Numerical Methods in Electromagnetics Many problems in electromagnetics cannot be solved analytically and require a numerical solution. The governing equations in these problems (Maxwell’s equations or related equations) can take the form of either a differential equation or an integral equation. Finite difference technique - a numerical method used to solve partial differential equations (PDE’s). Example PDE Y Poisson’s equation
Method of moments technique - a numerical method used to solve integral equations (IE’s), or integro-differential equations. Example IE Y Conductor held at a constant potential
Numerical solutions to electromagnetic problems invariably result in some type of numerical approximation. Care must be taken to ensure that the approximation used yields an accurate solution to the problem.
Finite Difference Technique Given the governing PDE for a particular electromagnetics problem, the steps involved in the application of the finite difference technique are: (1) Divide the domain of interest into a grid (usually rectangular) in one, two or three dimensions. For example, V(x,y,z) - three dimensional (3D) solution V(x,y) - two dimensional (2D) solution (no z variation) V(x) - one dimensional (1D) solution (no y or z variation) (2) Develop algebraic equations which approximate the partial derivatives in the governing equation (difference equations). (3) Solve the set of algebraic equations. Example (Poisson’s equation) The 3D Poisson’s equation in rectangular coordinates is
Consider a rectangular region defined by 0 # x # a and 0 # y # b enclosed by conductors of known potential [V(x,y)] and a known charge distribution throughout [D(x,y)]. Assuming no z-variation in the potential or charge yields a 2D form of Poisson’s equation.
The region of interest can be divided into a rectangular grid over which the difference equation approximations to the second-order derivatives are defined. The grid points located on the boundary represent fixed nodes where the potential is known. The grid points internal to the boundary are defined as free nodes where the potential must be computed. The indices on the grid nodes range from 0 to Ni+1 in the x-direction and 0 to Nj+1 in the y-direction. This indexing scheme yields a total of Ni Nj free nodes.
The required derivatives in the given 2D Poisson’s equation are:
The first order partial derivatives with respect to x and y can first be defined on either side of the grid point (i,j).
The approximations for the first derivatives on either side of grid point (i,j) can then be used to approximate the second order partial derivatives with respect to x and y.
Inserting the first order derivative approximations and collecting terms yields the second order derivative approximations.
Inserting the second derivative approximations into Poisson’s equation gives
(2D Poisson’s equation, rectangular grid) Given a square grid ()x = )y = h), the equation above reduces to
(2D Poisson’s equation, square grid) In a source-free region (D = 0), Poisson’s equation reduces to Laplace’s equation.
(2D Laplace’s equation, rectangular grid)
(2D Laplace’s equation, square grid) Note that the 2D Laplace’s equation on a square grid illustrates how the potential at any given point is described as the average of the four surrounding points. This principle can easily be extended to three dimensions where the potential is described as the average of the six surrounding points.
Direct Technique - The N equations in N unknowns (where N is the total number of grid points) are solved directly. This technique requires the solution of a linear system of equations (very inefficient as N grows large). Indirect Technique - (iterative technique) An initial guess is made for the node voltages. The finite difference equations are applied again and again, updating the node values, until the node values converge to a solution.
Example (Jacobi iteration) A square grid is shown below for the 2D solution of Laplace’s equation based on the given boundary conditions.
1st Iteration (k =1)
2nd Iteration (k =2)
!
25th Iteration (k = 25)
The number of iterations required for a given problem depends on the geometry of the conductor system, the number of grid points, and the required accuracy for the computed potentials. The voltages to which the node values converge are approximations based primarily on the accuracy of the difference equations used to approximate the derivatives in the PDE. More grid points (smaller grid point separation) yields more accurate solutions given better approximations for the derivatives in the PDE.
