Silindirik sargılı transformatörlerde alan dağılımının sonlu elemanlar metodu ile hesabı

Abstract

Geçmişte oldu.Su gibi günümüzde de tasarım problemleri ekonomik ve teknolojik güçlükler içerir. Teknolojinin gelişmesi, boyutların artması ve mühendislik sistemlerinin maliyetinin fazla oluşu daha genel ve daha doğru bilgisayar yöntemlerinde de paralel bir gelişmeyi gerekli kılmıştır. Son yıllarda sürekli olarak elektrik sistemlerinin veriminin arttırılmasının istenmesi tasarımcıları yeni sorunlarla karşı karşıya getirmiştir. Bu çalışmada, transformatör alan dağılımının hesap- lanması için sayısal bir yöntem olan "Sonlu Elemanlar Yön temi" tanıtılmaktadır. Bölüm 2' de varyasyonel hesap yön temleri ile Laplace ve Poisson denklemleri hakkında bilgi verilmiştir. Bölüm 3' de sonlu elemanlar yöntemi ayrıntılı bir şekilde anlatılmıştır. Bölüm 4, elektromagnetik alanlara ilişkin temel bilgileri içermektedir. Transformatör alan dağılımının belirlenmesinde yön temin kullanılışı Bölüm 5' de anlatılmıştır. Bu çalışmada kullanılan bilgisayar programına ilişkin bilgi Ek-A'da sunulurken, program çıktıları Ek-B bölümünde verilmektedir.

The calculation of transformer leakage flux is a prerequisite to the calculation of reaktance, short circu it forces and eddy current losses. It is therefore of fun damental importance for the transformer designer. Early methods were based upon simplifying assumpta- tions of the leakage field being unidirectional and with out curvature (not axi-symmetric). For normal cencentric windings this gives reasonably good results for reactance, radial forces, and eddy current losses in the conductors due to axial flux. But for axial forces and eddy current losses due to radial flux the methods are not applicable. Also for unusual winding arrangements with uneren ampere- turn distribution, the methods are normally too inaccura te. For flat two dimensional field applications consi derable attention has been given in recent years to fini te difference and finite element methods. The field regi on is covered by a grid, and magnetic vector potentials are calculated at the nodes, giving a numerical solution for the flux desnities. So for these methods do not appe ar to have gained any wide acceptance among transformer designers. The finite elements method is concerned with the solution of mathematical or physical problems which are usually defined in a continuous domain either by local differential equations or by equivalent global statements. To render the problem amenable to numerical treatmen, the infinite degrees of freedom of the system are dicretized or replaced by a finite number of unknown parameters, as indeed in the practice in other processes of approxima tions. The finite element method has its origin in the field of structural analysis. Although the earlier mathe matical treatment of the method was provided by Courant in 1943, the method was not applied to electromagnetic (EM) problems until 1963. The finite element analysis of any problem invol ves basially four steps: 1) discretizing the solution re gion into finite number of subregions or elements, 2) deriving governing equations for a typical element, v - 3) assembling of all elements in the solution region, 4) and solving the system of equations obtained. To construct an approximate solution by a simple finite element method, the problem region is subdivided into triangular elements (Chapter 3). The essence of the method lies in first approximating the potential V within each element in a standerdised fashion and thereafter interrelating the potential distributions in the various element so as the constrain in the potential to be conti nuous accross interelement boundaries. Within a typical triangular element, illustrated in Fig.l, it will be assumed that the potential is adequ ately approximated by the expression V = a+bx+cy (1) The true solution is thus replaced by a piecewise-planar function; the smoothly-curved, actual potential distribu tion over the x-y plane is replaced by a jevel-faceted approximation. It should be noted, however, that the po tential along any triangle edge is the linear interpolate between is two vertex values, so that if two triangles share the same vertices, the potential will be continuous across the interelement boundary. There are no gaps in the surface V(x,y) which approximates the true solution over the x~y plane; the approximate solution is piecewise- planar, but continuous everywhere y\ o Fig.l. Typical triangular finite element in x-y plane. The coefficients a, b, c in Eq, (1) may be found from the three independent simultaneous equations which are obtained by requiring the potential to assume vertex "? vi -*?. values Vl, V2' Y3 at the three vertice. Substituting the three vertex potentials and location into Eq, (1) in turn, there is obtained 1 X± y 1 X2 y. x. (2) The determinant of the coefficient matrix in Eq. (2) may be recognised on expansion as equal to twice the triangle area. Except in the degenerate case of zero area, the coefficients a, b, c are therefore readly determined by solving the simultaneous equation, Eq. (2). Substituti on of the result into Eq. (1) then yields. V = [lxy] 1 1 x. x. (3) Combining x, y and the elements of the inverted coeffici ent matrix into new functions of position Eq. (3) may be written V = E V± a± (x,y) i=l (4) where an = IK Dx2y3-x3y2) + (y2-y3)x+(x3-x2)y] (5) is a linear function of position only, and A represents the surface area of a triangle. The remaining two func tions are obtainable by cyclic interchange of subscripts, It is readly verified from (5) that the newly defined functions are interpolatory on the three vertices on the trinagle, i.e., that each function vanishes at all ver tices but one, and that is has unity value at that one: a.(x,, y.) « (6) - vxx - The potential itself iş governed by Laplace's equ ation, V2V = o (7) Principle of minimum potential energy requires that the potential distribution must be such as to minimize the stored field energy per unit length W(V) = \ J7|VV|2ds (8) The energy associated with a single triangular element may now be determined using Eq. (8) the region of integra tion being the element itself. The potential gradient within the element may be found from Eq. (4) as ^V = E V. Va. (9) ı=l so that the element energy becomes W(e) = | //|w|2ds (10) or from (9) W(e) = i Z XV. //7a. Va.ds V. (11) A i=ı j=ı i : : For brevity, define matrix elements C.(e) = // Va. Va. ds (12) ID ID where the superscript identifies the element. Equation (11) may thus be written as the matrix quadratic form W(e) = - VTC(e)V (13) Here V is the column vector of vertex values of potenti al; the superscript T denotes transposition. For any given triangle, the matrix C is evaluated, On substitution of the general equation, Eq. (5) into - vxxx - Eq. (12), a little algebra yields.. n(e) U12 and similarly for other entries of the matrix C. The calculation of transformer leakage flux is described based on the finite element method in Chapter 5. The section in which the magnetic field is to be calcu lated is divided up into small triangular elements. It is assumed that the transformer aperaters in a two winding connection. This implies that all winding segments carry ing implies that all winding segment carrying negative direction current are connected in series, and that all winding segments carrying positive direction current are connected in series. The leakage field for a typical two winding trans former shows that radial forces due to axial flux tend to compress the windings. The maximum axial force per unit valume occurs at the winding end. It tends to bend individual disk axially between radial spacers.