J 25 36,92 4 Pogam of expeimental veification In ode to veify the concluion emphaized befoe, the expeimental cicuit detailed in [10] ha been caied out. An oiginal pogam ha been conceived in Viual Baic fo acquiing and poceing the obtained data. Thi pogam can be un by a double click applied on the pictogam placed on the dektop (figue 5. Fig.5. Dektop. When unning the pogam, the peentation cove of the pogam appea on the diplay, figue 6 (the text coeponding to the following window ae witten in the Romanian language. Fig.6. Window Copeta. The main window of the pogam ae detailed in figue Fig.7. Window Meniu. 22 Iue 1, Volume 3, Januay 2008

5 Soin Enache, Auel Campeanu, Fig.8. Detail of menu Fiiee. Fig.13. Window Semnal. Fig.9. Detail of menu Editae. Fig.14. Window Editae. Fig.10. Detail of menu Monitoizae. Fig.15. Window Ajuto. Fig.11. Window Configuae. Fig.12. Window Achizitie. Thi pogam ha many facilitie: - allow the configuation of the data acquiition boad; - enue the wok aided by a help window. - enue the acquiition coeponding to the dynamic ignal we want; - allow the viualization in diffeent fom (line, ba, pie fo the acquied ignal; - allow to edit the file ASCII of the obtained data; - allow to ave and to type data; - enue the acce to a eie of acceoie ueful duing wok (pocket compute, clock; - allow the configuation coeponding to the wok inteface (backgound and text colou, text dimenion, icon on the dektop. 23 Iue 1, Volume 3, Januay 2008

6 5 Expeimental eult A eie of gaphic eult have been obtained with the help of the acquiition pogam detailed befoe; the following figue ae peented futhe on. Soin Enache, Auel Campeanu, 6 Concluion Thee gaphic lead to the following concluion: - when the value of the tato inductance inceae the tanient poce duation inceae (the tability deceae; - the inceae of the oto inductance alo involve the inceae of the tanient poce duation (the tability deceae; - the deceae of the main inductance value detemine a fate tabilization of the poce (tability inceae; Thee concluion confim the theoetical analyi pefomed befoe. Fig.16. Gaphic dependence coeponding to the cae L 0,529 H (1 and L 0,549 H (2. Fig.17. Gaphic dependence coeponding to the cae L 0,528 H (1 and L 0,548 H (2. Fig.18. Gaphic dependence coeponding to the cae L h 0,498 H (1 and L h 0,438 H (2. Refeence: [1] M. Alfidi And A. Hmamed, Robut Stability Analyi fo 2-D Continuou-Time Sytem Via Paamete-Dependent Lyapunov Function, WSEAS TRANSACTIONS on SYSTEMS and CONTROL, Iue 11, Volume 2, Novembe 2007, pp [2] A. Campeanu, M. Radulecu, I. Vlad, S. Enache, Influence of Co-Satuation fo Modeling and Simulation of the Dynamic Pocee in the Induction Moto, IEEE Region 8, EUROCON The Intenational Confeence on Compute a a Tool, Waaw, Poland, 9-12 eptembie 2007, IEEE Catalog Numbe: 07EX1617C, ISBN: X, Libay of Conge: , p [3] A. Campeanu, A. Ionecu, I. Vlad, S. Enache, Modelling and Simulation of Magnetic Satuation in Induction Moto, Intenational Electic Machine and Dive Confeence IEMDC 07, CD veion, AF , IEEE Catalog Numbe: 07EX1597C, ISBN: [4] A. Campeanu, I. Vlad, A. Ionecu, S. Enache, Etabliement of Electical Machine Optimum Contuctive Dimenion, ELECTROMOTION 2003, Maakeh, Maoc, vol. 1, p , INSPEC (Mateial Identity Numbe F , Inpec claification code B8310a.c. machine, Optimiation technique. [5] A. Campeanu, I. Vlad, S. Enache, Apect Regading Optimal Deign of High Powe Electical Machine, BaJov, The 5th Intenational Confeence on Optimization of Electic and Electonic Equipment, OPTIM '2002, INSPEC (Mateial Identity Numbe D , Inpec claification code B8310 a.c. machine. 24 Iue 1, Volume 3, Januay 2008

7 Soin Enache, Auel Campeanu, [6] S. Enache., I. Vlad, M.A. Enache, Apect Regading Dynamic Regime of Induction Moto Contol by the Stato Flux, Powe Engineeing, Enegy and Dive, 2007, POWERENG 2007, Setubal, Potugal, Intenational Volume, Iue, Apilie 2007, ISBN , IEEEXploe, DOI: POWERENG , pp [7] S. Enache, R. Pejbeanu, A. Campeanu, I. Vlad, Apect Regading Simulation of the Satuated Induction Moto Contol by the Voltage Invete Commanded in Cuent, IEEE Region 8, EUROCON The Intenational Confeence on Compute a a Tool, Waaw, Poland, 9-12 eptembie 2007, p [8] S. Enache, A. Campeanu, I. Vlad, M. Enache, A New Method fo Induction Moto Stability Analyi when Supplying at Vaiable Fecquency, Poceeding of WSEAS Intenational Confeence on SYSTEMS THEORY AND SCIENTIFIC COMPUTATION, Athen, 2007, ISSN: , ISBN: , pp [9] M.A. Enache, S. Enache, M. Dobiceanu, Conideation Regading Detemination of the Induction Moto Paamete, Intenational Electic Machine and Dive Confeence, Cete Iland, CD veion, AF , IEEE Catalog Numbe: 07EX1597C, ISBN: [10] M.A. Enache M.A., S. Enache, M. Dobiceanu, Influence of Induction Moto Paamete on Stability in Cae of Opeation at Vaiable Fequency, WSEAS Intenational Confeence on SYSTEMS THEORY AND SCIENTIFIC COMPUTATION, Vouliagmeny, Athen, Geece, Augut 24-26, 2007, pp [11] J. A. Feeia, A Review of Mathematical Model of Non-Linea Mechanical Sytem that Involve Fiction, WSEAS TRANSACTIONS on SYSTEMS, Iue 12, Volume 6, Decembe 2007, pp [12] N. Gilligh, C. Popp, V. Paiach, Stability Analyi of Nonlinea Dinamic Sytem, Intenational Confeence on Automation and Quality Contol, mai 1998, Cluj-Napoca, ISBN , vol. A1, pp. A61-A65. [13] A. Gace, A. J. Laub, J. N. Little, C. M. Thompon, Ue' Guide to the Contol Sytem Toolbox, The MathWok, Inc., [14] V. Ionecu, Teoia itemelo liniae, vol. I, Ed. Didactica i Pedagogica, Bucueti, [15] P. Kundu, Powe Sytem Stability and Contol, McGaw-Hill, Inc., [16] Y. V. Makaov, A New Rigidity Method fo Stabilizing Tanient Pocee By Coodinated Contol Action, Poc. of the Ikutk Intitute of Railway Enginee, Vol. 1, 2000, pp [17] J. H. Pak, S. Won, Stability Analyi fo Neutal Delay-diffeential Sytem, Jounal of The Fanklin Intitute Engineeing and Applied Mathematic, Vol. 337, No. 1, Philadelphia USA, Januay [18] N. Qaie, N. Iqbal, A Novel Nonlinea Contolle Deign fo Stabilization of the Acobot, WSEAS TRANSACTIONS on SYSTEMS and CONTROL, Iue 6, Volume 2, June 2007, pp [19] B. Shahian, M. Haul, Contol Sytem Deign uing MATLAB, Pentice Hall, [20] V. Subahmanyam, K. Suendam, Stability Analyi of Vaiable Fequency Induction Moto Uing D - Decompoition Method, Electic Machine and Powe Sytem, 11, 1996, pp [21], Guide d'etude de L'ATV 452, Edition CITEF, ISBN , [22], Matlab Ue' Guide, The MathWok, Inc., Natick, Maachuett, [23], Channel Iolation Amplifie Fou Opeating Manual, ELWE Lehyteme GmbH, 1998, Gemany. [24], Meauing Inteface Comeniu E Opeating Manual, ELWE Lehyteme GmbH, 1998, Gemany. 25 Iue 1, Volume 3, Januay 2008