ELEG 840

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ELEG 840

• Lecture 9
• Professor Dennis W. Prather

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• Last Time
• We discussed
• Equivalent medium parameters
• Ideal conducting surfaces
• Determination of steady state fields
• Today
• Transient sources
• - Raised cosine
• - Gaussian Pulse
• Conversion to frequency domain
• Source modeling
• - Initial condition
• - Hard and soft source
• - Scattered field formulation
• - Totalscattered field formulation

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• Incident Pulse and Its Spectrum
• In a transient analysis, one is generally
  interested in determining the scattered response
  over a particular band width of interest.
• To do this, the input source must incorporate all
  of the frequencies of interest.
• This can be achieved by using a pulsed source, as
  opposed to a sinusoidal source.
• Raised Cosine Pulse
• A raised cosine pulse consists of a single cycle
  of a cosine wave on a bias of 1

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The width of the raised cosine is 1/2Fb
The spectrum of this pulse is determined by
taking the Fourier transform of
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(1)
(2)
(3)
Combining, we get
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• Simplifying, we get
• This spectrum has basically a maximum value of f
  0, and a half-maximum at f Fb
• The first zero is located at 2Fb
• Therefore we can take 2Fb as the effective BW of
  the raise cosine pulse.
• As determined from the FDTD, the wavelength that
  corresponds to
• Fb?x min c/Fb

• And using

and
, and we have
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• Gaussian Pulse
• To get its BW, we take the Fourier transform

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This formation is best evaluated by completing
the square ?
Completing the square Use the substitution of
variables to get it in to this form
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g(t)
-t
t
-½ Fb
½ Fb

• In order to satisfy the zero initial condition,
  the origin must be shifted to at least ½ Fb
• At ½ Fb, g(t) 4.3 of its maximum
• This requires 40 sampling points in the duration
  of the pulse, as is in the raised cosine.

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• Conversion Time to Frequency Domain
• A pulse with a BW can be used to determine the
  scattering properties of an object over a finite
  BW so long as the pulse includes the BW of
  interest.

g(t)
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• If we are only interested in a discrete set of
  frequencies
• An efficient way to do this is to use a run-time
  Discrete Fourier Transform (DFT)
• One can compute the DFT while time marching is
  taking place
• This results from the definition of the Fourier
  Transform

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• Since the FDTD method produces a sampled set of
  values for E(t), with a time step ?t, we can use
  the DFT to compute the FT ?

Is the freq. of interest
The relationship between the time step and the
frequency resolution is
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• Impulse Response
• In addition to determining a single frequency
  response, this method can be used to determine
  the impulse response of the system.
• Recall that spectra of the rapid cosine and
  Gaussian pulses decrease sharply.
• The frequency components for fgtFb are relatively
  small.
• Thus, we can ignore those frequency components
  that are outside of this range.
• Suppose that in the frequency range of 0 to Fb,
  we sample at Nf points
• Fb4GHz, let ?t 0.125GHz, then Nf Fb/?t 32
• According to the relation between ?t and ?f,
• Then we need 40Nf 1280
• Note that Nt gtgtNf

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• Since the DFT computation can be computed on the
  fly we need only be concerned with values of m
  1, 2, , Nf
• Therefore, we can then divide the spectrum
  (discrete) at the scattered field response by the
  spectrum (sampled) of the incident pulse in order
  to determine the impulse response of the system
  over the BW of the incident pulse.

normalized
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• Source Modeling
• There are a variety of approaches for introducing
  the incident field into a FDTD computational
  region.
• The desired requirements for the source include
  the following
• The source field should appear to originate from
  the region external to the computational space
• At any time step, the incident wave must
  introduce no variations in the intended wave
  front.
• Should allow for flexibility I the polarization,
  time dependence, and phase behavior.
• The source should be transparent to any back
  reflections, so as not to introduce non-physical
  reflections.

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Source transparent
object

• Given these conditions,many types of sources have
  been developed
• Initial Condition
• The field is defined throughout the computational
  region prior to time marching.

E (i, j, k) H (i, j, k)
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• This source cannot be used to create a steady
  state condition.
• One of the common pulse used is the Gaussian
  Pulse.
• Where ß define the spatial BW of the pulse

x
Tw
?Fb
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• Hard and Soft Source Condition
• Continuous, sinusoidal source at all points along
  a given contour of the computational lattice.
• This lattice contour, usually taken as a plane,
  would then radiate the desired wave.
• However, specifications of field values at a
  lattice boundary, without consideration of the
  field values from adjacent lattice points,
  effectively represents a transaction.
• Such a transaction will cause undesired back
  reflections.
• In some applications only a small number of
  sinusoidal cycles needed

x
x
Hard source
x
object
0 lt t lt Ts ?sin (2pft)
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• So one can use a hard source over a given time
  frame, which may correspond to many cycles.
• After this time the source is turned off and the
  value of E and H at these points is calculated in
  the same way as every other point.
• If the duration is small enough, then the
  reflected field does not have time to interact
  with the source points.
• Soft Source
• If the hard source is not on long enough, it will
  interact with the reflected wave.
• However, an alternative source is a soft
  source, in which case the computer first
  calculates the field values according to the
  difference equation, but then adds on a
  sinusoidal element.

Source term
Determined from the difference equations
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• This method can be used to generate a desired
  sinusoidal wave. While undesirable non-physical
  back reflections are minimized.
• Back do still occur, but they are far less than
  that for the hard source.
• Two other sources
• Scattered field formulation
• Total scattered field formulation