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Elektrostatik alanların yük benzetim yöntemiyle incelenmesi

Elektrostatik alanların yük benzetim yöntemiyle incelenmesi

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##### Dosyalar

##### Tarih

1992

##### Yazarlar

Yıldırım, Hayri

##### 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, elektriksel alanların incelenmesinde kullanılan sayısal yöntemlerden biri olan ve temel ilkesi, elektrik sel alanı incelenecek olan bir elektrot sisteminin gerçek elektrik sel alanı yerine, alanın çözüleceği bölgenin dışarısına yerleştiri len ve değerleri sınır koşullarının sağlanmasıyla bulunan belirli sayıda ayrık elektriksel yükün oluşturduğu alanın alınmasına dayanan Yük Benzetim Yöntemi (YBY) ve bu yöntemin geliştirilmiş türleri ge niş bir şekilde incelenmiştir. Yöntemde kullanılan çeşitli yük tip lerinin potansiyel ve alan katsayılarına ilişkin ifadeler verilmiş tir. Ayrıca yöntemin doğruluğu ve buna etki eden faktörler ele alı narak yöntemin yüksek gerilimdeki uygulama alanlarından örnekler ve rilmiş ve bu yöntem kullanılarak yüksek gerilim yalıtım sistemleri nin bilgisayar destekli tasarımı ve analizinin etkinliğini arttırmak için ileride araştırma yapılabilecek konulara işaret edilmiştir. Son bölümde ise, Yük Benzetim Yöntemi kullanılarak bazı yük sek gerilim elektrot sistemlerinin elektrik alan dağılımlarına iliş kin çözümler yapılmıştır. Bu çözümlere ait sonuçlar verilerek yön temin kullanım kolaylığı, üstünlükleri ile doğruluğu gösterilmiştir. Bu yöntemle çözüm için genel kullanımlı bir bilgisayar programı oluşturmak mümkün olmadığından her bir elektrot sistemi için ayrı ayrı bilgisayar programı geliştirilmiştir.

Proper design of any high voltage device requires a complete knowledge of the electric field distribution. For a simple physical system, it is usually possible to find an analytical solution. However, in many cases, the physical systems are so complex that it is extremely difficult, if not impossible, to find analytical solu tions. In such cases, numerical methods are employed for electric field calculations. The existing numerical methods include the Finite Difference Method (FDM), the Finite Element Method (FEM), the Monte Carlo Method (MCM), the Moment Method (MM), the Method of Images (MI), the Charge Simulation Method (CSM) and the Surface Charge Simulation Method (SCSM). The charge simulation method, due to its favorable charac teristics, is very commonly used for field analysis of high voltage insulation systems. This thesis provides a comprehensive review of the basic charge simulation method and its various modified versions reported in the literature. Applications of the charge simulation method, alone as well as in combination with other methods, are considered. Different aspects of this method are critically examined and its potentials and limitations are identified. Electric field calcula tions of some practical high voltage electrode systems are made by using the basic charge simulation method, and the results are compared with those obtained by the existing approximate solutions. It is observed that some approximate methods may give rise to significant errors. Possible areas of future research are outlined in order to improve the overall effectiveness of computer-aided analysis and design of high voltage devices by using the charge simulation method. In the Charge Simulation Method, the actual electric field is simulated with a field formed by a number of discrete charges which are placed outside of the region where the field solution is desired. Values of the discrete charges are determined by satisfying the boundary conditions at a selected number of contour points. Once the values and positions of simulation charges are known, the poten tial and field distribution anywhere in the region can be computed easily. vx The basic principle of the charge simulation method is very simple. If several discrete charges of any type (point, line, or ring, for instance) are present in a region, the electrostatic potential at any point C can be found by summation of the potentials resulting from the individual charges as long as the point c does not reside on any one of the charges. Let q. be a number of n individual charges and Vi be the potential at any point c within the space. According to superposi tion principle n V. = E p...q. 1 j-1 1J J (1) where p.. are the "potential coefficients" which can be evaluated analytically for many types of charges by solving Laplaces or Poisson's equations. For example, in Figure 1 which displays three point charges q,, C. is given by q2 and q" in free space, the potential V. at point V = = - i 4tt£. r. 4ire.r0 o 2 4tt£.r0 o 3 = Pil.q1 + Pi2.q2 + Pi3.q3 (2) Thus, one the types of charges and their locations are defined, it is possible to relate V. and q. quantitatively at any boundary point. In the CSM, the simulation charges are placed outside the space where the field solution is desired (or inside any equipoten- tial surface such as metal electrodes). If the boundary point C. is located on the surface of a conductor, then V. at this "contour point" is equal to the conductor potential Ve. When this procedure is applied to m contour points, it leads to following system of m linear equations for n unknown charges Jll H.2 ' p21 p22... p2n ^ml ^2 * (3) Equation (3) is the basic foundation of the charge simulation method. For field calculations in a single dielectric medium, the actual charges on the surface of the conductors are replaced by nc fictitious charges placed inside (or outside) the conductors. The types and positions of these charges are assumed. In order to determine their magnitudes, njj contour points are selected on the surfaces of conductors, and it is required that at any one of these contour points the potential resulting from superposition of all the simulation charges is equal to the known conductor potential. In the conventional charge simulation method, the number of contour points is selected equal to the number of simulation charges i.e. Vll n,=n =n. Therefore, the charges are determined from trjn,n Ln4n u Jn (4) where [p] is the potential coefficient matrix, [q] is the column vector of values of unknown charges, and [v] is the potential of boundary points. Figure 1. Three point charges in free space. After solving equation (4) to determine the magnitudes of simulation charges, it is necessary to check whether the set of cal culated charges produces actual boundary conditions everywhere on the electrode surfaces. It must be emphasized that only n discrete contour points of the real electrode system have been used to solve equation (4), and thus the potential at any point other than the contour points might be different from the actual conductor potential. Thus, equation (1) is solved at a number of "check points" located on the electrodes where potentials are known to determine the simulation accuracy. If simulation does not meet the accuracy criterion, calcu lations are repeated by changing one or several of the following parameters : 1. The number of simulation charges, 2. The locations of simulation charges, 3. The types of simulation charges, 4. The locations of contour points. As soon as an adequate charge system has been developed, the potential and field at any point outside the electrodes can be calcu lated. Whereas the potential is found by equation (1), the field stress is calculated by superposition of magnitudes of various directional components. For example, for cartesian coordinate system, the net field E^ at point C^ is given by n 3pin. ~\+ [* n 3p4 1 ~h E. » i n I 3p.. 3x j=l ?q. j i + y n I 3z.q. Z (f..).q. j=l x3 x J i + X 2 (f..).q. ^ + Ü-1 . charges x contour points V=variable Figure 2. Simulation of multidielectric boundary. In formulating the equations at a given contour point, the charges which lie in the same dielectric as the contour point are ignored. For example, the potential at contour point 1 is calculated due to superposition of charges 1 to 5 only. Similary, the potential or field intensity at contour point 5 when viewed from dielectric A side will be due to superposition of charges 1 to 3 and 6 to 7 only. It is obvious from this discussion that simulation of multi- dielectric boundaries is more involved. Moreover, the accuracy of such a simulation deterirates when the dielectric boundary has a complex profile. Besides other factors, successful application of the CSM requires a proper choice of the types of simulation charges. In the IX initial attemps using the CSM, point and line charges of infinite and semi-infinite length were used. Singer et al. introduced ring charges, periodically variable density ring charges, and finite line charges. In recent years, a variety of other charge configurations have been introduced. In order to use any type of simulation charges, it is essential to determine the potential and the field coefficients. Expressions for such coefficients are given in this thesis for point, infinite line, finite line, ring, elliptic cylinder and spheroidal charges. In geneal, most of the high voltage systems can be successfully simulated by using the basic charge configura tions i.e. point, line and ring or a suitable combination of these charges. If the simulation charges completely satisfy the boundary conditions, the uniqueness theorem applied to the potential ensures that these charges give the correct potential not only on the boundary but also everywhere outside it. To determine the simulation accuracy, one or several of the following criterion can be used: 1. The potential error can be computed at a number of points for each conductor boundary. 2. The ratio of the tangential and the normal components of the field vector (Et/En) on the conductor boundary, or the angular deviation of the electric field vector on the electrode surfaces from its normal position. 3. In multidielectric systems, "potential discrepency" defined as the difference in solutions for potential at the dielectric boundary. 4. Comparisons of the CSM solution with solutions obtained from other numerical and analytical methods or experimental measurements, if possible. The error in the CSM depends upon the types, numbers as well as locations of simulation charges, the locations of contour points and the complexities of the profiles of electrodes as well as dielectrics. In general, for single dielectric system, the potential error can be reduced by increasing the number of simulation charges. It has been suggested that increasing the number of simulation charges beyond a certain limit does not necessarily have a favorable influence on the accuracy. In such cases, the application of one of the modified versions of the CSM of the Surface Charge Simulation Method. Modified versions of the CSM include Least Square Error Methods, Optimized Charge Simulation Method, Methods Using Complex Charges, and Hibrid (combination) Methods. Since the inital use of the CSM for computing the field distribution of a rod-plane gap, this method has been applied to a wide range of problems. These problems include various high voltage electrode systems, conductors and cables, insulators, bushings and spacers, and field optimization. In addition to these applications, the CSM has been used for field computations of a variety of devices such as electrostatic painting system, substations, laboratory test set-ups, air-insulated wall bushings, metalclad isolating switches, trigatron etc. v* +100 f y. Charge x Contour point "T V=0 V- 100 Figure 3. %100 2100 (a) (b) %100 %100 (c) (d) Figure 4. xx In this thesis, potential and field calculations of a sphere- plane gap and a cylindrical conductor-plane gap have been made by using the conventional charge simulation method. In the first problem, sphere-sphere electrode system has been considered. One of the spheres has the potential of V= +100 and the other has the poten tial of V=-100 (Fig. 3). Second sphere is used to simulate the earthed plane. First, point charges centred at the spheres are used to simulate the electrodes. Equipotential lines shown in Figure 4(a) belong to this solution. Because of this solution didn't have a required accuracy, calculations have been repeated by increasing the number of simulation charges or changing their locations (Figure 4(b), (c)). More accurate solution has been obtained by three point charges (Figure 4(d)). Figure 5 shows the variation of E/Emax with the function of the distance from the sphere electrode. w\ In the second problem, a cylindrical conductor-plane gap has been analized in a similar way. Line charges of infinite lenght have been used to simulate the electrodes. Required accuracy has been obtained by two line charges located suitably. Equipotential lines of the cylindrical conductor-plane electrode system are shown in Figure 6. Figure 6. The charge simulation method has undergone considerable development in recent years and has emerged as a very efficient and accurate numerical method for electric field calculations. The method has been successfully applied for the field analysis of a wide variety of high voltage insulation systems including 3-dimen- sional and multiple dielectric arrangements, electrode and insulator XII contour optimization and the analysis of space-charge modified fields. Recently, this method has been recommended as an attractive aid in teaching applied electrostatics to undergraduate students. Further work in the CSM modelling of complicated multidielectric systems, 3-dimensional fields, space charge modified fields, electrode and insulator contour optimization as well as computer aided design using this method is recommended. Research in these areas will further increase the applications and computational benefits of the CSM.

Proper design of any high voltage device requires a complete knowledge of the electric field distribution. For a simple physical system, it is usually possible to find an analytical solution. However, in many cases, the physical systems are so complex that it is extremely difficult, if not impossible, to find analytical solu tions. In such cases, numerical methods are employed for electric field calculations. The existing numerical methods include the Finite Difference Method (FDM), the Finite Element Method (FEM), the Monte Carlo Method (MCM), the Moment Method (MM), the Method of Images (MI), the Charge Simulation Method (CSM) and the Surface Charge Simulation Method (SCSM). The charge simulation method, due to its favorable charac teristics, is very commonly used for field analysis of high voltage insulation systems. This thesis provides a comprehensive review of the basic charge simulation method and its various modified versions reported in the literature. Applications of the charge simulation method, alone as well as in combination with other methods, are considered. Different aspects of this method are critically examined and its potentials and limitations are identified. Electric field calcula tions of some practical high voltage electrode systems are made by using the basic charge simulation method, and the results are compared with those obtained by the existing approximate solutions. It is observed that some approximate methods may give rise to significant errors. Possible areas of future research are outlined in order to improve the overall effectiveness of computer-aided analysis and design of high voltage devices by using the charge simulation method. In the Charge Simulation Method, the actual electric field is simulated with a field formed by a number of discrete charges which are placed outside of the region where the field solution is desired. Values of the discrete charges are determined by satisfying the boundary conditions at a selected number of contour points. Once the values and positions of simulation charges are known, the poten tial and field distribution anywhere in the region can be computed easily. vx The basic principle of the charge simulation method is very simple. If several discrete charges of any type (point, line, or ring, for instance) are present in a region, the electrostatic potential at any point C can be found by summation of the potentials resulting from the individual charges as long as the point c does not reside on any one of the charges. Let q. be a number of n individual charges and Vi be the potential at any point c within the space. According to superposi tion principle n V. = E p...q. 1 j-1 1J J (1) where p.. are the "potential coefficients" which can be evaluated analytically for many types of charges by solving Laplaces or Poisson's equations. For example, in Figure 1 which displays three point charges q,, C. is given by q2 and q" in free space, the potential V. at point V = = - i 4tt£. r. 4ire.r0 o 2 4tt£.r0 o 3 = Pil.q1 + Pi2.q2 + Pi3.q3 (2) Thus, one the types of charges and their locations are defined, it is possible to relate V. and q. quantitatively at any boundary point. In the CSM, the simulation charges are placed outside the space where the field solution is desired (or inside any equipoten- tial surface such as metal electrodes). If the boundary point C. is located on the surface of a conductor, then V. at this "contour point" is equal to the conductor potential Ve. When this procedure is applied to m contour points, it leads to following system of m linear equations for n unknown charges Jll H.2 ' p21 p22... p2n ^ml ^2 * (3) Equation (3) is the basic foundation of the charge simulation method. For field calculations in a single dielectric medium, the actual charges on the surface of the conductors are replaced by nc fictitious charges placed inside (or outside) the conductors. The types and positions of these charges are assumed. In order to determine their magnitudes, njj contour points are selected on the surfaces of conductors, and it is required that at any one of these contour points the potential resulting from superposition of all the simulation charges is equal to the known conductor potential. In the conventional charge simulation method, the number of contour points is selected equal to the number of simulation charges i.e. Vll n,=n =n. Therefore, the charges are determined from trjn,n Ln4n u Jn (4) where [p] is the potential coefficient matrix, [q] is the column vector of values of unknown charges, and [v] is the potential of boundary points. Figure 1. Three point charges in free space. After solving equation (4) to determine the magnitudes of simulation charges, it is necessary to check whether the set of cal culated charges produces actual boundary conditions everywhere on the electrode surfaces. It must be emphasized that only n discrete contour points of the real electrode system have been used to solve equation (4), and thus the potential at any point other than the contour points might be different from the actual conductor potential. Thus, equation (1) is solved at a number of "check points" located on the electrodes where potentials are known to determine the simulation accuracy. If simulation does not meet the accuracy criterion, calcu lations are repeated by changing one or several of the following parameters : 1. The number of simulation charges, 2. The locations of simulation charges, 3. The types of simulation charges, 4. The locations of contour points. As soon as an adequate charge system has been developed, the potential and field at any point outside the electrodes can be calcu lated. Whereas the potential is found by equation (1), the field stress is calculated by superposition of magnitudes of various directional components. For example, for cartesian coordinate system, the net field E^ at point C^ is given by n 3pin. ~\+ [* n 3p4 1 ~h E. » i n I 3p.. 3x j=l ?q. j i + y n I 3z.q. Z (f..).q. j=l x3 x J i + X 2 (f..).q. ^ + Ü-1 . charges x contour points V=variable Figure 2. Simulation of multidielectric boundary. In formulating the equations at a given contour point, the charges which lie in the same dielectric as the contour point are ignored. For example, the potential at contour point 1 is calculated due to superposition of charges 1 to 5 only. Similary, the potential or field intensity at contour point 5 when viewed from dielectric A side will be due to superposition of charges 1 to 3 and 6 to 7 only. It is obvious from this discussion that simulation of multi- dielectric boundaries is more involved. Moreover, the accuracy of such a simulation deterirates when the dielectric boundary has a complex profile. Besides other factors, successful application of the CSM requires a proper choice of the types of simulation charges. In the IX initial attemps using the CSM, point and line charges of infinite and semi-infinite length were used. Singer et al. introduced ring charges, periodically variable density ring charges, and finite line charges. In recent years, a variety of other charge configurations have been introduced. In order to use any type of simulation charges, it is essential to determine the potential and the field coefficients. Expressions for such coefficients are given in this thesis for point, infinite line, finite line, ring, elliptic cylinder and spheroidal charges. In geneal, most of the high voltage systems can be successfully simulated by using the basic charge configura tions i.e. point, line and ring or a suitable combination of these charges. If the simulation charges completely satisfy the boundary conditions, the uniqueness theorem applied to the potential ensures that these charges give the correct potential not only on the boundary but also everywhere outside it. To determine the simulation accuracy, one or several of the following criterion can be used: 1. The potential error can be computed at a number of points for each conductor boundary. 2. The ratio of the tangential and the normal components of the field vector (Et/En) on the conductor boundary, or the angular deviation of the electric field vector on the electrode surfaces from its normal position. 3. In multidielectric systems, "potential discrepency" defined as the difference in solutions for potential at the dielectric boundary. 4. Comparisons of the CSM solution with solutions obtained from other numerical and analytical methods or experimental measurements, if possible. The error in the CSM depends upon the types, numbers as well as locations of simulation charges, the locations of contour points and the complexities of the profiles of electrodes as well as dielectrics. In general, for single dielectric system, the potential error can be reduced by increasing the number of simulation charges. It has been suggested that increasing the number of simulation charges beyond a certain limit does not necessarily have a favorable influence on the accuracy. In such cases, the application of one of the modified versions of the CSM of the Surface Charge Simulation Method. Modified versions of the CSM include Least Square Error Methods, Optimized Charge Simulation Method, Methods Using Complex Charges, and Hibrid (combination) Methods. Since the inital use of the CSM for computing the field distribution of a rod-plane gap, this method has been applied to a wide range of problems. These problems include various high voltage electrode systems, conductors and cables, insulators, bushings and spacers, and field optimization. In addition to these applications, the CSM has been used for field computations of a variety of devices such as electrostatic painting system, substations, laboratory test set-ups, air-insulated wall bushings, metalclad isolating switches, trigatron etc. v* +100 f y. Charge x Contour point "T V=0 V- 100 Figure 3. %100 2100 (a) (b) %100 %100 (c) (d) Figure 4. xx In this thesis, potential and field calculations of a sphere- plane gap and a cylindrical conductor-plane gap have been made by using the conventional charge simulation method. In the first problem, sphere-sphere electrode system has been considered. One of the spheres has the potential of V= +100 and the other has the poten tial of V=-100 (Fig. 3). Second sphere is used to simulate the earthed plane. First, point charges centred at the spheres are used to simulate the electrodes. Equipotential lines shown in Figure 4(a) belong to this solution. Because of this solution didn't have a required accuracy, calculations have been repeated by increasing the number of simulation charges or changing their locations (Figure 4(b), (c)). More accurate solution has been obtained by three point charges (Figure 4(d)). Figure 5 shows the variation of E/Emax with the function of the distance from the sphere electrode. w\ In the second problem, a cylindrical conductor-plane gap has been analized in a similar way. Line charges of infinite lenght have been used to simulate the electrodes. Required accuracy has been obtained by two line charges located suitably. Equipotential lines of the cylindrical conductor-plane electrode system are shown in Figure 6. Figure 6. The charge simulation method has undergone considerable development in recent years and has emerged as a very efficient and accurate numerical method for electric field calculations. The method has been successfully applied for the field analysis of a wide variety of high voltage insulation systems including 3-dimen- sional and multiple dielectric arrangements, electrode and insulator XII contour optimization and the analysis of space-charge modified fields. Recently, this method has been recommended as an attractive aid in teaching applied electrostatics to undergraduate students. Further work in the CSM modelling of complicated multidielectric systems, 3-dimensional fields, space charge modified fields, electrode and insulator contour optimization as well as computer aided design using this method is recommended. Research in these areas will further increase the applications and computational benefits of the CSM.

##### 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,
Yük benzetim yöntemi,
Electrostatic field,
Charge simulation method