Notes 4

1
ECE 6345
Fall 2006
Prof. David R. Jackson ECE Dept.
Notes 5
2
Overview
This set of notes discusses improved models of
the probe inductance of a coaxially-fed patch
(accurate for thicker substrates). A
parallel-plate waveguide model is assumed.
3
Overview (cont.)
The following models are investigated

• cosine-current model
• gap-source model
• frill model

Reference
Comparison of Models for the Probe Inductance
for a Parallel Plate Waveguide and a Microstrip
Patch, H. Xu, D. R. Jackson, and J. T. Williams,
IEEE Trans. Antennas and Propagation, Vol. 53,
pp. 3229-3235, Oct. 2005.
4
Improved Probe Models
Cosine-current model
Note the derivative of the current is zero at
the top conductor (PEC).
Pc complex power radiated by probe current
5
Improved Probe Models (cont.)
Gap-source model
An ideal gap voltage source of height ? is
assumed at the bottom of the probe.
6
Improved Probe Models (cont.)
Frill model
A magnetic frill of radius b is assumed on the
mouth of the coax.
(TEM mode of coax, assuming 1 V)
7
Improved Probe Models (cont.)
Next, we investigate each of the improved probe
models in more detail

• cosine-current model
• gap-source model
• frill model

8
Cosine Current Model
Assume that
Note
Represent current as
9
Cosine Current Model (cont.)
Using Fourier-series theory
or
10
Cosine Current Model (cont.)
or
Result
(derivation omitted)
11
Cosine Current Model (cont.)
Note now we have both Ez and E? To see this
(Time-Harmonic Fields)
so
12
Cosine Current Model (cont.)
For Ez, we have
where
13
Cosine Current Model (cont.)
At
(BC1)
so
14
Cosine Current Model (cont.)
Also we have
(BC2)
To solve for E? , use
15
Cosine Current Model (cont.)
so
Hence we have
For the mth Fourier term
16
Cosine Current Model (cont.)
so that
where
Hence
17
Cosine Current Model (cont.)
For the mth Fourier term
where
18
Cosine Current Model (cont.)
Hence
or
19
Cosine Current Model (cont.)
or
or
(using the Wronskian identity)
or
20
Cosine Current Model (cont.)
We now find the complex power radiated by the
probe
21
Cosine Current Model (cont.)
so
Substituting for the Am coefficient
Hence we have
22
Cosine Current Model (cont.)
Circuit Model
23
Circuit Model (cont.)
Hence
Therefore,
Define
24
Circuit Model (cont.)
Also, use
so
Keep only the m 0 term
(same as previous result using uniform model)
25
Circuit Model (cont.)
The probe reactance is
26
Gap Model
From Fourier series analysis
27
Gap Model (cont.)
where
The magnetic field is found from Ez
where
28
Gap Model (cont.)
Final result
29
Frill Model
To find the current I (z) , use reciprocity.
Introduce a ring of magnetic current K 1 in the
? direction at z (the testing current B).
30
Frill Model (cont.)
31
Frill Model (cont.)
The ring may be replaced by a 1V gap source of
zero height (by the equivalence principle).
Let z ?0
The field of the gap source is then calculated as
was done in the gap-source model, using ? 0.
b
32
Frill Model (cont.)
Final result
33
Comparison of Models
Next, we show results that compare the various
models, especially as the substrate thickness
increases.
34
Comparison of Models
Models are compared for changing substrate
thickness
?r 2.2 a 0.635 mm f 2 GHz Z0 50 ? (b
2.19 mm)
35
Comparison of Models (cont.)
Models are compared for changing substrate
thickness
?r 2.2 a 0.635 mm f 2 GHz Z0 50 ? (b
2.19 mm)
36
Comparison of Models (cont.)
For the gap-source model, the results depend on ?.
?r 2.2 a 0.635 mm f 2 GHz Z0 50 ? (b
2.19 mm)
37
Comparison of Models (cont.)
The gap-source model is compared with the frill
model, for varying ?, for a fixed h.
h 20 mm
?r 2.2 a 0.635 mm f 2 GHz Z0 50 ? (b
2.19 mm)
R
X
38
Comparison of Models (cont.)
These results suggest the 1/3 rule The best ?
is chosen as
This rule applies for a coax feed that has a 50 ?
impedance.
39
Comparison of Models (cont.)
The gap-source model is compared with the frill
model, using the optimum gap height (1/3 rule).
?r 2.2 a 0.635 mm f 2 GHz Z0 50 ? (b
2.19 mm)
40
Comparison of Models (cont.)
The gap-source model is compared with the frill
model, using the optimum gap height (1/3 rule).
?r 2.2 a 0.635 mm f 2 GHz Z0 50 ? (b
2.19 mm)
41
Probe in Patch
A probe in a patch does not see an infinite
parallel-plate waveguide.
Exact calculation of probe reactance
Zin may be calculated by HFSS or any other
software, or it may be measured.
f0 frequency at which Rin is maximum
42
Probe in Patch (cont.)
Cavity Model Using the cavity model, we can
derive an expression for the probe reactance
This formula assumes that there is no z variation
of the probe current or cavity fields
(thin-substrate approximation), but it does
accurately account for the actual patch
dimensions.
43
Probe in Patch (cont.)
a probe radius
(x0, y0) probe location
This formula assumes that there is no z variation
of the probe current or cavity fields
(thin-substrate approximation), but it does
accurately account for the actual patch
dimensions.
44
Probe in Patch (cont.)
Image Theory Using image theory, we have an
infinite set of image probes.
45
Probe in Patch (cont.)
A simple approximate formula is obtained by using
two terms the original probe current and one
image. This should be an improvement when the
probe is close to an edge.
original
image
46
Probe in Patch (cont.)
As shown on the next plot, the best overall
approximation in obtained by using the following
formula
modified CAD formula
47
Probe in Patch (cont.)
Results show that the simple formula (modified
CAD formula) works fairly well.