EEE 431 Computational Methods in Electrodynamics

1. EEE 431Computational Methods in Electrodynamics Lecture 16 By Dr. Rasime Uyguroglu Rasime.uyguroglu@emu.edu.tr

2. Electrostatic Charge Distribution and The Method of Moments • Consider again the finite straight wire at a constant potential which was discussed in Lecture 15. • From statics we know that a linear electric charge distribution will create an electric potential, V(r):

3. Electrostatic Charge Distribution and The Method of Moments • This equation may be used to find potentials due to the known line charge densities. • However, for most of the practical cases the charge distributions are unknown even when the potential on the source is given.

4. Electrostatic Charge Distribution and The Method of Moments • The wire has a length of along the y direction. It’s radius is , and it is connected to a battery of 1 Volts. • Choosing the observation point along the wire axis:

5. Electrostatic Charge Distribution and The Method of Moments • Where:

6. Electrostatic Charge Distribution and The Method of Moments • The equation above is an integral equation to be solved for the unknown • Such problems are called inverse problems. • The solution of the integral equation may be obtained numerically by reducing it to a series of algebraic equations that may be solved by conventional matrix techniques.

7. Electrostatic Charge Distribution and The Method of Moments • Approximate the unknown charge distribution by an expansion of N known terms with constant, but unknown, coefficients, that is:

8. Electrostatic Charge Distribution and The Method of Moments • Thus the integral equation can be written as:

9. Electrostatic Charge Distribution and The Method of Moments • The above equation is a nonsingular integral, its integration can be changed to summation:

10. Electrostatic Charge Distribution and The Method of Moments • The functions in the expansion are chosen to model accurately the unknown quantity and minimize the computation. • These functions are called basis (or expansion) functions. • In this solution pulse function (subdomain piecewise constant) will be used.

11. Electrostatic Charge Distribution and The Method of Moments • The wire is divided into N segments having lengths of: y

12. Electrostatic Charge Distribution and The Method of Moments • These functions are defined to be of constant value over one segment and zero elsewhere. Or:

13. Electrostatic Charge Distribution and The Method of Moments

14. Electrostatic Charge Distribution and The Method of Moments • The matrix form:

15. Electrostatic Charge Distribution and The Method of Moments • Or: • Where: (T: transpose)

16. Electrostatic Charge Distribution and The Method of Moments • Where: • (NXN) matrix to be generated. • (NX1) excitation column vector (known). • (NX1) unknown response column vector to be found. • Then the solution is:

17. Electrostatic Charge Distribution and The Method of Moments • The integrals involved may be solved by using appropriate approximations. But this may not be possible for complicated problems. • Efficient numerical integration computer subroutines are available.

18. Electrostatic Charge Distribution and The Method of Moments • Summary: • The solution of the integral equation for the charge distribution on a wire has been discussed. • The unknown charge was approximated with some basis functions, dividing the wire into segments and then sequentially enforcing at the center of each segment to form a set of linear equations.

19. Method of Moments • Electromagnetic problems usually involve solution of linear partial differential equations or integral equations. • The general form is:

20. Method of Moments • Two electrostatic examples are: • Poisson’s equation, differential equation: • Coulomb’s Law:

21. Method of Moments/ Green’ Function • In general: • Where the kernel G(r,r’) is the Green’s function.

22. Method of Moments/ Green’ Function • Green’s functions offer a systematic way of converting a Differential Equation (DE) to an Integral Equation (IE). • A Green’s function is the solution of the DE corresponding to an impulsive (unit) excitation

23. Method of Moments/ Green’ Function • Green’s Function provides a method of dealing with the source term g. (in )

24. Method of Moments/ Green’ Function • To solve problems of type • The method of moments begins by approximating the unknown function by a linear combination of unknown functions in the form:

25. Method of Moments/ Green’ Function • The functions are called basis or expansion functions. • They are selected such that the appropriate values of the parameters on the right side of this equation is a reasonably accurate approximation to the left side.

26. Method of Moments/ Green’ Function • Basis Functions: One very important step in any solution is the choice of the basis functions. • 1)Entire domain functions (span the entire domain) • 2)Subdomain Functions (each one of is zero except in a subdomain.

27. Method of Moments/ Green’ Function • The subdomain approach involves subdivision of the structure into N non overlapping segments. The simplest one is the pulse. Piecewise continuous:

28. Inner Product • Inner (dot or scalar) Product of two functions w, g: • * indicates the complex conjugate.

29. Weighting (Testing Functions) • Here w’s are the weighting functions and s is the surface of the structure being analyzed (will be discussed). • Note that the functions w and g can be vectors. • This technique is known as the method of moments.( MoM)

30. Moment Methods (Method of Moments, MoM) • The procedure for applying MoM to solve the equation above usually involves four steps: • 1)Derivation of the appropriate integral equation (IE). • 2)Conversion (discretization) of IE into a matrix equation using basis (or expansions) functions and weighting functions.

31. Moment Methods (Method of Moments, MoM) • 3)Evaluation of the matrix elements. • 4)Solving the matrix equation and obtaining the parameters of interest. • The basic tools for step 2 will be discussed. • MoM will be applied to IEs rather than PDEs.