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Elektrostatik alanların sonlu farklar yöntemiyle incelenmesi

Elektrostatik alanların sonlu farklar yöntemiyle incelenmesi

##### Dosyalar

##### Tarih

1992

##### Yazarlar

Yarıcı, Suphi

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Bu tez çalışmasında, teknik uygulamalarda yaygın bir şekilde kullanılan sayısal yöntemlerden biri olan Sonlu Farklar Yöntemi ve yöntemin statik elektrik alan incelemelerinde kullanım olanakları ayrıntılı bir şekilde incelenmiştir. Sonlu Farklar Yönteminin ilkesi, incelenecek olan bölgenin kare, dikdörtgen,... şeklinde gözlerden oluşan bir ızgara şeklinde küçük alt bölgelere ayrılması ve ızgara üzerindeki düğümlere ilişkin potansiyel bağıntılarının yazılmasına dayanır. Çalışmada, öncelikle sonlu fark işleçleri tanıtılarak, çeşitli koordinat sistemlerinde Laplace denkleminin sonlu fark denklemleri şeklinde yazılması ele alınmıştır. Daha sonra, oluşturulan sonlu fark denklemlerinin çözümü için çeşitli çözüm yöntemleri ayrıntılı bir şekilde incelenmiştir. Son bölümde ise Sonlu Farklar Yöntemi (SFY) kullanılarak yüksek gerilim tekniğinde karşılaşılan temel elektrot sistemlerinden bazıları için alan incelemeleri yapılmıştır. Burada, incelenecek olan elektrot sistemine uygun algoritma çıkartılmış ve potansiyel değerleri, ha zırlanan bilgisayar programları yardımıyla hesaplanmış tır. Bulunan düğüm potansiyellerine lineer interpolasyon işlemi uygulanarak eşpotansiyel noktalar, dolayısıyla eş- potansiyel çizgiler elde edilmiştir. Bu şekilde düzlem- düzlem ve sivri uç-düzlem elektrot sistemleri ile içinde gaz boşluğu bulunan bir katı yalıtkan için elektrik alan incelemeleri yapılmış ve yöntemin alan incelemelerindeki kullanım zorlukları ve olanakları görülmüştür.

In recent years, several numerical methods for solving partial differential equations and thus also Laplace and Poisson equations have become available. There are inherent difficulties in solving partial dif ferential equations and thus in Laplace or Poisson equa tions for general two or three dimensional fields with sophisticated boundary conditions, or for insulating materials with different permittivities and/or conductiv ities. Each of the different numerical methods, however, has inherent advantages or disadvantages, depending upon the actual problem to be solved and thus the methods are to some extent complementary. Some of these numerical methods are Monte Carlo Method (MGM), Moment Method (MM), Charge Simulation Method (CSM), Boundary Element Method (BEM), Finite Element Method (FEM) and Finite Difference Method (FDM). FEM and FDM are based on the solution of Laplace equation in a differential form. CSM and MM are integral methods. The numerical electric field calculation by computer is still a subject of research and development. Before digital computers have been available, the elec tric field strength had been obtained either by approxi mating a given electrode by an analytically computable model or by field mapping, which estimates it from the equipotential distributions. The Finite Difference Method (FDM) has been used for field computations since before the computer age. However, without a computer, it is very painstaking work to treat regular calculations such as solving a simulta neous equation by iterative procedure even in a small model. Hence it is quite reasonable that the FDM was initially examined in a computer associated electric field simulation. Many important scientific and engineering prob lems fall into the field of partial differential equa tions. The mathematical formulation of most problems in science involving rates of change with respect to two or more independent variables, usually representing time, length or angle, leads either to a partial differential 9 vi equation or to a set of such equations. Special cases of the two dimensional second order equation dx2 dxdy dy2 dx oy (1) where a, b, c, d, e, f and g may be functions of the independent variables x and y and of the dependent vari able V, occur more frequently than any other because they are often the mathematical form of one of the conserva tion principles of physics. Equation (1) is said to be elliptic when b2-4ac<0, parabolic when b -4ac=0 and hy perbolic when b2-4ac>0. Finite Difference Method is found to be discrete technique wherein the domain of interest is represented by a set of points or nodes and information between these points is commonly obtained using Taylor series expan sions. To understand finite difference method it is first necessary to consider the nomenclature and funda mental concepts encountered in this form of approximation theory. The basic concepts are quite simple. The domain of solution of the given Laplace equation is first subdi vided by a net with a finite number of mesh points. An example of a square grid with its sides parallel to the x or y axis is shown in Figure 1. The derivative at each point is then replaced by a finite difference approxima tion. Also, boundary conditions must have known to solve the partial differential equations exactly. There are three types of boundary conditions. These are Dirichlet (first type) boundary condition, Neumann (second type) boundary condition and Robbins or hybrid (third type) boundary condition. yj+2 yj+i yj-i yj-2 X Ax- Ay i-2 Ai-1 Xi Xi+1 Xi+2 Figure 1. An example of a regular grid for finite differ ence method, indicating the node numbers. Vll Laplace equation for two dimensional field in Cartesian coordinates is a^ + üZ-o. (2) dx2 dy2 Replacing the derivatives by difference quotients which approximate the derivatives at the point {x.ityt) by tak ing Ax=Ay, Equation (2) yields (3) v2v^ = -^[v1 + v-2 + v3 + v4-4V0] =0 Equation (3) is the finite difference form of Laplace equation for a two dimensional field in Cartesian coordi nates. In this thesis, using of electrical engineering of Finite Difference Method which is commonly used in scientific and engineering problems was investigated. Electric field distributions with this method were exam ined for some of the basic electrode systems which are important in high voltage technology. In Chapter 2, finite difference operators, prin ciple of Finite Difference Method and the representation of Laplace equation by finite difference equations for regular and irregular regions with two and three dimen sional in some coordinates were investigated. Laplace equations in the different coordinate systems were given in Appendix A. The basis of Finite Difference Method is the replacement of a continuous domain representing the entire space surrounding the high voltage electrodes with a rectangular or polar grid of discrete " nodes " at which the value of unknown potential is to be computed. Thus, the derivatives are replaced describing Laplace equation with "divided difference" approximations ob tained as functions of the nodal values. In Chapter 3, the solution methods belonging to the obtained sets of finite difference equations were investigated. Usually, a convenient classification is direct and indirect (or iterative) methods. The term " direct methods " usually refers to techniques that in volve a fixed number of arithmetic operations to reach an answer. On the other hand, "indirect methods" involve the repetition of certain processes for an unknown number of times until a required accuracy in the answer is achieved. Some of these solutions methods are Gaussian Elimination, LU Decomposition, Gauss-Seidel Method and viii Successive Over Relaxation Method (SOR). In Chapter 4, studies made with Finite Difference Method were explained. During the studies, some comput ers programs were improved and these programs were given in Appendix B. Successive Over Relaxation Method (SOR) was used as a solution method. In developed programs, acceleration parameter or factor <& was varied between 1.0 and 1.9. Equipotential points were found by the linear interpolation process. The linear interpolation process was applied between calculated potential of nodes. These equipotential points were transferred to the potential file. This potential file was used to obtain the figure of potential distribution of the electrode system by a drawing program. In applications, first, Laplace equation was solved with Dirichlet boundary conditions on a rectangu lar region. Program SFY1 given in Appendix B was used to solve this problem. As second problem, plane-plane elec trode system was analyzed in Cartesian and Cylindrical coordinates. In these problems, Neumann boundary condi tions were used on the boundaries out of the electrodes. The results calculated with programs SFY2 and SFY3 given in Appendix B were confirmed by known analytical solu tions. Then, in a solid dielectric with a circular cylindrical gaseous cavity was considered, the axis of the cylinder being parallel to the electric field. The dielectric is contained between two infinite plane paral lel electrodes, whose distance apart is not great com pared with the dimensions of the cavity. The distribu tion of the potential in the solid dielectric has been calculated, in both alternating (er=2) and direct (cr=0°) voltage conditions by Finite Difference Method. The cross-section through the axis is divided into a rectan gular mesh of points, the mesh length being h, and the center of the cavity is taken as the origin of circular cylindrical coordinates. The potential V at the i,j node point is denoted by v\., the center of the cavity being assumed to be at zero potential. The potential V at a point r, 6, z is given by the Laplace equation dr2 r dr Bzz (4) which in finite difference form becomes 'i«.,j (5) IX From symmetry, only one quarter of the cross- section corresponding to the first quadrant need be considered. The treatment of the boundary conditions at z = 0 and z=+a/2 is fairly simple, since the potentials at the node points on these boundaries are known. There is no true boundary in the radial direction, but for unique ness of solution it has been assumed that, after a dis tance of 3rb in the radial direction, the voltage distri bution is uniform. Potential distributions were calcu lated for hb/a=0. 1-0. 25, rb/hb=l-3 and gr=2-<» by program SFY4 given in Appendix B. Potential distributions in a circular cylindrical gaseous cavity in a solid dielec tric, the axis of the cylinder being parallel to the field are shown in Figure 2. a) b) Fig. 2 Potential distributions in a solid dielectric with a circular cylindrical gaseous cavity for h./a=0.25, rb/hb=l. a-)er=2 (Alternating voltage condition) b)er=<» (Direct voltage condition) The last example is analyzing of the potential distribution in point-plane electrode system. In this example, to obtain a better solution, the region is subdivided into small rectangular meshes near the point electrode and large square meshes at a distance of the point electrode. The solutions belonging to this example were realized by using program SFY5 given in Appendix B. From the investigations and applications that made on Finite Difference Method have been obtained 1. FDM can be easily used for electrostatic field prob lems with closed regions, 2. Accuracy of electric field computation by Finite Difference Method mainly depends on size and regulari ty of the using meshes and adaptation of meshes to the boundaries. 3. Electrostatic field analysis by FDM can be used in order to achieve economy, reliability and well bal anced design of high voltage apparatus.

In recent years, several numerical methods for solving partial differential equations and thus also Laplace and Poisson equations have become available. There are inherent difficulties in solving partial dif ferential equations and thus in Laplace or Poisson equa tions for general two or three dimensional fields with sophisticated boundary conditions, or for insulating materials with different permittivities and/or conductiv ities. Each of the different numerical methods, however, has inherent advantages or disadvantages, depending upon the actual problem to be solved and thus the methods are to some extent complementary. Some of these numerical methods are Monte Carlo Method (MGM), Moment Method (MM), Charge Simulation Method (CSM), Boundary Element Method (BEM), Finite Element Method (FEM) and Finite Difference Method (FDM). FEM and FDM are based on the solution of Laplace equation in a differential form. CSM and MM are integral methods. The numerical electric field calculation by computer is still a subject of research and development. Before digital computers have been available, the elec tric field strength had been obtained either by approxi mating a given electrode by an analytically computable model or by field mapping, which estimates it from the equipotential distributions. The Finite Difference Method (FDM) has been used for field computations since before the computer age. However, without a computer, it is very painstaking work to treat regular calculations such as solving a simulta neous equation by iterative procedure even in a small model. Hence it is quite reasonable that the FDM was initially examined in a computer associated electric field simulation. Many important scientific and engineering prob lems fall into the field of partial differential equa tions. The mathematical formulation of most problems in science involving rates of change with respect to two or more independent variables, usually representing time, length or angle, leads either to a partial differential 9 vi equation or to a set of such equations. Special cases of the two dimensional second order equation dx2 dxdy dy2 dx oy (1) where a, b, c, d, e, f and g may be functions of the independent variables x and y and of the dependent vari able V, occur more frequently than any other because they are often the mathematical form of one of the conserva tion principles of physics. Equation (1) is said to be elliptic when b2-4ac<0, parabolic when b -4ac=0 and hy perbolic when b2-4ac>0. Finite Difference Method is found to be discrete technique wherein the domain of interest is represented by a set of points or nodes and information between these points is commonly obtained using Taylor series expan sions. To understand finite difference method it is first necessary to consider the nomenclature and funda mental concepts encountered in this form of approximation theory. The basic concepts are quite simple. The domain of solution of the given Laplace equation is first subdi vided by a net with a finite number of mesh points. An example of a square grid with its sides parallel to the x or y axis is shown in Figure 1. The derivative at each point is then replaced by a finite difference approxima tion. Also, boundary conditions must have known to solve the partial differential equations exactly. There are three types of boundary conditions. These are Dirichlet (first type) boundary condition, Neumann (second type) boundary condition and Robbins or hybrid (third type) boundary condition. yj+2 yj+i yj-i yj-2 X Ax- Ay i-2 Ai-1 Xi Xi+1 Xi+2 Figure 1. An example of a regular grid for finite differ ence method, indicating the node numbers. Vll Laplace equation for two dimensional field in Cartesian coordinates is a^ + üZ-o. (2) dx2 dy2 Replacing the derivatives by difference quotients which approximate the derivatives at the point {x.ityt) by tak ing Ax=Ay, Equation (2) yields (3) v2v^ = -^[v1 + v-2 + v3 + v4-4V0] =0 Equation (3) is the finite difference form of Laplace equation for a two dimensional field in Cartesian coordi nates. In this thesis, using of electrical engineering of Finite Difference Method which is commonly used in scientific and engineering problems was investigated. Electric field distributions with this method were exam ined for some of the basic electrode systems which are important in high voltage technology. In Chapter 2, finite difference operators, prin ciple of Finite Difference Method and the representation of Laplace equation by finite difference equations for regular and irregular regions with two and three dimen sional in some coordinates were investigated. Laplace equations in the different coordinate systems were given in Appendix A. The basis of Finite Difference Method is the replacement of a continuous domain representing the entire space surrounding the high voltage electrodes with a rectangular or polar grid of discrete " nodes " at which the value of unknown potential is to be computed. Thus, the derivatives are replaced describing Laplace equation with "divided difference" approximations ob tained as functions of the nodal values. In Chapter 3, the solution methods belonging to the obtained sets of finite difference equations were investigated. Usually, a convenient classification is direct and indirect (or iterative) methods. The term " direct methods " usually refers to techniques that in volve a fixed number of arithmetic operations to reach an answer. On the other hand, "indirect methods" involve the repetition of certain processes for an unknown number of times until a required accuracy in the answer is achieved. Some of these solutions methods are Gaussian Elimination, LU Decomposition, Gauss-Seidel Method and viii Successive Over Relaxation Method (SOR). In Chapter 4, studies made with Finite Difference Method were explained. During the studies, some comput ers programs were improved and these programs were given in Appendix B. Successive Over Relaxation Method (SOR) was used as a solution method. In developed programs, acceleration parameter or factor <& was varied between 1.0 and 1.9. Equipotential points were found by the linear interpolation process. The linear interpolation process was applied between calculated potential of nodes. These equipotential points were transferred to the potential file. This potential file was used to obtain the figure of potential distribution of the electrode system by a drawing program. In applications, first, Laplace equation was solved with Dirichlet boundary conditions on a rectangu lar region. Program SFY1 given in Appendix B was used to solve this problem. As second problem, plane-plane elec trode system was analyzed in Cartesian and Cylindrical coordinates. In these problems, Neumann boundary condi tions were used on the boundaries out of the electrodes. The results calculated with programs SFY2 and SFY3 given in Appendix B were confirmed by known analytical solu tions. Then, in a solid dielectric with a circular cylindrical gaseous cavity was considered, the axis of the cylinder being parallel to the electric field. The dielectric is contained between two infinite plane paral lel electrodes, whose distance apart is not great com pared with the dimensions of the cavity. The distribu tion of the potential in the solid dielectric has been calculated, in both alternating (er=2) and direct (cr=0°) voltage conditions by Finite Difference Method. The cross-section through the axis is divided into a rectan gular mesh of points, the mesh length being h, and the center of the cavity is taken as the origin of circular cylindrical coordinates. The potential V at the i,j node point is denoted by v\., the center of the cavity being assumed to be at zero potential. The potential V at a point r, 6, z is given by the Laplace equation dr2 r dr Bzz (4) which in finite difference form becomes 'i«.,j (5) IX From symmetry, only one quarter of the cross- section corresponding to the first quadrant need be considered. The treatment of the boundary conditions at z = 0 and z=+a/2 is fairly simple, since the potentials at the node points on these boundaries are known. There is no true boundary in the radial direction, but for unique ness of solution it has been assumed that, after a dis tance of 3rb in the radial direction, the voltage distri bution is uniform. Potential distributions were calcu lated for hb/a=0. 1-0. 25, rb/hb=l-3 and gr=2-<» by program SFY4 given in Appendix B. Potential distributions in a circular cylindrical gaseous cavity in a solid dielec tric, the axis of the cylinder being parallel to the field are shown in Figure 2. a) b) Fig. 2 Potential distributions in a solid dielectric with a circular cylindrical gaseous cavity for h./a=0.25, rb/hb=l. a-)er=2 (Alternating voltage condition) b)er=<» (Direct voltage condition) The last example is analyzing of the potential distribution in point-plane electrode system. In this example, to obtain a better solution, the region is subdivided into small rectangular meshes near the point electrode and large square meshes at a distance of the point electrode. The solutions belonging to this example were realized by using program SFY5 given in Appendix B. From the investigations and applications that made on Finite Difference Method have been obtained 1. FDM can be easily used for electrostatic field prob lems with closed regions, 2. Accuracy of electric field computation by Finite Difference Method mainly depends on size and regulari ty of the using meshes and adaptation of meshes to the boundaries. 3. Electrostatic field analysis by FDM can be used in order to achieve economy, reliability and well bal anced design of high voltage apparatus.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1992

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 1992

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 1992

##### Anahtar kelimeler

,Elektrostatik alan,
Sonlu farklar yöntemi,
Electrostatic field,
Finite differences method