ELEG 840

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ELEG 840
Lecture 5 Professor Dennis W. Prather
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• Last Time
• Presented a summary of the formulation for the
  scattering from a dielectric cylinder.
• Presented an overview of alternate computational
  methods which included MOM, FEM, BEM, FDFD, and
  FDTD methods.
• Did a survey of the FDTD method, beginning with
  the introduction of the technique by Kane Yee in
  1966
• Today
• Present the formulation of the FDTD method in
  1-Dimension
• Discuss issues related to stability
• Begin introduction of higher dimensional FDTD
  formulations

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• FDTD Process and Flow Chart
• The FDTD method begins by decomposing the
  scattering problem into a region that consists of
  man computational cells.
• Within these cells, the electric and magnetic
  fields are alternatively distributed.
• The cells or nodes are also interlaced in a time
  sequence.
• To solve the difference equations, we begin a
  time iteration process.
• ?given an initial condition, we solve for the
  distribution of the electromagnetic throughout
  the computational space.

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FDTD Flow Chart
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H
Connecting Boundary
Increment n
Compute the value of E
Recall the n increments time ? tn?t
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• Central Difference Expression
• Recall that Maxwells Equations in a source free
  isotropic, and linear medium are

• The FDTD method derives its name from a direct
  central finite-difference approximation to these
  equations.
• In the formulation we assume that f(u,v,w,t) is a
  component of either the electric or magnetic
  field.
• Assume (u, v, w) represent an octagonal
  coordinate system.
• We denote discrete points in space as
• (i?u, j?v, k?w)

in a discrete point in the comp. space.
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• We discretize time a
• tnDt
• Consequently,
• f (u, v, w, t) f(i?u, j?v, k?w, n?t)
• ?u, ?v, ?w are lattice increments in the u, v,
  w directions.
• Yee used the central finite-difference
  expressions for the space and time derivatives.
• For example, the first partial derivative of f in
  the u-direction, calculated at time tnn?t

• Note the /- ½ increments in the I subscript
  makes the electric and magnetic fields interlaced
  in space

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• An expression for

• Use these expression in Maxwells Equations
• In the 1-D case, we assume TEM polarization

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This represents a plane wave
? Propagate in x-direction
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Using the different formulations presented on the
previous page, we get
Computational cell in 1D ?
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Solve for the most recent time value we have
If the material is assumed to be non magnetic,
ie., , then we can define a new constant
This yields
The first equation that we code
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Interpolate
To implement steps
End
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Linear interpolation in time
Sub back in and combine terms and we get
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Solve for the most recent time step
Define a new variable
And multiply both sides by Rb ? we get
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Second equation that we code
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• ID FDTD Source
• The incident wave for the computational region
  can be either a transient pulse or a sinosoid.
• For the case of a transient pulse, we determine
  the response over the band width of the pulse.
• Transient Pulse
• A Gaussian pulse with the form

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Also, we define
impedance of the comp space

• The come from the spatial delay of
  as it follows from .

• Consider where the term results from the
  half time step.

• If we use a spatial sampling rate of
  , from stability.

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• Then we have

• Sinusoidal Pulse
• Can be thought of as a radiating dipole.

Desired frequency
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• ID Absorbing Boundary
• At the truncation Of the computational space, we
  must apply an Absorbing Boundary Condition (ABC),
  otherwise the fields will be reflected from that
  boundary and give rise to numerical error.
• There are many types of ABCs they include
• Radiation
• Bayliss-Turkel
• Enquist-Mojda
• Mar
• Higdon
• Liao
• Berenger, PMLs (perfectly matched layer)

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• Radiation ABC
• Consider the wave equation

• This can be factored into

Forward going operator
Backward going operator
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• The solution to the wave equation is a linear
  combination of sinusoids

• If we apply the operator

Vx
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Phase velocity

• Which results in x direction
• From the wave operator

One must equal to zero
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Faraday and Amperes
Backward going operator
Apply to forward going operator
else FDTD
2 equations
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In difference form we get
Do interpolation on n1/2
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In terms of spatial increments
Doing the same thing for the n-1/2 term we get
Last node in the right hand side
1 2 3 4 5 . . . . . . . . .
. N
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If we use the magic time step
This produces
x
N
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• Writing a ID FDTD Program
• Define physical constants

Calculated
2. Define program constants
3. If using a pulse source, define Ez and Hy,
otherwise implement a hard source for Ez (xi)
4. Start time marching -pulse n1Nt -sinusoid
n1Nt
5. Increment the spatial index
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Nodal Valves
Check ABCs
End
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Separate spatial loop
Increment time (4)
Store time values
Go to step (4)
? Change
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• Source code
• Snap shots of the pulse in time

tt0
tt1