Notes 6

1
ECE 6345
Fall 2006
Prof. David R. Jackson ECE Dept.
Notes 19
2
Overview
In this set of notes we introduce the Spectral
Domain Immitance (SDI) method, which is a
powerful method for solving for the fields due to
sources inside of layered media.

• The basic idea is developed here for an infinite
  phased current sheet.
• In the next set of notes, we apply the method for
  an arbitrary finite sheet of electric current.

3
Spectral Domain Immitance Method
We initially consider a planar source inside of a
layered structure. (The method can be extended to
include vertical sources as well.) The figure
shows a single layer, but the method works for
any number of layers.
4
SDI Method (cont.)
Introduce Fourier transform pair
Considering the integrals as limits of sums, we
can write
5
SDI Method (cont.)
6
SDI Method (cont.)
Denote
Then
The finite-size current sheet is thus expressed
as a superposition of infinite phased current
sheets.
7
SDI Method (cont.)
That is,
where
8
SDI Method (cont.)
Consider a single sheet of current of the form
The superscript p stands for phased current
sheet.
We wish to determine the amplitude of the plane
waves that this current source launches, and the
field at any point inside the structure.
9
SDI Method (cont.)
The current sheet launches a pair of plane waves
that propagate in a transverse direction
determined by the wavenumber.
top view
10
TMz PW
3D view
The upward-going TMz plane wave is shown here.
top view
11
TMz PW (cont.)
Denote
Note these new unit vectors depend on the values
of (m,n).
and
12
TMz PW (cont.)
For the TMz plane wave, we then have
13
TEz PW
The upward-going TEz plane wave is polarized as
shown here.
14
Wave Impedances
The wave impedances are calculated for waves
traveling upward.
TMz
TEz
15
TEN TMz PW
Consider the TMz plane wave that gets launched by
the current sheet
We wish to use a TEN model to find the plane-wave
field inside the layered structure.
16
TEN TMz PW (cont.)
We introduce the following modeling equations.
TEN model
Note that the voltage (tangential electric field)
must be continuous at the source location, so the
source model is a parallel element.
17
TEN TEz PW
We introduce similar modeling equations for the
TEz case.
TEN model
18
Source Model
Consider the current sheet in an infinite
homogeneous medium
By symmetry,
19
Source Model (cont.)
Denote
The TM current is that part of the total current
that launches only a TM plane wave, while the TE
current launches only a TE plane wave.
Conclusion the current that launches the TMz
plane wave is polarized in the u direction.
20
Source Model (cont.)
Hence
Similarly
21
Source Model (cont.)
We can now determine the source amplitude in the
TEN model
so
The zero subscript indicates that the current has
the exponential phase term suppressed.
or
Similarly,
22
TEN
The TEN models are shown below.
23
Example
Assume
for
Find
Note if we wanted to find Ez, we would need to
find the transverse magnetic field first, and
then apply Amperes law.
24
Example
In this example we have
The unit vectors are
25
Example (cont.)
26
Example (cont.)
27
Example (cont.)
(TMz or TEz)
From TL theory,
28
Example (cont.)
(TMz or TEz)
Hence
Therefore,
29
Example (cont.)
Similarly
Therefore,
30
Example (cont.)
Then we have
where
with