Computational Modelling of the Unbalanced Magnetic Pull by Finite Element Method

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1 Avalable onlne at Proceda Engneerng 48 (2012 ) MMaMS 2012 Computatonal Modellng of the Unbalanced Magnetc Pull by Fnte Element Method Martn Donát a * a Brno Unversty of Technology, Faculty of Mechancal Engneerng, Insttute of Sold Mechancs, Mechatroncs and Bomechancs, Techncká 2896/2, Brno , Czech Republc Abstract Ths paper presents a process of calculatng of the unbalanced magnetc pull n the rotatng electrcal machnes based on the electromagnetc coupled feld analyss. The computatonal model s created n the commercal fnte element software ANSYS and conssts of computatonal model of the electrcal crcut of the machne, whch s drectly coupled wth the two-dmensonal computatonal model of the magnetc crcut of the machne. The unbalanced magnetc pull, actng on the rotor, s calculated from results of nonlnear harmonc analyss by Maxwell stress tensor method and by Prncple of vrtual work. The obtaned results show that both methods gve dentcal value of unbalanced magnetc pull The Authors. Publshed by Elsever Ltd Publshed by Elsever Ltd.Selecton and/or peer-revew under responsblty of the Branch Offce of Slovak Metallurgcal Socety at Faculty Selecton of Metallurgy and/or peer-revew and Faculty under of Mechancal responsblty Engneerng, of the Branch Techncal Offce Unversty of Slovak of Košce Metallurgcal Open access Socety under at CC Faculty BY-NC-ND of Metallurgy lcense. and Faculty of Mechancal Engneerng, Techncal Unversty of Košce. Keywords: Rotatng electrcal machnes; Unbalanced magnetc pull; Fnte element method; Maxwell stress tensor method; Prncple of vrtual work Nomenclature A Area of the rng segment cross secton (m 2 ) L rng segment nductance (H) l length of rng segment (m) r b rotor bar rng segment resstance ( ) s slp of the rotor (-) U 0 voltage source Greek symbols geometrcal permeance of the rng segment cross secton (-) 0 permeablty of ar (Vs/Am) electrcal resstvty of materal ( m) electrcal resstvty of rng segment materal ( m) c * Correspondng author. Tel.: ; E-mal address: Donat.Martn@emal.cz Publshed by Elsever Ltd.Selecton and/or peer-revew under responsblty of the Branch Offce of Slovak Metallurgcal Socety at Faculty of Metallurgy and Faculty of Mechancal Engneerng, Techncal Unversty of Košce Open access under CC BY-NC-ND lcense. do: /j.proeng

2 84 Martn Donát / Proceda Engneerng 48 ( 2012 ) Introducton Rotatng electrcal machnes have a small ar gap between a rotor and a stator. Magnetc felds, operatng n ths ar gap, nduce electromagnetc forces, whch act on the machne structure. Ideally, the ar gap thckness s constant along a whole crcumference and the electromagnetc forces cancel out each other. In fact, the ar gap thckness s not constant along a whole crcumference, for example due to geometrc devatons of the stator and the rotor from deal cylndrcal shape, a statc eccentrcty or a dynamc eccentrcty, therefore the electromagnetc forces do not cancel out each other and the resultng electromagnetc force s called the unbalanced magnetc pull (UMP). The UMP s composed of a radal and a tangental component. The radal component of the UMP s usually sgnfcantly larger than the tangental one. Thus the UMP acts approxmately n drecton of mnmal ar gap thckness and ts magntude depends on the ar gap eccentrcty. Ths dependence s generally nonlnear. In the last decade, several publcatons, dealng wth the methodology of calculaton of the UMP, have been publshed. The computatonal models used n these publcatons can be dvded n two groups. The frst group are analytcal computatonal models that are typcally used n combnaton wth the de Laval model of the rotor. Examples of the analytcal computatonal models are shown for example n [1-3]. The advantage of these computatonal models s ther smplcty and thus short computatonal tme; therefore these computatonal models are used for study of the nfluence of the UMP on a stablty of the rotor moton. These computatonal models do not nclude some phenomena that can sgnfcantly affect the magntude of the UMP, for example the effect of slot harmoncs and saturaton effect of a ferromagnetc core of rotor and stator wndngs. The second group are complex computatonal models where the computatonal model of the magnetc crcut of the machne and the computatonal model of the electrcal crcuts of the machne are coupled drectly together and solved smultaneously by the fnte element method. These computatonal models allow to nclude a wde range of the effects, not only the slot harmoncs effect and the effect of saturaton but also losses caused by eddy currents etc. The dsadvantage of these computatonal models s hgh demands on computng power; therefore these computatonal models began to be used n the last few years. Examples of the fnte element computatonal models of the UMP are show for example n [3] and [4]. These authors are from the Department of Electrcal Engneerng at Helsnk Unversty of Technology and for calculatng of the UMP they used specal software FCSMEK, whch was developed n ths Department. The am of ths study s to create the fnte element computatonal model of the cage nducton motor, n the commercal software ANSYS, whch allows to study the nfluence of dfferent types of the ar gap eccentrcty on the UMP. 2. Methods The ANSYS software enables a drect couplng between the electrcal crcut model and the magnetc crcut model of the machne Magnetc crcut model of the machne It s assumed that the magnetc feld nsde the machne s constant n z-drecton; therefore two-dmensonal model of the magnetc crcut of the machne was used, as Fgure 1 shows. The magnetc crcut of the machne was dscretzed by PLANE 53 element. The rotor was rotated between the smulatons; therefore the fnte element meshes of the rotor and the stator were generated separately. On the outer dameter of the rotor wndng and on the nner dameter of the stator wndng the same number of nodes was generated. The ar gap between the rotor and the stator was dvded nto three parts. The part of the ar gap adjacent to the rotor was dscretzed wth the rotor, the part of the ar gap adjacent to the stator was dscretzed wth the stator and both fnte element mashes were connected by one layer of elements that have been generated drectly. Fgure 2 shows a detal of the mesh of the ar gap. Materal propertes of the ron core were descrbed by B-H curve. The materal propertes of the other parts of the model were descrbed by relatve permeablty and resstvty of the stator wndngs and the rotor bars.

3 Martn Donát / Proceda Engneerng 48 (2012 ) Fg. 1. Model of the magnetc crcut of the machne Fg. 2. Detal of the ar gap mesh 2.2. Electrcal crcut model of the machne The electrcal crcut model of the machne s composed of the electrcal crcut of the rotor and the electrcal crcut of the stator. The electrcal crcuts were created by CIRCU124 elements. The electrcal crcut of the stator s shown n Fgure 3. Ths crcut conssts of three phase AC voltage source - U 0, connected n the star, and stranded cols. Each stranded col represents one slot of the stator wndng. The electrcal crcut of the rotor conssts of massve conductors, whch represents the rotor bars r b, connected by shortng rngs. Elements represent the rng segment resstances and elements L represent the rng segment nductances. A part of the rotor crcut model s n Fgure 4. Accordng to [6], the rng segment resstance and the rng segment nductance can be calculated from equatons (1) and (2). R = ρ l A (1) L c μ 0 l = λ (2) Where c s the electrcal resstvty of the rng segment, l s the length of the rng segment, A s the area of the rng segment cross secton, 0 s the permeablty of ar and s the geometrcal permeance of rng segment. L L U 0 0 U 0 r b r b U 0 L L Fg. 3. The electrcal crcut model of the stator Fg. 4. Part of the electrcal crcut model of the rotor 2.3. Couplng of the magnetc crcut model wth the electrcal crcut model The ANSYS software allows a drect couplng between the electrcal crcut model and the magnetc crcut model of the machne. The elements that represent the stranded cols n the stator crcut model are determned by three nodes, as shown

4 86 Martn Donát / Proceda Engneerng 48 ( 2012 ) n Fgure 5. The nodes I and J are the nodes of electrcal crcut and have a degree of freedom of voltage. The node K s any node of the area that represents slot of stator wndng n the magnetc crcut model. Accordng to [7], the node K has the degrees of freedom CURR and EMF. CURR represents the current flowng per turn of the col and EMF represents the potental drop across the col termnals. Snce the col has only one unque current and one potental drop across the col termnals, a sngle value for each of these degrees of freedom unknowns s requred. Therefore, all nodes of the col regon n the magnetc crcut model must be coupled n the CURR degree of freedom and EMF degree of freedom. The same procedure was used for couplng the massve conductors of the rotor crcut model wth the rotor bars n the magnetc crcut model. Fg. 5. Prncple of couplng [7] 2.4. Calculaton of UMP UMP s the resultant of electromagnetc forces actng on the rotor of the machne. Ansys software allows to calculate the electromagnetc forces actng on the rotor by Maxwell stress tensor method or by Prncple of vrtual work. Both methods were used n ths paper. Applcaton of Maxwell stress tensor method n Ansys software requres desgnaton of Maxwell surface on whch the electromagnetc force dstrbuton s to be calculated. In ths case s that surface boundary between the rotor and the ar gap. Applcaton of Prncple of vrtual work n Ansys software requres prescrbng vrtual dsplacements equal to 1 at all nodes of the rotor and vrtual dsplacements equal to 0 at all nodes of the ar gap Boundary condtons A zero conductor, n the computatonal model of the electrcal crcut of the stator, was modeled by prescrbng zero voltage to the node 0, see Fgure 3. In order to ensure the convergence of the soluton of the magnetc vector potentals, enclosed surroundng area along the magnetc crcut model must be created. Ths enclosed surroundng area was modeled by INFIN110 elements, as shown n Fgure 6. Fg. 6. Magnetc crcut model wth enclosed surroundngs area

5 Martn Donát / Proceda Engneerng 48 (2012 ) Results The cage nducton motor wth eccentrc rotor was consdered, see Fgure 7. The nfluence of the ar gap eccentrcty on the UMP actng on the rotor of the three phase cage nducton motor was studed by nonlnear harmonc analyss. In order to consder the relatve moton between the stator and the rotor for fxed splt s, the rotor can be treated as statonary, and the resstvty of rotor bars and end rngs has to be scaled to /s, as t s wrtten n [8]. Due to consder the slot harmoncs effect, UMP was calculated n ten dfferent postons of the rotor bars due to the stator slots for each value of ar gap eccentrcty, as shown n Fgure 8. The parameters of the studed nducton motor are shown n Table I. Fg. 7. Type of ar gap eccentrcty Fg.8. Rotor postons Table 1. Parameters of cage nducton motor Item Value Unt Rated power 1.1 kw Rated lne voltage 230 V Supply frequency 50 Hz Rated speed 2845 rpm Number of stator slots 18 - Number of rotor slots 23 - Number of pole pars 1 - Number of turns of stator cols 83 - Stator outer dameter 62.5 mm Stator nner dameter mm Ar gap thckness 0.25 mm Stator core length 74 mm Connecton of stator cols star - Fgures shows a dependence of UMP on the poston angle for three values of the rotor eccentrcty x ex =0.01, 0.06 and 0.12mm. It can be seen that f the rotor eccentrcty s small the dependence of UMP on the poston angle of the rotor s approxmately snusodal. Increasng of the rotor eccentrcty leads to ncreasng of the magntude of UMP and the effect of saturaton affects strongly the dependence of UMP on the poston angle of the rotor.

6 88 Martn Donát / Proceda Engneerng 48 ( 2012 ) Fg. 9. UMP as a functon of the poston angle of the rotor, xex=1 10-5m Fg. 10. UMP as a functon of the poston angle of the rotor, xex=6 10-5m Fg. 11. UMP as a functon of the poston angle of the rotor, xex= m 4. Concluson An effcent way to assess the nfluence of the ar gap eccentrcty on UMP, based on electromagnetc coupled feld analyss n Ansys software, has been presented. The UMP actng on rotor was calculated by Prncple of vrtual works and by Maxwell stress tensor method. The obtaned results show that both methods gve dentcal value of UMP for dfferent

7 Martn Donát / Proceda Engneerng 48 (2012 ) value of ar gap eccentrcty and we can also see that the saturaton effect of ferromagnetc core affects the UMP ampltude at hgher value of ar gap eccentrcty The next step wll be to couple ths computatonal model wth the structural model of the rotor and examnng the nfluence of the ar gap eccentrcty on the dynamc behavor of the rotor of the rotatng electrcal machne. Acknowledgement The work has been supported by the grant project FSI-S-11-11/1190. References [1] Gerlando, A., et al., Analytcal modelng of unbalanced magnetc pull n sotropc electrcal machnes, Proceedngs of the 18 th Conferennce on Electrcal Machnes, p.1. [2] Guo, D., et al., The Unbalanced Magnetc Pull and ts Effects on Vbraton n Three-phase generator wth eccentrc rotor. Journal of Sound and Vbraton, 254, p [3] Lundström, N., Adanpää, J. O., Dynamc consequences of electromagnetc pull due to devatons n generator shape. Journal of Sound and Vbraton, 301, p [4] Holopanen, T. P., Electromechancal Interacton n Rotordynamcs of Cage Inducton Motors. Julkasja-Utgvare Publsher, Helsnk [5] Tenhunene, A., Electromagnetc Forces Actng Between the Stator and Eccentrc Cage Rotor. Espoo 2003, Helsnk [6] Gmden, M. H., Trabels, H., Calculaton of Two-axs Inducton Motor Model Usng Fnte Elements wth Coupled Crcut, Proceedngs of the 6 th Internatonal Mult-Conference on Systems, Sgnals and Devces, p. 207 [7] ANSYS, Inc, Documentaton for ANSYS13.0, Canonsburk [8] Yang, T., et al., Parameters and Performance Calculaton of Inducton Motor by Non-lnear Crcut Coupled Fnte Element Analyss, Proceedngs of the Internatonal Conference PEDS 2009, p.979

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