ECE 1100 Introduction to Electrical and Computer Engineering

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ECE 6341
Spring 2009
Prof. David R. Jackson ECE Dept.
Notes 34
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Example
line current
By using a Fourier-transform method, the solution
is
where, for y gt 0,
(see the appendix)
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Example (cont.)
The vertical wavenumber is
The wavenumber ky is interpreted as
A convenient change of variables is the
steepest-descent transformation
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Example (cont.)
Then
The path C in the complex ?-plane is not unique
until we choose either or here.
This is because the path is not uniquely
determined by only
To see this in more detail, write
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Example (cont.)
Because kx is real,
Hence
or
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Example (cont.)
There are four possible paths.
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Example (cont.)
kx will vary from -? to ? along each of these
paths.
The path must be chosen so that along the path
Assume we choose the sign (an arbitrary
choice)
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Example (cont.)
Correct path C
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Example (cont.)
Now proceed with the change of variables
Hence, we have
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Example (cont.)
Next, let
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Example (cont.)
The integral then becomes
Hence, we can identify
Hence
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Example (cont.)
SDP
so
Hence
(SDP or SAP)
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Example (cont.)
Using
we see that
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Example (cont.)
(SDP or SAP)
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Example (cont.)
Examination of the original path allows us to
determine the direction of integration along the
SDP.
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Example (cont.)
Calculate
so
or
From the figure we see that the correct choice is
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Example (cont.)
Then we have
The exact solution is
It can easily be verified that
for
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Appendix
Derivation of formula
TMz
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Appendix (cont.)
We then have
Define
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Appendix (cont.)
Choose - sign for
Boundary Conditions at y 0
(satisfied automatically)
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Appendix (cont.)
Hence
We then have
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Appendix (cont.)
Hence
and