EEE 431 Computational Methods in Electrodynamics Lecture 15 By Dr. Rasime Uyguroglu Rasime.uyguroglu@emu.edu.tr

Integral Equations and The Moment Method

Integral Equation Method/ Electrostatic Charge Distribution Finite Straight Wire (Charged) at a Constant Potential Formulation of the Problem (In terms of the integral eqn.) For a given charge distribution, the potential is:

Electrostatic Charge Distribution Now consider a wire of length along the y direction. The wire has radius , and connected to a battery of 1 Volts. To have 1Volts everywhere on the surface (actually inside too), a charge distribution is set up. Let this charge be

Electrostatic Charge Distribution Then, Position vector of any point in space. Position vector of any point on the surface of the wire. Surface charge density.

Electrostatic Charge Distribution Simplifying assumptions: Assume . Also assume the wire is a solid conductor. Then: And:

Electrostatic Charge Distribution Integral Equation: If the observation point is brought onto the surface (or into the wire) the potential integral must reduce to 1 volt for all on S or in S. Choose along the wire axis.

Electrostatic Charge Distribution Then: or

Electrostatic Charge Distribution And This is the integral equation. Solve the integral equation for .

Electrostatic Charge Distribution Numerical Solution: Transforming the integral equation into a matrix equation: The inverse of the integral equation for will be achieved numerically by discretizing the integral equation.

Electrostatic Charge Distribution Let us divide the wire uniformly into N segment each of width . If is sufficiently small we may assume that is not varying appreciably over the extent , and we can take it as a constant at its value at the center of the segment.

Electrostatic Charge Distribution Now take a particular and utilize the property of the segmentation.

Electrostatic Charge Distribution There are N unknowns above, namely: We need N linearly independent equations. Take k=1,2,3,…,N.

Electrostatic Charge Distribution Then:

Electrostatic Charge Distribution Or: Where: (T: transpose)

Electrostatic Charge Distribution Where: (NXN) matrix to be generated. (NX1) excitation column vector (known). (NX1) unknown response column vector to be found. Then the solution is:

Electrostatic Charge Distribution Evaluation of the Matrix Elements:

Electrostatic Charge Distribution Where, is the distance between the m th matching point and the center of the n th source point.

Electrostatic Charge Distribution Exercise: Consider a wire with , a=0.001m, V=1 Volt. Determine the charge distribution for N=5.

Moment Methods (Method of Moments, MoM) The MoM is the name of the technique which solves a linear operator equation by converting it to a matrix equation.

Moment Methods (Method of Moments, MoM) Consider the differential equation Where L is a differential operator, is the unknown field and is the known given excitation. The Method of Moments is a general procedure for solving this equation.

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.

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.

Differential Equations Vs. Integral Equations Integral equations may take several forms, e.g, Fredholm equations.

Moment Methods (Method of Moments, MoM) Where is a scalar (or possibly complex) parameter. Functions K(x,t) and f(x) are known. K(x,t) is known as the kernel of the integral equation. The limits a and b are also known, while the function is unknown.

Moment Methods (Method of Moments, MoM) The second class of integral equations, with a variable upper limit of integration, Volterra equations:

Moment Methods (Method of Moments, MoM) If f(x)=0 the integral equations become homogeneous. All above equations are linear. An integral equation becomes non-linear when appears in the power of n>1 under the integral sign.

Differential Equations Vs. Integral Equations Most differential equations can be expressed as integral equations, e. g., This can be written as the Voterra integral equation.

Differential Equations Vs. Integral Equations Solve the Voterra integral equation: In general given an integral with variable limits: It differentiated by using the Leibniz rule:

Differential Equations Vs. Integral Equations It differentiated by using the Leibnitz rule: Differentiating We obtain:

Differential Equations Vs. Integral Equations Or: Integrating gives: Where is the integration constant. Or From the given integral equation: