EEE%20431%20Computational%20Methods%20in%20Electrodynamics

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EEE 431Computational Methods in Electrodynamics

• Lecture 13
• By
• Dr. Rasime Uyguroglu
• Rasime.uyguroglu_at_emu.edu.tr

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Numeric Dispersion

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• A dispersion relation gives the relationship
  between the frequency and the speed of
  propagation.
• Consider a plane wave propagating in the positive
  z direction in a lossless medium.
• In time harmonic form the temporal and spatial
  dependence of the wave are given by

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Where is the frequency and is the phase
  constant.
• The speed of the wave can be found by determining
  how fast a given point on the wave travels.
• Let constant

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Setting this equal to a constant and
  differentiating with respect to time gives

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Where is the speed of the wave,
• propagating in the z-direction.
• Therefore the phase velocity yields

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• This is apparently a function of frequency, but
  for a plane wave the phase constant is
  given by .
• Thus, the phase velocity is
• Where c is the speed of light in free space.

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Note that in the continuous world for a lossless
  medium, the phase velocity is independent of
  frequency. The dispersion relationship is

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Since c is a constant, as for any
  given material, all frequencies propagate at the
  same speed.
• Unfortunately this is not the case in the
  discretized FDTD worlddifferent frequencies have
  different phase speeds.

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Governing Equations
• Define the spatial shift-operator
• And the temporal shift-operator
• Let a fractional superscript represent a
  corresponding fractional step.

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• For example

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Using these shift operators the finite-difference
  version of
• Can be written

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Rather than obtaining an update equation from
  this, the goal is to determine the phase speed
  for a given frequency.
• Assume that there is a single harmonic wave
  propagating such that

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Where is the phase constant which exists in
  the FDTD grid and are constant
  amplitudes.
• We will assume that the frequency
• is the same as the one in the continuous
  world.
• Note that one has complete control over the
  frequency of excitation, however, one does not
  have control over the phase constant, i.e., the
  spatial frequency.

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Assume, the temporal shift-operator acting on the
  electric field

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FINITE DIFFERENCE TIME DOMAIN METHOD( Numerical
Dispersion)

• Similarly, the spatial shift-operator acting on
  the electric field yields

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Thus, for a plane wave, one can equate the shift
  operators with multiplication by an appropriate
  term

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Now the scalar equation

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Employing Eulers formula to convert the complex
  exponentials to trigonometric functions

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Canceling the exponential space-time dependence
  which is common to both sides produces

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Solving for the ratio of the electric and
  magnetic field amplitudes yields

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABCs)

• Another equation relating can be
  obtained from

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FINITE DIFFERENCE TIME DOMAIN METHOD

• Expressed in terms of shift operators, the
  finite-difference form of the equation

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• As before, assuming plane-wave propagation, the
  shift operators can be replaced with
  multiplicative equivalents. The resulting
  equation is

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FINITE DIFFERENCE TIME DOMAIN METHOD

• Simplifying and rearranging yields

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numeral
Dispersion)

• Equating equations and
  cross-multiplying

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Taking the square root
• This equation gives the relation between
• Which is different than the one obtained for the
  continuous case.

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numeral
Dispersion)

• However, the two equations do agree in the limit
  as the discretization gets small.

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FINITE DIFFERENCE TIME DOMAIN METHOD (numerical
Dispersion)

• The first term in the Taylor series expansion of
  for small
• Assume that the spatial and temporal steps are
  small enough so that the arguments of
  trigonometric functions are small in the
  dispersion relation.

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• From this
• Which is exactly the same as in the continuous
  world. However, this is only true when the
  discretization goes to zero.
• For finite discretization, the phase velocity in
  the FDTD grid and in the continuous world differ.

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FINITE DIFFERENCE TIME DOMAIN METHOD ( Numerical
Dispersion)

• In the continuous world
• In the FDTD world the same relation holds

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FINITE DIFFERENCE TIME DOMAIN METHOD

• For the one dimensional case a closed form
  solution for is possible.
• A similar dispersion relation holds in two and
  three dimensions, but there a closed-form
  solution is not possible.

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• For a closed form solution
• The factor
• Where,

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Thus

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• Consider the ratio of the phase velocity in the
  grid to the true phase velocity

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• For the continuous case

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)

• The ratio becomes
• This equation is a function of the material, the
  Courant number, and the number points per
  wavelength.

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FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical
Dispersion)