Particular Matrix in the Study of the Index Hour Mathematical Model

Transcription

1 ANALELE UNIVERSITĂŢII EFTIMIE MURGU REŞIŢA ANUL XXI, NR. 2, 214, ISSN Mihel Poienr, Dn Ştefn Georgescu, Ovidiu Mgdin Ţnţ, Niţn Ilie, Olriu Elen Dniel, Dorel Cernomzu Prticulr Mtrix in the Study of the Index Hour Mthemticl Model The three phse trnsformer clock hour figure mthemticl model cn be conceived in his regulr form s 3X3 squre mtrix, clled mtrix code, or s mtrix eqution, clled code eqution nd is conceived through the elementry mtrices: M, M b, M c or by defining mtrices: M 1, M 1, M 1. The code eqution expression is dependent on the definition function: the sgn function or the trivlent vrible function. Interrelted with the two possibilities re shown, defined nd explined the following prticulr mtrix: trnsfer mtrix T. Finlly re presented the interreltion between these prticulr mtrixes nd highlighted the possibilities of exploittion. Keywords: clock hour figure, mthemticl model, prticulr mtrix 1. Generl considertions on the clock hour figure mthemticl model The clock hour figure mthemticl model is represented through squre mtrix with three lines nd three columns [4, 6, 9]: G i The Gi mtrix is clled code mtrix. At the sugesstion of cdemicin prof. Ph.D. Emnuel DIACONESCU, to emphsize the leding role of the Electrotechnics Deprtment of USV, in mtrix discovery, the uthors propose the cceptnce of mtrix CEUS nme. The mtrix code η ij element cn be defined through the sgn function or through the trivlent element lgebr [6], [9], [1]. In the first cse: 11 (1)

2 η ij = sgn (2) sgn = (3) sgn 1 1 (4) All the possible code mtrix obtined through this representtion re presented in Tble 1. For the defining through the trivlent element lgebr is vlid the reltion: η ij = k; where: k (5) k 2 1 (6) When defining through trivlent element lgebr the mtrix code configurtion is the one presented in the Tble 2. The studies, highlight the interdependence between mtrix code configurtion (the lines nd columns configurtion) nd the configurtion on the terminl connection fixed on the trnsformer cover. b c η η η A η η η B η η η C Figure 1. The interdependence between mtrix code configurtion nd the configurtion on the terminl connection [6], [9] 12

3 Tble 1 Tble 2 The lines position reflect the modifictions in the high voltge windings terminl nd the columns position reflect the modifiction in the low voltge windings terminl [2], [3], [4], [5], [6], [9]. These conclusion represent the most remrcble nd interesting spects. The connection between the mtrix code nd the configurtion on the terminl connection, is expressed in sugestive mnner in figure 1. The code mtrix reflects fithfully the following chnges in the trnsformer connection digrm [6], [9]: - circulr permuttion of terminl connection; - circulr permuttion of the trnsformer terminl nottion; - the convesion between them of two terminl connection t the primry winding nd secondry winding; - the inversion of the trnsformer supply power(from the high voltge to the low voltge or vice vers); - the connection vrition from the N in Z t the delt connection nd zigzg connection; - reverse direction for the phse winding wrpping; 13

4 - reversl the beginning with the end of phse winding; - the terminl nottion reversl for winding phse. Concerning to the mentioned modifiction re recommended nd used the following nottions nd symbols [6], [9]: A - for the mtrix obtined through the lines permuttion in direct orienttion (rrow orienttion): A A (7) A - for the mtrix obtined through the lines reverse permuttion (rrow orienttion): A A (8) A - for the mtrix obtined through the column permuttion in direct orienttion (rrow orienttion) A A (9) A - for the mtrix obtined through the column permuttion in reverse orienttion (rrow orienttion) A A (1) A invlc - for the mtrix obtined through the inversion of the first line with the second line nd through the inversion of the second column with the third column. 14

5 A A invlc A T for the trnspose mtrix (by chnging lines with columns): A A T () 2. Prticulr mtrix identified within the study of the clock hour figure mthemticl model Bsed on the mtrix code decomposition fcility in three elementry mtrices M, M b, M c, we obtin the generl [6], [9]: G i = M +M b + M c () where: M M c ;M b 15 ; (14) In connection with elementry mtrix re identified nd defined three defining mtrices M 1, M 1 nd M 1, : M 1 = ; M 1 = ; M 1 = This form is vlid if the mthemticl model is defined through the signum function or trough the trivlent element lgebr. The clock hour figure mthemticl model cn be configured in mthemticl terms through severl equtions code. The equtions generl form depends on (15)

6 the defining mnner (through the signum function or through the trivlent element lgebr). When defining through the sgn function the code eqution generl form hs the form [6]: M1sgnsgnTsgnT M G i = c1 (16) Tb,, 16 1where sgnt, sgnt b, sgnt c coefficient represents the first line of the code mtrix defined trough the sgn function [9]: sgn T = η ; sgn T b = η ; sgn T c = η ; where η ij Є (-1, 1, ). When defining through the trivlent element lgebr the code eqution generl form hs the form [9]: M1K,K M1G i = c(17) T M,KTTb 1where k T, k Tb, k Tc coefficients represent trivlent constnt defined through the reltion: k T = η ; k Tb = η ; k Tc = η; unde η ij Є (, 1, 2).. The code eqution expression for the sgn function defining re presented in tble 3 nd for the trivlent element lgebr re presented in tble 4. The uthors propose unified form for the code eqution Mexpression. 1qq M G i = c1 (18) qb,, M M 1where: q = η ; q b = η ; q c = η ; η, η, η being defined, by cse, through the sgn function or through the trivlent element lgebr. Given the connection between the defining mtrices M 1, M 1, M 1 nd the trnsfer mtrix T nd tht the connection is nlyticlly expressed through the reltions: T 1 = M 1 ; T 2 = M 1 ; T 3 = M 1 (19) result the generl expression of the code eqution tht Ths the form: 3 qq T2 G i = bc (2) q,, The unitry form of the code eqution cretes the conditions of expression, in the sme embodiment connected to the sgn function (figure 2) nd other connected to the trivlent element lgebr (figure 3). 1 T

7 Tble 3 Tble 4 ) b) Figure 2. The connection between the grph code configurtion nd the three phse trnsformer terminl connection configurtion ) through sgn function; b) through trivlent element lgebr [6,9] 17

8 Figure 3. The connection between the grph code configurtion nd the threephse trnsformer terminl connection configurtion q, q b, q c coefficients re defined in the sme form s in the code eqution. 3. Conclusions 1. One of the most importnt discovery of the USV s Reserch Centre EMAD is the clock hour figure mthemticl model represented in the first phse through squre mtrix with three lines nd three columns. The lines position reflect the modifiction in the high voltge windings terminl nd the columns position reflect the modifiction in the low voltge winding terminl. 2. Strting from the eqution code, the reserch hs mrked out the possibility of clock hour figure mthemticl modelling, through code eqution, respectively through code grph. 3. Modelling the clock hour figure in the three forms of expression bove mentioned, is relised through the sgn function either through trivlent element lgebr, obtining distinct mthemticl expression for either mnner of defining. 4. The unifiction possibility of mthemticl model expression form ws confirmed by the fct tht the elementry mtrices M, M b, M c, defining mtrices M 1, M 1, M 1, lgrnge mtrix L nd the trnsfer mtrix T hve the sme form nd in the cse of defining through sgn function either trough trivlent element lgebr. 5. At the end of the pper re presented uniform expression of the mthemticl model through the code eqution nd code grph tht hve the sme form through sgn function either trough trivlent element lgebr. Acknowledgment This pper hs been finncilly supported within the project entitled SOCERT. Knowledge society, dynmism through reserch, contrct number POSDRU/159/1.5/S/146. This project is co-finnced by Europen Socil Fund 18

9 through Sectorl Opertionl Progrmme for Humn Resources Development Investing in people! References [1]. Cernomzu D., Considerţii privind verificre grupei de conexiuni ferentă trnsformtorelor electrice trifzte, prin metod curentului continuu, In: Energetic, Nr. 9, Septembrie 1981, p [2]. Cernomzu D., Considerţii privind proprietăţile codurilor de conexiuni determinte prin metod curentuui continuu, Energetic, Nr. 9, Septembrie 1982, p [3]. Cernomzu D., Considerţii privind legătur între proprietăţile codurilor grupelor de conexiuni determinte prin metod curentului continuu şi posibilităţile de utilizări pentru schimbre grupelor de conexiuni l trnsformtorele trifzte, Volumul Conferinţei Nţionle de Electrotehnică şi Energetică, Timişor, 1982, vol. 3, p [4]. Cernomzu D., Delrue Ph., Milici M., Le schém équivlent de l mtrice de code de l indice horire d un trnsformtor triphsé, Anlele Universităţii Ştefn cel Mre Sucev, nul III, 1996, nr. 5, p [5]. Cernomzu D., Delrue Ph., Milici D., Les codes de lindice horire d un trnsfromteur triphsé. L éqution des codes, Anlele Universităţii Ştefn cel Mre, Sucev, nul III, 1996, Nr. 6, p [6]. Prodn C., Contribuţii teoretice şi experimentle privind conexiunile şi grupele de conexiuni l trnsformtorele electrice de forţă Teză de doctort, Sucev: Universitte Ştefn ce Mre, Fcultte de Inginerie Electrică și Științ Clcultorelor, 28. [7]. Prodn Cr., Poienr N., Cernomzu D., Contribution concerning the code equtions used to identify the clock hour figure of the vector group t three-phses trnsformer, 5th Interntionl on Electricl nd Power Engineering, EPE 28, Tomul LIV(LVIII), Fsc.4, Işi, 3-5 October 28, p [8]. Prodn C., Poienr N., Cernomzu C., The nlysis of the signls of the codes of the hour indices of hourly to t the emergence of shortcircuit in trnsformers, Interntionl Journl of Arts nd Sciences - Mediterrnen Conference for Acdemic Disciplines 2, Gozo, Mlt, Mrch 6-1, 2. [9]. Poienr N., Contribuții teoretice și experimentle privind conexiunile și grupele de conexiuni specile l trnsformtorele trifzte Teză de doctort. Sucev: Universitte Ştefn ce Mre, Fcultte de Inginerie Electrică și Științ Clcultorelor, 2. [1]. Poienr N., Prodn Cr., Cretu N.C., Olriu E.D., Cernomzu D., Considertions upon the connections estblished between the dependent vribles ssocited to the clock hour figures tht re possible t the 19

10 three-phses trnsformer, Buletinul Institutului Politehnic Işi, Tomul LIV (LVIII), Fsc.4. Electrotehnică, Energetică, Electronică. 5-th Interntionl Conference on Electricl nd Power Engineering- EPE 28, Işi, Români, 3-5 Octombrie, 28, p []. Poienr N., Creţu N.C., Cernomzu D., Introducing the mthemticl model of the clock hour figure through trivlent elements lgebr, 7th Interntionl Conference on Electromechnicl nd Power Systems SIELMEN 29, Işi, 8th - 9th of October 29, p Addresses: Prof. dr. ing. Dorel Cernomzu, Ştefn cel Mre University of Sucev, str. Universităţii, nr., 729, Sucev, dorelc@eed.usv.ro Şef lucrări dr. ing. Elen Dniel Olriu, Ştefn cel Mre University of Sucev, str. Universităţii, nr., 729, Sucev, eleno@eed.usv.ro Drd. ing. Mihel Poienr, Ştefn cel Mre University of Sucev, str. Universităţii, nr., 729, Sucev, mihel_poienr@yhoo.com Drd. ing. Dn Ştefn Georgescu, Ştefn cel Mre University of Sucev, str. Universităţii, nr., 729, Sucev, dngrig@eed.usv.ro Drd. ing. Ovidiu Mgdin Ţnţ, E-ON Servicii Tehnice SRL, str. Prcului, nr.2, 7237 Sucev, ovidiu.tnt@yhoo.com Dr. ing. Ilie Niţn, Ştefn cel Mre University of Sucev, str. Universităţii, nr., 729, Sucev, nitn@eed.usv.ro