Computational electromagnetics


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Computational electromagnetics

  1. 1. Module 04 Computational Electromagnetics Awab Sir ( 8976104646
  2. 2. Computational Electromagnetics • The evaluation of electric and magnetic fields in an electromagnetic system is of utmost importance. • Depending on the nature of the electromagnetic system, Laplace or Poisson equation may be suitable to model the system for low frequency operating conditions. • In high frequency applications we must solve the wave equation in either the time domain or the frequency domain to accurately predict the electric and magnetic fields. • All these solutions are subject to boundary conditions. • Analytical solutions are available only for problems of regular geometry with simple boundary conditions. Awab Sir ( 8976104646
  3. 3. Computational Electromagnetics When the complexities of theoretical formulas make analytic solution intractable, we resort to non analytic methods, which include • Graphical methods • Experimental methods • Analog methods • Numerical methods Awab Sir ( 8976104646
  4. 4. Computational Electromagnetics Graphical, experimental, and analog methods are applicable to solving relatively few problems. Numerical methods have come into prominence and become more attractive with the advent of fast digital computers. The three most commonly used simple numerical techniques in EM are • Moment method • Finite difference method • Finite element method Awab Sir ( 8976104646
  5. 5. Computational Electromagnetics EM Problems Partial differential equations Finite Difference Method Finite Element Method Integral equations Method of Moments Awab Sir ( 8976104646
  6. 6. Computational Electromagnetics We now use numerical techniques to compute electric and magnetic fields In principle, each method discretizes a continuous domain into finite number of sections and then requires a solution of a set of algebraic equations instead of differential or integral equations. Awab Sir ( 8976104646
  7. 7. Computational Electromagnetics Consider the Laplace equation which is given as follows: And a source free equation given as Where, u is the electrostatic potential. Awab Sir ( 8976104646
  8. 8. Why do we need to use numerical methods? If we take an example of a parallel plate capacitor. When we neglect the fringing field we get the following equation. Where, V is the closed form analytical solution (can see the effects of varying any quantity on the RHS by significant change in the LHS). For solving the equation 3 we do not need the use of numerical techniques.Awab Sir ( 8976104646
  9. 9. Why do we need to use numerical methods? However if we now take into account the fringing field the solution for every point x & y such that equation 4 is satisfied is not manually possible To find the field intensity at any point we need to use numerical techniques. Awab Sir ( 8976104646
  10. 10. Step 1: Divide the given problem domain into sub-domains It is a tough job to approximate the potential for the entire domain at a glance. Therefore any domain in which the field is to be calculated is divided into small elements. We use sub domain approximation instead of whole domain approximation. Considering a one dimensional function Awab Sir ( 8976104646
  11. 11. Step 2: Approximate the potential for each element The approximate potential for an element can be given as OR Where, a, b and c are constants. Considering the first equation for each element, the potential distribution will be approximated as a straight line for every element. Awab Sir ( 8976104646
  12. 12. Step 3: Find the potential u for every element in terms of end point potentials Assuming The above equation can also be written as We can write Awab Sir ( 8976104646
  13. 13. Step 3: Find the potential u for every element in terms of end point potentials We can write equation 1 and 2 in matrix form as follows Rearranging the terms we get Awab Sir ( 8976104646
  14. 14. Step 3: Find the potential u for every element in terms of end point potentials Substituting equation 5 in equation 1 we get Similarly we can find the electrostatic potential for each element. Awab Sir ( 8976104646
  15. 15. Step 4: Find the energy for every element The energy for a capacitor is given as follows: The field is distributed such that the energy is minimized. The electric field intensity is related to the electrostatic potential as follows Awab Sir ( 8976104646
  16. 16. Step 5: Find the total energy The total energy is the summation of the individual electrostatic energy of every element in the domain. Awab Sir ( 8976104646
  17. 17. Step 6: Obtain the general solution The field within the domain is distributed such that the energy is minimized. For minimum energy the differentiation of electrostatic energy with respect to the electrostatic potential is equated to zero for every element. Solving this we get a matrix Awab Sir ( 8976104646
  18. 18. Step 6: Obtain the general solution The matrix [K] is a function of geometry and the material properties. The curly brackets denote column matrix. Equation 9 does not lead to a unique solution. For a unique solution we have to apply boundary conditions. Awab Sir ( 8976104646
  19. 19. Step 7: Obtain unique solution We need to apply boundary conditions to equation 9 to obtain a unique solution. For example we assume u1=1 V and u4=5 V. On applying boundary conditions the RHS becomes a non zero matrix and a unique solution can be obtained. Awab Sir ( 8976104646
  20. 20. Steps for Finite Element Method 1. Divide the given problem domain into sub- domains 2. Approximate the potential for each element 3. Find the potential u for every element in terms of end point potentials 4. Find the energy for every element 5. Find the total energy 6. Obtain the general solution 7. Obtain unique solution Awab Sir ( 8976104646
  21. 21. FINITE ELEMENT METHOD Awab Sir ( 8976104646
  22. 22. FINITE ELEMENT METHOD Awab Sir ( 8976104646
  23. 23. FINITE ELEMENT METHOD Awab Sir ( 8976104646
  24. 24. FINITE ELEMENT METHOD Awab Sir ( 8976104646
  25. 25. FINITE ELEMENT METHOD The finite element analysis of any problem involves basically four steps: 1. Discretizing the solution region into a finite number of sub regions or elements 2. Deriving governing equations for a typical element 3. Assembling of all elements in the solution region 4. Solving the system of equations obtained. Awab Sir ( 8976104646
  26. 26. 1. Finite Element Discretization Awab Sir ( 8976104646
  27. 27. 1. Finite Element Discretization We divide the solution region into a number of finite elements as illustrated in the figure above, where the region is subdivided into four non overlapping elements (two triangular and two quadrilateral) and seven nodes. We seek an approximation for the potential Ve within an element e and then inter-relate the potential distributions in various elements such that the potential is continuous across inter-element boundaries. The approximate solution for the whole region is Awab Sir ( 8976104646
  28. 28. 1. Finite Element Discretization Where, N is the number of triangular elements into which the solution region is divided. The most common form of approximation for Ve within an element is polynomial approximation, namely for a triangular element and for a quadrilateral element. Awab Sir ( 8976104646
  29. 29. 1. Finite Element Discretization The potential Ve in general is nonzero within element e but zero outside e. It is difficult to approximate the boundary of the solution region with quadrilateral elements; such elements are useful for problems whose boundaries are sufficiently regular. In view of this, we prefer to use triangular elements throughout our analysis in this section. Notice that our assumption of linear variation of potential within the triangular element as in eq. (2) is the same as assuming that the electric field is uniform within the element; that is, Awab Sir ( 8976104646
  30. 30. 2. Element Governing Equations Awab Sir ( 8976104646
  31. 31. 2. Element Governing Equations The potential Ve1, Ve2, and Ve3 at nodes 1, 2, and 3, respectively, are obtained using eq. (2); that is, Awab Sir ( 8976104646
  32. 32. 2. Element Governing Equations We can obtain the values of a, b and c. Substituting these values in equation 2 we get Awab Sir ( 8976104646
  33. 33. 2. Element Governing Equations And A is the area of the element e Awab Sir ( 8976104646
  34. 34. 2. Element Governing Equations The value of A is positive if the nodes are numbered counterclockwise. Note that eq. (5) gives the potential at any point (x, y) within the element provided that the potentials at the vertices are known. This is unlike the situation in finite difference analysis where the potential is known at the grid points only. Also note that α, are linear interpolation functions. They are called the element shape functions. Awab Sir ( 8976104646
  35. 35. 2. Element Governing Equations The shape functions α1 and α2 for example, are illustrated in the figure below Awab Sir ( 8976104646
  36. 36. 2. Element Governing Equations The energy per unit length can be given as Where and Awab Sir ( 8976104646
  37. 37. 2. Element Governing Equations The matrix C(e) is usually called the element coefficient matrix. The matrix element Cij (e) of the coefficient matrix may be regarded as the coupling between nodes i and j. Awab Sir ( 8976104646
  38. 38. 3. Assembling of all Elements Having considered a typical element, the next step is to assemble all such elements in the solution region. The energy associated with the assemblage of all elements in the mesh is Where n is the number of nodes, N is the number of elements, and [C] is called the overall or global coefficient matrix, which is the assemblage of individual element coefficient matrices. Awab Sir ( 8976104646
  39. 39. 3. Assembling of all Elements The properties of matrix [C] are 1. It is symmetric (Cij = Cji) just as the element coefficient matrix. 2. Since Cij = 0 if no coupling exists between nodes i and j, it is evident that for a large number of elements [C] becomes sparse and banded. 3. It is singular. Awab Sir ( 8976104646
  40. 40. 4. Solving the Resulting Equations From variational calculus, it is known that Laplace's (or Poisson's) equation is satisfied when the total energy in the solution region is minimum. Thus we require that the partial derivatives of W with respect to each nodal value of the potential be zero; that is, To find the solution we can use either the iteration method or the band matrix method. Awab Sir ( 8976104646
  41. 41. Advantages FEM has the following advantages over FDM and MoM 1. FEM can easily handle complex solution region. 2. The generality of FEM makes it possible to construct a general-purpose program for solving a wide range of problems. Awab Sir ( 8976104646
  42. 42. Drawbacks 1. It is harder to understand and program than FDM and MOM. 2. It also requires preparing input data, a process that could be tedious. Awab Sir ( 8976104646
  43. 43. FINITE DIFFERENCE METHOD Boundary Conditions A unique solution can be obtained only with a specified set of boundary conditions. There are basically three kinds of boundary conditions: 1. Dirichlet type of boundary 2. Neumann type of boundary 3. Mixed boundary conditions Awab Sir ( 8976104646
  44. 44. Boundary Conditions Dirichlet Boundary Condition Consider a region s bounded by a curve l. If we want to determine the potential distribution V in region s such that the potential along l is V=g. Where, g is prespecified continuous potential function. Then the condition along the boundary l is known as Dirichlet Boundary condition. Awab Sir ( 8976104646
  45. 45. Boundary Conditions Neumann Boundary Condition Neumann boundary condition is mathematically represented as Where, the conditions along the boundary are such that the normal derivative of the potential function at the boundary is specified as a continuous function. Awab Sir ( 8976104646
  46. 46. Boundary Conditions Mixed Boundary Condition There are problems having the Dirichlet condition and Neumann condition along l1 and l2 portions of l respectively. This is defined as mixed boundary condition. Awab Sir ( 8976104646
  47. 47. FINITE DIFFERENCE METHOD A problem is uniquely defined by three things: 1. A partial differential equation such as Laplace's or Poisson's equations. 2. A solution region. 3. Boundary and/or initial conditions. Awab Sir ( 8976104646
  48. 48. FINITE DIFFERENCE METHOD A finite difference solution to Poisson's or Laplace's equation, for example, proceeds in three steps: 1. Dividing the solution region into a grid of nodes. 2. Approximating the differential equation and boundary conditions by a set of linear algebraic equations (called difference equations) on grid points within the solution region. 3. Solving this set of algebraic equations. Awab Sir ( 8976104646
  49. 49. Step 1 Suppose we intend to apply the finite difference method to determine the electric potential in a region, shown in the figure below. The solution region is divided into rectangular meshes with grid points or nodes as shown. A node on the boundary of the region where the potential is specified is called a fixed node (fixed by the problem) and interior points in the region are called free points (free in that the potential is unknown). Awab Sir ( 8976104646
  50. 50. Step 1 Awab Sir ( 8976104646
  51. 51. Step 2 Our objective is to obtain the finite difference approximation to Poisson's equation and use this to determine the potentials at all the free points. Awab Sir ( 8976104646
  52. 52. Step 3 To apply the following equation, to a given problem, one of the following two methods is commonly used. ) Awab Sir ( 8976104646
  53. 53. Iteration Method We start by setting initial values of the potentials at the free nodes equal to zero or to any reasonable guessed value. Keeping the potentials at the fixed nodes unchanged at all times, we apply eq. (1) to every free node in turn until the potentials at all free nodes are calculated. The potentials obtained at the end of this first iteration are not accurate but just approximate. To increase the accuracy of the potentials, we repeat the calculation at every free node using old values to determine new ones. The iterative or repeated modification of the potential at each free node is continued until a prescribed degree of accuracy is achieved or until the old and the new values at each node are satisfactorily close. Awab Sir ( 8976104646
  54. 54. Band Matrix Method Equation (1) applied to all free nodes results in a set of simultaneous equations of the form Where: [A] is a sparse matrix (i.e., one having many zero terms), [V] consists of the unknown potentials at the free nodes, and [B] is another column matrix formed by the known potentials at the fixed nodes. Awab Sir ( 8976104646
  55. 55. Band Matrix Method Matrix [A] is also banded in that its nonzero terms appear clustered near the main diagonal because only nearest neighboring nodes affect the potential at each node. The sparse, band matrix is easily inverted to determine [V]. Thus we obtain the potentials at the free nodes from matrix [V] as Awab Sir ( 8976104646
  56. 56. The concept of FDM can be extended to Poisson's, Laplace's, or wave equations in other coordinate systems. The accuracy of the method depends on the fineness of the grid and the amount of time spent in refining the potentials. We can reduce computer time and increase the accuracy and convergence rate by the method of successive over relaxation, by making reasonable guesses at initial values, by taking advantage of symmetry if possible, by making the mesh size as small as possible, and by using more complex finite difference molecules. One limitation of the finite difference method is that interpolation of some kind must be used to determine solutions at points not on the grid. One obvious way to overcome this is to use a finer grid, but this would require a greater number of computations and a larger amount of computer storage. Awab Sir ( 8976104646
  57. 57. METHOD OF MOMENTS Awab Sir ( 8976104646
  58. 58. METHOD OF MOMENTS Like the finite difference method, the moment method or the method of moments (MOM) has the advantage of being conceptually simple. While the finite difference method is used in solving differential equations, the moment method is commonly used in solving integral equations. MoM uses integral method. The advantage of this method is that the order of the problem is reduced by one. For example a parallel plate capacitor is a 3-D domain. However, we will be working only on the surface of the capacitor plates, it becomes reduced to 2-D domain. Awab Sir ( 8976104646
  59. 59. Steps for MoM The potential at any point on the plate is a function of the charge distribution Awab Sir ( 8976104646
  60. 60. Step 1 The charge can be given as Awab Sir ( 8976104646
  61. 61. Step 2 To determine the charge distribution, we divide the plate section into smaller rectangular elements. The charge in any section is concentrated at the centre of the section. The potential V at the centre of any section is given by Awab Sir ( 8976104646
  62. 62. Step 3 Assuming there is uniform charge distribution on each subsection we get This equation can be rearranged to obtain Where, [B]: column matrix defining the potentials [A]: square matrix In MoM, the potential at any point is the function of potential distribution at all points, this was not done in FDM and FEM. Awab Sir ( 8976104646
  63. 63. Finite Difference Method (FDM) Finite Element Method (FEM) Method of Moments (MoM) Basic principle Based on differentiation i.e. the differential equation is converted to a difference equation. 1. Energy based (energy minimization) 2. Weighted residual (reducing the error) Based on integral method Advantage 1. Simplest method 2. Taylor series based 1. Computationally easier than MoM 2. Can be applied to unisotropic media. 3. [K] matrix is sparse 1. More accurate (errors effectively tend to cancel each other) 2. Ideally suited for open boundary conditions 3. Popular for antennas Disadvantage 1. Need to have uniform rectangular sections (not possible for real life structures) 2. Outdated 1. Difficult as compared to FDM 1. Mathematically complex Awab Sir ( 8976104646