numerical eddy stuff with T

COMPUTATIONAL PROBLEMS OF ELECTRICAL ENGINEERING Vol. 1, No. 1, 2011 NUMERICAL MODELING OF EDDY CURRENTS AND HEATING IN NONMAGNETIC STEEL STRUCTURAL ELEMENTS OF POWERFUL TRANSFORMERS AND ELECTRIC REACTORS A. Basova, V. Ivankov, S. Kokoshyn , I. Khimjuk , M. Myslovich Institute of Electrodynamics of Nationa l Academy of Sciences of Ukraine [email protected] Abstract. The method of computation of eddy currents, losses and heating in structural nonmagnetic steel elements of powerful electric equipment is discussed. Keywords: powerful electric, transformers, reactors. 1. Introduction Growth of power of electrical systems and expansion of system interconnection is put by the promoted requirements to reliability and economy of powerful power electric equipment. Creation of powerful unique transformers and reactor s limited in their size and weight results in the sharp rise of specific electro-magnetic and, consequently , thermal and electrodynamic load on active and structural elements of this equipment. In this connection, at pl anning and making powerful electrical equipment there are some intricate engineering problems, which can be solved by the methods of computation of electromagnetic fields, additional losses and heating in the windings and in the details of constructions, inductive parameters of windings, electrodynamic forces, methods of electromagnetic protection, questions of opt imization of structural decisions and a number of others [1, 6, 8, 10, 11]. Knowledge of the electromagnetic field distribution (eddy currents) is the basis for solution of the indicated tasks, with all other electrodynamic sizes being determined. Both analytical and numerical methods of computation of the electromagnetic fields were developed during the last de cades [2 – 6, 8, 11]. A number of algorithms were developed and on their basis the software was created for computation of electromagnetic fields and parameters of powerful engineering and electrical e quipment [6 – 8, 11 - 13]. In spite of the fact that the level of computing engineering allows now to solve numerically intricate spatial problems, however there are problems related to geometrical nonlinear tasks. As an example it is possible to present pressing plates on the bars of a magnetic system, yoke beam, insertions in walls and lids and other structural elements the thic kness of which is 200 – 300 times less than their length. In such cases complications consist not only in construction of net when the method of finite elements is used, but also in the necessity to increase computation time. In the given work the step-by-step approach method of numerical simulation of eddy currents and heating in structural elements made of nonmagnetic steel of powerful transformers and reactors is offered, which allows a three-dimensional task to convert into a two-dimensional task. 2. Problem formulation It is assumed that the el ement of construction of a transformer or reactor (pressing plate on the bar of a magnetic system, involute of yoke beam, insertions in walls and lids of tanks) is the conducting body of thickness of h and limited by the flat surfa ce S. The stream of induction ()t Bz falls on the S surface athwart [4, 5]. Thus ()t i z e B t Bω 0= , (1) where 0B is magnetic induction at 0=t , and ω is angular frequency. Passing a magnetic stream through the S surface induce eddy currents in the conducting body. It is accepted that the surface of the S conducting body is located in plane XOY, and the magnetic stream is parallel to the axis OZ. Th erefore, in future we drop the index of z. At such approach, structural elements made of nonmagnetic steel will be the sheets of arbitrary geometrical form with the cuts of arbitrary form or without them, thus their thickness of h will be less than depth of penetration ()5 . 0/ 2σωμ δ ≈ . 3. Numerical solution and results Taking into account the accepted assumptions, the task of computation of eddy currents and losses in a three-dimensional construction is converted into the task for the sheet of complicated form at the given induction of the magnetic field. Lviv Polytechnic National University Institutional Repository http://ena.lp.edu.ua A. Basova et al. 6 Using the second Maxwell’s equation [6] t B E∂ −∂=× ∇ /r r (2) and also taking into account that σ/j Err= (3) we will get tBj∂∂− = × ∇rrrσ . (4) Because 0=j divr , using electric vector potential Tr , in accordance with determination [6, 11] T jrrr× ∇ = (5) and taking into account (4), we get tB yT xT ∂∂=∂∂+∂∂σ22 22 . (6) In the equation (6) T – normal component of vector Tr . At the given normal stream of induction (1) determination of the T potential and, consequently, the vector of current density jr is converted with consideration of (5) and (6) into the boundary problem for the Poisson equation at the zero Dirichlet conditions: 02B j Tωσ− = ∇, 0=ГT. (7) The values of the components of eddy current are determined in accordance with (5) yTJx∂∂= ; xTJy∂∂− = . (8) The boundary problem (7) can be solved by means of the ANSYS software [7, 12, 13]. At the second stage it is possible to solve the task of estimation of heating, as three-dimensional task for determination of exceeding of temperature of body. The task is converted into the boundary problem for the Poisson equation with the boundary condition of convective heat exchange on the border of the Г by a volume region Ωwith permanent heat conductivity λ and permanent coefficient of heat emission α [9] () Q grad div − = θ λ, (9) ()0θ θ αθλ − =∂∂−n . (10) In (9) the Q is equal to the losses from eddy currents xJ and yJ. In accordance with these considerations the compu- tation of eddy currents and losses has been conducted in the nonmagnetic yoke beam of the three-phase power transformer AT ДТН – 250000/345. Frequency of current – 50 Hz, electric conductivity of beam of 710 11 , 0⋅ s/m, length of beam - 7150 mm, width of the area – 1312 mm, thickness of the area – 12 mm. In Fig.1 and Fig.2 the three-dimensional construction of the yoke beam and its involute is presented. Magnetic induction has been determined by the program in the computation pl ane that coincides with the central plane of the yoke beam [8]. As a result, for the three-phase system two tables of values of induction in a computation net covering the projections of involute of central plane of beam on a plane have been determined. One table contains the values of projection of vector of normal constituent of induction on the real temporal axis ()y x Bn, Re (Fig.3), the second table contains values of projection of normal cons tituent of induction on the imaginary axis ()y x Bn, Im (Fig.4). Values of vectors of eddy current density induced by ()y x Bn, Re and ()y x Bn, Im are depicted in Fig.5 and Fig.6. The distribution of total losses from eddy currents is shown on Fig. 7, and the distribution of temperature of the proper to the calculated values of total losses is shown on the Fig. 8. Fig. 1. Three-dimensional construction of the yoke beam Fig. 2. Involute of yoke beam Lviv Polytechnic National University Institutional Repository http://ena.lp.edu.ua Numerical Modeling of Eddy Currents and Heating… 7 Fig. 3. Distribution of induction of the actual making ReBn Fig. 4. Distribution of inducti on of the imaginary making ImBn Fig. 5. Distribution of vectors of eddy current density for ReBn Fig. 6. Distribution of vectors of eddy current density for ImBn . Fig. 7. Distribution of local losses Fig. 8. Distribution of temperature 4. Conclusions The proposed method of computation of eddy currents, losses and heating in structural elements made of nonmagnetic steel can be recommended at the stage of planning powerful electric equipment. References 1. Analiza i synteza pol elektromagnetycznych, Praca zbiorowa pod redakcia J. Turowskiego, Wyd. PAN, 1990. 2. Y. R. Crutzen, G. Molinari and G. Rabinacci, eds. Industrial Application of Electromagnetic Codes, vol. 1, Dordrecht, Germany, Kluwer, 1990. 3. A. Krawczyk and J. A. Tegopoulos. Numerical Modeling of Eddy Currents, Oxford, UK, Clareudon, 1995. 4. N. J. Siakavellas. Analytical modeling of eddy currents induced by time – varying magnetic field in a conducting plate. COMPEL – Int. J. Comput. Math. Electr. Electron. Eng., vol. 13, № 3, pp. 497 – 508, 1994. 5. N. J. Siakavellas. Two simple Models for Analytical Calculation of Eddy Currents in thin Conducting Plates, IEEE Trans. of Magn., vol. 33, № 3, pp. 2245 – 2257, 1997. 6. J. Turowski. Electrodynamik a techniczna, Warshawa, WNT, 1993. 7. Vector Fields Software for Electromagnetic Design, Oxford, Vector Fields Limited, 1992. 8. Иванков В. Ф. Расчет магнитного поля , потерь в баках трансформаторов и электрических реакторов . Праці інституту електродинаміки НАН України , № 1(10), 2005 р . 9. Березовский А. А. Лекции по нелинейным краевым задачам математической физики. Киев , 1974. 10. Лейтес Л. В. Электромагнитные расчеты трансформаторов и реакторов . М.: Энергия , 1981. 11. Моделирование электромагнитных полей в электротехнических устройствах , Под. ред. Сикоры Р. и Степанова А., Киев , Техника , 1990. 12. http://www.ans.com.ru 13. http://www.femm.berlios.de ЧИСЕЛЬНЕ МОДЕЛЮВАННЯ ВИХРОВИХ СТРУМІВ ТА НАГРІВАННЯ У НЕМАГНІТНИХ СТАЛЕВИХ СТРУКТУРНИХ ЕЛЕМЕНТАХ СИЛОВИХ ТРАНСФОРМАТОРІВ ТА РЕАКТОРІВ А. Басова , В. Іванков , С.Кокошин , І. Хім’юк, М. Мислович Запропоновано метод обчислення вихрових струмів та нагрівання у структурних елементах силового електро - технічного обладнання , виготовлених з немагнітної сталі . Lviv Polytechnic National University Institutional Repository http://ena.lp.edu.ua