EEE 431 Computational Methods in Electrodynamics

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EEE 431Computational Methods in Electrodynamics

• Lecture 15
• By
• Dr. Rasime Uyguroglu
• Rasime.uyguroglu_at_emu.edu.tr

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Integral Equations and The Moment Method
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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

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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

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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.

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Electrostatic Charge Distribution

• Simplifying assumptions
• Assume . Also assume the wire is
  a solid conductor. Then
• And

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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.

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Electrostatic Charge Distribution

• Then
• or

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Electrostatic Charge Distribution

• And
• This is the integral equation. Solve the integral
  equation for .

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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.

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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.

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Electrostatic Charge Distribution

• Now take a particular and utilize the
  property of the segmentation.

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Electrostatic Charge Distribution

• There are N unknowns above, namely
• We need N linearly independent equations. Take
  k1,2,3,,N.

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Electrostatic Charge Distribution

• Then

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Electrostatic Charge Distribution

• Or
• Where (T transpose)

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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

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Electrostatic Charge Distribution

• Evaluation of the Matrix Elements

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Electrostatic Charge Distribution

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

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Electrostatic Charge Distribution

• Exercise Consider a wire with ,
  a0.001m, V1 Volt. Determine the charge
  distribution for N5.

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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.

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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.

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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.

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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.

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Differential Equations Vs. Integral Equations

• Integral equations may take several forms, e.g,
  Fredholm equations.

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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.

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Moment Methods (Method of Moments, MoM)

• The second class of integral equations, with a
  variable upper limit of integration, Volterra
  equations

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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 ngt1 under the integral
  sign.

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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.

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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

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Differential Equations Vs. Integral Equations

• It differentiated by using the Leibnitz rule
• Differentiating
• We obtain

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Differential Equations Vs. Integral Equations

• Or
• Integrating gives
• Where is the integration constant.
• Or
• From the given integral equation