ELECTRIC FIELD CALCULATIONS BY NUMERICAL

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ELECTRIC FIELD CALCULATIONS BY NUMERICAL,TECHNIQUES. A THESIS SUBMITTED IN FUFILLMENT OF THE REQUIREMENTS FOR. THE DEGREE OF,BACHELOR OF TECHNOLOGY,ELECTRICAL ENGINEERING. BISWANATH MALIK,ROLL NO 10502034,UNDER THE GUIDANCE OF. PROF SARADINDU GHOSH,PROF SANDIP GHOSH,DEPARTMENT OF ELECTRICAL ENGINEERING. NATIONAL INSTITUTE OF TECHNOLOGY,ROURKELA 769008,Certificate.
This is to certify that the project progress report entitled Electric field. calculation by numerical techniques submitted by Biswanath malik in. partial fulfillment of the requirements for the award of bachelor of. technology degree in electrical engineering at national institute of. technology Rourkela is an authentic work carried out by him under my. supervision and guidance,To the best of my knowledge the matter. embodies in the project work has not submitted to any other. university institute for the award of any degree diploma. Date Prof Sandip Ghosh,Department of Electrical engineering. Place National institute of technology,Rourkela 769008. ACKNOWLEDGMENT, I would like to articulate our deep gratitude to our project guide Prof Saradindu. Ghosh Prof Sandip Ghosh who has always been our motivation for carrying out. the project I am thanking to C programmer Susanta Kumar Rout for his sincere. help to write programs It is our pleasure to refer Microsoft Word exclusive of. which the compilation of this report would have been impossible An assemblage. of this nature could never have been attempted with out reference to and. inspiration from the works of others whose details are mentioned in reference. section We acknowledge out indebtedness to all of them Last but not the least. our sincere thanks to all of our friends who have patiently extended all sorts of help. for accomplishing this undertaking,Biswanath Malik.
Roll no 10502034,Chapter Topic Page,Abstract 6,Chapter 1 Introduction 7. Chapter 2 Finite difference method,2 1 fundamental of FDM 8. 2 2 Two dimensional electric field calculations by FDM 12. Chapter 3 Finite elements method,3 1 Fundamentals of FEM 15. 3 2 Two dimensional electric field calculations by FEM 20. Chapter 4 Three dimensional electric field calculations. 4 1 Three dimensional Laplace s equation 34, 4 2 Non uniformly distributed dielectric of a capacitor 36. 4 3 Electric fields near a dc busbar 39,Chapter 5 Finite Element Analysis using ANSYS.
5 1Fundamentals of ANSYS 41, 5 2 Motor analysis using finite elements methods in ANSYS workbench 53. Conclusion and future works 66,References 67,Appendix A MATLAB Programming 68. Appendix B C programming 73, Objective of the study of electric field calculations by numerical. techniques is to use different numerical techniques to find electric. field distributions which are inevitable tool in various electricity. concerned technologies in particular for analyzing discharge. phenomenon and designing high voltage equipment In this thesis. two numerical methods are discussed e g finite difference method. and finite element method Both methods are used to find two. dimensional electric field distributions with given boundary. conditions using MATLAB Electric field distributions in more. practical three dimensional cases with non uniformly distributed. dielectric of a capacitor in a DC busbar has found using C. programming Also electromagnetic field calculations of electric. motor have been done in ANSYS,Introduction, Calculation of electric fields with the aid of an computer is now a inevitable tool in various. electricity concerned technology in particular for analyzing discharge phenomenon and. designing high voltage equipments Electric and magnetic fields comprises two components dealt. with in one of the classical physics electromagnetism Calculation of electric fields is usually. considered easier than that of magnetic ones from two reasons First the electric field is. expressed with a scalar potential at least in simple low frequency problems Secondly non linear. characteristics are more often involved in magnetic fields Compared with magnetic field. however the calculation electric fields generally require higher accuracy because the highest. electric field stress on insulator is usually the most important and decisive value in insulation. design or discharge study This is one of reason why the boundary dividing methods are. preferred to the region dividing ones such as finite difference method FDM or finite element. method FEM Usually former method does not need numerical differentiation to obtain field. A fundamental equation for the electric field is Laplace s equation or Poisson s equation. perhaps the simplest among many partial differential equations that express physical phenomena. among various numerical calculation methods FDM and FEM is very unique as it is applied. exclusively to electric field calculations Fundamental difference between FDM and FEM is that. FDM can be used for calculation of potential at nodes only but FEM can be used for calculation. of potential at nodes as well as within the elements Calculation of electric field in 3D. arrangement poses no essential problem by of the numerical methods if the field is given by. Laplace s equation The difficulty is that it usually required the tedious work preparing the input. of a large amount of errorless data associate with 3D conditions. Numerical solution of EM problems started in the mid 1960s with the availability of modern. high speed digital computers Since then considerable effort has been expended on solving. practical complex EM related problems for which closed form analytic solutions are either. intractable or do not exist The numerical approach has the advantage of allowing the actual. work to be carried out by operates without a knowledge of higher mathematics with a resulting. economy of labor on the part of the highly trained personnel. THE Finite difference method,2 1 fundamentals of FDM.
The finite difference method is a powerful numerical method for solving partial differential. equations In applying the method of finite differences a problem is defined by. A partial differential equation such as Poisson s equation. A solution region,Boundary and or initial conditions. An FDM method divides the solution domain into finite discrete points and replaces the partial. differential equations with a set of difference equations Thus the solutions obtained by FDM are. not exact but approximate However if the discretization is made very fine the error in the. solution can be minimized to an acceptable level,The Poisson s equation in 3 D is given by. For 2 D case Poisson s equation simplifies to, In applying the methods of finite differences we define the solution region into a finite number. of meshes as shown in Fig2 1, Fig2 1 Division of solution region into grid points. The meshes can be various shapes we shall only consider the rectangular and square meshes. only First we consider a mesh configuration having five nodes and unequal arms as the Fig2 2. Fig2 2 A mesh with unequal arms, With reference to Fig2 1 Vo corresponds to the voltage Vij For the five node mesh configuration.
of Fig2 2 the voltages are defined as, Let P1 P2 P3 andP4 represent the midpoint of the arms as shown in Fig 2 2 In order to replace. the Poisson equation 2 2 by difference equations we obtain the approximate first derivatives at. the points P1 to P2 and use these first derivatives to approximate the second derivative. The first derivatives at P1and P2 are,In the same manner. The first derivative at P1and P2 is,In the same manner. Further for Laplace equation s and equation 2 8 simplifies to. Thus we see that voltage at the central node is the mean of the voltages at the other four nodes. With reference to Fig2 1 equation 2 8 can be written as. Equation 2 8 equation 2 9 can be used to solve Poisson s and Laplace s equation. respectively when uniform grids are used These equations along with the specified boundary. conditions can be used to solve a problem, 2 2 Two dimensional electric field calculations by finite difference method. The solution region is divided into square meshes Here boundary is regular Total 21 nodes has. unknown potentials We have marked that and 24 nodes are known potential Here using. difference elements we have found the potential at each nodes whose potential are not known up. to 17th iteration,Figure 2 3,RESULT No of iteration.
1 2 3 4 5 6 7 8 9,V1 0 0 0 0 1 861 3 38 4 51 5 312. V2 0 0 0 0 3 125 6 645 9 10 6 11 68,V3 0 0 0 6 25 11 23 14 66 16 86 18 3 19 24. V4 0 0 12 5 19 14 23 044 25 3487 26 82 27 73 28 34. V5 0 25 32 81 36 22 38 08 39 1381 39 8 40 21 40 51. V6 50 56 25 58 2 59 055 59 52 59 79 59 95 60 05 60 127. V7 0 0 0 0 4 321 6 87 9 063 10 63 11 69,V8 0 0 0 6 25 13 49 17 95 21 1 23 15 24 64. V9 0 0 12 5 22 656 28 96 33 064 35 63 37 625 38 8, V10 0 25 37 5 44 726 48 655 51 25 52 85 53 92 54 625. V11 50 0 67 58 70 2365 71 684 72 6 73 1625 73 531 73 784. V12 0 0 0 11 035 12 1271 14 92 17 18 833 19 164, V13 0 0 12 5 23 8525 29 3303 33 23 35 72 37 625 38 82.
V14 0 25 40 625 46 39 50 998 53 32 55 97 57 411 58 33. V15 50 62 5 69 53 72 78 74 9137 76 33 77 205 77 833 78 239. V16 0 0 12 5 20 3368 23 49 25 565 26 98 27 853 28 43. V17 0 25 37 5 45 3151 49 05 51 447 53 05 54 058 54 71. V18 50 62 5 69 535 72 926 75 76 2 77 255 77 867 78 185. V19 0 25 32 815 36 52 38 27 39 46 40 0325 40 5 40 7. V20 50 62 5 67 57 70 4587 72 73 3 73 5 73 73 73 85. V21 50 56 25 58 203 59 13 59 5675 59 85 60 60 125 60 18. 10 11 12 13 14 15 16 17,V1 5 92 6 22 6 47 6 644 6 76 6 84 6 885 6 91. V2 12 46 12 94 13 27 13 52 13 7 13 77 13 81 13 9,V3 19 89 20 33 20 62 20 82 20 97 21 02 21 1 21 13. V4 28 76 29 29 21 29 357 29 4 29 46 29 5 29 52,V5 40 72 40 79 40 91 40 94 40 96 40 98 41 41. V6 60 17 60 2 60 22 60 23 60 24 60 245 60 25 60 25. V7 12 44 12 95 13 3 13 51 13 65 13 77 13 82 13 88,V8 25 63 26 3 26 73 27 27 24 27 35 27 43 27 5. V9 39 62 40 16 40 55 40 8 40 89 41 41 11 41 16,V10 55 1 55 43 55 7 55 79 55 86 55 92 55 96 56.
V11 73 955 74 74 15 74 2 74 22 74 23 74 23 74 25,V12 19 94 20 36 20 66 20 82 20 95 21 21 1 21 12. V13 39 65 40 2 40 56 40 8 40 96 41 41 1 41 16,V14 58 93 59 42 59 67 59 85 59 9 60 60 05 60 08. V15 78 6 78 7 78 84 78 91 78 95 78 98 79 79 02,V16 28 84 29 29 227 29 34 29 42 29 47 29 5 29 523. V17 55 13 55 44 55 67 55 78 55 87 55 91 55 97 56,V18 78 51 78 83 78 841 78 91 78 95 79 79 79 02. V19 40 72 40 81 40 87 40 93 40 96 40 98 41 41,V20 74 74 06 74 13 74 18 74 2 74 25 74 24 74 25.
V21 60 19 60 2 60 22 60 23 60 24 60 25 60 25 60 25. MATLAB PROGRAM, APPENDIX A A 1 MATLAB program for 2D problems using finite difference method. 0 0 0 0 0 0 0 0 50 0000, 0 6 9652 13 9304 21 1778 29 5534 41 0167 60 2542 100 0000 0. 0 13 9304 27 5788 41 2271 56 0194 74 2590 100 0000 0 0. 0 21 1778 41 2271 60 1326 79 0380 100 0000 0 0 0,0 29 5534 56 0194 79 0380 100 0000 0 0 0 0. 0 41 0167 74 2590 100 0000 0 0 0 0 0,0 60 2542 100 0000 0 0 0 0 0 0. 0 100 0000 0 0 0 0 0 0 0,50 0000 0 0 0 0 0 0 0 0,The finite element method.
3 1 fundamentals of FEM, The finite element method has its origin in the field of structural analysis The method. was not applied to EM problems until 1968 Like the finite difference method the finite element. method is useful in solving differential equations As finite difference method represents the. solution region by array of grid points its application becomes difficult with problems having. irregularly shaped boundaries Such problems can be handled more easily by using the finite. element method The finite elements analysis of any problem involves basically four steps A. discretizing the solution region into a finite number of sub regions or elements B deriving. governing equations for a typical element C assembling all the elements in the solution region. and D solving the system of equations obtained,A FINITE ELEMENTS DISCRETIZATION. We divide the solution region into a number of finite elements as illustrated in figure3 1. Fig3 1 A typical finite element subdivision of an irregular domain. Where the region is subdivided into four non overlapping elements and seven nodes We seek an. approximation for the potential Ve within an element e and then interrelate the potential. electric field calculations by numerical techniques a thesis submitted in fufillment of the requirements for the degree of bachelor of technology in electrical engineering by biswanath malik roll no 10502034 department of electrical engineering national institute of technology rourkela 769008 2009 2 electric field calculations by numerical techniques a thesis submitted in fufillment of the

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