CONDUCTOR AND DIELECTRIC PROPERTY EXTRACTION USING MICROSTRIP TEE
RESONATORS
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classified information.
________________________________________
Andrew Richard Fulford
Certificate of Approval:
______________________________ ______________________________
Lloyd S. Riggs Stuart M. Wentworth, Chair
Professor Associate Professor
Electrical and Computer Engineering Electrical and Computer Engineering
______________________________ ______________________________
R. Wayne Johnson Stephen L. McFarland
Samuel Ginn Distinguished Professor Dean
Electrical and Computer Engineering Graduate School
CONDUCTOR AND DIELECTRIC PROPERTY EXTRACTION USING MICROSTRIP TEE
RESONATORS
Andrew Richard Fulford
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
August 8, 2005
iii
CONDUCTOR AND DIELECTRIC PROPERTY EXTRACTION USING MICROSTRIP TEE
RESONATORS
Andrew Richard Fulford
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon the request of individuals or institutions and at their expense. The author reserves
all publication rights.
______________________________
Signature of Author
______________________________
Date
Copy sent to:
Name Date
iv
VITA
Andrew Richard Fulford, son of Charles Richard Fulford and Burdette Hughes,
was born February 28, 1979 in Enterprise, Alabama. He graduated from Enterprise High
School in 1997. For the next two years he was enrolled at Enterprise State Junior College
before transferring to Auburn University?s engineering program in the Fall of 1999. He
completed a co-op program with Milliken and Company and an internship with the Army
Aviation Technical Testing Center during his undergraduate studies at Auburn
University. In May of 2003 he received a Bachelor?s of Electrical Engineering. He
continued his education becoming a graduate student in the Department of Electrical and
Computer Engineering at Auburn University. While completing his graduate degree, his
research effort has been in the area of microwave material characterization.
v
THESIS ABSTRACT
CONDUCTOR AND DIELECTRIC PROPERTY EXTRACTION USING MICROSTRIP TEE
RESONATORS
Andrew Richard Fulford
Master of Science, August 8, 2005
(B.E.E., Auburn University, 2003)
95 Typed Pages
Directed by Stuart M. Wentworth
The growth and resultant competitive nature of the wireless industry has
prompted the need for high frequency electronics to be provided at a low cost. An
efficient method of component characterization is necessary to ensure that circuit designs
meet the specifications for desired performance in as few design iterations as possible.
The use of board ?real estate? is also a consideration in the choice of a test structure to be
used for component characterization. The open-ended quarter wave tee resonator in a
microstrip configuration is such a test structure.
This thesis describes a method using tee resonator measurements to extract line
attenuation and further to extract dielectric relative permittivity and loss tangent if the
characteristics of the conductor are assumed. Furthermore, a novel approach is presented
where a pair of microstrip tee resonators are employed, each with a different impedance
vi
resonating element, and the behavior of the resonances is used to extract conductor sheet
resistance as well as the dielectric properties. The method is demonstrated for screen-
printed silver epoxy lines on an alumina substrate. This methodology is further examined
by way of method of moments modeling. This modeling is used to validate the procedure
that was developed for use on physical test boards.
vii
ACKNOWLEDGEMENTS
The author would like to express his sincere appreciation to his family for
providing continued love and support and for their patience throughout his life. To his
advisor, Dr. Stuart M. Wentworth, the author would like to sincerely thank him for the
opportunity to study under him and for patience, guidance, and selflessness throughout
the years of study. The author appreciates Mr. Mike Palmer and Mr. Joe Haggerty for
their guidance and assistance in the board fabrication efforts. The author wishes to
acknowledge the assistance and support of the Alabama Microelectronics Science and
Technology Center and the Auburn University Information Technology Peak of
Excellence. Lastly, the author would like to thank everyone who helped make his time at
Auburn University a productive and enjoyable one.
viii
Style manual of journal used Graduate School: Guide to preparation and
submission of theses and dissertations _________________________________________
Computer software used Microsoft Office 2003, Matlab 6.0, ADS, Flowmerics
Micro-Stripes 6.0_________________________________________________________
ix
TABLE OF CONTENTS
LIST OF TABLES????????.??????????...?????...................xi
LIST OF FIGURES??????????????????????????...?..xii
CHAPTER 1: INTRODUCTION?..??????????????????????1
1.1 Motivation ?????????????????????????..?.....1
1.2 Test structures??????????????????????.?.??....2
1.2.1 Scattering parameter tutorial????.??????????..??...?.2
1.2.2 Meander lines????????????????????..??........5
1.2.3 Gap-coupled resonator??????????...?????????.....6
1.2.4 Open-ended stub tee resonator???????????????...........7
1.3 Presented technique???..???????????????????..?.8
CHAPTER 2: PROPERTY EXTRACTION APPROACH???.????????..............10
2.1 Operation of tee resonator??????....???...???..????...?.....10
2.2 Permittivity and attenuation extraction????..................................................12
2.3 Dielectric and conductor property extraction?????????????....18
CHAPTER 3: BOARD FABRICATION?.............................................................................23
3.1 Fabrication??...????????????..????..?..??????23
CHAPTER 4: MEASUREMENTS????????????????????.........31
4.1 Measurement procedure?.....................................................................................31
4.1.1 Standard procedure..........................................................................?.........32
4.1.2 Variability study??????????????????????...35
4.2 Property extraction???????..????????????..???.?37
4.2.1 Permittivity extraction??????????????????..??37
4.2.2 Total attenuation extraction?????????..???????..?..40
4.2.3 Sheet resistance and loss tangent extraction????????????.43
CHAPTER 5: SIMULATION?.??????????????????????...45
5.1 Board design verification???????????????????..??.45
5.2 Modeling approach?????????????.????????..??.47
5.2.1 ADS ?momentum??...????????????????????.48
5.2.2 Flowmerics micro-stripes?????????????????.??.48
5.2.3 Property extraction verification approach?????????????.49
5.3 Results??????????????????.???????.??......51
x
5.3.1 Model meshing resolution???????????????????.51
5.3.2 Delta L calculation??????????????????????53
5.3.3 Relative permittivity extraction?????????????????55
5.3.4 Sheet resistance and loss tangent extraction?.???????????57
CHAPTER 6: CONCLUSIONS?.??????????????????????61
6.1 Technique summary??.????????????????????..?61
6.2 Future work?????.?????????????.??????.?.....62
6.3 Conclusions?????????????????????????........64
REFERENCES?????.???????????????????????.?.65
APPENDIX: MATLAB CODE ??...?????????...???...????.???.68
1 Plot of sheet resistance versus loss tangent code???????...???.??..68
2 Code for extracting the resonant frequency for the first line???...???.??71
3 Effective permittivity calculation code?????????...??????.?.71
4 Characteristic impedance calculation code??????????????.......73
5 Code for calculating total attenuation for the first line?????...?????..73
6 Relative permittivity calculation code?????????????????..75
7 Code used in the permittivity calculation????????????????.76
8 Code used to compute ?L??????????????????????.76
9 Code for the calculation of attenuation due to radiation??????????...77
10 Sheet resistance calculation code..???..???????????????77
11 Conductor attenuation calculation code????????????????..78
12 Code for extracting the resonant frequency for the second line???????.79
13 Code for calculating total attenuation for the second line?????????..80
xi
LIST OF TABLES
3.1 Tee resonator and calibration line design dimensions??????????...23
4.1 Average value and standard deviation of total attenuation with and without using
TDR to place the probes ??????????...???.??????.......35
4.2 Measurement variability??????????????????????36
4.3 Typical data for the test board??????????????...?????38
4.4 Typical permittivity extraction results??????????...??????38
4.5 Typical total attenuation extraction results???????????????41
4.6 Interpolation addition results????????????????????.42
5.1 Results of delta L experiment????????????????????54
5.2 Design parameters for model set???????????????????56
5.3 Extracted relative permittivity and resonant frequency values???????..57
xii
LIST OF FIGURES
1.1 Two-port network?????????.?????????????.?.?.2
1.2 Two-port scattering matrix description?..?????????????.??.3
1.3 Two-port network terminated in a matched load??????.???????4
1.4 Meander lines????????????????????????...??5
1.5 Gap-coupled resonator??????...????.????????????.7
1.6 Open-ended stub tee resonator????.??????????????...?8
2.1 Basic illustration of the microstrip tee resonator??????????...??10
2.2 Diagram indicating the 2 ports and the point of input impedance??????.11
2.3 Resonant S21 dips for the tee resonator????????????????..12
2.4 Cross-section of microstrip line???????????????????.13
2.5 The S21 measurement at a resonant dip for a tee resonator??????...??.13
2.6 Permittivity calculation flowchart?????...??????????..??.16
2.7 Total attenuation calculation flowchart..........................................................?...17
2.8 Extraction of RS and tan? flowchart??????????????.???..21
2.9 Ideal intersection plot for a pair of tee resonators that shows how tan? and RS are
extracted??????????...?????????????????..22
3.1 Lavenir layout for tee resonators and calibration line?????????.??....24
3.2 Mask for tee resonators and calibration through line???...?????.??.25
3.3 Screens for the resonators and groundplane??????????????..26
3.4 Screenprinter????????????????????????.??.27
xiii
3.5 Drying oven??...?????????????????????...?.?28
3.6 Firing oven???????..?????????????????..?.?28
3.7 Completed microstrip tee resonator test structure??????????..?...29
4.1 Probe in contact with board during testing?????????????..?..31
4.2 Testing equipment?????????????????????.?.??32
4.3 Test board during measurement on the Alessi probe station?????.???33
4.4 TDR signatures of correct and incorrect probe placement?????????34
4.5 Plot of long 30 ohm resonator relative permittivity vs. long 50 ohm resonator
relative permittivity????????????????????????39
4.6 Single dip used for attenuation calculation???????????????40
4.7 Chart comparison of 30-sample study of long 50 ? tee resonator with (black) and
without (grey) the MATLAB interpolation addition???????????.42
4.8 Plot of sheet resistance vs. loss tangent using measurements on a 30 ? and 50 ?
tee resonator pair?????????????????????????43
5.1 ADS schematic layout used in the design of the test board????????...46
5.2 Resonant dips used for confirmation of the design of the four resonators???47
5.3 Property extraction verification approach using modeling?????????50
5.4 Progression of resonant frequency towards target value using micro-stripes??51
5.5 Progression of resonant frequency towards target value using ADS
?momentum???????????????????????????..52
5.6 Definition of resonator length with consideration of ?L?????????...53
5.7 Extraction of relative permittivity??????????????????..55
5.8 300 cells per wavelength sheet resistance vs. loss tangent plot???????.58
5.9 500 cells per wavelength sheet resistance vs. loss tangent plot???????.59
5.10 700 cells per wavelength sheet resistance vs. loss tangent plot???????.60
xiv
6.1 Sheet resistance vs. loss tangent plot for test board with copper lines and a
microwave laminate substrate????????????????????63
1
CHAPTER 1
INTRODUCTION
1.1 Motivation
The emergence of wireless communications has prompted the need for cheap and
reliable microwave and rf circuit components. This need is evident in both commercial
and military applications. For example, wireless local area networks (LAN)?s rely on
high frequency circuitry to provide service for data communication in the commercial
and military environment. Another example is the demand for inexpensive but robust
circuitry in the cellular phone industry. The ability to provide reliable microwave and rf
circuitry while observing a low manufacturing cost is of utmost importance due to the
competitive nature of these markets. In order to keep manufacturing costs at a minimum
it is important for high frequency designers to have accurate characterization of the
components used in the circuit design. This will lessen the number of costly
manufacturing iterations required to reach the level of performance that is desired from
the circuit, thus lowering overall manufacturing cost. The wide variety of substrates and
metallizations in use and projected for use in microwave and rf circuits drives the need
for accurate microstrip loss measurements. In many cases, it is difficult to accurately
model losses because of uncertainties in surface roughness and in the behavior of the
dielectrics as a function of frequency or environmental conditions [1,2].
2
1.2 Test structures
A number of test structures have been developed to measure the line attenuation
of microstrip circuits [3-6]. Three commonly used test structures are meander lines, gap-
coupled resonators, and open-ended stub tee resonators. The advantages and
disadvantages of these test structures will be discussed in the following sections. A brief
tutorial in scattering parameters will first be given to serve as an aid in the
comprehension of the following discussions.
1.2.1 Scattering parameter tutorial
Scattering parameters (S-parameters) and the associated scattering matrix can be
used to characterize multiport networks at high frequencies [7]. Scattering matrices are
viewed as a representation more in accord with actual measurements because they deal
with the principles of incident, reflected, and transmitted waves [8]. This concept is
important because the measurements taken with a network analyzer in this work are best
explained by the concept of scattering parameters.
Consider a device of the type shown in Figure 1.1. Here, the device is commonly
referred to as a two port network. It is important to note that the scattering matrix can be
2-port network,
[S]
Zo Zo
Figure 1.1 Two-port network
3
formed for a network possessing more than 2 ports. But, for the purposes of this work,
only an understanding of a two port network is required. This two port network can be
used to represent the ratios of voltage amplitude entering and leaving a port as shown in
Figure 1.2. Upon observation, it is evident that V1- is composed of the portion of V1+ that
is reflected and the portion of V2+ that is transmitted. This behavior can be written in
terms of the scattering parameters as
1 11 1 12 2V S V S V? + += + . (1.1)
V2- can be written in a similar fashion as
2 21 1 22 2V S V S V
? + += + . (1.2)
These expressions can then be expressed in the matrix form as
1 11 12 1
21 222 2
V S S V
S SV V
? +
? +
? ? ? ?? ?=
? ? ? ?? ?? ? ? ?? ?
? ? ? ?
. (1.3)
A particular scattering parameter, Sij, can be defined as the fraction of the voltage
wave entering port j that exits port i [7]. This can be written in general terms as
0, k
i
ij
j V k j
VS
V +
?
+
= ?
= . (1.4)
S22S11
S12
S21
V1+ V2-
V1- V2+
Port 1 Port 2
Figure 1.2 Two-port scattering matrix description
4
This can also be thought of as the source of the voltage being at the jth port and the
destination of the voltage being at the ith port.
We can determine the transmission scattering matrix, S21, by terminating port 2 in
a matched load as shown in Figure 1.3. This will result in V2+= 0 and the resultant S-
parameter is
2
2
21
1 0V
VS
V +
?
+
=
= . (1.5)
The scattering parameter, S21, is most commonly used in this work to represent the
amount of power transferred from port 1 to port 2 during the measurement procedure.
This translation of S21 into a form of power listed in units of dB can be performed by
multiplying the log magnitude of S21 by 20:
2
21
1
21 21
( ) 20log
( ) 20log
VS dB
V
S dB S
?
+=
=
. (1.6)
2-port network,
[S]
Zo Zo Zo
Figure 1.3 Two-port network terminated in a matched load
5
1.2.2 Meander lines
Meander lines are two different length transmission lines that meander across the
surface of the substrate as illustrated in Figure 1.4. Attenuation in microstrip line is
found by taking the difference in the measured log magnitude of the scattering parameter
S21 for two different length meandering transmission lines [9]. This method is useful in
removing losses associated with induced discontinuities such as ports and connectors [5].
A method of calculating the total attenuation in terms of decibels per unit length involves
dividing the difference in the S21 log magnitude data by the difference of the two line
lengths. This method requires that the impedance of the lines used closely match the
testing equipment impedance (50?). Otherwise, standing waves established in the lines
will overwhelm the difference data. With this approach it is often necessary to apply
averaging techniques on the data to ensure accuracy in the extraction procedure due to the
characteristic ripple of the measured data [5]. Smoothing, a linearly moving average of
adjacent points, is often used to remove some of the ripple in the data during
measurement. It is important to note that smoothing is applied to the line measurement
rather than to the difference data since the ripple is a function of line length. Smoothing
Figure 1.4 Meander lines
6
must be used carefully as to not mask the true behavior at low frequency ranges [9].
1.2.3 Gap-coupled resonator
A gap-coupled resonator is essentially a structure consisting of a transmission line
separated by gaps on either side. A diagram depicting the typical microstrip gap-coupled
resonator structure is shown in Figure 1.5. The property extraction approach involves
measuring the resonant S21 peaks that occur when the resonator length is integral
multiples of a half guide wavelength [5]. These peaks are used to find the quality factor.
This quality factor can then be used to compute total attenuation. This type of resonator
exhibits the most productive measurements with low loss lines [5,3]. This is in contrast
to the meander line approach. Care in the collection of S21 measurement data must be
exercised due to the nature of the resonant peaks. This method requires that
measurements be taken with a frequency span in close proximity to the peak of interest to
allow a high resolution of the collected data. This high resolution of collected data helps
to ensure an accurate calculation for attenuation. It is also apparent that this type of test
structure is susceptible to capacitive loading at the gaps. This behavior leads to increased
resonator electrical length and subsequent inaccuracy in the extracted properties at high
frequencies [4].
7
1.2.4 Open-ended stub tee resonator
Tee resonators have been used to determine the relative permittivity and loss
tangent of dielectric substrates when the conductor properties were known [4,6,10]. A
diagram depicting a typical tee resonator microstrip test structure is shown in Figure 1.6.
The basic principal is that an open-ended transmission line stub appears as a short circuit
at periodic frequencies. Transmission measurements reveal resonant dips at these
frequencies. The resonant frequencies are used to extract relative permittivity, while the
quality factor of the dips is used to extract line attenuation. The collected data has been
touted as being more accurate than a gap-coupled half-wavelength resonator because
there is one open end for radiative loss instead of four open ends [4,6].
Figure 1.5 Gap-coupled resonator
8
1.3 Presented technique
In this work a technique is presented for the extraction of the permittivity and loss
tangent of the substrate and the sheet resistance of the conductor using the open-ended tee
resonator structure. The total line attenuation consists of conductor, dielectric and
radiative loss. The radiative loss is a small but appreciable component that can be
calculated at the frequency range of interest [11,12]. The dielectric loss is a function of
loss tangent, while the conductor loss is a function of metal conductivity and roughness at
the conductor-dielectric interface. Both types are also a function of line geometry. It is
common practice to assume the conductor properties in order to estimate the dielectric
properties [3,6]. But since the attenuation is often dominated by conductor loss, small
error in the assumed conductor properties can result in large errors in the extracted loss
tangent.
For both the through line and the open-ended stub of a tee line resonator, the
characteristic impedance is customarily chosen to match the system impedance (e.g. 50
Figure 1.6 Open-ended stub tee resonator
9
?). But the open-ended stub will present resonant frequency dips regardless of its
characteristic impedance. Because the losses are also a function of line dimensions,
different width stubs will have attenuations consisting of different contributions from
conductor and dielectric loss. The method proposed in this work describes how
measurements on a pair of tee resonators can be used to extract both conductor sheet
resistance and the dielectric loss tangent. The method is demonstrated for screen-printed
Ag/Pd microstrip lines on an alumina substrate. An interesting result is that the
extraction approach did not work as well as expected. This may be a result of the dated
equations used in the process, minor flaws in the board preparation, or inaccuracies in the
extraction routine itself. Method of moments modeling was used in an effort to
understand why the approach was not as accurate as expected.
10
Dielectric:
(tan ?) Loss Tangent
(?r) Relative Permittivity
Conductor:
(?) Conductivity
(Rs) Sheet Resistance
Resonator Stub
Groundplane
Figure 2.1 Basic illustration of the microstrip tee resonator
CHAPTER 2
PROPERTY EXTRACTION APPROACH
2.1 Operation of tee resonator
The quarter wavelength open ended tee resonator is presented in the microstrip
configuration as shown in Figure 2.1. The tee resonator test structure is formed by a
dielectric material sandwiched between a conductor in the form of a tee on top and a
groundplane conductor covering the bottom surface of the dielectric. This complete
structure forms the microstrip tee resonator test vehicle. The conductor forming the tee
11
and the groundplane possess the associated parameters conductivity and sheet resistance.
The dielectric material has the associated parameters loss tangent and relative
permittivity.
The tee resonator ideally appears as an electrical short circuit at the frequency
where the stub is one quarter wavelength long. This behavior is observed by computing
the impedance looking into the stub of the tee resonator as indicated in Figure 2.2. This
characteristic can be expressed by evaluating the input impedance Zin [13]:
0
0
0
tan
tan
L
in
L
Z jZ lZ Z
Z jZ l
?
?
+=
+
2where pi?
?
? ?=? ?
? ?
. (2.1)
For the quarter wavelength stub we have 4l ?= , so tan tan 2l pi? ? ?= =?? ?
? ?
. Inserting this
result into (2.1) gives 00in
L
jZZ Z
jZ=
20
L
Z
Z= . Now, since the stub is open-ended, ZL appears
infinite, so
20
in
ZZ =
? = 0. No power would be transferred from port 1 to port 2 in this
Port 1
Port 2
Zin
Figure 2.2 Diagram indicating the 2 ports and the point of input impedance
12
ideal case. But because of losses, the line does not quite appear as a perfect short, and
sharp dips in the measured transmitted power (S21) occur at the resonant frequency and at
integer multiples of a half wavelength higher in frequency. An example of the dips
caused by this characteristic behavior is shown in Figure 2.3.
2.2 Permittivity and attenuation extraction
Consider a microstrip line with dimensions shown in the Figure 2.4 cross section.
Here, the metal is characterized as having a conductivity, ?, and the dielectric has a loss
tangent, tan?. In Figure 2.5, the frequency-dependent parameter S21 corresponds to the
transmitted power from port 1 to port 2 of the tee resonator (inset).
Figure 2.3 Resonant S21 dips for the tee resonator
13
-35
-30
-25
-20
-15
-10
-5
0
1.45 1.46 1.47 1.48 1.49 1.5
frequency (GHz)
S2
1 (
dB
)
Port 1 Port 2
fr
BW
S21r
L
3 dB
Figure 2.5 The S21 measurement at a resonant dip for a tee resonator
Figure 2.4 Cross-section of microstrip line
14
The guide wavelength ?G is related to the length of the stub L at a resonance
condition by
[2( 1) 1]4 GNL L ?? ++? = , (2.2)
where N = 1, 2, 3? and is termed the resonance number. Here, N = 1 corresponds to the
fundamental resonance where the stub length is equal to a quarter of a guide wavelength.
For a nonmagnetic material the guide wavelength is related to resonance frequency and
the effective relative permittivity ?e by [14]
G
r e
c
f? ?= . (2.3)
Manipulation of (2.4) and (2.5) yields
2(2 1)
4 ( )e r
N c
f L L?
? ?
? ?
? ?
? ?? ?
?=
+? . (2.4)
The correction length ?L is a small but appreciable length resulting from fringing fields
at the open end of the microstrip stub. It can be estimated using the approach of
Kirshchning et al [15]:
1 3 6
5
kkkL h
k? = , (2.5)
where
0.8544
0.81
1 0.85440.81
0.2360.26
0.434907 0.189
0.87
e
e
w
hk
w
h
?
?
? ?? ?
? ?? ?
? ?? ?
? ?? ?
? ?? ?
? ?? ?
? ?
? ?? ?
? ?
? ? +? ?
+ ? ?=
? ? ?
+? ?? ?
15
0.371
2 1 2.358 1
r
w
hk
?
? ?? ?
? ?= +
+
1.9413/ 2
1
3 0.9236
0.5274tan 0.084
1
k
e
w
hk
?
? ? ?? ?? ?? ?
? ?? ?? ?= +
( )( )0.036 14 0.0377 6 5 rk e ??= ?
1.456
1
5 41 tan 0.067
wk k
h
? ? ?? ?= + ? ?? ?
? ?? ?? ?
7.5 /6 1 0.218 .w hk e?= ?
If ?e is known, then ?r can be calculated using more complex equations that fit numerical
quasi-static solutions within 0.2% [16].
1 1(1 10 / )2 2r r abe h w? ?? ?+ ?= + + (2.6)
where
0.0530.9
0.564 3r
r
b ??+? ?= ? ?+
? ?
and
4 2
3
4
1 1521 ln ln 1
49 18.7 18.10.432
w w
wh ha
hw
h
? ?? ? ? ?+
? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ?
? ?= + + +? ?? ? ? ? ? ?? ?
? ?? ? ? ? ? ?? ? ? ?+
? ?? ?? ?? ?
It is apparent that since an accurate calculation of ?e requires ?L be known, and the
calculation of ?L requires both ?e and ?r be known, an iterative procedure must be used to
find ?e, ?r, and ?L. This iterative procedure is shown in Figure 2.6. First, an initial value
of ?L = 0 is assumed and (2.4) is solved for ?e. Next, this value of ?e is used with
equation (2.6) to determine ?r. Then, the Kirschning approach (2.5) is used to find ?L.
?L is inserted into (2.4) to calculate a new ?e and so on until the changes in ?e, ?r and ?L
are insignificant. This has only taken a few iterations in our trials.
16
The quality factor, or Q, provi0des a measure of energy loss in a resonator. The
loaded quality factor QL is found from the BW3dB and the resonance frequency fr by [4]
3
r
L
dB
fQ
BW= . (2.7)
The unloaded quality factor Qu is [4]
(( )/10)21 211 (2)10
L
U S S
r t
QQ
e ?
=
?
. (2.8)
Here, S21r is the value of S21, in dB, at the resonance frequency. S21t is the transmission
parameter for an equivalent length microstrip line without the tee stub. Subtracting this
Compute ?e: (2.4)
Compute ?r: (2.6)
Compute ?L: (2.5)
Final
iteration:
No
Compute frusing measured S21 data
?e and ?r value Yes
Assume ?L = 0
Input known values: N, L, w, and h
Figure 2.6 Permittivity calculation flowchart
17
term from S21r in (dB) therefore corrects for losses in the through line. In practice, S21t is
accounted for in the calibration and can therefore be left out of the equation. For low loss
lines, the loaded and unloaded Q-factors are approximately equal.
With an accurate determination of ?e, we can relate the total attenuation to the
unloaded Q-factor by [4]
8.686( / ) r eT
u
fdB m
cQ
pi ?? = . (2.9)
A summary for the process of computing total attenuation for a given line at the
resonance mode, N, is described by the flowchart indicated in Figure 2.7. Here the
measured data file representative of an S21 dip is used to extract fr and the 3 dB
bandwidth. This data is then used to compute QL using (2.7). QL is then used to compute
Qu using (2.8). Finally, the total attenuation is computed by (2.9) using the extracted fr,
computed Qu, and the accurate determination of ?e.
Compute QL: (2.7)
Compute QU : (2.8)
Compute ?? : (2.9) ?evalue: Figure 2.6
Read collected S21 data file
?T value
Figure 2.7 Total attenuation calculation flowchart
18
2.3 Dielectric and conductor property extraction
The total attenuation can be found for microstrip stubs of approximately equal
length but different characteristic impedance. The total attenuation is
T c d r? ? ? ?= + + (2.10)
where ?c , ?d and ?r are conductor, dielectric and radiation attenuation, respectively.
The conductor attenuation is a function of metal conductivity as well as surface
roughness at the metal-dielectric interface, lumped together in the term sheet resistance,
RS. We use the relationship between Rs and ?c developed by Pucel et al [17]:
c 1 2
0
1 3
0
2
3 4 5
0
8.686w 1For 0< , ( / )
h 2
8.6861For 2, ( / )
2
8.686wFor 2, ( / )
h
s
s
c
s
c
RdB m bb
Z h
Rw dB m bb
h Z h
RdB m bbb
Z h
?pi
?pi
?
? =
? ? =
? =
(2.11)
where
2
1
1 1
2 4
ewb
hpi
? ?? ?= ?
? ?? ?? ?? ?
? ?
2 41 ln
e e
h h w tb
w w t w
pi
pi
? ?? ?= + + +
? ?? ?? ?? ?
3 21 ln
e e
h h h tb
w w t wpi
? ?? ?= + + ?? ?
? ?? ?? ?
4
0.942
e
e
e
w
w hb
wh
h
pi= +
+
19
5 2 ln 2 0.942e ew wb eh hpipi ? ?? ?= + +? ?? ?
? ?? ?
In the equation for b5, e is the natural log base (2.718). The effective width, we, corrects
width to account for metal thickness [18]:
1.25 4 w 11 ln for ,
h 2
1.25 2 w 11 ln for .
h 2
e
e
t ww w
t
t hw w
t
pi
pi pi
pi pi
? ?? ?= + + ?? ?
? ?? ?? ?
? ?? ?= + + ?? ?
? ?? ?? ?
(2.12)
Finally, Zo is the characteristic impedance of the microstrip line accounting for metal
thickness [19]:
0
0
60 8 wln 0.25 for 1,
h
-1120 w
1.393 0.667ln 1.444 for 1.h
e
ee
e e
e
whZ
w h
w wZ
h h
?
pi
?
? ?= + ?
? ?? ?
? ?? ?= + + + ?
? ?? ?? ?? ?
(2.13)
The dielectric attenuation is a function of loss tangent, tan?, related by the
expression [20]
( )08.686 ( 1)tan( / ) 2 1r ed
e r
kdB m ? ? ??
? ?
?=
? . (2.14)
Here, ko is the free space phase constant,
0
0
2 2 fk
c
pi pi
?= = .
The attenuation resulting from power radiated from an open-ended quarter wave
resonant stub is adapted from [11]:
20
( ) ( )
2 3
3
0
8.686 480
/ ,r er
h Rf
dB m Z c
pi ??
= (2.15)
where
[ ]
2
3/2
1 11 ln
2 1
e ee
e e e
R ? ??? ? ?? ?? ++= ? ? ??
? ?? ?
.
This expression matches experimental results by Denlinger who asserts that no more than
25% of the total attenuation will be attributed to radiation loss for a microstrip
transmission line resonator if h/? < 0.01 and ?r > 2.5 [12]. The values obtained for
radiation loss in our efforts were significantly less than the stated maximum 25%, usually
below 0.01%, and met the above conditions as h/? = 0.002 and ?r = 9.5.
Since ?T is known for two lines, the two properties RS and tan? can be extracted.
This is done by plotting a line of all possible combinations of RS and tan? corresponding
to the ?T for each stub, and then locating the point of intersection for the resultant pair of
plots. Figure 2.8 outlines the approach for the extraction of loss tangent and sheet
resistance. First, the attenuation due to radiation, ?r, is computed using (2.15). Then, a
compensated value of total attenuation, ?TCompensated, is calculated by subtracting ?r from
?T. A value of sheet resistance, RSMax, that assumes all radiation compensated attenuation
comes from ?c is calculated using ?c = ?TCompensated in (2.11). A vector, RS vector, is then
formed for values from 0 to RSMax. This vector is then used to compute a vector for ?c, ?c
vector, by entering values computed in the RS vector into (2.11). A vector for ?d, ?d
vector, is computed by subtracting each value of the ?c vector from ?TCompensated. Then a
tan? vector is formed by entering each value of the ?d vector and accurate ?e value into
21
(2.14). Finally, the RS vector and the tan? vector are used to plot the line representative
of all possible combinations of RS and tan? for a single resonator stub operating at a
specific resonance mode, N. This entire process is repeated for the second stub resulting
in two lines of differing slopes. The intersection point of these two lines indicates the
Compute RS Max with ?T Compensated: (2.11)
Compute a tan? vector using ?d vector and ?e : (2.14)
Compute ?r: (2.15)
?T value: Figure 2.7
?T ??r = ?T Compensated
Form a RS vectorfrom 0 to RS Max
Compute an ?c vector using RS vector : (2.11)
Compute an ?d vector = ?T Compensated ? ?c vector
?evalue: Figure 2.6
Plot RS versus tan?
Locate intersection point for plot of multiple lines
Begin routine for a single line
All
lines
plotted
Yes
No
Figure 2.8 Extraction of RS and tan? flowchart
22
value of RS and tan?. Figure 2.9 serves as a demonstration of the type of plot generated
by the aforementioned methodology.
RS
tan?
50 ? line
30 ? line
Figure 2.9 Ideal intersection plot for a pair of tee resonators that shows
how tan? and RS are extracted
23
CHAPTER 3
BOARD PREPARATION
3.1 Fabrication
Test boards in the microstrip configuration were developed using a ceramic
substrate as the dielectric and a conductive paste as the conductor. Materials and the
associated data sheets needed to construct these test boards were then obtained from the
manufacturers. Alumina (96% alumina, 4.5 inches square, 25 mils thick, from Coors
Ceramic, Co.) was used as the substrate material. Silver/Palladium conductive paste (6/1
Ag/Pd from Heraeus) was used to form the conductors. The modeling software suite
used in the design of the tee resonator test vehicle was the very efficient and powerful
Agilent Advanced Design System (ADS). This was done with the intention to use the
board space as efficiently as possible. The design consisted of four resonators and a
Table 3.1 Tee resonator and calibration line design dimensions
fr (GHz), (N=1) line length L (mil) line width w (mil)
calibration line * 3000 24.87
through line ** 3000 24.87
50 ? stub 0.8 1460.98 24.86
30 ? stub 0.8 1396.25 60.88
50 ? stub 2.4 485.55 24.84
30 ? stub 2.4 463.29 60.83
* line used for calibration (see Figure 3.1)
** each resonator stub has a through line attached to it (see Figure 3.1)
24
calibration line. The design parameters of the resonators were a first resonance (N=1) of
800 MHz for the two longer resonators, and a first resonance (N=1) of 2.4 GHz for the
two shorter resonators. The resultant dimensions representative of the design of the four
resonators and the calibration line are shown in Table 3.1.
The resonances are chosen so that the 2rd (N=2) resonance of the longer
resonators would match the 1st (N=1) resonance of the shorter resonators as a further
attempt to confirm results. Both the long set and short set of resonators were chosen to
have resonator stubs of differing characteristic impedances. This effectively provided
one long stub and one short stub at 50 ? and one long and one short stub at 30 ?. The
physical parameters, width and length, of each resonator and the through line were then
Alignment marks
50 ? stub 30 ? stub
30 ? stub
50 ? stub
vias
Calibration line
Through lines
Figure 3.1 Lavenir layout for tee resonators and calibration line
25
recorded. This data was then used to create the board layout using Lavenir CAD
software. The resultant layout created using this software is shown in Figure 3.1. It is
important to note that there are actually two layouts created. The layout shown is for the
resonators and thru line on the top of the alumina substrate. The other layout created is a
single black square representative of the groundplane covering the bottom of the
substrate. This layout is not shown due to its self explanatory nature.
A gerber file representative of the layout shown in Figure 3.1 was created to be
Figure 3.2 Mask for tee resonators and calibration through line
26
used in the photoplotting process. The mask for the resonators and calibration line is
shown in Figure 3.2. The masks were then used to expose photo-sensitive screens.
Figure 3.3 Screens for the resonators and groundplane
27
This process is performed by placing the mask over the area of the screen covered with
photo-resist. The areas that are not covered by the mask are exposed to intense
ultraviolet light. This causes the area exposed to solidify and become a permanent
structure in the mesh of the screen. The areas that are covered are not solidified and are
removed by a stream of water. These screens for both the tee resonator structures and the
groundplane are shown in Figure 3.3. Once the screens are dry they can be used in the
screenprinter shown in Figure 3.4. The appropriate settings are used to ensure that the
proper amount of conductive paste is applied through the open sections of the mesh and
onto the alumina substrate. The paste is first applied to the resonator side of the alumina
substrate. After the paste is applied to the substrate it is immediately placed in
Figure 3.4 Screenprinter
28
the drying oven shown in Figure 3.5. The substrate is then taken from the drying oven
and placed in the firing oven shown in Figure 3.6 for 90 minutes at
Figure 3.5 Drying oven
Figure 3.6 Firing oven
29
a central zone set-point temperature of 900 degrees Celsius. The screen printing, drying
oven and firing oven processes were repeated for the groundplane. The boards were then
drilled at high rpm using a diamond tip drill bit in the areas defined as vias in Figure 3.1.
These holes were then filled with the Silver/Palladium conductive paste using a syringe.
Figure 3.7 Completed microstrip tee resonator test structure
30
The boards were then cured in the firing oven at the above stated conditions. The filling
procedure allowed a uniform coating along the interior of the via holes. This uniform
coating allows a conductive path through the substrate from the via pad to the ground
plane surface. The final result is a complete microstrip tee resonator test structure as
shown in Figure 3.7.
31
CHAPTER 4
MEASUREMENTS
4.1 Measurement procedure
The transmission scattering parameter S21, which corresponds to the amount of
power transmitted through a two-port network, is measured using a Hewlett-Packard
8510C Vector Network Analyzer in conjunction with an Alessi Probe Station fitted with
GGB Picoprobe 40A-GSG-1250-LP-HT probes. A probe contacting the board during
measurement is shown in Figure 4.1. The complete test equipment setup is shown in
Figure 4.2.
GGB Picoprobe probe
Through line
Via
Figure 4.1 Probe in contact with board during testing
32
The general approach to the data collection process involves taking an initial
wideband measurement to locate the resonant frequencies, and then narrow band
measurements are made to get sufficient accuracy in order to extract total attenuation.
The measurements are performed with averaging on (32 points), smoothing off, and data
output set to 201 points. The test vehicle in position for measurement on the Alessi probe
station is shown in Figure 4.3.
4.1.1 Standard procedure
A standard procedure for performing network analyzer measurements was
developed to ensure the accuracy of data collected. One procedure used to provide
Alessi probe station
HP8510C vector network analyzer
Figure 4.2 Testing equipment
33
assurance in the accuracy of data collected is a ?through-response? calibration. This
calibration is performed on the through line prior to the measurement of the resonators.
Performing a calibration before resonator measurements allows the effects of the through
line to be minimized or taken out of the tee resonator measurement. This allows a
collection of data that is more representative of the resonator stub itself. It is important to
note at this point that even though the through line for the resonators and the calibration
line are of identical design, the small fabrication variances do not allow a 100% removal
of the effects of the through line from the resonator measurement.
In addition, measures must be taken to ensure that the probes are adequately
situated on the contacts. This is done in an effort to minimize measurement variance
Calibration through line
Long 50 ? resonator
Long 30 ? resonator
Short 50 ? resonator Short 30 ? resonator
Heat chuck
Probe 1 Probe 2
Figure 4.3 Test board during measurement on the Alessi probe station
34
induced by human error when making repeated probe contacts with the test pads. To
minimize error in this step, we have employed time domain reflectometry (TDR) to
assure a good contact between probe and pad. This step effectively produces a visible
fully error-corrected transmission response of the structure in near real time [21]. TDR
signatures for the cases of both good and poor probe and pad contacts are shown in
Figure 4.4. Total attenuation was calculated using data from 30 repeated measurements
on a single board with and without using TDR to determine the validity of this step in the
measurement process. Note that this data was collected using the long 50 ohm resonator
(N=1 => ~800 MHz). The results of this procedure can be viewed in Table 4.1. The
standard deviation for the results using data collected with TDR was found to be 2 orders
-1
-0.5
0
0.5
1
1.5
-1E-09 -5E-10 0 5E-10 1E-09 2E-09 2E-09 3E-09 3E-09 4E-09 4E-09
sec
mU
Correct pad and probe contact
Poor pad 2 and probe 2 contact
Poor probe 1 and pad 1 contact
Figure 4.4 TDR signatures of correct and incorrect probe placement
35
of magnitude less than the results not using TDR. This study confirmed the use of TDR
as an important step in the measurement process.
The standard procedure for taking a measurement consists of five different steps.
First, the connection of the probes and the vehicle calibration through line is validated
using time domain reflectometry (TDR). Second, the calibration is set by performing a
?through-response? calibration using the through line. Third, the probes are moved to the
resonator where TDR is once again used to validate the proper connection between the
probes and the resonator through line. Fourth, the calibration setting is enabled so that
the proper dip is visible. An example of this would be the single dip at N=1 for the 50 ?
long resonator. Last, the data is stored on a floppy disc so that the data can be later
transferred to a PC for processing.
4.1.2 Variability study
A measurement procedure variability study was performed to quantify expected
error margins with the collected data. This study was performed for repeated
measurements on a single board and measurements on multiple boards. This provided
both same board and board-to-board variability. These measurements were performed at
Table 4.1 Average value and standard deviation of total attenuation with and
without using TDR to place the probes
Without TDR With TDR
Number of Samples 30 30
Average Value 0.148 0.155
Standard Deviation 0.0022 0.000087
36
room temperature and at 100 degrees Celsius. The results for both repeated single board
measurement variability and board-to-board measurement variability are listed Table 4.2.
The repeated measurements on a single board were done in an effort to quantify
the variability of the current testing procedure for a single test vehicle. This repeated
measurement testing process consisted of 15 measurements on three different test
vehicles at room temperature. The same process was performed for 5 measurements on
board 1 at 100 degrees Celsius in order to quantify the effect of elevated temperature on
measurement variability. Each of these measurements was taken using the standard five
step measurement taking procedure outlined in the previous sub-section. The long 50 ?
resonator, operating at the resonance mode N=1 (785 MHz), was the test structure used.
The average variability of plus or minus 0.4 percent at room temperature and 0.1 percent
at 100 degrees Celsius shows that the testing procedure itself results in a promising total
average variability of 0.8 percent at room temperature and 0.2 percent at 100 degrees
Celsius.
Table 4.2 Measurement variability
Test Procedure variability (+/-)**
Board #1 (15 samples, same board) @ room temp 0.6%
Board #2 (15 samples, same board) @ room temp 0.2%
Board #3 (15 samples, same board) @ room temp 0.4%
Average same board variability @ room temp 0.4%
Board #1 (5 samples, same board) @ 100 deg C 0.1%
Board to Board (8 different boards) @ 100 deg C 2.0%
Board to Board (11 different boards) @ room temp 2.0%
**computed as the variability of the average S21 dip calculated over a 200
MHz bandwidth centered at the fundamental resonance frequency
37
Board-to-board variability was the next topic examined. Eleven different boards
at room temperature and eight boards heated to 100 degrees Celsius were tested in order
to measure the variability of measurements from board to board. Each test vehicle was
examined using the standard five step measurement taking process. The 50-ohm long tee
resonator operating at the first resonance was the structure used in this board-to-board
testing. The average board-to-board variability was found to be plus or minus 2 percent.
It was discovered that the variability for board-to-board measurements was higher than
the variability for repeated measurements on the same board
4.2 Property extraction
The substrate permittivity, total attenuation, sheet resistance, and loss tangent can
be calculated using MATLAB coding based on the approach discussed in Chapter 2.
This coding is given in the appendix for further explanation. The physical dimensions of
the test boards were measured in order to improve the accuracy of the extracted
properties. The thickness of the conductive paste was measured using a profilometer as
approximately 0.4 mils. The physical dimensions of the tee resonators on a finished test
board were also measured using a microscope. This and all other data used in the
property extractions are listed in Table 4.3.
4.2.1 Permittivity extraction
The effective and relative permittivity are computed using the typical physical
parameters listed in Table 4.3 and the measured resonant frequency for a resonator
operating at a specified resonance mode. The resonant frequency is the most negative
38
point of a S21 dip collected using the standard measurement procedure. The typical
extracted permittivity values for the long and short pairs of resonators operating at their
respective first resonance modes are listed in Table 4.4. The ?r for the long 50 ?
Table 4.3 Typical data for the test board
Z0 (?) 50 30
through line width (mil) 25 25
t (mil) 0.4 0.4
h (mil) 25 25
tan? (Coors data sheet) 4.0 x 10-4 4.0 x 10-4
short resonator pair
w (mil) 25 61
L (mil) 486 463
long resonator pair
w (mil) 25 61
L (mil) 1461 1396
Table 4.4 Typical permittivity extraction results
Z0 (?) 50 30
short resonator
pair
fr (GHz) 2.37 2.44
?e 6.34 6.57
?r 9.44 8.92
long resonator
pair
fr (GHz) 0.785 0.782
?e 6.55 7.21
?r 9.78 9.83
39
resonator was 9.78, while the long 30 ? resonator had ?r = 9.83. These values are
slightly higher than the ones on the data sheet provided by the substrate manufacturer
(Coors Ceramics, ?r = 9.5). The short pair of resonators had a value of relative
permittivity that was slightly lower than the manufacturer?s stated value.
The variability of this extraction process was also examined using the extraction
of relative permittivity. The testing of this process involved the plotting of relative
permittivity for the long 50 ohm resonator versus the long 30 ohm resonator. This
procedure was performed for the modes N=1 through N=3 on 8 separate test vehicles.
The purpose of this type of plot, shown in Figure 4.5, was to quantify the amount of
variability in the extraction of relative permittivity for the two different resonators. The
Relative Permittivity Computed With Measured Physical Length
and Resonant Frequency
9
9.2
9.4
9.6
9.8
10
10.2
10.4
9 9.2 9.4 9.6 9.8 10 10.2 10.4
Relative permittivity 50 ohm resonator
Re
lat
ive
pe
rm
itt
ivi
ty
30
oh
m
res
on
ato
r
N=1
N=2
N=3
Figure 4.5 Plot of long 30 ohm resonator relative permittivity vs. long 50 ohm
resonator relative permittivity
40
tighter grouping of the values in the horizontal direction suggests that the data for the 50
ohm resonator has less variability than that of the 30 ohm resonator.
4.2.2 Total attenuation extraction
Data points corresponding to a dip of the type shown in Figure 4.6 form the data
file collected during measurement that is used to extract total attenuation. A MATLAB
routine pulls resonant frequency, 3 dB bandwidth and S21R from this file. This data and
the previously computed effective permittivity are used to calculate a value of total
attenuation. The typical results of this process are shown in Table 4.5.
Improvement in the attenuation extraction routine was realized. Attenuation is
calculated by feeding the collected S21 data into a MATLAB program. The analyzer
Frequency (GHz)
S 21
(dB
) 3 dB bandwidth
3 dB
Figure 4.6 Single dip used for attenuation calculation
41
allows selection of number of measurement data points. We selected 201 data points, (as
opposed to 401 or 801) for most rapid measurement and data recording. However, this
led to discretization error in the MATLAB routine as frequency bandwidth was
determined. To get more accurate results, an interpolation routine was added to the
attenuation extraction program. For each of 30 measurements, Figure 4.7 compares
attenuation using the original MATLAB routine, without interpolation, to the more
accurate approach using interpolation. We see that without interpolating, the MATLAB
routine was rounding the data and losing some accuracy. The addition of interpolation
Table 4.5 Typical total attenuation extraction results
Z0 (?) 50 30
short resonator
pair
fr (GHz) 2.37 2.44
S21R (dB) -30 -37
BW3dB (GHz) 0.046 0.035
QL 51.6 69.5
QU 51.7 69.5
?T (dB/in) 0.267 0.208
long resonator
pair
fr (GHz) 0.785 0.782
S21R (dB) -26 -31
BW3dB (GHz) 0.026 0.025
QL 30.1 30.9
QU 30.2 30.9
?T (dB/in) 0.155 0.158
42
improves the measurement accuracy, but by doing so it also results in more variability as
shown in Table 4.6.
Table 4.6 Interpolation addition results
with
interpolation
without
interpolation
Number of Samples 30 30
Average value (total attenuation (dB/in)) 0.155 0.155
Standard Deviation 0.00029 0.000087
0.1538
0.154
0.1542
0.1544
0.1546
0.1548
0.155
0.1552
0.1554
0.1556
0.1558
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Measurement Number
At
ten
ua
tio
n (
dB
/in
)
attenuation without interpolation
attenuation with interpolation
Figure 4.7 Chart comparison of 30-sample study of long 50 ? tee resonator with
(black) and without (grey) the MATLAB interpolation addition
43
4.2.3 Sheet resistance and loss tangent extraction
For a given impedance stub, a combination of all possible values of tan? versus
RS that give the measured total attenuation can be plotted. The resulting line will have a
distinctive slope for each impedance stub. A typical RS versus tan? plot is shown in
Figure 4.8. Here, the intersection of the lines for the 30 ? and 50 ? stubs gives tan? =
13 x 10-4 and RS = 67 milliohms per square. Assuming a very smooth copper line of the
same thickness as our lines, a simple best-case estimate of RS would be 14 milliohms per
square. This value is calculated as [3]:
22 rS fR pi ??= . (4.1)
0
50
100
150
200
250
0 10 20 30 40 50 60 70 80
Sheet resistance (milliohms/square)
Lo
ss
tan
ge
nt
(x
10
-4 )
measured value = 13
data sheet value = 4 computed value = 14
measured value = 67
50 ohm
30 ohm
DC value = 19
Figure 4.8 Plot of sheet resistance vs. loss tangent using measurements on a 30
? and 50 ? tee resonator pair
44
For comparison, a DC sheet resistance value of 19 milliohms per square was measured on
the calibration line of the test board. Our extracted value, using less conductive Ag/Pd
paste on a substrate that is not perfectly smooth has a significantly higher RS than this.
Finally, the substrate data sheet indicated a loss tangent, at 1 MHz, of about 0.0004. Our
value is slightly over three times this value.
45
CHAPTER 5
SIMULATION
Simulation using available software suites was used to further examine the
accurateness of the extracted values obtained using data from measurements on the test
board. These software suites were used to both validate the design of the test board and
to examine the extraction routine itself. This allowed the confirmation of the operation of
the resonators on the test board and the performance of the extraction routine void of
physical constraints such as measurement variability and possible flaws in the finished
test board.
5.1 Board design verification
Tee resonators can be designed with current computer aided design (CAD)
simulation software to meet the established particular electrical design parameters within
the limits of physical fabrication tolerances [15,22-29]. For this reason, modeling
software suites were employed in the design of the resonator structure. This allowed the
microstrip tee resonator test vehicle to be effectively tested to ensure proper operation
before the time consuming fabrication processes were undertaken.
The modeling software suite used in the design of the tee resonator test vehicle
was ADS. This software package allows a very wide range of options with regards to the
46
actual computation method used. ?Schematics? is a symbolic environment that allows
the placement of representative parts from libraries to form the structure of interest. This
environment has both its advantages and disadvantages. The environment allows a user
with limited experience to accurately design circuits for simulation. This type of
software also allows the user to employ tools that generate quick design parameters and
results for a model. A significant disadvantage of this design environment is that it relies
upon quasi-static approximations for the computation of results [24-28]. The final design
of the four resonators and the calibration through line can be viewed in Figure 5.1. The
use of this software also allowed a simulation to be performed for each resonator to
Figure 5.1 ADS schematic layout used in the design of the test board
47
confirm that the resonant dips occurred at the correct locations. These resonant dips
generated by ADS are shown in Figure 5.2.
5.2 Modeling Approach
The modeling of the microstrip tee resonator structure was also performed in an
effort to examine the property extraction process discussed in Chapter 2. Performing
simulations for models of the tee resonator designed to match the design criteria of the
test board allows a generation of data that is similar to the data collected by
measurements. This S21 data in the form of the characteristic dip can be generated with
the parameters known. This gives an advantage over the data collected using the test
Figure 5.2 Resonant dips used for confirmation of the design of the four resonators
48
board in that the model parameters can be directly compared with the results computed
using the extraction routine. It is important to note that the accuracy of the generated
data relies heavily on the model design itself [15,22-29]. This modeling process was
performed using two different software suites.
5.2.1 ADS ?momentum?
?Momentum? is the environment in ADS that was used to study the property
extraction procedure discussed in Chapter 2. Momentum is very versatile in that it allows
several different types of simulation sweep schemes and meshing strategies.
?Momentum? operates on the principle of finite element modeling (FEM). This type of
numerical computation is stated as being less accurate at replicating the behavior of an
exact physical model when compared to other modeling schemes [22,23]. This, however,
is compensated for by the fact that ADS uses techniques such as adaptive sweep and
single layer approximation to give very accurate results with a significantly lower amount
of required computing power.
5.2.2 Flomerics micro-stripes
Flomerics micro-stripes is another powerful modeling suite used in the
examination of the property extraction process. Micro-stripes is a software package
developed to simulate electromagnetic phenomena in a consistent and reliable manner
[22,23]. This is accomplished by using the Transmission Line Modeling (TLM) method
of numerical analysis. The use of this method of computation is very accurate in
modeling structures and cavities at microwave frequencies [22]. This is currently the
49
only computational method that allows a user to create a physical model with an exact
solution [22,23]. The TLM method is a three-dimensional modeling scheme that
involves the computation of both shunt and series elements resulting in a true three-
dimensional representation of the model [23]. The application of this theory in the use of
the micro-stripes software package has both advantages and disadvantages. The
advantage is that it uses the TLM method described above and thus can give very
accurate results when compared to physical models. The disadvantage is that this type of
computation requires very tightly meshed models to give these highly accurate results.
The tightly meshed model requires the number of cells to range on the order of millions
to achieve highly accurate results. The time required to compute results for each model is
dependant upon both the total number of cells and the physical size of the cells. This is a
significant factor due to the time required to perform each successive simulation.
5.2.3 Property extraction verification approach
The use of these software packages in the examination of the property extraction
process involves an iterative approach shown in Figure 5.3. First, the values that define
the behavior of the model are decided and recorded for later use. These values include
parameters such as relative permittivity and loss tangent for the dielectric, the
conductivity of the metal used for the conductors, and the physical dimensions of the
components that define the complete microstrip test structure. Then, the model is
designed using these values. The model is then simulated at the decided conditions.
These conditions include the meshing resolution, frequency span, and type of simulation
sweep plan. The meshing resolution is the number of cells per wavelength used in the
50
model design. The frequency span is the start and end point in frequency that the
simulation is performed. The type of simulation sweep span is either adaptive or linear.
In our experience it was found that the adaptive sweep plan in ADS is used for results
that are quick but less precise. The linear sweep plan takes much more time to complete
but it also provides the necessary accuracy. The results of interest are in the form of a
single S21 dip. This single dip provides the resonant frequency and 3dB bandwidth
information for the model of the tee resonator that was simulated. Then the properties are
extracted using MATLAB coding that was developed based on the process described in
Chapter 2. The results from this coding routine are then compared to the values used in
the design of the model that was simulated. This process is repeated for a number of
Input parameters
Design model
Run simulation Obtain results
Perform value extraction
Obtain results
Compare
Figure 5.3 Property extraction verification approach using modeling
51
different resonators at different conditions. This is done in an effort to understand the
behavior of the property extraction procedure defined in Chapter 2.
5.3 Results
5.3.1 Model meshing resolution
Meshing resolution or the number of cells per wavelength contained in the
structure of interest was found to have a great influence on the generated data. The
resonant frequency generated for a single resonator modeled using micro-stripes was
observed in an effort to understand the influence of model meshing resolution. Several
simulations for this model were performed. In each simulation, the size of the cells was
decreased resulting in an increase in the resolution of the model?s meshing. The
0.769
0.748
0.791 0.788 0.782
0.712
0.726
0.7
0.72
0.74
0.76
0.78
0.8
0 5 10 15 20 25
Cell size (mil)
Re
son
an
t f
req
ue
nc
y (
GH
z) Target Value
Figure 5.4 Progression of resonant frequency towards target value using micro-stripes
52
variables were the meshing resolution of the model and the resultant resonant frequency.
Figure 5.4 demonstrates the influence of meshing resolution on the extraction of the
resonant frequency. Each data point is indicative of a simulation performed at a set
meshing resolution. As the size of the cells decrease, resulting in a higher meshing
resolution, the resonant frequency value generated moves closer to the target value. This
target value was the resonant frequency observed in the simulation with ADS during the
board design. This model meshing resolution study was performed in a similar fashion
using ADS Momentum. The observed resonant frequency also moves toward the target
value with each successively higher meshed simulation. Figure 5.5 further supports the
importance of model meshing on the resultant generated resonant frequency.
0.798
0.7958
0.79596
0.79654
0.7955
0.796
0.7965
0.797
0.7975
0.798
0.7985
50 100 150 200 250 300
Number of cells per wavelength
Re
son
an
t f
req
ue
nc
y (
GH
z)
Target Value
Figure 5.5 Progression of resonant frequency towards target value using ADS
?momentum?
53
5.3.2 Delta L calculation
The fringing field at the end of the resonator stub is also an important topic that
can be examined by modeling with CAD software packages. This fringing field is
important because the length of the stub is a critical factor in the characterization of
material properties. The length of the resonator is specified as the physical length of the
conductor that forms the resonating stub and the additional length generated by the
fringing field. This criterion is shown in Figure 5.6. The ADS ?schematics?
environment was used in a specific example in an effort to confirm the physical
computation of the fringing field with the closed form end expression found in
Kirschning et al [15,27]. The computation of relative permittivity was used as a
reference for this part of the experiment. The relative permittivity was entered into the
Figure 5.6 Definition of resonator length with consideration of ?L
54
microstrip substrate (MSUB) block in schematics, the length of the resonator (L) was
computed, and then the scattering parameter S21 was generated. The combination of
these parameters were then used with an independently developed relative permittivity
calculation algorithm that allowed the determination of the resonant frequency needed to
produce the same relative permittivity value as the one used in the schematics model.
This relative permittivity value was arrived at with a computation that incorporated the
closed form end expression and a computation that did not incorporate the closed form
end expression. The results are significant because the microstrip line open circuit
(MLOC) symbolic circuit does not account for the end effect. An examination of Table
5.1 validates the closed form end expression given in Kirschning et al. The value of the
?L compensated length generated by the extraction routine is in close proximity to the
value given by schematics using MLOC when the resonant frequency and value of
relative permittivity are held constant for both methods. The slight difference in final
length is due to the variability caused by unknown variables such as the loss tangent of
the dielectric and the conductivity of the resonator and groundplane.
Table 5.1 Results of delta L experiment
Determination
Method
Relative
Permittivity
Value
Resonator
Length (mil)
Resonant
Frequency
(GHz)
Computed Delta
L (mil)
Schematics using
MLOC 9.5 1469.16 (L) 0.795
Relative
permittivity
extraction routine 9.5 1461.0 (L`) 0.795 7.954 (L``)
Results : (L~ =L`+L``) => (L=1461.0+7.954) => (L=1468.954 ~= 1469.16)
55
5.3.3 Relative permittity extraction
Figure 5.7 indicates the resonant frequency generated by the 50 ohm resonator
model designed to have a first resonance of 800 MHz. This model was simulated using
ADS ?momentum?. The value calculated using the resultant resonant frequency is within
a value of 0.0007 of the target value of 9.5 for relative permittivity. This is significant
because the model was created using the value of 9.5 for relative permittivity. The
extraction routine uses the variables w (width of resonator), h (height of the substrate), N
(number of resonance), L (length of resonator stub), and fr (resonant frequency) to
calculate relative permittivity. The variables w, h, L, and N are assumed to remain the
same in the model and in the calculation. The variable fr is the variable that changes as
Frequency (MHz)
S 21
(dB
)
800790780770 810 820 830
-20
-30
-40
fr = 795.8 MHz
Figure 5.7 Extraction of relative permittivity
56
the model simulation changes. The target value for fr is the value used in the extraction
routine to arrive at the calculated value of 9.5 for relative permittivity. This relative
permittivity value of 9.5 is also used in the design of the model. Therefore, the goal for
confirming the extraction procedure is to arrive at a point of resonance in the model
simulation that produces a computed value of relative permittivity that is the same value
used in the simulation. It is important to note that this model was created using values for
conductivity (1.4e7 (S/m)) and loss tangent (2e-4) that allowed this resonant frequency to
be obtained.
A set of models with the design parameters listed in Table 5.2 were developed to
further examine the extraction of relative permittivity. These models were meshed at
300, 500, and 700 cells per wavelength. The loss tangent and conductivity values were
held constant in all of the performed simulations. The resulting variance in the extracted
relative permittivity values can be observed in Table 5.3. Simulating different impedance
resonator models shows the deviation from the relative permittivity value used in the
Table 5.2 Design parameters for model set
Microstrip Parameters Stub Parameters
h 25 mil Model w (mil) L (mil)
?r 9.5 40 ? 37.67 1149.21
?r 1 50 ? 24.54 1173.79
? 5x107 S/m 60 ? 16.3 1193.78
t 0.4 mil 70 ? 10.89 1210.2
tan? 4x10-4 design N=1=1GHz for all models
57
design of the model. It is interesting to note that our extracted value of relative
permittivity increases as the impedance of the resonator stub increases.
5.3.4 Sheet resistance and loss tangent extraction
The procedure for the extraction of the dielectric and conductor properties was
also performed using ADS ?momentum?. In particular, the process for the extraction of
loss tangent and sheet resistance was examined. This extraction process was
implemented by way of MATLAB coding that was developed based on the technique
Table 5.3 Extracted relative permittivity and resonant frequency values
300 cells/wavelength
Zo ?r fr
70 9.57 0.987
60 9.48 0.991
50 9.39 0.995
40 9.28 1.001
500 cells/wavelength
Zo ?r fr
70 9.59 0.986
60 9.48 0.991
50 9.39 0.995
40 9.30 1.000
700 cells/wavelength
Zo ?r fr
70 9.59 0.986
60 9.49 0.991
50 9.40 0.995
40 9.30 1.000
model design value ?r = 9.5
58
explained in section 3 of Chapter 2. To accomplish this, 4 models were developed, each
producing its own set of data. The design parameters for these models are listed in Table
5.2. The stub parameters indicated were designed for a first resonant frequency (N=1) of
1GHz. These models were chosen in order to observe the behavior of the intersection
plot routine for the extraction of loss tangent and sheet resistance in terms of symmetry
about the 50 ? characteristic impedance of the thru line. The models were then meshed
at resolutions of 300 cells per wavelength, 500 cells per wavelength, and 700 cells per
wavelength. Then the models were all simulated to provide the S21 versus frequency data
necessary to demonstrate the extraction process. This data was then used to form the
three sets of intersection plots shown in Figures 5.8, 5.9, and 5.10. This was
accomplished by plotting a line representative of all possible combinations of sheet
0
20
40
60
80
100
120
140
0 2 4 6 8 10 12 14
Sheet resistance (milliohms/square)
Lo
ss
tan
ge
nt
(x
10
-4 )
measured value = 10.8
70 ohm
40 ohm
50 ohm
60 ohm
simulation value = 8.9
measured value = 7
simulation value = 4
Figure 5.8 300 cells per wavelength sheet resistance vs. loss tangent plot
59
resistance versus loss tangent for each different impedance model meshed at 300 cells per
wavelength. This process was then repeated for the other meshing resolutions. Figures
5.8, 5.9, and 5.10 indicate the intersection plots for the model sets meshed at resolutions
of 300, 500, and 700 cells per wavelength, respectively.
A comparison of Figures 5.8, 5.9, and 5.10 indicates that the intersections of the
lines for the different impedance lines move towards a single intersection as the
resolution of the models increase. This behavior indicates that the extraction of sheet
resistance and loss tangent is greatly influenced by the meshing resolution of the model.
The target value for RS indicated in the figures is calculated using (4.1) and the target
value for tan? is the value used in the design of the models. The extraction procedure
0
20
40
60
80
100
120
140
0 2 4 6 8 10 12 14
Sheet resistance (milliohms/square)
Lo
ss
tan
ge
nt
(x
10
-4 )
measured value = 8
simulation value = 4
simulation value = 8.9
measured value = 10.6
70 ohm
40 ohm
50 ohm
60 ohm
Figure 5.9 500 cells per wavelength sheet resistance vs. loss tangent plot
60
consistently overestimates RS while the resultant value for tan? is in close proximity to
the value used in the design of the models.
0
20
40
60
80
100
120
140
0 2 4 6 8 10 12 14
Sheet resistance (milliohms/square)
Lo
ss
tan
ge
nt
(x
10
-4 )
measured value = 11.4
simulation value = 8.9
simulation value = 4
measured value = 5
40 ohm
50 ohm
60 ohm
70 ohm
Figure 5.10 700 cells per wavelength sheet resistance vs. loss tangent plot
61
CHAPTER 6
CONCLUSIONS
6.1 Technique summary
A technique for extracting both conductor and dielectric properties using a pair of
microstrip tee resonators has been presented. This technique has been demonstrated on
transmission lines realized with Ag/Pd conductor on alumina substrate. Characteristic
dips in the scattering parameter S21 are generated by each tee resonator. These dips can
then be used to extract the parameters necessary to compute total attenuation and
effective permittivity. This data can then be further used to extract the sheet resistance of
the conductor and the loss tangent of the substrate. This effectively defines the properties
of the conductor and dielectric that are useful in the design of microwave devices.
A measurement variability study was conducted to estimate the margins of error
in data collection. The low variability of same board testing suggests that the
measurements process is adequate for the purposes of accurate data collection. The
higher variability of board-to-board measurements suggests more variability in the
fabrication process than in the measurement process. More test vehicles with tighter
fabrication tolerances are desired to ensure that the data collected from measurements is
as accurate as possible.
The extracted value of loss tangent for the test board is slightly higher than the
value stated in the manufacturer?s data sheet while the sheet resistance is expectedly
62
much greater than that of perfectly smooth copper. Assuming the method is sound, this
discrepancy between the expected and extracted loss tangent and sheet resistance values
may be a result of board preparation flaws or it could be that the method does not work so
well if the conductor loss is much greater than the dielectric loss. It may also be a result
of inaccuracies in the equations used, a possibility suggested by the work of Thompson et
al [30].
6.2 Future work
There is room for improvement in the presented dielectric and conductor
extraction process. The equations used for finding the total attenuation and substrate
permittivity work reasonably well based on results obtained using both measured and
simulated data. The iterative technique and equations used for the extraction of loss
tangent and sheet resistance produce results that are reasonably close to expected results.
Further investigation into the equations for extracting sheet resistance and loss tangent
should be conducted.
Currently the method is being further examined using data measured on a test
board with copper etched lines on a microwave laminate substrate. This configuration
exhibits a more balanced ratio of conductor and dielectric loss. This is in contrast to the
printed Ag/Pd paste on alumina substrate, where the conductor loss was much greater
than dielectric loss at our measurement frequencies. Figure 6.1 indicates that the loss
tangent and sheet resistance values for the laminate board obtained using the tee
resonator approach agree quite favorably with the expected results. More testing on this
63
type of configuration should be done as a comparison to the high loss conductor low loss
dielectric configuration to determine the limitations of the technique.
The influence of permittivity calculation on the method should be further
examined. A trend currently exists whereby the value of extracted relative permittivity
increases as the impedance of the resonator stub increases. Accounting for dispersion in
the calculation of permittivity could possibly improve the accuracy of the method by
eliminating this occurrence. The technique might also benefit from understanding the
fringing field at the end of the resonator stub. One way to study this may be to short the
end of the half-wavelength resonator stub by way of a via to the groundplane. This
would effectively eliminate ?L from the fringing field. Another way to further study ?L
is by using the common resonances of the long and short resonator pair to extract the
0
10
20
30
40
50
60
70
80
0 2 4 6 8 10 12 14 16 18
Sheet resistance (milliohms/square)
Lo
ss
tan
ge
nt
(x
10
-4 )
measured value = 10.35
measured value = 21
data sheet value = 23
computed value = 10.05
50 ohm
30 ohm
Figure 6.1 Sheet resistance vs. loss tangent plot for test board with copper lines and a
microwave laminate substrate
64
length. These methods would possibly allow a more accurate estimation of delta L. This
better understanding of the delta L effect would possibly result in a more accurate
extraction of the properties for the conductor and dielectric.
Modeling efforts should also be continued. The modeling of the tee resonators is
an effective way to quickly alter parameters of the test vehicle without the burden of
fabricating a new test vehicle. Models with high meshing resolutions are needed for the
accurate generation of data. These highly meshed models take a great deal of time to
simulate due to the large computing requirements. Further work in this area should be
conducted to better understand the intricacies of the extraction process.
6.3 Conclusions
The conductor and dielectric properties of microstrip can be extracted using a pair
of differing impedance resonators. The technique, however, needs to be further
developed to achieve a high level of confidence in the obtained results. The dated
equations, limited number of test samples, and imperfect modeling are all areas that
should be worked on to improve the extraction technique. The obtainment of a greater
set of test samples will allow confidence in the measured data. This will allow a better
examination of the equations currently being used. More investment in the time
consuming modeling efforts will also help to better understand the strengths and
weaknesses of the properties extraction process.
65
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[1] E. J. Denlinger, ?A Frequency Dependent Solution for Microstrip Transmission
Lines?, IEEE Trans. Microwave Theory Tech, vol. MTT-19, pp. 30-39, Jan. 1971.
[2] J. D. Woermke, ?Soft Substrates Conquer Hard Designs,? Microwaves, Jan, 1982.
[3] D. A. Rudy, J. P. Mendelson, and P. J. Munz, ?Measurement of RF Dielectric
Properties with Series Resonant Microstrip Elements,? Microwave Journal, Mar.
1998, pp. 22-41.
[4] J. Carroll, M. Li, and K. Chang, ?New Technique to Measure Transmission Line
Attenuation,? IEEE Trans. Microwave Theory Tech., vol. MTT-43, No. 1, Jan.
1995, pp. 219-222.
[5] S. M. Wentworth, ?Measuring the Microwave Attenuation in Microstrip,? The
31st Southeastern Symposium on System Theory, Auburn, AL, 1999, pp.200-204.
[6] D. I. Amey and S. J. Horowitz, ?Microwave Material characterization,? ISHM ?96
Proc., pp. 494-499.
[7] S. M. Wentworth, Fundamentals of Electromagnetics with Engineering
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[9] B. L. Dillaman (1996), High Frequency Characterization of Materials Used In
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[10] R. L. Peterson and R. F. Drayton, ?A CPW T-Resonator Technique for Electrical
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[11] E. Belohoubek and E. Denlinger, ?Loss Considerations for Microstrip
Resonators,? IEEE Trans. Microwave Theory Tech., vol. MTT-23, 1975, pp. 522-
526.
66
[12] E. J. Denlinger, ?Radiation from Microstrip Resonators,? IEEE Trans. Microwave
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[14] M. N. O. Sadiku, Elements of Electromagnetics, 3rd Ed., Oxford University Press,
Inc., 2001, pp. 526-527.
[15] M. Kirschning, R. H. Jansen, and N. H. L. Koster, ?Accurate Model for Open End
Effect of Microstrip Lines,? Electronics Letters, Vol. 17, No. 3, 5 February 1981,
pp. 123-125.
[16] D. H. Schrader, Microstrip Circuit Analysis, Prentice-Hall, Inc., 1995, p.31
[17] R. A. Pucel, D. J. Masse and C.P. Hartwig, ?Losses in Microstrip,? IEEE Trans.
Microwave Theory Tech, June 1968, pp. 342.
[18] C. Nguyen, Analysis Methods for RF, Microwave, and Millimeter-Wave Planar
Transmission Line Structures, John Wiley & Sons, Inc., 2000, pp. 68-70.
[19] I. J. Bahl, and R. Garg, ?Simple and Accurate Formulas for Microstrip with Finite
Strip Thickness,? Proc. IEEE, Vol. 65, 1977, pp. 1611-1612.
[20] D. M. Pozar, Microwave Engineering, 2rd Ed., John Wiley & Sons, Inc., 1998, pp.
163.
[21] ?HP 8510C Network Analyzer: Operating and Programming Manual?, 1st ed.,
Hewlett-Packard, 1991.
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[23] S. Akhtarzad and P. B. Johns, ?Three-Dimensional Transmission-Line Matrix
Computer Analysis of Microstrip Resonators,? IEEE Trans. Microwave Theory
Tech, vol. MTT-23, pp.990-997, Dec. 1975.
[24] W. J. Getsinger, "Measurement and Modeling of the Apparent Characteristic
Impedance of Microstrip," IEEE Trans. Microwave Theory Tech, vol. MTT-31,
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[25] E. Hammerstad and F. Bekkadal. Microstrip Handbook, ELAB Report STF44
A74169, February 1975.
[26] E. Hammerstad and O. Jensen. "Accurate Models for Microstrip Computer-Aided
Design," MTT Symposium Digest, 1980.
67
[27] M. Kirschning and R. H. Jansen. Electronics Letters, January 18, 1982.
[28] H. A. Wheeler, "Formulas for the Skin Effect," Proc. IRE, Vol. 30, September,
1942, pp. 412-424.
[29] J. C. Rautio, ?Microstrip Conductor Loss Models for Electromagnetic Analysis,?
IEEE Trans. Microwave Theory Tech, vol. MTT-51, pp. 915-921, March 2003.
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Papapolymerou, ?Characterization of Liquid Crystal Polymer (LCP) Material and
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68
APPENDIX
MATLAB CODE
1 Plot of sheet resistance versus loss tangent code
%This program is used to plot two lines for all possible combinations of Sheet
%Resistance versus Loss Tangent for a pair of tee resonators of different impedances.
%The intersection point of these two lines indicates the values of loss tangent
%and sheet resistance.
%initialize the workspace and set constants
clc
clear
%speed of light
c=2.998e8;
%thickness of the substrate
h=25;
%thickness of the conductor
t=0.3937;
%resonance mode
N=1;
%number of points in the vectors
Np=2000;
%data for 1st line
%width of the resonator stub
w1=25;
%length of the resonator stub
L1=1461;
%calls the function frcompute1.m, returns the resonant frequency
%(lowest point of S21 dip)
fr1=frcompute1;
%calls the function eecompute.m to compute the effective permittivity
ee1=eecompute(N,L1,fr1,h,w1);
69
%calls zcompute.m returns the characteristic impedance of the resonator stub(ohms)
z1=zcompute(ee1,h,t,w1);
%calls the function atcompute1.m, returns total attenuation(dB/in)
at1=atcompute1(ee1,c)
%calls the function ercompute.m, returns relative permittivity
er1=ercompute(N,L1,fr1,h,w1)
%calls the function ar_compute.m, returns radiative attenuation(dB/in)
ar1=ar_compute(c,ee1,fr1,z1,h);
%total attenuation minus radiative attenuation
at_r1=at1-ar1;
%calls the function Rscompute.m, returns the sheet resistance
%NOTE: Initially total attenuation - radiative attenuation is assumed for
%the computation of sheet resistance.
Rsmax1=Rscompute(fr1,w1,h,t,er1,z1,at_r1);
%divides the sheet resistance calculated with the compensated total attenuation
%by the number of specified points
dRs1=Rsmax1/Np;
%forms a vector of values for sheet resistance
%(minimum value=~0 to maximum value=max Sheet resistance)
Rs1=dRs1:dRs1:Rsmax1;
%calls the function accompute.m, returns the conductor attenuation (dB/in)
ac1=accompute(fr1,w1,h,t,er1,z1,Rs1);
%compute dielectric loss by subtracting radiative compensated total loss from
%conductor loss
ad1=at_r1-ac1;
%compute loss tangent as found in Pozar for all possible values
tand1=ad1*c*sqrt(ee1)*(er1-1)/(27.3*er1*(ee1-1)*fr1*.0254);
%data for 2nd line
%width of the resonator stub
w2=61;
%length of the resonator stub
L2=1396;
70
%calls the function frcompute2.m, returns the resonant frequency
%(lowest point of S21 dip)
fr2=frcompute2;
%calls the function eecompute.m to compute the effective permittivity
ee2=eecompute(N,L2,fr2,h,w2);
%calls zcompute.m returns the characteristic impedance of the resonator stub(ohms)
z2=zcompute(ee2,h,t,w2);
%calls the function atcompute1.m, returns total attenuation(dB/in)
at2=atcompute2(ee2,c)
%calls the function ercompute.m, returns relative permittivity
er2=ercompute(N,L2,fr2,h,w2)
%calls the function ar_compute.m, returns radiative attenuation(dB/in)
ar2=ar_compute(c,ee2,fr2,z2,h);
%total attenuation minus radiative attenuation
at_r2=at2-ar2;
%calls the function Rscompute.m, returns the sheet resistance
%NOTE: Initially total attenuation - radiative attenuation is assumed for
%the computation of sheet resistance.
Rsmax2=Rscompute(fr2,w2,h,t,er2,z2,at_r2);
%divides the sheet resistance calculated with the compensated total attenuation
%by the number of specified points
dRs2=Rsmax2/Np;
%forms a vector of values for sheet resistance
%(minimum value=~0 to maximum value=max Sheet resistance)
Rs2=dRs2:dRs2:Rsmax2;
%calls the function accompute.m, returns the conductor attenuation (dB/in)
ac2=accompute(fr2,w2,h,t,er2,z2,Rs2);
%compute dielectric loss by subtracting radiative compensated total loss from
%conductor loss.
ad2=at_r2-ac2;
%compute loss tangent as found in Pozar for all possible values
tand2=ad2*c*sqrt(ee2)*(er2-1)/(27.3*er2*(ee2-1)*fr2*.0254);
%plots the two lines that form the intersection point
71
plot(Rs1,tand1,Rs2,tand2)
legend('50 ohm','30 ohm')
grid on
xlabel('Rs')
ylabel('tand')
2 Code for extracting the resonant frequency for the first line
function fr=frcompute1
%This function reads a file containing frequency (GHz)
%versus S21(dB)and returns the resonant frequency.
%loads the file into the variable 'resp'
resp = load('50_avg.txt');
%determines the number of values in the file
s = size(resp);
%vector used to index through the values in the columns
%of the text(data) file
m = (1:s(1));
%finds the lowest value in the S21(dB) column
[min_dB,i] = min(resp);
%designates the corresponding frequency as the
%resonant frequency
fr = resp(i(2));
%converts frequency in GHz to Hz
fr =fr*1e9;
3 Effective permittivity calculation code
function ee=eecompute(N,L,fr,h,w)
%effective permittivity calculation
% This program is used to perform the extraction of effective permittivity
% given the necessary values of N(mode number), L(length of resonator stub),
% F(the operating frequency of the corresponding mode number N, h(height of
% substrate), and w(width of the resonator).
%
% You must have the functions improved_er, and Lcorrect in the same
% working directory to use this program.
72
%
% The program will compute a converged value of effective permittivity with the
% inclusion of a calculated value for the additional length of the resonator stub
% created by the fringing fields at the end of the stub.
%
% Note: the values stabilize after the second loop through, therefore the final
% value is assumed the adequately compensated value.
%Constants to be used throughout the program, converts
%mils to meters
c = 2.998e8;
L_m = L*25.4e-6;
h_m = h*25.4e-6;
w_m = w*25.4e-6;
%Calculate the effective permittivity of tee resonator
e_eff = (((2*N-1)*c)/(4*fr*L_m))^2;
%functions that perform the extraction of er
er_i = improved_er(e_eff,h_m,w_m);
delL = Lcorrect(w_m,h_m,er_i,e_eff);
e_eff2 = (((2*N-1)*c)/(4*fr*(L_m+delL)))^2;
e_eff = e_eff2;
er_i2 = improved_er(e_eff,h_m,w_m);
er_i = er_i2;
delL2 = Lcorrect(w_m,h_m,er_i,e_eff);
e_eff3 = (((2*N-1)*c)/(4*fr*(L_m+delL2)))^2;
e_eff = e_eff3;
er_i3 = improved_er(e_eff,h_m,w_m);
er_i = er_i3;
delL3 = Lcorrect(w_m,h_m,er_i,e_eff);
%This is the last necessary loop for the convergence of the final value
e_eff4 = (((2*N-1)*c)/(4*fr*(L_m+delL3)))^2;
e_eff = e_eff4;
er_i4 = improved_er(e_eff,h_m,w_m);
er_i = er_i4;
delL4 = Lcorrect(w_m,h_m,er_i,e_eff);
ee=e_eff;
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4 Characteristic impedance calculation code
function z_o = zcompute(ee,h,t,w)
%This function computes the characteristic impedance
%for a particular resonator stub.
%Computing the effective width "we"
if (w/h)<=(1/(2*pi))
we=w+((1.25*t)/pi)*(1+log((4*pi*w)/t));
end
if (w/h)>=(1/(2*pi))
we=w+((1.25*t)/pi)*(1+log((2*h)/t));
end
%computing the characteristic impedance "z_o"
if (w/h)<=1
z_o=(60/sqrt(ee))*log((8*h)/we+0.25*(we/h));
end
if (w/h)>=1
z_o=((120*pi)/sqrt(ee))*(we/h+1.393+0.667*log(we/h+1.444))^-1;
end
z_o=z_o
5 Code for calculating total attenuation for the first line
function at = atcompute1(ee,c)
%This function reads a file containing frequency (GHz)
%versus S21(dB)and returns the total attenuation (dB/in).
%loads the file into the variable 'resp'
resp = load('50_avg.txt');
%determines the number of values in the file
s = size(resp);
%vector used to index through the values in the columns
%of the text(data) file
m = (1:s(1));
%finds the lowest value in the S21(dB) column
[min_dB,i] = min(resp(m,2));
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%designates the corresponding frequency as the
%resonant frequency
f_res = resp(i,1)
%determines the 3dB value used to find the
%corresponding frequency for the computation
%of the 3dB bandwidth
three_dB = min_dB+3;
%finds the S21 values that correspond to the 3dB
%bandwidth limits(upper frequency and lower frequency)
span = find(resp(m,2) <= three_dB);
%sets the upper and lower limit frequencies found
%from the step directly above into variables
%lower limits
x2 = resp(min(span),2);
y2 = resp(min(span),1);
%upper limits
xx2 = resp(max(span),2);
yy2 = resp(max(span),1);
%find lower value just past the <= three_dB point
x1 = resp(min(span)-1,2);
y1 = resp(min(span)-1,1);
%find upper value just past the <= three_dB point
xx1 = resp(max(span)+1,2);
yy1 = resp(max(span)+1,1);
%perform the linear interpolation
y_low = y1+(y2-y1)*((three_dB-x1)/(x2-x1));
y_high = yy1+(yy2-yy1)*((three_dB-xx1)/(xx2-xx1));
%computes the 3dB bandwidth by subtracting the corresponding
%frequency values of the upper and lower S21 3dB limits respectively
BW = y_high-y_low;
%computes the loaded quality factor
QL = f_res/BW;
%intermediate step
T=1-2*10^(min_dB/10);
%calculate the unloaded Q
Qu=QL/sqrt(T);
75
%computes f_res in Ghz to f_res in Hz
fr=f_res*1e9;
%total attenuation calculated in dB/in
at=(8.686*pi*fr*sqrt(ee))/(c*Qu*39.37)
6 Relative permittivity calculation code
function er=ercompute(N,L,fr,h,w)
%relative permittivity calculation
% This program is used to perform the extraction of relative permittivity
% given the necessary values of N(mode number), L(length of resonator stub),
% F(the operating frequency of the corresponding mode number N, h(height of
% substrate), and w(width of the resonator).
%
% You must have the functions improved_er, and Lcorrect in the same
% working directory to use this program.
%
% The program will compute a converged value of relative permittivity with the
% inclusion of a calculated value for the additional length of the resonator stub
% created by the fringing fields at the end of the stub.
%
% Note: the values stabilize after the second loop through, therefore the final
% value is assumed the adequately compensated value.
%Constants to be used throughout the program, converts
%mils to meters
c = 2.998e8;
L_m = L*25.4e-6;
h_m = h*25.4e-6;
w_m = w*25.4e-6;
%Calculate the effective permittivity of t resonator
e_eff = (((2*N-1)*c)/(4*fr*L_m))^2;
%functions that perform the extraction of er
er_i = improved_er(e_eff,h_m,w_m);
delL = Lcorrect(w_m,h_m,er_i,e_eff);
e_eff2 = (((2*N-1)*c)/(4*fr*(L_m+delL)))^2;
e_eff = e_eff2;
er_i2 = improved_er(e_eff,h_m,w_m);
er_i = er_i2;
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delL2 = Lcorrect(w_m,h_m,er_i,e_eff);
e_eff3 = (((2*N-1)*c)/(4*fr*(L_m+delL2)))^2;
e_eff = e_eff3;
er_i3 = improved_er(e_eff,h_m,w_m);
er_i = er_i3;
delL3 = Lcorrect(w_m,h_m,er_i,e_eff);
%This is the last necessary loop for the convergence of the final value
e_eff4 = (((2*N-1)*c)/(4*fr*(L_m+delL3)))^2
e_eff = e_eff4;
er_i4 = improved_er(e_eff,h_m,w_m)
er_i = er_i4;
delL4 = Lcorrect(w_m,h_m,er_i,e_eff);
er=er_i;
7 Code used in the permittivity calculation
function y = improved_er(e_eff,h_m,w_m)
%Perform iteration to determine relative permittivity (e_r).
%Create a vector representing possible values for the relative permittivity constant.
n = linspace(0.0, 20.0, 100000);
%Loop that performs the actual extraction.
for(x = 1:1:100000)
b = 0.564*((n(x)-0.9)/(n(x)+3))^0.053;
a =
1+(1/49)*log(((w_m/h_m)^4+(w_m/(52*h_m)^2))/((w_m/h_m)^4+0.432))+(1/18.7)*log
(1+(w_m/(18.1*h_m)))^3;
e_trial = ((n(x)+1)/2)+((n(x)-1)/2)*(1+10*(h_m/w_m))^-(a*b);
if (e_trial >= e_eff)
break
end
end
%value of relative permittivity
y = n(x);
8. Code used to compute ?L
function y = Lcorrect(w_m,h_m,er_i,e_eff)
77
%This function computes the length added to the physical
%length of the microstrip tee resonator due to fringing fields.
%Uses Kirschning et al in Elect. Letters, Feb 5, 1981 Vol 17, No.3, p.123
E1 = 0.434907*((e_eff^.81+0.26)/(e_eff^.81-
.189))*(((w_m/h_m)^0.8544+0.236)/((w_m/h_m)^.8544+.87));
E2 = 1+(((w_m/h_m)^0.371)/(2.358*er_i+1));
E3 = 1+((0.5274*atan(0.084*(w_m/h_m)^(1.9413/E2)))/e_eff^0.9236);
E4 = 1 + 0.0377*atan(0.067*(w_m/h_m)^1.456)*(6-5*exp(0.036*(1-er_i)));
E5 = 1-.218*exp(-7.5*(w_m/h_m));
y = ((E1*E3*E5)/E4)*h_m;
9 Code for the calculation of attenuation due to radiation
function ar=ar_compute(c,ee,fr,z,h)
%This function is used to compute radiative losses in dB/in.
%Converts height in mil to height in meters
h_m=h*25.4e-6;
%Computing the constant R
R=((ee+1)/ee)-((ee-1)^2/(2*ee^(3/2)))*log((sqrt(ee)+1)/(sqrt(ee)-1));
%Computing the radiation loss in dB/in
ar=(8.686*pi*h_m^2*R*fr^3)/(z*c^3*39.37)
10 Sheet resistance calculation code
function Rs=Rscompute (fr,w,h,t,er,z,ac)
%Function calculates sheet resistance for microstrip.
%w(stub width), h(substrate height), & t(conductor thickness)
%are in units of mil.
%x1-x5 are temporary variables.
%conductivity
sigma=5e7;
%calculate an intial value for Rs (ohms/square)
Rs_i=sqrt((pi*4*pi*10^(-7)*fr)/sigma)
%calculate the effective line width
x1=(t/h)*(coth(sqrt(6.517*w/h)))^2;
78
dwp=(t/pi)*(log(1+10.8731/x1));
dw=(dwp/2)*(1+1/cos(h*(sqrt(er-1))));
we=w+dw;
%calculate some temporary constants
x2=1+(h/we)+(h/(pi*we))*(log(2*h/t)-(t/h));
x3=1-(we/(4*h))^2;
x4=((we/h)+(2/pi)*log(2*pi*2.7183*((we/(2*h))+0.94)))^-2;
x5=(we/h)+(we/(pi*h))/((we/(2*h))+0.94);
x6=1+(h/we)+(h/(pi*we))*(log((4*pi*w)/t)+(t/w));
%convert alpha c form dB/in to dB/mil
acmil=ac/1000;
%calculate sheet resistance
if (w/h)>(1/(2*pi))
if (we/h)<=2
Rs=acmil/((8.686/(2*pi*z*h))*x3*x2);
else if (w/h)>2
Rs=acmil/((8.686/(z*h))*x2*x5*x4);
end
end
else if (w/h)<(1/(2*pi))
Rs=acmil/(((8.686/(2*pi*z*h))*x3*x6));
end
end
11 Conductor attenuation calculation code
function ac=accompute(fr,w,h,t,er,z,Rs)
%Function calculates conductor attenuation(dB/in).
%w(stub width), h(substrate height), & t(conductor thickness)
%are in units of mil.
%x1-x5 are temporary variables.
%conductivity
sigma=5e7;
%calculate an intial value for Rs (ohms/square)
Rs_i=sqrt((pi*4*pi*10^(-7)*fr)/sigma)
%calculate the effective line width
x1=(t/h)*(coth(sqrt(6.517*w/h)))^2;
dwp=(t/pi)*(log(1+10.8731/x1));
dw=(dwp/2)*(1+1/cos(h*(sqrt(er-1))));
79
we=w+dw;
%calculate some temporary constants
x2=1+(h/we)+(h/(pi*we))*(log(2*h/t)-t/w);
x3=1-(we/(4*h))^2;
x4=((we/h)+(2/pi)*log(2*pi*2.7183*(((we/(2*h))+0.94))))^-2;
x5=(we/h)+(we/(pi*h))/((we/(2*h))+0.94);
x6=1+(h/we)+(h/(pi*we))*(log(4*pi*w/t)+(t/w));
%calculate alpha c
if (w/h)>(1/(2*pi))
if (w/h)<=2
ac=(8.686*Rs/(2*pi*z*h))*x3*x2;
else if (w/h)>2
ac=(8.686*Rs/(z*h))*x2*x5*x4;
end
end
else if (w/h)<=(1/(2*pi))
ac=((8.686*Rs/(2*pi*z*h))*x3*x6);
end
end
%converts from (dB/mil) to (dB/in)
ac=ac*1000;
12 Code for extracting the resonant frequency for the second line
function fr=frcompute2
%This function reads a file containing frequency (GHz)
%versus S21(dB)and returns the resonant frequency.
%loads the file into the variable 'resp'
resp = load('30_avg.txt');
%determines the number of values in the file
s = size(resp);
%vector used to index through the values in the columns
%of the text(data) file
m = (1:s(1));
%finds the lowest value in the S21(dB) column
[min_dB,i] = min(resp);
80
%designates the corresponding frequency as the
%resonant frequency
fr = resp(i(2));
%converts frequency in GHz to Hz
fr =fr*1e9;
13 Code for calculating total attenuation for the second line
function at = atcompute2(ee,c)
%This function reads a file containing frequency (GHz)
%versus S21(dB)and returns the total attenuation (dB/in).
%loads the file into the variable 'resp'
resp = load('30_avg.txt');
%determines the number of values in the file
s = size(resp);
%vector used to index through the values in the columns
%of the text(data) file
m = (1:s(1));
%finds the lowest value in the S21(dB) column
[min_dB,i] = min(resp(m,2));
%designates the corresponding frequency as the
%resonant frequency
f_res = resp(i,1)
%determines the 3dB value used to find the
%corresponding frequency for the computation
%of the 3dB bandwidth
three_dB = min_dB+3;
%finds the S21 values that correspond to the 3dB
%bandwidth limits(upper frequency and lower frequency)
span = find(resp(m,2) <= three_dB);
%sets the upper and lower limit frequencies found
%from the step directly above into variables
%lower limits
x2 = resp(min(span),2);
y2 = resp(min(span),1);
81
%upper limits
xx2 = resp(max(span),2);
yy2 = resp(max(span),1);
%find lower value just past the <= three_dB point
x1 = resp(min(span)-1,2);
y1 = resp(min(span)-1,1);
%find upper value just past the <= three_dB point
xx1 = resp(max(span)+1,2);
yy1 = resp(max(span)+1,1);
%perform the linear interpolation
y_low = y1+(y2-y1)*((three_dB-x1)/(x2-x1));
y_high = yy1+(yy2-yy1)*((three_dB-xx1)/(xx2-xx1));
%computes the 3dB bandwidth by subtracting the corresponding
%frequency values of the upper and lower S21 3dB limits respectively
BW = y_high-y_low;
%Computes the loaded quality factor
QL = f_res/BW;
%intermediate step
T=1-2*10^(min_dB/10);
%calculate the unloaded Q
Qu=QL/sqrt(T);
%computes f_res in Ghz to f_res in Hz
fr=f_res*1e9;
%total attenuation calculated in dB/in
at=(8.686*pi*fr*sqrt(ee))/(c*Qu*39.37)