ELEG 840

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ELEG 840

• Lecture 8
• Professor Dennis W. Prather

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• Last Time
• Structure of a 2D FDTD program
• Stability in higher dimensions
• Derived the 3D FDTD formulation
• Today
• Equivalent medium parameters
• Ideal conducting sources
• Determination of steady state fields
• Transient sources

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• Equivalent Medium Parameters
• The discretization of Maxwells equations becomes
  invalid at the boundary, or interface, between
  two media.
• In this case, we use special boundary conditions
  to deal with this problem.
• One simple method is to simply let all of the
  cell be homogenous cubes ?ignore the interface.
• A better method is to introduce an equivalent
  medium parameter at the surface of scattering
  object.
• Once the medium is discretized, the boundaries,
  or surfaces align perfectly with the boundaries
  of the FDTD cells.

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y
1
x
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• The E field components at each node are
  tangential to the surface of the cells.
• The H-Field components are normal to the surface
• ?Along x constant surface
• ?TE, 2D
• The arrangement of cells and surface is
  consistent with the boundary conditions that the
  E-field and normal B-field are continuous across
  the boundary.

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• To see this consider

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Consider them across an x constant surface
Ey
z
Hz
Ez
Ez
y
Ex
Hx
Ex
Hy
x
Ey
Hy
Ey
Hz
2D Space
Ez
Ez
Hx
Ey
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• At one side of the surface, and at
  the other side , so we need to decide
  what value of we should use.
• Therefore we make a table of modification to
  at different constant surfaces

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Draw a two cell interface
z
y
x
x constant surface
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The corresponding form of (5) (1) medium
1 (2) medium 2 If we add these two equations,
we get
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• At the x constant surface, the tangential
  Hz-field is continuous
• Also, the normal Bx-field is continuous

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Combining, we get
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Substituting in we get This form must be
representative of Maxwells Equations
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Similar to two inductors in series. ?We can
follow a similar procedure to determine the
equivalent permitivity and conductivity
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This then requires that
Series capacitor
So the resulting expression is
This is the same as eqn. 5, but
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• Ideal Conducting Surfaces
• In a highly conducting media, there are two
  common methods for dealing with this kind of
  boundary.
• The first way us to assume the surface to a PEC
• In this case, the tangential electric field
  components are set equal to ZERO!
• The second approach we use the medium parameters
  for a real conductor.
• For example, if the medium has mhos/m ?
  copper
• ? aluminum
• If the frequency is then the corresponding skin
  depth

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• If the FDTD cell was
• Then the skin depth is only a small fraction of
  the edge of one cell!
• Determination of Steady State Fields
• One of the more significant advantages of the
  FDTD is its ability to perform both steady state
  and transient field analysis.
• A steady state condition can be achieved using
  the FDTD method by either Fourier transforming
  the output response of the EM and the extracting
  the frequency of interest.
• ? Pulse based incident field
• ? Store 1N,
• As incident source use was a sinwave and let a
  steady condition build up.

Frequency domain representations of E-field at
(i, j, k)
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• A key factor in achieving a steady state
  condition is to time-step long enough to ensure
  that all of the transient fields have decayed
  away.
• Typically, this requires 2-5 transient periods,
  T5, which are defined as the number of time-steps
  required for one cycle of the incident field to
  transverse the maximum extent of the scattering
  object.

T5
2-5 periods of this to reach s.s
x
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• In this case, the incident field is a sinusoidal
  wave, having a frequency of the desired response.
• Note that the source may also be different from a
  plane wave.
• In this approach the value of the E and the H
  field components are zero at t 0, throughout
  the entire computational domain.
• Once introduced, the source as time marched
  according to the Yee algorithm.
• The question then arises how do we extract the
  S.S. field in both amplitude and phases at each
  node?
• Generally speaking, there are three ways to do
  this
• 1. Peak value method
• 2. Quarter wave method
• 3. Least squares method

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t

• Peak Value Method
• In a steady state field, the derivative of the
  field at its maximum value is 0, in other words,
  the sign of the derivative on either side is
  different.
• Therefore, if we look at the field values at 3
  consecutive time steps, fn-1, fn, fn1, we can
  determine if we are at a maximum or not.

-

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• More succinctly as
• When at peak value that is the magnitude of the
  frequency response at that point.
• We can also determine the phase at any point in
  computational region from the sinusoidal behavior
  of the steady state field.
• Consider the the scattered field at any point in
  space can be written as
• refers to the phase of the scattered field
  relative to the origin of the computational
  region.
• If we let the time when the field is at a
  positive peak be n?t, then we have

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t
t
T
2T
has a value between
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• Complex Quarter Wave Transform
• An alternate method for determining the amplitude
  and phase of the steady state field is to sample
  the field at a particular node for a given
  instant in time.
• By doing this, we can define the electric field
  as
• If we now sample the same node at a given number
  of time steps, later, ?n, we have
• If we chose (which
  corresponds to a quarter wave delay) then we have
• We can combine E1 anf E2 to form a complex field
  value.

Embers Formula
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• So we divide both sides by (we know this value
  tn?t)
• We can uniquely determine both

• Least Squares Fit Method
• In contrast to the two previous methods, this
  method offers unlimited precision or accuracy.
• This method stores the field values at selected
  nodes for a given number of steady state cycles.

I
peak
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• Note that the data only has to be stored during
  the remaining steady state cycles.
• Once the field values have been stored, once can
  extract the magnitude and phase according to the
  following algorithm
• We first note the filed at any node can be
  written as a sinusoidal field with a magnitude c,
  and phase
• Using trigonometric identities, we can rearrange
  this as
• where
• Using the FDTD method, we can store the values of
• for the last few cycles of time marching.

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• Now using a least squares approach, we can find
  the values of A and B that minimize the error
  between
• To minimize this we perform
• 1)
• 2)
• Where
• As a result

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• And
• Using this to form a system of equations, we have

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Then simply invert the 2 x 2 matrix and solve for
A B ? C Ø