The modified nodal analysis method applied to the modeling of the thermal circuit of an asynchronous machine

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1 IOP Conference Series: Maerials Science and Engineering PAPER OPEN ACCESS The modified nodal analysis mehod applied o he modeling of he hermal circui of an asynchronous machine To cie his aricle: O Nedelcu e al 2017 IOP Conf. Ser.: Maer. Sci. Eng Relaed conen - PCB-level Elecro hermal Coupling Simulaion Analysis Runjing Zhou and Xuchen Shao - Comparison of wo analysis mehods on bubble size disribuion X.C. Fu, J.J. Xia, Y. Chen e al. - Analysis mehods for he pracical applicaion of fracure mechanics R A Ainsworh View he aricle online for updaes and enhancemens. This conen was downloaded from IP address on 31/03/2018 a 23:34

2 The modified nodal analysis mehod applied o he modeling of he hermal circui of an asynchronous machine O Nedelcu 1, C I Saliseanu 1, F Popa 1, B Saliseanu 2, C V Oprescu 2 and V Dogaru 3 1 Valahia Universiy of Targovise, Deparmen of Elecronics, Telecommunicaions and Energy Engineering, Targovise, Romania 2 Valahia Universiy of Targovise, PhD Suden, Targovise, Romania 3 Valahia Universiy of Targovise, Docoral School, Targovise, Romania oilia.nedelcu@valahia.ro Absrac. The complexiy of elecrical circuis or of equivalen hermal circuis ha were considered o be analyzed and solved requires aking ino accoun he mehod ha is used for heir solving. Choosing he mehod of solving deermines he amoun of calculaion necessary for applying one of he mehods. The heaing and venilaion sysems of elecrical machines ha have o be modeled resul in complex equivalen elecrical circuis of large dimensions, which requires he use of he mos efficien mehods of solving hem. The purpose of he hermal calculaion of elecrical machines is o esablish he heaing, he overruns of emperaures or over-emperaures in some pars of he machine compared o he emperaure of he ambien, in a given operaing mode of he machine. The paper presens he applicaion of he modified nodal analysis mehod for he modeling of he hermal circui of an asynchronous machine. 1. Inroducion By modeling he heaing and venilaion sysems of he elecrical machines, equivalen complex elecrical circuis ha have large dimensions are achieved and hey require a complex analysis. The analysis of hese circuis, in saionary and dynamic regime, requires he use of effecive mehods of solving, [1]. Due o he simpliciy and flexibiliy of he modified nodal analysis mehod, i is one of he mos frequenly used as i formulaes equaions for elecrical circuis and your compuer simulaes, [2]. Generally, circui analysis in a dynamic regime is reduced o he analysis of a sring of resisive circuis, by replacing coils and condensers wih discree paerns resisive of circui, associaed o an implici numerical algorihm of numerical inegraion. Developing mehods for analyzing nonlinear resisive elecrical circuis, as effecive as possible, has a special imporance, resuling from subsiuing dynamic circui elemens wih discree resisive models associaed wih an implici algorihm of inegraion. The wo heorems of Kirchhoff, ogeher wih he consiuive equaions of he sides, are he base of nodal analysis, [3], [4]. Currens on cerain sides canno always be expressed according o he parameers ha are characerisic of he respecive sides and he poenial nodes, siuaion ha does no allow he use of a classical nodal analysis mehod, for his reason is exension was appealed, o yield he modified nodal analysis mehod. Conen from his work may be used under he erms of he Creaive Commons Aribuion 3.0 licence. Any furher disribuion of his work mus mainain aribuion o he auhor(s) and he ile of he work, journal ciaion and DOI. Published under licence by Ld 1

3 By doing he analogy beween a nework of cooling venilaion and a non-linear resisive circui i can be deermined he flow values of he cooling fluid of an elecrical machine. 2. Descripion of modified nodal mehod The modified nodal analysis mehod is a generalizaion of he classical nodal mehod, which allows he analyzed circui o conain circui elemens ha are incompaible wih he classical mehod. The srucure of circuis o which hey apply he nodal mehod consiss in he following circui elemens: resisors, linear capaciors and non-coupled magneically coils; nonlinear resisors commanded in volage; nonlinear capaciors; independen sources of curren; independen sources of power ha do no form by hemselves a side. The difference beween he classical nodal mehod and he modified nodal mehod is ha in he modified nodal mehod currens of he circui elemens ha are non-compaible wih he classical nodal mehod (such as currens of non-linear coils commanded in curren and currens of linear coils magneically coupled) appear addiionally, as independen variables. Linear coils ha are non-coupled do no inroduce addiional variables, [4-7]. For he modified nodal analysis mehod, he reference is considered he poenial of one of he nodes of he circui for which i will be deemed V = 0, his node of reference in he marix of incidence A in he node which is missing. I is applied he Kirchhoff's firs heorem o he oher nodes, i.e. n 1 nodes, wriing he currens of sides depending on he poenials a nodes. The currens of sides, having null resisance, canno be expressed by hemselves depending on he poenials a nodes, bu shall be considered as independen variables found in he vecor V n1 1,, ogeher wih poenials a nodes. Depending on poenials, all nodes and ensions on he sides will be expressed as shown in relaion (1): U U U The firs Kirchhoff heorem shall be wrien as in relaion (2): P E J A P AP V AE V AE V. (1) A V J A J AP IP AE IE AJ IJ 0 n1 1,. (2) The ensions of he passive sides will deermine he currens of he passive sides, as shown in (3): I p G UP G AP V. (3) Afer hey deermined he currens of he passive sides, he expression obained will replace he firs Kirchhoff heorem, and hen he characerisic equaions of sides wih null resisance will be added, affording a corresponding marix sysem for he modified nodal mehod, relaions (4) - (6): A P P G A A I A J, (4) E E J U E AE V; U E E, (5) AP G AP V AE IE AJ J. (6) AE V E 2

4 3. Applying he modified nodal mehod in modeling an asynchronous machine wih roor in cage In Figure 1 i is presened a secion of an asynchronous machine wih a roor in a cage, in which case he roor is he hermal source and calculaions mus be made for he hermal resisance s corresponding hea ransmission, developed in he roor, ouwards, [8]. Figure 1. Asynchronous machine wih roor in cage 1 ax; 2 roor yoke; 3 roor noch; 4 roor windings; 5 air gap; 6 saor noch; 7 saor ooh; 8 saor yoke; 9 radiaor; 10 housing Figure 2 shows he equivalen circui diagram wih heaing sysems (in saionary regime). Using he program PDCEN Program of Roing Decompose of Elecrical Circui afer he Nodes, i will decompose afer he nodes he circui from Figure 2, [9]. The program, [10], is wrien in C++ and implemened on a microcompuer ype PC compaible wih IBM, used for he numerical inegraion of an ordinary differenial equaion generalized wih he firs order regressive mehod. Dynamic circui elemens are replaced wih discree resisive circuis associaed wih hese numerical mehods, and for he analysis of elecrical circuis he program used he modified nodal mehod, [10], [11]. Running he program he resuls presened in ables 1 and 2 were obained. The sub-circuis srucure is presened in Table 1. Table 1. Sub-circuis srucure SUBCIRCUITS=8. SUBCIRCUIT 1 of size SUBCIRCUIT 2 of size 1 2 3

5 SUBCIRCUIT 3 of size SUBCIRCUIT 4 of size SUBCIRCUIT 5 of size 1 25 SUBCIRCUIT 6 of size SUBCIRCUIT 7 of size SUBCIRCUIT 8 of size The nodes of inerconnecion are: {1,3,8,10,12,20,24,26,32}. Table 2 shows he disribuion of 9 inerconnecion nodes o he 8 sub-circuis obained. Table 2. Disribuion he 9 inerconnecion nodes o he 8 sub-circuis obained 9 INTERCONNECTION NODES INTERCONNECTION NODE 32 IS ADJACENT TO: CLUSTER 8 OF SIZE 4 (5 CONNECTIONS) CLUSTER 7 OF SIZE 2 (2 CONNECTIONS) CLUSTER 6 OF SIZE 3 (2 CONNECTIONS) CLUSTER 5 OF SIZE 1 (1 CONNECTIONS) CLUSTER 4 OF SIZE 4 (3 CONNECTIONS) CLUSTER 3 OF SIZE 4 (2 CONNECTIONS) BEST CLUSTER FOR NODE 32: 8 OF SIZE 4 (5 CONNECTIONS) INTERCONNECTION NODE 12 IS ADJACENT TO: CLUSTER 8 OF SIZE 4 (1 CONNECTIONS) CLUSTER 4 OF SIZE 4 (1 CONNECTIONS) BEST CLUSTER FOR NODE 12: 8 OF SIZE 4 (1 CONNECTIONS) INTERCONNECTION NODE 20 IS ADJACENT TO: CLUSTER 8 OF SIZE 4 (1 CONNECTIONS) CLUSTER 6 OF SIZE 3 (1 CONNECTIONS) CLUSTER 3 OF SIZE 4 (1 CONNECTIONS) BEST CLUSTER FOR NODE 20: 6 OF SIZE 3 (1 CONNECTIONS) INTERCONNECTION NODE 26 IS ADJACENT TO: CLUSTER 7 OF SIZE 2 (1 CONNECTIONS) CLUSTER 4 OF SIZE 4 (1 CONNECTIONS) BEST CLUSTER FOR NODE 26: 7 OF SIZE 2 (1 CONNECTIONS) INTERCONNECTION NODE 3 IS ADJACENT TO: CLUSTER 6 OF SIZE 3 (1 CONNECTIONS) CLUSTER 2 OF SIZE 1 (1 CONNECTIONS) CLUSTER 1 OF SIZE 4 (1 CONNECTIONS) BEST CLUSTER FOR NODE 3: 2 OF SIZE 1 (1 CONNECTIONS) INTERCONNECTION NODE 24 IS ADJACENT TO: CLUSTER 5 OF SIZE 1 (1 CONNECTIONS) CLUSTER 4 OF SIZE 4 (1 CONNECTIONS) BEST CLUSTER FOR NODE 24: 5 OF SIZE 1 (1 CONNECTIONS) 4

6 INTERCONNECTION NODE 10 IS ADJACENT TO: CLUSTER 4 OF SIZE 4 (2 CONNECTIONS) CLUSTER 1 OF SIZE 4 (1 CONNECTIONS) BEST CLUSTER FOR NODE 10: 4 OF SIZE 4 (2 CONNECTIONS) INTERCONNECTION NODE 8 IS ADJACENT TO: CLUSTER 3 OF SIZE 4 (1 CONNECTIONS) CLUSTER 1 OF SIZE 4 (1 CONNECTIONS) BEST CLUSTER FOR NODE 8: 3 OF SIZE 4 (1 CONNECTIONS) FINAL NUMBER OF INTERCONNECTION NODES: 9; 0 COULD BE MOVED DIRECTLY TO CLUSTERS _INTERCONN CLUSTERS=8 CLUSTERS CLUSTER 1 OF SIZE CLUSTER 2 OF SIZE 1 2 CLUSTER 3 OF SIZE CLUSTER 4 OF SIZE CLUSTER 5 OF SIZE 1 25 CLUSTER 6 OF SIZE CLUSTER 7 OF SIZE CLUSTER 8 OF SIZE _CLUSTERS RENUM

7 _RENUM Figure 2. The decomposiion of hermal circui of he asyncronous machine in subcircuis Afer he decomposiion circui, he nodal marix has he bordered block diagonal srucure as in Figure 3. 6

8 Figure 3. Nodal marix srucure afer decomposiion circui 4. Conclusions The modified nodal mehod does no require resricions of he circui srucure o be analyzed, [12]. For he analysis, in a saionary regime and also in a dynamic regime, of he equivalen complex elecrical circuis wih large dimensions which are obained from modeling heaing and venilaion sysems of he elecrical machines, he modified nodal analysis mehod is used. The bordering nodal marix consiss of rows and columns corresponding wih poenial inerconnecion nodes. In he case of he modified nodal mehod, he bordering is formed wih rows and columns which correspond o he nodes of inerconnecion and he command currens of inerconnecion. The purpose of a hermal calculaion for elecrical machines is o esablish heaing, overruns of emperaures or over-emperaure in some pars of he machine compared o he ambien emperaure, a a given operaing mode of he machine. References [1] Hănţilă F 1979 A Mehod for Solving Nonlinear Resisive Neworks, Roum. Sci. Techn. Élecroechn. e Énerg. 24(2) [2] Iordache M and Dumiriu L 2004 Compuer-aided analysis of non-linear analogue circuis (in Romanian: Analiza asisaă de calculaor a circuielor analogice neliniare), Poliehnica Press Publishing House, Buchares, Romania [3] Iordache M, Dumiriu L and Maei I 1999 Symbolic analysis based on Modified NOdal Mehod, User s Manual (in Romanian: ASINOM Analiza SImbolică bazaă pe Meoda Nodală Modificaă, Manual de uilizare), Deparmen of Elecrical Engineering, Universiy Poliehnica of Buchares [4] Schwarz A E 1987 Compuer-aided design of microelecronic circuis and sysems, Academic Press, London, UK 7

9 [5] Chua L O, Desoer C A and Kuh E S 1987 Linear and nonlinear circuis, McGraw-Hill, N.Z. [6] Topan D and Mandache L 2002 Mehods of analysis in complex elecric circuis (in Romanian: Meode de analiză în circuie elecrice complexe), Universiaria Publishing House, Craiova, Romania [7] Iordache M 2007 Modified Nodal Mehod (in Romanian: Meoda Nodală Modificaă), Simpozionul Naional de Elecroehnica Teoreica SNET 07, Buchares, Romania, Ocober 12-14, pp [8] Magureanu R and Vasile N 1990 Brushless synchrounous servomoors (in Romanian: Servomooare fara perii ip sincron), Technical Publishing House, Buchares, Romania [9] Iordache M and Perpelea M 1995 Compuer-aided analysis of large complex linar and nonlinear elecric circuis (in Romanian: Analiza asisaă de calculaor a circuielor elecrice şi elecronice neliniare complexe de mari dimensiuni), Didacic and Pedagogical Publishing House, Buchares, Romania [10] Dumiriu L and Iordache M 1998 Modern heory of elecric circuis- Vol. I: Theoreical background, Applicaions, Algorihms and Compuaional sofwares (in Romanian: Teoria modernă a circuielor elecrice Vol. I: Fundamenare eoreică, Aplicaţii, Algorimi şi Programe de calcul), All Educaional S.A. Publishing House, Buchares, Romania [11] McCalla W J 1988 Fundamenals of compuer-aided circui simulaion, Kluwer Academic Publishers, Boson, USA [12] Nedelcu O, Iordache M and Enescu D 2006 An efficien mehod for compuing of he elecric machine venilaion, The Sixh WESC, Torino, Ialy, July 10-12, p 251 8