ECE 1100 Introduction to Electrical and Computer Engineering

1
ECE 6341
Spring 2009
Prof. David R. Jackson ECE Dept.
Notes 39
2
Finite Source
For phased current sheet
Recall that
3
Finite Source (cont.)
Hence
Note
4
TEN Model for
We can also write
Comparing with the previous result, we have
Similarly,
This motivates the following identifications
5
TEN Model (cont.)
6
TEN Model (cont.)
7
TEN Model (cont.)
8
Example
Find
9
Example (cont.)
Hence
10
Example (cont.)
11
Dyadic Greens Function
where
due to the unit-amplitude electric dipole at
From superposition
We assume that the currents are located on a
planar surface.
where
12
Dyadic Greens Function (cont.)
This is recognized as a 2D convolution
Taking the 2D Fourier transform of both sides,
where
13
Dyadic Greens Function (cont.)
Assuming we wish the x component of the electric
field due to an x-directed current, we have
In order to indentify , we use
14
Dyadic Greens Function (cont.)
Recall that
Hence
15
Dyadic Greens Function (cont.)
The other eight components could be found in a
similar way.
16
Microstrip Line
Microstrip Line
Dominant mode
We assume a purely x-directed current.
17
Microstrip Line (cont.)
Fourier transform of current
18
Microstrip Line (cont.)
Hence we have
19
Microstrip Line (cont.)
Integrating over the ?-function, we have
where we now have
20
Microstrip Line (cont.)
Enforce EFIE using Galerkins method
The EFIE is enforced on the red line.
where
(testing function basis function)
Recall that
Substituting into the EFIE integral, we have
21
Microstrip Line (cont.)
Since the testing function is the same as the
basis function,
Since the Bessel function is an even function,
22
Microstrip Line (cont.)
Hence we have
This is a transcendental equation of the
following form
Note
23
Microstrip Line (cont.)
Branch points
The wavenumber kz0 is the wavenumber that causes
branch points to arise.
Hence
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Microstrip Line (cont.)
Poles
or
25
Microstrip Line (cont.)
Branch points
Poles
26
Microstrip Line (cont.)
Note on wavenumber
We must have that
Otherwise, there would be poles on the real axis,
and this would correspond to leakage into the TM0
surface-wave mode of the grounded substrate.
The mode would then be a leaky mode.
Hence
27
Microstrip Line (cont.)
We next proceed to calculate the voltage
functions explicitly.
28
Microstrip Line (cont.)
29
Microstrip Line (cont.)
30
Microstrip Line (cont.)
At
Hence
31
Microstrip Line (cont.)
low-frequency results
"A frequency-dependent solution for microstrip
transmission lines," E. J. Denlinger, IEEE Trans.
Microwave Theory and Techniques, Vol. 19, pp.
30-39, Jan. 1971.
32
Microstrip Line (cont.)
"A frequency-dependent solution for microstrip
transmission lines," E. J. Denlinger, IEEE Trans.
Microwave Theory and Techniques, Vol. 19, pp.
30-39, Jan. 1971.
frequency variation
33
Microstrip Line (cont.)
Characteristic Impedance Quasi-TEM Method
original problem
equivalent problem (TEM)
The equivalent homogeneous problem has (by
definition) the same wavenumber as the original
problem. This defines the effective
permittivity.
34
Microstrip Line (cont.)
Characteristic impedance we use the equivalent
TEM problem
(The zero subscript denotes the value when using
an air substrate.)
Simple CAD formulas may be used for the Z0 of an
air line.
35
Microstrip Line (cont.)
Characteristic Impedance Voltage-Current Method
Note It is possible to perform the spatial
integrations for the voltage in closed form
(details are omitted).
36
Microstrip Line (cont.)
Characteristic Impedance Power-Current Method
Note it is possible to perform the spatial
integrations in closed form (details are omitted).
37
Microstrip Line (cont.)
Characteristic Impedance Power-Voltage Method
Note it is possible to perform the spatial
integrations in closed form (details are omitted).
38
Microstrip Line (cont.)
Comparison of methods

• At low frequency all three methods agree well.
• As frequency increases, the VI, PI, and PV
  methods give a Z0 that increases with frequency.
• The Quasi-TEM method gives a Z0 that decreases
  with frequency.
• The PI method is usually regarded as being the
  best one for high frequency.

39
Microstrip Line (cont.)
Quasi-TEM Method
?r 15.87, h 1.016 mm, w/h 0.543
"A frequency-dependent solution for microstrip
transmission lines," E. J. Denlinger, IEEE Trans.
Microwave Theory and Techniques, Vol. 19, pp.
30-39, Jan. 1971.
40
Microstrip Line (cont.)
A novel approach for calculating the
characteristic impedance of printed-circuit
lines, F. Mesa and D. R. Jackson, IEEE Microwave
and Wireless Components Letters, Vol. 4, pp.
283-285, April 2005.
41
Microstrip Line (cont.)
Ve effective voltage (average taken over
different paths).