ELEG 840

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ELEG 840

• Lecture 13
• Professor Dennis W. Prather

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• Last Time
• Discussed Mur 1st and 2nd order ABCs.
• Discussed Liaos Interpolation ABC
• Today
• Introduce the PML ABC
• Propagation of the E and H fields

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• Impedance Matching of Lossy Media
• In the previous couple of lectures, we discussed
  a number of ABCs.
• In general, the ABCs have been based on
  analytically derived operators. Such operators
  serve to estimate the field valve on the ABC
  boundary by referring to field values either
  inferior to the ABC, at previous values of time,
  BOTH.
• However, these ABCs do not represent the current
  state of the art!
• An alternative approach to using such ABCs is to
  include a lossy region, that surrounds the
  computational region.
• In this case as the waves propagate, they get
  absorbed due to presence of a lossy media.
• Although such ABCs have been around for some
  time, it was not until Jean-Pierre Berenger in
  1994 overcame these limitations.

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• Semi-Infinite Reflections
• Consider a semi-infinite medium
• Medium 1 Medium 2
• In medium 1, the total magnetic field consists of
  both the forward and backward going waves.
• In medium 2, we have only the forward going wave.

X 0
X - axis
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Reflection Coefficient
Transmission Coefficient
To determine the form of the electric field we
can simply use Amperes Law
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• Recall that for a plane wave , where z is free
  space impedance and is defined as
• Using this MEs in 2D becomes
• So we have

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• Snells Law
• Snells Law relates the angle of refraction that
  a wave undergoes as it propagates from one medium
  to another.

Total Tangential Electric Field is continuous
across the boundary
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If region is a PEC, then , in this case
use But because we have Law
of Reflection
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So now if we evaluate for lossless dielectric,
at x 0, we have
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• Snells Law
• Maxwells Equations in Lossy Media
• For lossy media MEs, take the following form

Magnetic Conductivity
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To derive the name equation in a lossy
media If we are in a source free media
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• If we consider plane propagation in the x and y
  directions, we have
• Substituting in we get

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• Impedance Matching Conduction for Lossy Media
• When an EM wave exits one material and enters
  another, it undergoes back reflections.
• However, when one of the materials is lossy, it
  is possible to use the conductivity values to
  balance the input and output impedances!
• When two impedances are equal, there are zero
  back reflections.
• In this case the wave is fully transmitted into
  the lossy media, where it is absorbed.
• To see how this comes about, consider the
  evaluation of Hz1, Hz2, Ey1, and Ey2 at x0.

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By equating we get We also know
that (Snells Law) (Normal Incidence)
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If we combine these and solve for , we
get
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Recall that and , therefore if
, then according to Snells Law, in which case
we have
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And noting that Then Where
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If in this equation we let
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• So as long as ? 0, there will be no back
  reflections.
• Unfortunately, in general, this is not the case,
  so this ABC has very limited application.
• However, as discussed next, this limitation was
  overcome by Berenger in 1994.

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• Perfectly Matched Layer
• As presented above, a free space, yet absorbing,
  medium makes an ideal ABC, except for the fact
  that it only works for normal incidence.
• However, Berenger recognized that by splitting
  the transverse field into two orthogonal
  components, one traveling parallel to the
  boundary and the other traveling boundary, that
  the absorbing medium approach could then work for
  oblique angles of incidence.
• TM Polarization
• For TM polarization, the magnetic field is
  parallel to the material interface

Sy
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• Note in this case that ? and ? have been
  normalized by the relative permittivity and
  permeability values.
• Berenger then modified these as follows

• In that case,

and
represent the split field components of the
magnetic field.

• We can simplify this notation by introducing

where
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Using this, we have
We then derive the PML wave equation by taking
the appropriate spatial derivation.
Dividing these Eqns. by
and
And adding them together we arrive at the
PML wave equation.
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The solutions to this are
Then from these equations we can obtain the
transverse E field component.
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So the objective now is to determine the value for
such that there
are no back reflections from the boundary
interface.
To do this, we, again, enforce the continuity of
the tangential field components at x0.
due to the requirements, the phase in

• From this equation, we can see that

• both exponentials must be equal.
• The second equation from the tangential magnetic
  field components

or
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So if
and
then,
Note that this is true for all incidents lt s
TE Polarizations
For TE Polarizations we have Ez, Hx, Hy In this
case, ?Es reduce to
where
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• Working through the TE case, as we did for the TM
  case, we get that

Eyconstant
yb
Ex constant
y0
xa
x0
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1
3
3
2
2
3
3
1
PMC zones
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To derive the difference equations, Substituti
ng in we get
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Now extract and get Now let
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Then the equations can be written as In
this case, the condition in the different zones
are
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As a result the difference equations can be
written as
Now define some alternative material parameters
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Where K depends on where the computation point is
being computed. Now re-expressing the difference
equation For the rest of the fields use a
similar procedure And lastly we can recover
the E and H fields from
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To avoid the sudden change of medium, Berenger
proposed that the losses should increase
gracefully with the depth, ?, in each pm
layer Thickness is the pm thickness Using
this formula it was shown that the reflection
factor was
conductivity
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Ez, Hx, Hy
On Boundary