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EEE 431 Computational Methods in Electrodynamics

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

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EEE 431 Computational Methods in Electrodynamics

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

  2. Integral Equations and The Moment Method

  3. 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:

  4. 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

  5. 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.

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

  7. 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.

  8. Electrostatic Charge Distribution • Then: • or

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

  10. 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.

  11. 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.

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

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

  14. Electrostatic Charge Distribution • Then:

  15. Electrostatic Charge Distribution • Or: • Where: (T: transpose)

  16. 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:

  17. Electrostatic Charge Distribution • Evaluation of the Matrix Elements:

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

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

  20. 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.

  21. 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.

  22. 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.

  23. 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.

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

  25. 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.

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

  27. 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.

  28. 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.

  29. 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:

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

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

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