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Considerations in the simulation of large monolithic microwave integrated circuits enclosed in a conducting package

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O rder N u m b er 9 31 66 2 7
Considerations in the simulation of large monolithic microwave
integrated circuits enclosed in a conducting package
Burke, Jo h n Joseph, P h .D .
University of Massachusetts, 1993
C opyright © 1993 b y B urke, Joh n Josep h . A ll rig h ts reserved.
UMI
300 N. Zeeb Rd.
Ann Arbor, MI 48106
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CONSIDERATIONS IN THE SIMULATION OF
LARGE MONOLITHIC MICROW AVE
INTEGRATED CIRCUITS ENCLOSED IN A
CONDUCTING PACKAGE
A Dissertation Presented
by
JOHNJ. BURKE
Submitted to the Graduate School of the
University of Massachusetts in partial fulfillment
of the requirement for the degree of
DOCTOR OF PHILOSOPHY
February 1993
Department of Electrical and Computer Engineering
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©
Copyright by John J. Burke 1993
All Rights Reversed
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CONSIDERATIONS IN THE SIM ULATION OF
LARGE M ONOLITHIC M ICROW AVE
INTEGRATED CIRCUITS ENCLOSED IN A
CONDUCTING PACKAGE
A Dissertation Presented
by
JO H N J. BURKE
Approved as to content and style by:
Robert W. Jackson, Chair
Daniel H. Schaubert, Member
/ s j f-x Q j jCt—
Robert E. McIntosh, Member
Donald F. St. Mary, Member
.____
Lewis E. Franks, Department Head
Department of Electrical and Computer Engineering
College of Engineering
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This dissertation is dedicated to the memory of my grandfathers,
John J. Burke and Joseph P. Bambera.
iv
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ACKNOWLEDGMENTS
I wish to express my sincere thanks to Dr. Robert W. Jackson for his guidance,
support and encouragement throughout the preparation of this dissertation. I would like
to thank Dr. Robert E. McIntosh for the teaching experience I gained while acting as a
TA for his Microwave Engineering course, and for his input to the research. Professor
Daniel H. Schaubert for the suggestions and insights he contributed to my work at the
monthly seminars and Professor Donald F. St. Mary for serving on my dissertation
committee. Many thanks to Dr. El-Badawy El-Sharawy, Jason Gerber and the other
members of LAMMDA for their assistance and friendship. Jean Sliz is also greatly
acknowledged for her friendship and assistance.
I would like to thank the great game of hockey and my teammates for helping me
keep my sanity, especially over the past two years.
I would like to thank my parents, sisters and in-laws for their support and
encouragement over what seemed like an everlasting endeavor. Last but not least, I
would like to thank my wife, Angela, for her love, support and perseverance throughout
this long process.
v
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A B ST R A C T
CO NSIDERATIONS IN TH E SIM ULATIO N
OF LARGE M ONOLITHIC M ICROW AVE
INTEG RATED CIRCUITS ENC LO SED IN A
CO ND UCTING PACK AG E
FEBRUARY 1993
JOHN J. BURKE
B.S., NORTHEASTERN UNIVERSITY
M.S., UNIVERSITY OF CALIFORNIA AT LOS ANGELES
Ph.D., UNIVERSITY OF MASSACHUSETTS
Directed by: Professor Robert W. Jackson
This thesis presents a numerical and experimental analysis of MMIC circuits
enclosed in a conducting package. Three different simulation techniques are developed: a
full-wave method o f moments (MOM) procedure, a simple circuit model and the enhanced
diakoptic method.
In the design of MMICs, the effect of enclosing the circuit is typically assumed to
be negligible. However, as the electrical size of enclosures increase package resonances
are possible. If the system operates at frequencies near one o f these resonances, coupling
between the fundamental microstrip mode and the resonant mode is possible. This
phenomena is referred to as parasitic coupling to a resonant mode and its importance is
emphasized in this thesis.
vi
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An experimentally verified full-wave MOM procedure is used to examine some of
the fundamental aspects o f resonant mode coupling. In addition, methods of reducing this
coupling will also be investigated. For example, both the addition of loss to an enclosure
or layout modifications can be used to reduce resonant mode coupling.
Since a full-wave analysis, although rigorous, is also very complex to implement, a
simple circuit model is developed to describe resonant mode coupling. Simple analytical
expressions for the entire model are easily evaluated, making this is a very attractive feature
for implementation into a CAD package. In addition, it requires several orders of
magnitude less CPU time than the MOM.
As the size o f MMIC circuits increase they become too complicated to analyze using
a straightforward full-wave approach. A full-wave analysis of a typical MMIC of moderate
complexity may require the solution of a large system of equations. Therefore, as second
alternative to the MOM the diakoptic method is modified to analyze an MMIC in an
enclosure. For very large circuits significant CPU savings result. A new spectral filtering
technique, called the enhanced diakoptic method, is developed to improve accuracy.
vii
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TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS........................................................................................................... v
ABSTRACT................................................................................................................................. vi
LIST OF TABLES...................................................................................................................... xi
LIST OF FIGURES.................................................................................................................... xii
CHAPTER
1.
INTRODUCTION..............................................................................................................1
Chapter 1 References...................................................................................................... 5
2.
FULL-WAVE ANALYSIS OF PACKAGED M M ICs.............................................. 8
2.1
2.2
2.3
Introduction......................................................................................................... 8
Spectral Green's Function....................................................................................8
Enclosure Resonances........................................................................................20
2.3.1
2.3.2
2.3.3
2.4
2.5
2.6
2.7
Method of Moments Solution.......................................................................... 31
Efficient Computation of the Impedance Matrix Elements......................... 32
Network Parameters...........................................................................................37
Expansion Functions..........................................................................................40
2.7.1
2.7.2
2.7.3
2.8
Introduction......................................................................................... 20
Evaluation of the Resonant Frequencies......................................... 22
Summary o f the Enclosure Resonances Determination................ 30
Region 1-Discontinuities................................................................... 42
Region 2-Transmission Lines........................................................... 44
Region 3 -Sources.............................................................................. 44
Conclusion........................................................................................................... 56
Chapter 2 R eferences...................................................................................................57
3.
RESONANT MODE COUPLING IN PACKAGED MMICs: A
THEORETICAL AND EXPERIMENTAL INVESTIGATION.......................... 60
3.1
3.2
Introduction......................................................................................................... 60
Experimental Verification of the MOM for Circuits in Resonant
Enclosures..........................................................................................................61
3.2.1
Introduction......................................................................................... 61
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3.2.2 Large Gap.............................................................................................63
3.2.3 Shunt Stub............................................................................................74
3.3
Parasitic Coupling to Resonant M odes..........................................................99
3.3.1 Large Gap........................................................................................... 102
3.3.2 Shunt Stub.......................................................................................... 108
3.4
C onclusioa.......................................................................................................119
Chapter 3 R eferences................................................................................................120
4.
A SIMPLE CIRCUIT MODEL FOR RESONANT MODE COUPLING
IN PACKAGED M M ICs.......................................................................................... 122
4.1
4.2
Introduction......................................................................................................122
Development of the Circuit M odel...............................................................123
4.2.1 Circuit Mutual Impedance................................................................123
4.2.2 Full-wave Mutual Impedance.......................................................... 125
4.2.3 Comparison of the Mutual Impedances.........................................129
4.3
4.4
Numerical Evaluation of Circuit components............................................. 130
Results.............................................................................................................. 132
4.4.1
4.4.2
4.4.3
4.4.4
4.5
Small Gap........................................................................................... 132
Shunt Stub.......................................................................................... 136
Shunt Stub in a Larger Box..............................................................140
Band Pass Filter................................................................................. 146
Conclusioa.......................................................................................................150
Chapter 4 R eferences................................................................................................152
5.
ANALYSIS OF MMICs IN RESONANT ENCLOSURES WITH THE
DIAKOPTIC M ETHOD........................................................................................... 153
5.1
5.2
Introduction ................................................................................................. 153
Diakoptic Method............................................................................................154
5.2.1 Theory................................................................................................ 154
5.2.2 The Relationship Between the Diakoptic and Moment
Methods..............................................................................................157
5.2.3 Results................................................................................................ 161
5.3
Enhanced Diakoptic Method......................................................................... 168
5.3.1 Theory of Spectral Filtering............................................................ 168
5.3.2 Determination of the Cutoff Mode Numbers................................ 170
5.3.3 Results................................................................................................ 174
5.4
Conclusioa.......................................................................................................189
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Chapter 5 R eferences.................................................................................................193
6.
CONCLUSION............................................................................................................. 194
Chapter 6 R eferences.................................................................................................198
APPENDICES
A.
B.
C.
D.
DERIVATION OF dYv /d co.......................................................................... 199
LARGE ARGUMENT SPECTRAL GREEN’S FUNCTION.................. 201
THE SCATTERING M ATRIX .................................................................... 203
THE ELECTRICAL CHARACTERISTICS OF THE
DIELECTRIC ABSORBER........................................................................ 206
BIBLIOGRAPHY.................................................................................................................... 209
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LIST OF TABLES
Table
Page
2.1
Summary of the resonant frequencies and Q's for various enclosures............31
2.2
Summary of the transition models employed for all substrates used
in this dissertation...................................................................................................56
3.1
The theoretical resonant frequency and Q for each mode in cavity B ........... 63
3.2
The resonant frequency and Q for each mode in the low 2-dielectric
cover cavity B......................................................................................................... 92
3.3
The resonant frequency o f each mode in the low 2 -S i cover cavity B.......... 92
3.4
The ratio of the TM i io mode powers to the incident power for the
large gap................................................................................................................ 108
3.5
The ratio of the TM i io mode powers to the incident power for the stub.... 118
4.1
Transformer turns ratios for the transmission lines on both sides of
the gap....................................................................................................................133
4.2
Transformer turns ratios for the transmission lines on both sides of
the stub...................................................................................................................136
4.3
Transformer turns ratios for the stub..................................................................138
4.4
The lumped elements of the resonant circuits for each mode in the
high Q package......................................................................................................142
4.5
Transformer turns ratios for the transmission lines on both sides of
the stub...................................................................................................................145
4.6
Transformer turns ratio for the stu b ...................................................................145
4.7
The lumped elements o f the resonant circuits for each mode in the
low Q package.......................................................................................................145
4.8
The lumped elements of the resonant circuits for each mode in the
high Q package......................................................................................................146
4.9
Transformer turns ratios for the bandpass filter...............................................148
5.1
kc for
5.2
Comparison of the operation counts for the M OM and enhanced
diakoptic m ethods................................................................................................ 191
a few representative substrates................................................................ 172
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LIST OF FIGURES
Figure
Page
2.1
Geometry of the MMIC package used in the derivation of the
Green's function...................................................................................................... 9
2.2
The transmission line equivalent circuit for the geometry shown
in F ig u re 2 .1 ................................................................................................... 16
2.3
The transmission line equivalent circuit for determining Y[$
(i = 1,
k ) ........................................................................................................17
2.4
The transmission line equivalent circuit for determining Yil),
(i = Jc+1, ..., N ) ................................................................................................. 18
2.5
Im ( I ^ ) and Im(Qtm ) versus frequency for the lossless enclosure
with the mode numbers, n and m , are equal to 1....................................23
2.6
Im (} ^ ) and Im(<2rAf) versus frequency for tiie high Q enclosure with
the m ode numbers, n and m , are equal to 1.............................................25
2.7
Im (J ^ ) and QTM versus frequency for the moderate Q enclosure with
the m ode numbers, n and m, are equal to 1 .............................................28
2.8
Im(YM) and Qtm versus frequency for the low Q enclosure with
the m ode numbers, n and m, are equal to 1.............................................29
2.9
Geometry of a transmission line with a single shunt open circuit stub
attached. The stub, located at x c = 7.5 mm (a/2), has a length
L = 1.8 mm and is attached to a transmission line, located at
y c = 12 mm (b!2) of width w = 1.4 mm.............................................................36
2.10
Comparison of the predicted transmission response of the stub using
three different values of a and s A; a = 4 and e A = 0 .0 1 , a = 4 and
s A = 0.2, and a = 2 and £A = 0.2.................................................................... 38
2.11
Schematic of an
port microstrip circuit with a voltage generator
and a terminating impedance connected to each external port.................... 39
2.12
A simple microstrip circuit consisting o f a step embedded between two
transmission lines.................................................................................................. 41
2.13
The arrangement of the rectangular cells used near a discontinuity..................43
2.14
The arrangement of the rectangular cells used along a uniform section of
transmission line.................................................................................................... 45
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2.15
Geometry of a transmission line with a gap in the center. The transmission
line, located at y c = 12 mm (b!2), has a width of w = 1.4 mm and a gap of
g = 10.5 m m ................................................................................................47
2.16
Comparison of the predicted transmission response of the large gap
in the high Q package using the MOM without the region 3 configuration
versus the m easured resu lts...................................................................... 49
2.17
The region 3 configuration of basis functions used to simulate the coaxial
to microstrip transition......................................................................................... 50
2.18
Comparison of the predicted transmission response of the large gap
in the high Q package using the MOM with the region 3 configuration
versus the m easured results.......................................................................51
2.19
Geometry of a transmission line with a single shunt open circuit stub
attached. The stub, located a tx c = 15 mm (a/2), has a length L = 1.9 mm
and is attached to a transmission line, located aty c = 24 mm (b/2),
o f width w = 1.4 m m ................................................................................ 52
2.20
Comparison of the predicted transmission response of the stub in the
high Q package using the MOM without the region 3 configuration versus
the measured results............................................................................................. 54
2.21
Comparison of the predicted transmission response of the stub in the
high Q package using the MOM with the region 3 configuration versus
the measured results............................................................................................. 55
3.1
Basic geometry of the brass cavity with the following inner dimensions:
a , b and c ...........................................................................................................62
3.2
Geometry of a transmission line with a gap in the center. The transmission
line, located at y c, has a width of w = 1.4 mm and a gap of
g = 10.5 m m ................................................................................................ 64
3.3
Comparison of the calculated and measured transmission coefficient of
the large gap in the high Q cavity A for two different locations:
y c = 12 mm and yc = 5 mm................................................................................ 6 6
3.4
Comparison of the calculated and measured transmission coefficient
of the large gap located at y c = 5 mm in the high Q cavity A. The MOM
simulation was performed with the circuit located at y c = 5.35 mm.............. 6 8
3.5
Comparison of the calculated and measured transmission coefficient
of the large gap in the low Q cavity A for two different locations:
y c = 12 mm and yc = 5 mm................................................................................ 70
3.6
Comparison of the calculated and measured transmission coefficient of
the large gap located at yc = 5 mm in the low Q cavity A. The MOM
simulation was performed with the circuit located aty c = 5.35 mm.............. 72
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3.7
Computed 77 for the large gap in the low Q cavity A for two different
locations: yc = 12 mm and y c = 5 mm............................................................... 73
3.8
Geometry of a transmission line with a single shunt open circuit stub
attached. The stub, located at x c (a/2), has a length L = 1.9 mm and is
attached to a transmission line, located at y c, of width w = 1.4 mm..............75
3.9
Ideal theoretical response of the shunt open circuit stub................................... 76
3.10
Comparison of the calculated and measured transmission coefficient
of the shunt stub in the high Q cavity A for two different locations:
y c = 12 mm and y c = 5 mm.................................................................................77
3.11
Comparison of the calculated and measured transmission coefficient
of the shunt stub circuit of length L = 1.9 mm located aty c = 1 2 mm
in the high Q cavity A .......................................................................................... 80
3.12
Comparison of the calculated and measured transmission coefficient of
the shunt stub in the high Q cavity A for two different locations:
y c = 12 mm and y c = 5 mm. For the circuit positioned at
y c = 12 mm, the MOM simulation was performed with y c = 11.45 mm,
L = 1.8 mm and er =10.75. For the circuit positioned at y c = 5 mm,
the MOM simulation was performed with y c = 5.35 mm, L = 1.9 mm
and er =10.75........................................................................................................ 82
3.13
Comparison of the calculated and measured transmission coefficient of the
shunt stub in the low Q cavity A for two different locations: y c = 12 mm
and y c = 5 mm....................................................................................................... 84
3.14
Computed 77 for the shunt stub in the low Q cavity A for two different
locations: yc = 12 mm and yc = 5 mm. The MOM simulation was
perform ed with L = 1.9 mm and er = 1 0 .5 .................................................87
3.15
Comparison of the calculated and measured 77 for the shunt stub in
the low Q cavity A for two different locations: y c = 12 mm and
y c = 5 mm.............................................................................................................. 8 8
3.16
Comparison o f the calculated and measured transmission coefficient
of the shunt stub in the high Q cavity B............................................................. 91
3.17
Comparison o f the calculated and measured transmission coefficient
of the shunt stub in the low Q-dielectric cover cavity B ...................................93
3.18
Comparison of the calculated and measured transmission coefficient
of the shunt stub in the low Q-Si cover cavity B...............................................95
3.19
Computed 77 for the shunt stub enclosed in the dielectric and Si cover
cavities....................................................................................................................96
xiv
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3.20
Comparison of the calculated and measured 77 for the shunt stub enclosed
in the: low Q dielectric cover cavity B and low <2-Si cover
cavity B................................................................................................................... 97
3.21
Summary of the calculated transmission coefficient of the large gap
in the high Q cavity A for two different locations: yc = 12 mm and
y c = 5 mm............................................................................................................. 103
3.22
Current, at/ = 10.8129 GHz, on the two strips of the large gap in
the high Q cavity A . ............................................................................................ 104
3.23
Power lost per unit length to the T M i 1 0 mode via the x component
o f the electric field a t / = 10.8129 GHz for the large gap in the high Q
cavity A..................................................................................................................104
3.24
Summary of the calculated transmission coefficient of the large gap in
the low Q cavity A for two different locations: yc = 12 mm and
yc = 5 mm............................................................................................................. 106
3.25
Current, at/ = 10.8129 GHz, on the two strips of the large gap in
the low Q cavity A................................................................................................107
3.26
Power lost per unit length to the T M i 1 0 mode via the x component
of the electric field at/ = 10.8129 GHz for the large gap in the low Q
cavity A ..................................................................................................................107
3.27
Ideal currents on the: transmission line and stub.............................................. 109
3.28
Summary of the calculated transmission coefficient of the shunt stub
in the high Q cavity A for two different locations: yc = 12 mm and
y c = 5 mm ............................................................................................................. I l l
3.29
Currents, at/ = 11.1 GHz, on the: transmission line and stub.
The circuit is enclosed in the high Q cavity A.................................................. 112
3.30
Power lost per unit length along the transmission to the T M i 10 mode
via the x component o f the electric field at/ = 11.1 GHz. The circuit is
enclosed in the low Q cavity A .......................................................................... 113
3.31
Summary of the calculated transmission coefficient of the shunt stub
in the low Q cavity A for two different locations: yc = 12 mm and
yc = 5 mm ............................................................................................................. 115
3.32
Currents, at/ = 11.1 GHz, on the: transmission line and stub.
The circuit is enclosed in the low Q cavity A .................................................. 116
3.33
Power lost per unit length to the T M i 1 0 mode via the x component of
the electric field, at / = 11.1 GHz, along the transmission line and
along the stub. The circuit is enclosed in the low Q cavity A....................... 117
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4.1
Schematic of a small length of a microstrip transmission line and
the resulting circuit m odel........................................................................ 124
4.2
Geometry of the MMIC package used to determine the full-wave mutual
impedance between a u directed current element located at
) and
a v directed current element located at ( x j , y j , Z k ) ............................................126
4.3
Geometry o f a transmission line with a gap in the center. The transmission
line, located at y c - 1.55 mm (b/2), has a width of w = 0.1 mm and a gap
of g = 0.1 mm...................................................................................................... 134
4.4
Comparison of the predicted transmission response of the small gap in
the high Q package using the circuit model versus the conventional
MOM.....................................................................................................................135
4.5
Geomeuy of a transmission line with a single shunt open circuit stub
attached. The stub, located at xc (a/2), has a length L = 1.9 mm and
is attached to a transmission line, located at y c (b/2), of width
w = 1.4 mm..........................................................................................................137
4.6
Comparison of the predicted transmission response of the stub in the
high Q package using the circuit model versus the conventional MOM
139
4.7
Comparison of the predicted transmission response of the stub in the
low Q enclosure using the proposed circuit model versus the conventional
MOM..................................................................................................................... 141
4.8
Comparison of the predicted transmission response of the stub in the
larger high Q package using the circuit model versus the conventional
M OM .................................................................................................................... 143
4.9
Comparison of the predicted transmission response of the stub in the
larger low Q package using the circuit model versus the conventional
MOM.......................................................... :......................................................... 144
4.10
Geometry of a two resonator coupled line bandpass filter. The width
and length of the resonators are w = 0.64 mm L = 5.0 mm, respectively.
The spacing of the resonators are
= 0.13 mm and S2 - 0.64 mm............147
4.11
Comparison of the predicted transmission response of the bandpass
filter in the high Q package using the circuit model versus the conventional
MOM..................................................................................................................... 149
5.1
Schematic of center fed dipole and the resulting diakopted circuit................ 155
5.2
Geometry of a center fed dipole located at xc = 7.0 mm and
y c = 2.85 mm.......................................................................................................162
5.3
Comparison o f the computed input reactance of the dipole in the high
Q package using the diakoptic method versus the input reactance computed
using the conventional MOM............................................................................. 163
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5.4
Comparison of the predicted input reactance of the dipole in the high
Q package using the diakoptic method versus the conventional MOM
165
5.5
Geometry of a transmission line with a single shunt open circuit stub
attached. The stub, located at xc (a/2), has a length L = 1.9 mm and is
attached to a transmission line, located atyc (b/2), of width
w = 1.4 mm.........................................................................................................166
5.6
Comparison of the predicted transmission response of the stub in the high
Q package using the diakoptic method versus the conventional MOM
167
5.7
Comparison of the predicted input reactance of the dipole in a high
Q package using the enhanced diakoptic method for different values of
Kc versus the conventional M OM ............................................................. 171
5.8
Comparison of the computed input reactance of the dipole using the
enhanced diakoptic method with the optimum value of k c ( 9.625) versus
the conventional MOM.......................................................................................173
5.9
Comparison of the predicted transmission response of the stub in the high
Q package using the enhanced diakoptic method ( k c = 13.5) versus the
conventional MOM............................................................................................. 175
5.10
Comparison of the predicted transmission response of the stub in the low
Q package using the enhanced diakoptic method versus the conventional
MOM....................................................................................................................177
5.11
Comparison of the predicted transmission response of the stub in the
larger high Q package using the enhanced diakoptic method ( k c = 13.5)
versus the conventional M OM .................................................................. 179
5.12
Comparison of the predicted transmission response of the stub in the
larger low Q package using the enhanced diakoptic method versus the
conventional MOM............................................................................................. 180
5.13
Geometry of a two resonator coupled line bandpass filter. The width
and length of the resonators are w = 0.64 mm L = 5.0 mm, respectively.
The spacing of the resonators are s j = 0.13 mm and S2 = 0.64 mm
182
5.14
Comparison of the predicted transmission response of the bandpass filter
in the high Q package using the enhanced diakoptic method versus the
conventional M O M ............................................................................................183
5.15
Comparison of the predicted transmission response of the bandpass filter
in the low Q package using the enhanced diakoptic method ( k c = 14.25)
versus the conventional M OM ...............................................................184
5.16
Geometry of a transmission line with a gap in the center. The transmission
line, located at yc = 1.55 mm (b/2), has a width of w = 0.1 mm and a gap
of g = 0 .1 mm..................................................................................................... 186
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5.17
Comparison of the predicted transmission response of the small gap in
the high Q package using the enhanced diakoptic method ( k c = 15.75)
versus the conventional M OM ............................................................... 187
5.18
Comparison of the predicted transmission response of the stub in the high
Q package using the enhanced diakoptic method ( kc = 9.625) versus the
conventional MOM..............................................................................................190
D. 1
Geometry of a transmission line with a gap in the center. The transmission
line, located at y c = 12 mm (b/2), has a width of w = 1.4 mm and a gap of
g = 10.5 m m ...............................................................................................207
D.2
Comparison of the calculated and measured transmission coefficient of
the large gap located at y c = 12 mm in the low Q cavity A......................208
xviii
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CH APTER 1
IN T R O D U C T IO N
Large microwave systems are generally made up of smaller modules, each of which
contains smaller sub-modules or individual circuits. These sub-modules are usually
enclosed in a conducting package to reduce outside electromagnetic interference and to
isolate one sub-module or circuit from another.
In recent years, GaAs technology has matured to the point where many submodules or even whole modules are being replaced by a single monolithic microwave
circuit (MMIC). MMICs are potentially low cost circuits because they are fabricated using
batch processing. MMICs offer improved reliability and reproducibility through
minimization o f wire bonds and have the added benefit of small size and weight. Because
circuit tuning of MMICs is difficult and expensive, there is a critical need for accurate
microwave CAD packages.
In the design of MMICs, the effect of enclosing the circuit is typically assumed to
be negligible. This has not been considered a very serious problem in current designs
because circuit packages have not been electrically large. As the level of integration and/or
the frequency of operation increases, the electrical size of the enclosure will also increase.
For a moderately sized enclosure, one or two packaged resonances are possible. In a large
enclosure, many resonances may occur in the frequency band of operation. If the system
operates at frequencies near one of these resonances, coupling between the fundamental
microstrip mode and the resonant mode is possible. This phenomena is referred to as
parasitic coupling to a resonant mode. Resonant mode coupling can result in catastrophic
coupling between different elements of a circuit that might otherwise have been isolated.
Note that it differs from proximity coupling where two or more areas of a circuit in close
proximity couple to each other. An example of proximity coupling is a section of parallel
coupled microstrip transmission lines.
1
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A number of full-wave techniques have been developed for the very accurate
modeling of microstrip structures [1] - [13]. Many of these techniques neglect the parasitic
coupling of widely spaced circuit elements and also neglect the coupling between the circuit
and the package that encloses them. Although [6 ], [7], [11] model a microstrip circuit in an
enclosure and analyze at least one circuit with a resonant mode, none of them specifically
examine the causes of resonant mode coupling and ways to reduce i t
The first goal o f this dissertation is to examine some of the fundamental aspects of
resonant mode coupling. There is still much to be learned about the way a circuit interacts
with the resonant modes of an enclosure. In a microstrip circuit with no cover plate and
side walls, it is usually assumed that most of the power lost to radiation and surface waves
occurs at a discontinuity. But Lewin [14] has shown that the interaction of circuit with
space wave radiation and surface waves can occur at more than a guided wavelength from
the discontinuity. For a circuit in an enclosure a similar effect has been observed [15].
Using a full-wave analysis it will be shown that the largest coupling to a resonant mode
occurs where large current standing waves are present in regions where the co-polarized
mode field is largest For example, an area of a circuit with a large x-directed current will
couple strongly to a resonant mode in the areas where the Ex mode field is large.
In addition to examining the fundamental aspects of resonant mode coupling,
methods o f reducing this coupling will also be investigated. For example, the addition of
loss to an enclosure may reduce resonant mode coupling, but may not eliminate it [15],
[16], [17]. Repositioning a circuit in an enclosure to reduce resonant mode coupling is
another technique that will be investigated in this dissertation. Little work has been done
on the effect that circuit layout has on resonant mode coupling with the exception o f [15]
and [18]. However, this technique is best applied to a relatively simple circuit consisting of
only a few discontinuities in a moderately sized enclosure. Since most current MMICs
2
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contain more than a few discontinuities, repositioning the circuit in an enclosure is often not
practical.
The next two goals of this dissertation are to develop alternatives to the MOM for
analyzing a circuit in an enclosure. A full-wave analysis although rigorous it is also very
complex to implement. In addition, as the size of MMIC circuits increase they become too
complicated to analyze using a straightforward full-wave approach. A full-wave analysis
of a typical MMIC of moderate complexity may require the solution of a large system of
equations to accurately model the complete circuit. The need for new ways to analyze
MMICs in an enclosure is therefore apparent
The second goal of this dissertation is to develop a circuit model to describe
resonant mode coupling for use on commercially available CAD packages. Commercially
available full-wave CAD packages model each discontinuity in isolation, and then obtain
the overall performance by combining the models using a circuit CAD program such as
SUPERCOMPACT. Parasitic coupling is neglected using this technique.
Toward that end, Jansen and Wiener [18] have developed a simple circuit theory
model to describe coupling of circuit junctions to a resonant mode. The details of their
formulation are not completely clear, but their model is based on the assumption that
resonant mode coupling occurs only at a discontinuity. As previously discussed, this
picture of resonant mode coupling is incomplete. Therefore, the circuit model developed in
this thesis will incorporate the theory of resonant mode coupling derived from the rigorous
analysis described in the first part of the thesis.
Implementing this circuit model in a CAD package for a complex MMIC in an
enclosure may be very tedious. Consequently, this circuit model is suited for MMIC
circuits of moderate complexity. For complex MMIC circuits in an enclosure, a different
modeling technique is necessary.
3
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The third goal of this dissertation is to develop a method to analyze a complex
MMIC circuit in an enclosure. The diakoptic method will be used to analyze complex
MMIC circuits. Goubau et al [19] developed the diakoptic method to analyze complex
multi-element antennas. Until very recently [20] the diakoptic method had not been applied
to MMIC problems and even now has not been reported in sufficient detail for an
assessment of it to be made. Also, it has not been applied to MMIC circuits in an
enclosure. Therefore, the proposed solution is to modify the diakoptic method to analyze
MMIC circuits in an enclosure by introducing the concept o f spectral filtering.
The topics which must be addressed to study an MMIC circuit in a conducting
enclosure are treated in the Chapters that follow. Chapter 2 is devoted to the full-wave
analysis of a packaged MMIC. Although many of the techniques presented in Chapter 2
are not new, they provide an essential basis for the remaining Chapters of the dissertation.
In Chapter 3, the full-wave analysis developed in Chapter 2 is experimentally verified. In
addition, some of the fundamental aspects of resonant mode coupling and methods of
reducing this coupling by relocating a circuit in an enclosure are discussed. Chapter 4
discusses the development and implementation of a simple circuit model on a commercially
available CAD package that describes resonant mode coupling. In Chapter 5, the diakoptic
method is modified to analyze a complex MMIC in an enclosure. The thesis is concluded
in Chapter 6 .
4
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Chapter 1 References
[1]
R.H. Jansen, "Hybrid M ode Analysis of End Effects of Planar Microwave and
Millimeter W ave Transmission Lines," Proc. Inst. Elec. Eng., Vol. 128, pt. H. pp.
77-86, April 1981.
[2]
J. Boukamp and R.H. Jansen, "The High Frequency Behavior of Microstrip Open
Ends in Microwave Integrated Circuits Including Energy Leakage," 14th European
M icrowave Conf. Proc., pp. 142-147, 1984.
[3]
R.W. Jackson and D.M. Pozar, "Full-Wave Analysis of Microstrip Open-End and
Gap Discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT-33, pp.
1036-1042, October 1985.
[4]
P.B. Katehi and N.G. Alexopoulos, "Frequency-Dependent Characteristics of
Microstrip Discontinuities in Millimeter-Wave Integrated Circuits," IEEE Trans.
M icrowave Theory Tech., Vol. MTT-33, pp. 1029-1035, October 1985.
[5]
R.H. Jansen, "The Spectral-Domain Approach for Microwave Integrated Circuits,"
IEEE Trans. Microwave Theory Tech., Vol. MTT-33, pp. 1043-1056, October
1985.
[6 ]
J.C. Rautio and R.F Harrington, "An Electromagnetic Time-Harmonic Analysis of
Shielded Microstrip Circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT35, pp. 726-730, August 1987.
[7]
R.H. Jansen, "Modular Source-Type 3D Analysis of Scattering Parameters for
General Discontinuities, Components and Coupling Effects in (M)MICs," 17th
European Microwave Conf. Proc., pp. 427-432, 1987.
[8 ]
J.R. Mosig, "Arbitrarily Shaped Microstrip Structures and their Analysis with a
Mixed Potential Integral Equation," IEEE Trans. Microwave Theory Tech., Vol.
MTT-36, pp. 314-323, February 1988.
5
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[9]
N.H.L. Koster and R.H. Jansen, "The Microstrip Step Discontinuity: A Revised
Description," IEEE Trans. Microwave Theory Tech., Vol. MTT-34, pp. 213-223,
February 1986.
[10]
W.P. Harokopus and P.B. Katehi, "Characteristics of Microstrip Discontinuities on
Multilayer Dielectric Substrates Including Radiation Losses," IEEE Trans.
Microwave Theory Tech., Vol. MTT-37, pp. 2058-2066, December 1989.
[11]
L.P. Dunleavy and P.B. Katehi, "A Generalized Method for Analyzing Shielded
Thin Microstrip Discontinuities," IEEE Trans. Microwave Theory Tech., Vol.
MTT-37, pp. 1758-1766, December 1988.
[ 12]
R.W. Jackson, "Full-Wave, Finite Element Analysis of Irregular Microstrip
Discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT-37, pp. 81-89,
January 1989.
[13]
H.Y. Yang and N.G. Alexopoulos, "A Dynamic Model for Microstrip-Slotline
Transition on Related Structures," IEEE Trans. Microwave Theory Tech., Vol.
MTT-36, pp. 286-293, February 1988.
[14]
L. Lewin, "Spurious Radiation From Microstrip," Proc. IEE, Vol. 125, No. 7, pp.
633-642, July 1978.
[15]
J.J. Burke and R.W. Jackson, "Reduction of Parasitic Coupling in packaged
MMICs," IEEE MTT-S Int. Microwave Symp. Dig., pp. 255-258, May 1990.
[16]
D.F. Williams, "Damping of the resonant modes of a rectangular metal package,"
IEEE Trans. Microwave Theory and Tech., Vol. MTT-37, pp. 253-256, January
1989.
[17]
A.F. Armstrong and P.D. Cooper, "Techniques for Investigating Spurious
Propagation in Enclosed Microstrip," The Radio and Electronic Engineer, Vol. 48,
No. 1/2, pp. 64-72, Jan/Feb, 1978.
6
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[18]
R.H. Jansen and L. Wiemer, "Full-Wave Theory Based Development of MM-wave
Circuit Models for Microstrip Open End, Gap, Step, Bend and Tee," IEEE MTT-S
Int. Microwave Symp. Dig., pp. 779-782, June 1989.
[19]
Goubau, G., Puri, N.N. and Schwering, F.K., "Diakoptic theory for multielement
antennas," IEEE Transactions on Antennas and Propagation, Vol. AP-30, No. 1,
pp. 15-26, January 1982.
[20]
Howard, G.E. and Chow, Y.L., "A high level compiler for the electromagnetic
modeling of complex circuits by geometrical partitioning," IEEE MTT-S
International Microwave Symposium Digest, pp. 1095-1098,1991.
7
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CH APTER 2
FULL-W AVE A N A LY SIS OF PACK AG ED M M ICs
2.1
Introduction
The full-wave analysis of packaged MMICs is developed in this chapter. First, the
derivation of the spectral Green's function is presented. Following this is a discussion of
locating the resonances of a dielectric loaded cavity. The next section is a review of the
method of moments (MOM) which is followed by the development of an acceleration
technique for the efficient evaluation of MOM impedance matrices. The solution for the
unknown current distribution under the condition that each port is terminated in its
characteristic impedance is then given. Lastly, the different basis functions used to expand
the currents are discussed.
2.2
Spectral Green's Function
Figure 2.1 shows the geometry of the MMIC package for which a Green's function
will be derived. There are N dielectric layers of thickness dj, relative permittivity Erf and
relative permeability (in'- The MMIC circuitry is located at z = zjc. The side walls of the
enclosure are assumed to be perfect conductors. The surface impedance of the top cover is
designated as Z s j and the bottom cover as Z5 5 .
An electric cunrent source is located at z = zk
J (x ,y ,z ) = Js ( x ,y ) 8 ( z - z k )
(2 . 1 )
The resulting electromagnetic fields in general can be expressed as the superposition of TM
and TE fields. For the TM and TE fields in each layer let:
A(i)= z ^
(2 .2 a)
8
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Figure 2.1
Geometry o f the MMIC package used in the derivation of the Green's
function.
9
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2 . 2 b)
Fi i ) =z<P^
where
i = 1,
N
The potential functions are the solutions of the scalar Helmholtz equation:
(2.3)
V 24>tu + kf ’Ptu = 0
where
kf = £ ^ 1 $
U = E otM
The total electric and magnetic field components in each layer are given by:
E ji\ x , y , z ) = ^ - V t
JEriko
£ ? (W
dz
+ z x v r0 g
( 32
) = - ^ - — + t2
■
JSrik0 & 2
H ^ ( x ,y ,z ) = - z x V M +
1
M r i k olo
tf f( W
2
where E ^ and
(2.4a)
(2.4b)
V, d$TE
dz
(
) = —. . J _ [ df zj + k f 0%
jf^rik0^lo
/
(2.4c)
(2.4d)
are the tangential components of the electric and magnetic fields,
respectively.
The boundary conditions for equation (2.3) are given as follows. At z = zi (i = 1,
..., k -1, £+1,..., AM), the tangential components of the electric and magnetic fields are
continuous:
Et(0(x,y,Z i) = £?'+1)(x,y,Zi)
(2.5a)
H ^ \ x , y , Zi) = H ji+l\ x , y , Zi)
(2.5b)
The tangential component of the electric field at z = zk is also continuous. However, there
is a jump discontinuity in the tangential magnetic field. These two boundary conditions can
be expressed as:
% k)(x ,y ,z k ) = E ^ +1\ x , y , z k )
10
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(2.6a)
Js(x,y) = z x (H <*+1 W , ^ ) - £ ,(*W , ^ ) }
(2.6b)
The tangential component of the electric field must vanish on the side walls of the enclosure
because they are assumed to be perfect conductors. These boundary conditions can be
expressed by the following:
y - % i>(x,y) = 0
fo rx = 0 ,a
(2.7a)
fory = 0 ,b
(2.7b)
M.i\ x , y ) = 0
x -E ^ i\ x , y ) = 0
£ f ( x ,y ) = 0
The solution to equation (2.3) is obtained by the separation of variables, in general
the solution can be written as:
(2.8)
Substituting equation (2.8) into (2.3) yields the following pair of equations:
V,2* ® + **?(*> = o
(2.9)
d 2mH L + k2P ^ - f t
2
zi TU ~ u
(2. 10)
dz
where
—p 11 a.2_i 2
Kz i ~ c r ir LriK0
Im (£,,)< 0
p
Vf = V - l - j oz
(2 . 11)
(2. 12)
The solution to equation (2.9) can be obtained by enforcing the boundary conditions given
by equation (2.7)
(*’>') = Ctm s^n (kxnx)sin(kymy)
(2.13a)
^ T E ^ y ) = CTE cos(kxnx)cos(kymy)
(2.13b)
where
,
niz
kxn= —
a
, _ mn
ym~~b~
k 2p = kln+ k$m
(2.14a)
(2.14b)
(2.14c)
11
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The total solution to equation (2.3) can be expressed as a summation over all
possible modes such th a t:
oo oo
<1>%(x,y,z) = X
m=0n=0
(2.15)
Substituting equation (2.15) into (2.4), the electric and magnetic field components in each
layer can be expressed by the following:
771=077=0 LJ vi 0
+ />^ )(z)[£xVr'Fr£ (x,y)]}
oo
(2' 16)
oo
771=0 7J=0
(2.17)
+~ 7
^ - [-V ,^ fe 7 )]}
jH rikQTiQ dz
JJ
4 W( W ) = X X - T T - , S f e > ,I'rM fc )’)
m-On-QJ8^
OO
oo
£ f ( x ,y ,z ) = X
n=Qn=QJ£rik0
m'2
oo
oo
(2.18)
H ? H x,y,z) = X X ^ " T
*@ « )* ie(* * 3 0
£> £> M ri*orio
(2-19)
In what follows it will be convenient to express the transverse fields in each region
by the following [ 1 ], [2 ]:
oo
oo
E ^ \ x ,y ,z ) = X X { Vr M ( ^ ™ ( ^ > 0 +
(z)?te(*30}
(2.20)
771=0 71=0
OO
H ^ \ x ,y ,z ) =
OO
X X ^ l^ ^ r M ^ y J +z ia fe U y )}
( 2 .2 1 )
771=0 77=0
Where eTM \eTE) is the transverse electric field mode vector, V j^ 0 ® ) the mode voltage
and I jm {Ij e } the mode current for the TMnm (TEnm) mode. Comparing equation (2.20)
to (2.16) and (2.21) to (2.17), one may write
® 7 ) = i§ fe )
12
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(2.22)
_ ~Hn d^TM
_ " ^ fi
WSfe) = T
*
j £ r ik 0
(2.23)
*
J £ rik 0
for the TM mode
(2.24)
V&(z> = P & (z)
jHrikoTlo dz
-1
dVj£
jUrikoVo
dz
(2.25)
and for the TE mode. The mode vectors eTM and eTE are given by:
£TM(x,y) = - ^ t ^ m ( x , y ) = CTM{ - x k xnTx { x ,y ) - y k ymTy {x,y)]
(2.26)
eTE(x ,y ) = z x V j'FreC ^y) = Cr e {-xA:ym7;(x,y) + y^„Ty(x,y)}
(2.27)
where
= c o s(^ „ x )sin (^ TO>-)
(2.28a)
Ty(x,y) = sm (kxnx)cos(kymy)
(2.28b)
In addition the mode vectors are normalized such that:
ba
ba
rr-
JJ
00
e TUnm
j \ e m -eTEd xdy = 0
oo
ri f l
nn-- 1 and m = k
' e TUlk * ^ ~ ) q
.^ lo r m ^ k
(2.29)
(2.30)
‘
where
U = E orM .
Substituting equation (2.26) into (2.30) yields:
1
CTM - \ k ,
4s s
1 -n^
ab
m and n * 0
0
m or n = 0
(2.31)
and substituting equation (2.27) into (2.30) yields:
—
Cte -
m^ Oo r n ^ O
aft
0
m = 0 and n = 0
where
13
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(2.32)
r 0 .5
P =o
1 .0
P* o
e^ " t
After some algebraic manipulation, the normal components of the electric and
magnetic fields, Ez and Hz , can be expressed as:
oo
oo
E ^ ( x ,y ,z ) = X X ^ T - /(2 ^ ) { V • * T M ^ y )}
n=0
m
=
oo
(2.33)
oo
Hz \ x , y , z ) = £ X “— 7-----V ^(z){V -[zx?r£(x,j)]}
(2.34)
m=0n=QJf1ri'c0rl0
Substituting equations (2.20), (2.21), (2.33) and (2.34) into Maxwell's equations
and applying equations (2.29) and (2.30), yields the equivalent transmission-line equations
and the modal current, 1 $ , [2 ]:
in the ^-direction for the modal voltage,
( 2 ' 3 5 )
- f - =- ^ r ^ ( z )
dz
(2.36)
where
U = E orM .
and the wave admittances for the TM and TE modes are defined as:
(2.37)
^O^Zi
4 E = -rzh r
(2-38)
Note that equations (2.35) and (2.25) are equivalent if U = E. Likewise, equations (2.36)
and (2.23) are equivalent if U = M.
The boundary conditions for the solution of equations (2.35) and (2.36) are
obtained by substituting (2.20) and (2.21) into (2.5) and (2.6). Applying the orthonormal
properties of the mode vectors, the boundary conditions can be expressed as:
for z = zi (i = 1 ,...,
Vm(Zi) = v w l)(Zi)
(2.39a)
^ffi(Zi) = 4 u 1)(z«)
(2.39b)
k -1, £+1, ..., N - 1), and
14
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V T V = V « \z k ) = V&+1\ z k )
(2.40a)
lTU - I tu (zk ) _ ^TUl)(zk )
(2.40b)
h u = j p i (*>>’)• e-ru (x ,y )d x dy
(2.41)
for z = zk, where
We can now draw an equivalent circuit for the TU mode (£/ = £ or A/) as shown in
Figure 2.2 [2], [3]. By using transmission line theory the current, i j y , and the voltage,
vTU, at z = Zk are related via
P -42)
xu
where Yu is the driving point admittance at z = Zk. From the equivalent circuit shown in
Figure 2.2, Yu, is given by:
y
_ y ( * ) . y ( * + l)
(2.43)
*U ~ *LU + *RU
where
y ( i ) * LU ^ + J ^ T V ^
y(i') _
~ I t U (z i )
_
^T U
+J
^ L U
tan &i
L U ~ v & ( Zi)
(2.44)
for / = 0
-SB
y(0 Yr U1'*+ jY jV ^
for i = k + l,...,N
y ( 0 _ I r u ( Zi - 1) _
(2.45)
for i = N + \
-ST
d i= k zidi
(2.46a)
di = zi - z i. i
(2.46b)
Figure 2.3 shows the equivalent circuit for determining Y[y (i = 1 ,..., k) and Figure 2.4
shows the equivalent circuit for determining Y^u (i = k+1, ..., N ).
15
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+
+
t?
tv?
+
q?
=S?
j?a
"^T
ts?
gS)
CB*
«?I
N
The transmission line equivalent circuit for the geometry shown in Figure 2.1
it
Figure 2.2
CM
ig
'S ?
•4^
TS?
16
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(/ = k+1,
The transmission line equivalent circuit for determining
Figure 2.4
18
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For the subsequent method of moments solution, the tangential component of the
electric field at z = zjt due to a surface current on the same surface will be needed.
Substituting equation (2.42) into (2.20) yields:
oo oo
(2.47)
El(* \x ,y ,z Jt) = X X j - S M-h M ^x ,y) + - ^ E-eTE^x,y)\
w=o«=ol
£
J
Expressing the surface current, J s (x ,y ), as
J s(x,y)= xJx (x,y) + yJy (x,y)
(2.48)
and substituting equations (2.26), (2.27) and (2.48) into (2.41) yields:
4s„£m I km
-f L J x
ab j
1tm ~ ~
kyn
^ x n , k y m ) + - f - J y { k x n , k ym)
_ _ l^ £n£m ^ym
r 1 ^ ^x(kxnikym)
ab 1 kp
lTE ~ a/
^ •^yikxn,kym) ,
(2.49)
(2.50)
where
Jx(kxn->kym) —I f Jx (x,y)Tx (x,y)d xd y
s
(2.51a)
Jy^xn'kym ) ~ I I Jy(x,y)Ty (x,y)d xd y
(2.51b)
Substituting equations (2.26), (2.27), (2.49) and (2.50) into (2.47), the tangential
component of the electric field at z = zk after some algebraic manipulation can be
represented as:
oo
oo
Elk\ x , y , z k )= X X
.
- ^ T (x ^ & k ^ k y J -Z ik ^ k y j]
(2.52)
772=0/1=0
where
Js(kxn1kym ) = xJx (kxn,kym) + yJy (kxn,kym)
'Tx (x,y)
0
T(x,y) =
0
Ty (x,y)
Q(kxn,kym) =
'x x Q n ik ^ k y J
xyQ ^k^kyJ
yxQ y X (k x rrkym )
y y Q y y ik ^ k y J
Qxx (kxn, kym) —
-jfQ rM + ^rrQ T E
v p
v
(2.53)
(2.54)
(2.55)
(2.56a)
19
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Q x y ^ k y J = QyX(kxn,kym) = J ^ y a L ( Q m - Qt e )
"P
(2.56b)
(2.56c)
V P
P
(2.57)
U = E or M.
The poles of the spectral Green's function, Q(kxn,kym), correspond to surface
waves (no enclosure), parallel-plate waves (with cover and no side walls) or cavity modes
(in an enclosure). Since all the circuits analyzed in this dissertation are in an enclosure, the
frequency at which a pole occurs is referred to as a resonant frequency of the cavity.
Locating poles of the spectral Green's will be discussed in the next section.
2.3
Enclosure Resonances
2.3.1
Introduction
In this section locating the poles of the spectral Green's function, Q(kxn,kym), will
be discussed. The frequency at which a pole occurs is referred to as a resonant frequency
of the cavity. A pole of Qm (QTE) corresponds to a TM (TE) resonant mode of the cavity.
These poles are located by finding the zeros of Yu and Ye .
For a typical enclosure housing an MMIC chip the cover height is low enough that
only TM modes are resonant over the frequency of operation. For this reason, locating
only the zeros of YM will be discussed. However, everything discussed in this section is
also applicable to TE modes.
20
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After determining the resonant frequencies of an enclosure, a useful figure of merit
is the quality factor or Q. In what follows three definitions for the Q of a lossy enclosure
are developed.
The first expression for the Q is given by [4], [5]:
2 tonW ™
<2 -58>
n
'T’JUf
where the average electric energy stored in the resonant TM mode, We , is given by:
W ™ = i - R e ^ J J J |f :™ ( x , y , z f dxdyckj
(2.59)
the average power loss to the resonant TM mode, P ™ , is given by:
= ± R e ( J J E ™ (x ,y ) ■J* (x,y) dxdyj
(2.60)
and (Qq is the real resonant frequency. The solution of the real resonant frequency will be
discussed shortly.
An alternative expression for the Q can be found from the equivalent circuit for the
TM mode (Figure 2.2) developed in section 2.2. By using transmission line theory the
current, iTM, and the voltage, vm , at z = zk are related via
%
=
(2.61)
XM
where YM is the driving point admittance at z = Zjfc. From the equivalent circuit shown in
Figure 2.2, it
i is easily shown that the average power loss to the resonant TM mode, P ™ ,
is given by:
r™ = j M
2 Re[r«(a»o>]
( 2 -6 2 )
Applying Foster's reactance theorem [1], [4], [6 ],[7] to the equivalent circuit, the
average electric energy stored in the resonant TM mode is given by:
(2.63)
CO=COq
Substituting equations (2.62) and (2.63) into equation (2.58) yields a second expression
for the Q:
21
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Q)0 Im
<2i =
d(0
0-0
o
(2.64)
2Re[rM(©0)]
where dYM/do) is given by equation (A.1).
The third expression for the Q of an enclosure is given by [1], [4]:
Re(<pr )
2
2 Im(<or )
where (Or is the complex resonant frequency of the enclosure. The solution of the complex
resonant frequency will be discussed shortly.
2.3.2
2.3.2.1
Evaluation of the Resonant Frequencies
Lossless
For a lossless cavity, QTM and YM are purely imaginary and (0r is purely real. The
poles o f Qtm are located by finding the zeros of Ym - The zeros of YM can be located by
searching for the real frequency at which I m (I^ ) = 0. The poles of QTM and the zeros of
Ym are simple. Conversely, the zeros of Qtm and the poles o f YM are simple.
Furthermore, the poles and zeros of QfM alternate along the co axis and the poles and zeros
of Ym alternate along the 6 ) axis. Because the poles and zeros alternate, the slopes of QfM
and Ym versus frequency must be positive [1], [4], [6 ].
For example, consider a cavity of the following dimensions: a = 15 mm, b = 24
mm and c = 10 mm. There are three dielectric layers o f thickness d \ - 1.27 mm, d j =
7.967 mm, and d 3 = 0.762 mm with relative permittivities of er l = 10.5, er2 = 1-0, and
er3 = 11.7. Figure 2.5a shows I m ( l^ ) versus frequency and Figure 2.5b shows
Im(<2rM) versus frequency. For Figure 2.5, the mode numbers, n and m, are equal to 1.
Note that the slopes of QTM and YM are positive. In the band 8-13 GHz, there is one
22
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Lossless
2.50e-2
> 0.00e+0
-5.00e-2
8
9
10
11
12
13
F (GHz)
(a)
Lossless
2000
1000
S
H
0
1
-1000
-
-2000
8
9
10
11
12
13
F (GHz)
(b)
Figure 2.5
(a) Im(yM) and (b) ]m(QTM) versus frequency for the lossless enclosure
with the mode numbers, n and m, are equal to 1.
23
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
frequency, f 0, for which ~kn.(YM) = 0. This frequency has been determined to be f 0 =
10.3106 GHz.
23.2.2 High O
If loss is present in the cavity, YM is complex and the resonant frequency of the
enclosure, cor, is also complex. The zeros of YM are located by searching for a complex
CD
where the real and imaginary parts of YM simultaneously equal zero. Searching for the real
CD where
Im(yM) = 0 may result in a solution which is not an actual zero of Ym (i.e.
R e ( i^ ) * 0). Therefore, generally the zeros should not be located by searching for a
frequency where only one part of YM equals zero. However, for cavities with small to
moderate loss, the real part of the resonant frequency can be determined with good
accuracy if the approximate solution satisfies both of the following conditions:
=0
(2.66)
(2.67)
The condition imposed by equation (2.67) may not seem obvious at first. In order
to better understand this condition, let us reexamine the example given above with a small
amount of loss added to the enclosure. Loss is introduced to the enclosure through a
complex permittivity £r l = 10.5(l-j0.0023). This cavity is referred to as the high Q or low
loss enclosure.
Figure 2.6a shows lm(YM) versus frequency and Figure 2.6b shows Im (Qtm )
versus frequency for the high Q enclosure. For Figure 2.6, the mode numbers, n and m,
are equal to 1. Comparing Figure 2.5b to Figure 2.6b shows that adding loss to the cavity
results in
not becoming infinite for any real a>. The poles of QTM have been
24
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
S.OOe-2
2.50e-2
> 0.00e+0
-5.00e-2
8
9
10
11
12
13
F (GHz)
(a)
High Q
2000
1000
S
H
O'
w
S
-1000
-2000
8
9
10
11
12
13
F (GHz)
(b)
Figure 2.6
(a) Im(yM) and (b) Im(QrM) versus frequency for the high Q enclosure
with the mode numbers, n and m, are equal to 1.
25
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
shifted a small amount from the real co axis to points slightly above it. In another words,
the poles have a small imaginary part.
Examination of Figure 2.6a shows that there are two solutions where I m ( l^ ) = 0
for a real co. The first zero of Im(yM), 10.3106 GHz, occurs at the same frequency as the
resonant frequency of the lossless enclosure. This was expected because the addition of a
small amount of loss will result in the addition of a small imaginary part to the resonant
frequency. The second zero of Im(7M) , 11.15 GHz, occurs at the same frequency as the
pole of I m ( l^ ) for the lossless enclosure. Both solutions satisfy equation (2.66).
However, only the first solution satisfies equation (2.67). Therefore, in the band 8-13
GHz, there is one real frequency, / 0, that satisfies equations (2.66) and (2.67). This
frequency has been determined to be / 0 = 10.3106 GHz.
To verify that / 0 is equal to the real part of the complex resonant, a complex root
finder was employed. In the band 8-13 GHz, there is one complex frequency, f r , for
which 1 ^ = 0. This frequency has been determined to be f r = 10.3106+j0.0103 GHz.
The real part o f the complex resonant frequency, f r, does equal f 0 ; therefore,
equations (2.66) and (2.67) can be used to find the real part of the resonant frequency for
high Q enclosures.
The Q for the high Q enclosure was calculated using equations (2.58), (2.64) and
(2.65). All three equations give the same value o f 5013.6.
2.3.2.3 Moderate O
To examine the effect of increasing the loss has on the resonant frequency and Q,
the permittivity of layer 3 for the example above was changed to £r3 = 11.7(l-j0.5). This
cavity is referred to as the moderate Q or moderate loss enclosure.
26
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 2.7a shows Im (I^ ) versus frequency and Figure 2.7b shows
and Re.(QTM) versus frequency for the moderate Q enclosure. For Figure 2.7, the mode
numbers, n and m, are equal to 1. Comparing Figure 2.6b (high Q) to Figure 2.7b shows
that adding more loss to the cavity results in
becoming smaller in the vicinity of
the pole.
In the band 8-13 GHz, there is one real frequency, f 0, that satisfies equations
(2.66) and (2.67). This frequency has been determined to be f 0 = 10.3076 GHz. In the
band 8-13 GHz, there is also one complex frequency, f r, for which YM = 0. This
frequency has been determined to be f r = 10.3052+j0.0369 GHz.
The real resonant frequency, / 0, and the real part of the complex resonant
frequency, f r, differ by only 0.4 MHz; therefore, equations (2.66) and (2.67) can be used
to find the real part of the resonant frequency for moderate Q enclosures.
The Q for the moderate Q enclosure was calculated using equations (2.58), (2.64)
and (2.65). Equation (2.58) gives Qq = 139.65, equation (2.64) gives Q1 = 138.64, and
equation (2.65) gives Q2 = 139.67. The Q's calculated by equations (2.58) and (2.65)
give essentially the same result, while equation (2.64) is slightly different
2.3.2.4 Low O
To examine the effect of a large loss has on the resonant frequency and Q, the
permittivity of layer 3 for the example above was changed to £r3 = 11.7(l-j 10.0). This
cavity is referred to as the low Q or large loss enclosure.
Figure 2.8a shows Im (YM) versus frequency and Figure 2.8b shows lm (Q TM)
and Re(<2rAf) versus frequency for the low Q enclosure. For Figure 2.8, the mode
numbers, n and m, are equal to 1. Comparing Figure 2.6b (high Q) to Figure 2.8b shows
that a large loss to the cavity results in Im(<2rM) becoming much smaller in the vicinity of
27
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Medium Q
5.00e-2
2.50e-2
/•“•s
E
b 0.00e+0
E
-2.50e-2
-5.00e-2
9
8
10
11
12
1?
F (GHz)
(a)
Medium Q
2000
Im(QTM)
Re(QTM)
3.00e+3
-1000
1.00e+3
Rc(QTM)
1000
0.00e+0
-2000
8
9
10
11
12
13
F (GHz)
(b)
Figure 2.7
(a) Im(7M) and (b) QTM versus frequency for the moderate Q enclosure
with the mode numbers, n and m, are equal to 1.
28
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Q
b 0.00e+0
-2.50e-2
-5.00e-2
9
8
10
11
12
13
F (GHz)
(a)
Low Q
600
800
Im(QTM)
Re(QTM)
400
600
200
400
E
200
—■
—^~l*‘
-200
8
9
10
11
12
13
F (GHz)
(b)
Figure 2.8
(a) Im(YM) and (b) QTM versus frequency for the low Q enclosure with the
mode numbers, n and m, are equal to 1.
29
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the pole. Comparing Figure 2.7b (moderate Q) to Figure 2.8b shows that adding more
loss to the cavity results in Re(QTM) spreading out versus frequency.
In the band 8-13 GHz, there is one real frequency, / 0, that satisfies equations
(2.66) and (2.67). This frequency has been determined to be / 0 = 10.5091 GHz. In the
band 8-13 GHz, there is also one complex frequency, f r, for which YM = 0. This
frequency has been determined to be f r = 10.4810+j0.1394 GHz.
The real resonant frequency, / 0, and the real part of the complex resonant
frequency, f r, differ by only 28.1 MHz; therefore, equations (2.66) and (2.67) can be
used to find the real part of the resonant frequency for low Q enclosures.
The <2 for the low Q enclosure was calculated using equations (2.58), (2.64) and
(2.65). Equation (2.58) gives <2b = 38.784, equation (2.64) gives Ql = 35.467, and
equation (2.65) gives Q2 = 37.587. The Q calculated by equation (2.58) results in a value
about 3.2 % higher than equation (2.65). The Q calculated by equation (2.64) results in a
value about 5.6 % higher than equation (2.65).
2.3.3
Summary of Enclosure Resonance Determination
In this section, the location of the real and complex resonant frequencies was
discussed. Comparing the real resonant frequencies to the real part of the complex resonant
frequency shows that difference is on the order of tens of mega-Hertz for Q's as low as 40.
Three equations for the Q of an enclosure were developed. Comparing the results
of the three equations shows that difference between any of the two is less than 10 % for
Q's as low as 40. Table 2.1 summarizes the results of this section for an enclosure
containing different amounts of loss.
30
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Table 2.1
Summary of the resonant frequencies and Q's for various enclosures.
Loss
fo (GHz)
fr (GHz)
None
Small
Moderate
Large
10.3106
10.3106
10.3076
10.5091
10.3106
10.3106+i0.0103
10.3052+i0.0369
10.4810+i0.1394
2.4
Qo
a
q2
Eq. (2.58) Eq. (2.64) Eq. (2.65)
oo
OO
OO
5013.6
5013.6
5013.6
138.64
139.65
139.67
38.784
35.467
37.587
Method of Moments Solution
The electric field integral equation (EFIE) representing the boundary condition that
the total tangential electric field, E\ot, must vanish on the microstrip line can be written as
E \ ° \x ,y ) = Einc(x ,y ) + Estcat{x,y) = 0
(2.68)
where
y) = £<*>(W i ) = S
-7,
m=0n=0 aD
and E f10 is the incident electric field and Js is the surface current on the microstrip line.
The surface current on the microstrip line is expanded into a set of basis functions
as follows:
Nx
Ny
Js (x,y) = x £ l xiJxi (x, y) +
IyjJyj (x,y)
(2.69)
j =l
/= i
where Ixi and Iyj are the unknown current coefficients. Following Galerkin's method,
equation (2.68) is tested with Jxi and Jyj which results in a set of algebraic equations for
the unknown current coefficients, such that
v* =
N*
«=i
+
Ny,
^
.Iyj
j =i
k = l,...,N x
J
31
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(2.70a)
The elements of the excitation vector are given by [8], [9], [10], [11], [12]:
Vx k = j j E ‘xnc(x,y)Jxk(x ,y )d x d y
Vy l = \\E ™ { x ,y )J yl(x ,y )d x d y
Wr
[0
if (**Ofc) = ( * f o ^ )
otherwise
Vrp
if( x k ,yk ) = ( x ? ,y { )
0
otherwise
(2.71a)
(2.71b)
In the above the expressions, ( x f , y f ) is the position of the rth excitation and r = 1
Np, where Np is the total number of ports. The superscript P is used to designate a port.
A typical element of the impedance matrix is in the form:
J
a _ __rv
m=0n=0
where
Jpifcxn’kym) = JJ Jpi(x,y)Tp (x ,y )d x d y
(2.73)
Si
p ,q = x o r y
2.5
Efficient Computation of the Impedance Matrix Elements
For an MMIC with no side walls, efficient techniques have been developed to
calculate the elements of the impedance matrix [13], [14], [15]. When the MMIC is
contained is a conducting enclosure, the evaluation of the impedance matrix is numerically
intensive. Hill and Tripathi [16] developed an efficient technique using the two
dimensional fast Fourier transform to determine the elements of the impedance matrix.
However, the fast Fourier transform requires the circuit to be discretized using a uniform
grid. The use of a uniform grid can lead to a large number of unknowns for a relatively
simple circuit that requires a high spatial resolution in a particular region.
32
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In this section an efficient technique for the evaluation of the impedance matrix
elements is developed using Kummar's transform [17]. Jansen and Sauer [18]
independently developed a similar technique using the spectral operator method [19]. The
slowly converging series in equation (2.72) is decomposed into two series. The first term
is a rapidly converging series that is evaluated at each frequency. The second term is a
slowly converging series; however, it is only evaluated at a single frequency.
In order to implement equation (2.72) on a computer, the summations over n and m
are truncated at N°° and M°° such that
m=0n=0
The choice o f N°° and M°° and the convergence behavior of equation (2.74) will be
discussed shortly.
From the discussion in Appendix B, the spectral Green's Function for large kp can
be represented as:
(2.75)
As a result, we can rewrite equation (2.74) as:
(2.76)
where
(Z m \ j = ^
^
m=0n=0
ab
Ob
&pq^xn.kym)Jpi(kxn,kym ) 7 ^ (Icxn,kym)
(2.77a)
(2.77b)
m=0n=0
Zpq).. decays more rapidly for increasing kp than does
therefore, the limits for
the summations in equation (2.77a) are truncated at N A and M A where:
Na «
N°°
33
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
MA «
M°°
The choice of N A and M A and the convergence behavior of equation (2.77a) will be
discussed shortly.
The frequency dependence of equation (2.78) can be separated into two parts by
substituting equation (B. 10) into (2.78). The first term is inversely proportional to
frequency, while the second term is proportional to frequency. Equation (2.78) can now
be expressed as:
(2 ™
(Z 7 9 )
where
k k
AT .
( z j f ( /)).. = £
1
"T" rJ
£
m = 0n= 0
a°
M°° N°° . „
( z ™ ( /) )., = £ £
J
m = 0n= 0
y
aD
<2-80>
Kp
jr h
W. 2P * e g i W J p l ^ . k y n V ^ . k y J
(2 .8 1 )
kP
_ \ k xn
^ P = X
_f+l
if /7= y
if p = q
[-1
if p * q
pq
Although the evaluation of equations (2.80) and (2.81) still require intensive computational
effort, they need only be evaluated at a single frequency ( f R ). After evaluating equations
(2.80) and (2.81) at f R, ( z ^ ) .. as a function of frequency is simply given by equation
(2.79).
As stated previously, ZA is a rapidly converging series. In order for this to be true,
QA(kxn,kym) must be approach zero for small to moderate values of kxn and kym. In
another words, Qq
s
approaches Q(kxn,kym) as k ^ and kym increase. A useful
figure of merit used to determine how well Q q s (kxn,kym) approximates Q(kxn,kym) is
defined as:
34
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e4 =
(2.82)
Q fcm ’kyn,)
In order to determine N A and M A for a given e A, first assume kA = kx„ + kym and
kjm = Km- Next, kA is varied until equation (2.82) is satisfied. After determining kA, N A
can be found by substituting kA into equation (2.14) which yields:
(2.83a)
II
o
Cl ,
=~K
Tt
3
and M
II
1'
b .
(2.83b)
ii
o
tT
n
Note that as e A becomes smaller, N A and M A become larger.
In order to implement equations (2.72) of (2.76) on a computer, The summations in
Z and Z°° over n and m are truncated at N°° and M “ . Dunleavy [11] investigated the
convergence of Z and determined the optimum truncation of the series to be:
at
, roo oca
N =—
A,
a t
=—
where 1.5 < a < 4.0 and A x is the length of the x-subsection and
(2.84a)
(2.84b)
is the length of the y-
subsection. A^ and A y are described in more detail in section 2.7.
To examine the effect of varying a and s A has on the solution obtained using the
method of moments, consider a transmission line with a single shunt open circuit stub
attached. The stub is located atx c = 7.5 mm (a/2) and has a length of 1.8 mm. The
transmission line is located aty c = 12 mm (bJ2). The width of the transmission line and
the stub is 1.4 mm (Figure 2.9). The circuit is enclosed in a cavity of the following
dimensions: a = 15 mm, b = 24 mm and c = 12.7 mm. The substrate thickness is d\ =
1.27 mm and the relative permittivity is er i = 10.5(l-y0.0023). An enclosure of this size
has only one resonant mode, the T M no (10.8129 GHz), in the band 9-12 GHz.
35
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b
Figure 2.9
Geometry of a transmission line with a single shunt open circuit stub
attached. The stub, located at x c - 7.5 mm (a/2), has a length L = 1.8 mm
and is attached to a transmission line, located at y c = 12 mm (jb/2), of width
w = 1.4 mm.
36
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 2.10 compares the predicted transmission response of the stub using three
different values of a and e A; a = 4 and e A = 0.01, a = 4 and e A = 0.2, and a = 2 and
e A = 0.2. The MOM was performed using 123 expansion functions. Agreement between
the three different values of a and e A is excellent; therefore, a = 2 and e A = 0.2 can be
used and still obtain excellent convergence.
2.6
Network Parameters
In this section, the unknown current coefficients will be determined under the
conditions that each port is terminated in its characteristic impedance. After finding the
unknown current coefficients, the s-parameters are then determined.
In order to determine the current coefficients with the ports terminated in its
characteristic impedance, equation (2.72) will have to be modified to include the port
terminations. Before doing this modification, equation (2.70) is written in a more compact
form:
V = ZI
(2.85)
The z-matrix in equation (2.85) can be viewed as the open circuit impedance matrix for a
- N p terminals are internal nodes. Equation (2.85) can be rewritten as:
_ z IP
N
1
z pn
i
’z pp
i
i v i
1
the remaining
1---- ---- 1
terminals (N^, = N x + Ny ). Np o f the terminals are external ports and
i—
network with
where the superscript P denotes a port and the superscript / denotes an internal node.
Examining equation (2.71), it is easily seen that the matrix V7 is equal to the null matrix.
A voltage generator u f with an internal impedance Z0l- is connected to each
external port as illustrated in Figure 2.11. The voltage, V p , and the current, i f , at each
port are related by:
(2.87)
37
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 12 mm
-10
-20
-30
- Ar = 2 / Er = 0.20
- Ar = 4 / Er = 0.20
‘ Ar = 4 /E r = 0.01
-40
9
10
11
12
F (G H z)
Figure 2.10
Comparison of the predicted transmission response of the stub using three
different values of a and e A; a = 4 and e A = 0 .0 1 , a = 4 and e A = 0.2,
and a = 2 and s A = 0.2.
38
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 2.11
Schematic of an Np port microstrip circuit with a voltage generator and a
terminating impedance connected to each external port.
39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
0
.1
'zpp z pr
ZIP z n_
(2 . 88 )
I
---
1
Substituting equation (2.87) into (2.86) and simplifying yields:
where
yPP
ij
_ j ZPP+ZQi
for i —j
for i * j
[Z f
Equation (2.88) is then solved for the unknown current coefficients, l p and I 7.
This yields the current under the conditions that each port is terminated in its characteristic
impedance.
The network described by equation (2.88) is designated as an augmented network.
The short circuit admittance matrix of the augmented network is given by:
jP
Ya — ‘
~ UP
rrP
1 U {= 0
(2.89)
for k * j
where Ya is the short circuit admittance matrix of the augmented network.
The relationship between the augmented y-matrix and the s-matrix is given by (see
Appendix C):
S = U - r 1/2Y V /2
(2.90)
where U is the identity matrix and
r7i/2
C01
rV2 =
(2.91)
z l/2
Z0Ar
2.7
Expansion Functions
Figure 2.12 shows a simple microstrip circuit consisting of a step embedded
between two transmission lines. The circuit is divide into three regions. Different types of
basis function are used to expand the current in each of the three regions [20]. Rooftop
functions are used for expansion functions in areas of discontinuities (region 1). Functions
40
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 2.12
A simple microstrip circuit consisting of a step embedded between two
transmission lines.
41
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
that satisfy the edge conditions are used for expansion functions in uniform sections of
transmission lines (region 2). For a source near the side o f the enclosure (region 3),
rooftop functions are used in a special configuration.
2.7.1
Region 1 - Discontinuities
The microstrip conductor near a discontinuity is divided into rectangular cells along
the x and y axes as shown in Figure 2.13. Although the cells in this formulation do not
have to be of the same size, typically the same cell size occurs over small sections. Mosig
and Gardiol [21] recommend that the linear size of a cell should not exceed one tenth of a
guided wavelength. The functions used to expand the current near a discontinuity are
rooftop functions which are described by:
J Xi ( x , y ) = f xt i(x)gPi{y)
(2.92a)
Jyj(x,y) = gPj(x)f$j(y)
(2.92b)
where
1 + ~ n r ^ u ~ uk )
Ui-A^^u^Uk
A uk
/ 4 («)=
1
R
1 - 7 7 r ( “ ~ uk )
Uk < u < uk + A,*
0
otherwise
1
(2.93)
A vk
(2.94)
Sv*(v) = - \ k
0
otherwise
The center of the x-directed currents are marked with a cross and the center of the ydirected currents are marked with a circle. The x-directed currents overlap each other in the
x-direction, but not in the y-direction. For y-directed currents the reverse is true.
42
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
r-
XI
Figure 2.13
XI
The arrangement of the rectangular cells used near a discontinuity.
43
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2.7.2
Region 2 - Transmission Lines
The microstrip conductor along a uniform section of transmission line is also
divided into rectangular cells. The arrangement of these cells is shown in Figure 2.14.
The center of the x-directed currents are marked with a cross and the center o f the ydirected currents are marked with a circle. Functions that satisfy the edge condition are
along the width of the transmission line. The functions used to expand the current along
uniform sections of transmission lines are described by:
JXi(x,y) = fxi(x)g]i(y)
(2.95a)
Jyj(x,y) = gPj(x)f^(y)
(2.95b)
where /*,(x) and g^-(x) are given by equations (2.93) and (2.94), respectively and
/iw=- -wJy l -
(2.96)
0
otherwise
-.21-V2
Itw'
1-
(v —V;.)
w
u
0
2.7.3
|v -v ,|sf
(2.97)
otherwise
Region 3 - Sources
For a source near the side of the enclosure (region 3), the current could be
expanded with the basis functions used in region 1 or 2. However, if the results are to be
compared to measured data, the effect of the coaxial to microstrip transition must be either
removed or simulated. The method most often used to remove the effect of the coaxial to
microstrip transition is called de-embedding. Several de-embedding techniques have been
developed to remove the effect of the coaxial to microstrip transition from the measured sparameters. Two of the most popular de-embedding techniques are the time domain [22]
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C w ;)
t
w
0
X
0
I
XI
Figure 2.14
iR .
*Xl
The arrangement of the rectangular cells used along a uniform section of
transmission line.
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and the thru-short-delay [23] methods. Similar techniques are used to remove the effect of
the source from the simulated s-parameters [24]. However, in order for these de­
embedding algorithms to work, the enclosure containing the MMIC can not have any
package resonances. In Chapter 3, the effect that cavity resonances have on packaged
MMICs is investigated. Since the s-parameters of these circuits can not be de-embedded,
an effort will be made in this section to simulate the physical coaxial to microstrip transition
used in the circuits presented in Chapter 3.
2.7.3.1 Transition Model for 1.27 mm Thick Duroid
A new arrangement of the basis functions used to model the current in the vicinity
o f the source region is developed for a 1.27 mm thick Duroid 6010 substrate (er = 10.5)
with SMA coax to microstrip launches (Omni Spectra OSM 2052-1215-00). This new
arrangement will be referred to as the region 3 configuration. Two circuits will be
simulated with and without the region 3 configuration and the results will be compared to
measurements. These measurements are discussed in more detail in Chapter 3; therefore,
the measured results will for now only be presented without discussion.
For a circuit simulated using the region 3 configuration, region 1 and 2 basis
functions are used in conjunction with the region 3 configuration as illustrated in Figure
2.12. For a circuit simulated without using the region 3 configuration, region 2 basis
functions are used over the entire source region in lieu of the region 3 configuration.
As a first example, consider a transmission line with a large gap in the center as
shown in Figure 2.15. The transmission line, located aty c = 12 mm, has a width of w =
1.4 mm and a gap of g = 10.5 mm as shown in Figure 2.15. The circuit is enclosed in a
cavity of the following dimensions: a = 15 mm, b = 24 mm and c = 12.7 mm. The
substrate thickness is d\ = 1.27 mm and the relative permittivity is er l = 10.5(l-y'0.0023).
46
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
b
Figure 2.15
Geometry of a transmission line with a gap in the center. The transmission
line, located at yc = 12 mm (b/2), has a width of w = 1.4 mm and a gap of g
= 10.5 mm.
47
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Since this cavity contains only a small amount of loss it will be referred to as the high Q
package.
Figure 2.16 compares the transmission response of the large gap in the high Q
package using the MOM without the region 3 configuration versus the measured results.
The MOM was performed using 144 expansion functions. The response of the large gap
predicted by the MOM has the same characteristics and is qualitatively similar to the
measured results between 10.5 GHz and 11.5 GHz. However, below 10.5 GHz and
above 11.5 GHz, the slope of the MOM solution is different than the measured response.
The discrepancy between the MOM simulation and the measured response is as large as 7
dB at 9 GHz. Further investigation indicates that a better arrangement of the basis
functions is needed to simulate the coaxial to microstrip transition. This new arrangement,
shown in Figure 2.17, will be referred to as the region 3 configuration. By using the
region 3 configuration in the vicinity of the source, it has been found that very good
agreement between the simulated and measured results can be obtained. For a 1.27 mm
thick Duroid 6010 substrate, a 1.4 mm wide transmission line near the source should be
divided up into
= 4 sub-sections along its length and Nys = 5 sub-sections along its
width. Rooftop functions are used to expand the current on the shaded areas shown in
Figure 2.17. However, in the cross-hatched regions, the current is not modeled. Thus, no
expansion functions are necessary. The optimum length of the source region, Ls, was
determined by trial and error to be 1.5 mm.
Figure 2.18 compares the transmission response of the large gap in the high Q
package using the MOM with the region 3 configuration versus the measured results. The
MOM was performed using 56 expansion functions. Agreement between the MOM with
the region 3 configuration and the measurements is excellent
As a second example, consider a transmission line with a single shunt open circuit
stub attached as shown in Figure 2.19. The stub, located at xc = 15 mm (a/2), has a length
48
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 12 mm
•20
Without region 3
Measured
■30
-40
9
11
10
12
F (G H z)
Figure 2.16
Comparison of the predicted transmission response of the large gap in the
high Q package using the MOM without the region 3 configuration versus
the measured results.
49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 2.17
The region 3 configuration of basis functions used to simulate the coaxial to
microstrip transition.
50
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 12 mm
o
-10
-20
With region 3
Measured
-30
-40
9
10
11
12
F (G H z)
Figure 2.18
Comparison of the predicted transmission response of the large gap in the
high Q package using the MOM with the region 3 configuration versus the
measured results.
51
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
b
Figure 2.19
Geometry of a transmission line with a single shunt open circuit stub
attached. The stub, located at x c = 15 mm (a/2), has a length L = 1.9 mm
and is attached to a transmission line, located a ty c = 24 mm (b/2), of width
w = 1.4 mm.
52
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L = 1.9 mm and is attached to a transmission, located a ty c = 24 mm (fc/2), of width w =
1.4 mm. The stub is enclosed in a cavity o f the following dimensions: a = 30 mm, b = 48
mm and c = 10.0 mm. The substrate thickness is d i = 1.27 mm and the relative
permittivity is £r l = 10.5(l-_/0.0023). Since this cavity contains only a small amount of
loss it will be referred to as the high Q package.
Figure 2.20 compares the transmission response of the stub in the high Q package
using the MOM without the region 3 configuration versus the measured results. The MOM
was performed using 227 expansion functions. Agreement between the MOM and the
measured is terrible. Next, the region 3 configuration that was developed for the large gap
(Ls = 1 .5 mm,
= 4 and Nys = 5) is employed to better model the coaxial to microstrip
transition. Figure 2.21 compares the transmission response of the stub in the high Q
package using the MOM with the region 3 configuration versus the measured results. The
MOM was performed using 179 expansion functions. Agreement between the MOM with
the region 3 configuration and the measured is very good.
2 .13 .2 Transition Model For The Other Substrates Used in This Dissertation
When developing the transition model for the 1.27 mm thick Duroid 6010 it was
determined that good agreement between the simulated and measured results was obtained
with
= Xg/ 25, Ays ~ A ^/20,
= 4 and Nys = 5 where Ls = N ^ A ^ and
Ays = W / N ys. However, it was discovered that satisfactory agreement between the
simulated and measured results can be obtained with
only necessary to use
= 2 and A ^ ~ A.g/15. It was
- 4 and A ^ = A ^/25 to fine tune the simulated results to the
measured results. However, for the remaining substrates used in this dissertation no
measured results are available. Therefore, for these substrates the following criteria will be
used:
= 2, A ^ = A^/15 and A ys ~ A ^/20. Using the above criteria, the transition
53
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High Q
Yc = 24 mm
-10
-20
-30
■ Without region 3
‘ Measured
40
9
10
11
12
13
F (G H z)
Figure 2.20
Comparison of the predicted transmission response of the stub in the high Q
package using the MOM without the region 3 configuration versus the
measured results.
54
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 24 mm
-10
a
■c
-20
-30
" With region 3
‘ Measured
^0
9
10
11
12
13
F (G H z)
Figure 2.21
Comparison of the predicted transmission response of the stub in the high
package using the MOM with the region 3 configuration versus the
measured results.
55
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
models employed for those substrates are given in Table 2.2. Table 2.2 also summarizes
the model used for 1.27 mm thick Duroid 6010. Note, only the transition model for the
1.27 mm thick Duroid 6010 substrate has been verified experimentally.
Table 2.2
Summary of the transition models employed for all substrates used in this dissertation.
2.8
Material
£r
Thickness (mm)
Ls (mm)
W (mm)
Duroid
10.5
1.27
1.5
1.4
4
5
Duroid
10.5
0.635
1.5
0.64
2
3
Quartz
4.5
0.127
0.498
0.24
2
3
GaAs
12.9
0.1
0.18
0.1
2
3
Nys
Conclusion
In this chapter the full-wave analysis of packaged MMICs was discussed. The
derivation of the spectral Green's function using and equivalent transmission line was
presented. Following this was a discussion of locating the resonances o f a dielectric loaded
cavity. Next a review of the method of moments (MOM) was given and an acceleration
technique for the efficient evaluation of MOM impedance matrices was presented. The
solution for the unknown current distribution under the condition that each port is
terminated in its characteristic impedance was also given. Finally, the various basis
functions used to expand the currents were discussed. A special basis function
arrangement was developed to more accurately model the coaxial to microstrip transition.
56
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Chapter 2 References
[1]
Harrington, R.F., Time harmonic electromagnetic fields. McGraw-Hill Book
Company, New York, 1961.
[2]
Felsen, L.B. and Marcuvitz, N., Radiation and scattering of waves. Prentice-Hall,
Inc., New Jersey, Chapter 2, 1973.
[3]
Itoh, T., "Spectral domain immitance approach for dispersion characteristics of
generalized printed transmission lines," IEEE Transactions on Microwave Theory
and Techniques, vol. MTT-28, pp. 33-736, July 1980
[4]
Collin, R.E., Foundations for microwave engineering. McGraw-Hill Book
Company, New York, 1966.
[5]
Ramo, S., Whinnery, J. and Van Duzer, T., Fields and waves in communications
electronics. John Wiley and Sons, New York, 1984.
[6]
Beringer, R. "Resonant cavities as microwave circuit elements". In Montgomery,
C.G., Dicke, R.H., and Purcell, E.M. (eds) Principles of microwave circuits. MIT
Radiation Laboratory Series, Vol. 8., McGraw-Hill Book Company, Inc., New
York, pp. 207-239, 1948.
[7]
Kajfez, D., Notes on Microwave Circuits Volume 1. Kajfez Consulting,
University, MS, 1984.
[8]
Rautio, J.C. and Harrington, R.F., "An electromagnetic time-harmonic analysis of
shielded microstrip circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT-35,
pp. 726-730, August 1987.
[9]
Dunleavy, L.P. and Katehi, P.B., "A generalized method for analyzing shielded
thin microstrip discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT37, pp. 1758-1766, December 1988.
57
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[10]
Rautio, J.C., "A time-harmonic electromagnetic analysis of shielded microstrip
circuits," Ph.D. Thesis, Syracuse University, Syracuse, NY, 1986.
[11]
Dunleavy, L.P., "Discontinuity characterization in shielded microstrip: a theoretical
and experimental study," Ph.D. Thesis, University of Michigan, 1988.
[12]
Katehi, P.B., "Radiation losses in mm-wave open microstrip filters,"
Electromagnetics, Vol. 7, pp. 137-152,1987.
[13]
Jackson, R.W. and Pozar, D.M., "Full-wave analysis of microstrip open-end and
gap discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT-33, pp.
1036-1042, October 1985.
[15]
Katehi, P.B. and Alexopoulos, N.G., "Frequency-dependent characteristics of
microstrip discontinuities in millimeter-wave integrated circuits," IEEE Trans.
Microwave Theory Tech., Vol. MTT-33, pp. 1029-1035, October 1985.
[16]
Hill, A. and Tripathi, V.K., "An efficient algorithm for three-dimensional analysis
of passive microstrip components and discontinuities for microwave and millimeterwave integrated circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT-39,
pp. 83-91, January 1991.
[17]
Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions. Dover.,
New York, 1972.
[18]
Jansen, R.H. and Sauer, J., "High-speed 3D electromagnetic simulation for
MIC/MMIC cad using the spectral operator expansion (soe) technique," 1991 IEEE
MTT-S Digest, pp. 1087-1090, 1991.
[19]
Jansen, R.H., "Recent advances in the full-wave analysis of transmission lines for
the application in MIC and MMIC design," 1987 SBMO International Microwave
Symposium Digest, pp. 467-475, 1987.
58
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[20]
Jansen, R.H., "Modular source-type 3D analysis of scattering parameters for
general discontinuities, components and coupling effects in (M)MICs," 17th
European Microwave Conf. Proc., pp. 427-432, 1987.
[21]
Mosig, J.R.and Gardiol, F.E., "General integral equation formulations for
microstrip antennas and scatters," Proc.Inst. Elec. Eng., Part H, Vol. 132, pp.
425-432, December 1985.
[22]
Stinehelfer, H.E., "Discussion of de-embedding techniques using time-domain
analysis," IEEE Proceedings, Vol. 74, No. 1, pp. 90-94, Jan. 1986.
[23]
Franzen, N.R. and Speciale, R.A., "A new procedure for system calibration and
error removal in automated s-parameter measurements," 5th European Microwave
Conf. Proc., pp. 69-73, 1975.
[24]
Rautio, J.C., "A de-embedding algorithm for electromagnetics," International
Journal of Microwave and Millimeter-Wave Computer-Aided Engineering, Vol. 1,
No. 3, pp. 282-287, July 1991.
59
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CH APTER 3
RESO NANT M ODE CO UPLING IN PACK AG ED M M IC s: A
TH EO R ETIC A L A N D EX PER IM EN TA L IN V E STIG A TIO N
3.1
Introduction
A number of full-wave techniques have been developed for the very accurate
modeling of microstrip structures [1] - [13]. However, many o f these techniques neglect
the effect o f enclosing the circuit in a package. This has not been a very serious problem in
current designs because circuit packages have not been electrically large. For sufficiently
large enclosures package resonances are possible. If the system operates at frequencies
near one of these resonances, catastrophic coupling can occur between different elements of
a circuit. Although [6], [7], [11] model a microstrip circuit in an enclosure and analyze at
least one circuit with a resonant mode, none of them specifically examine the causes of
resonant mode coupling and ways to reduce i t
In this chapter the coupling of power to resonant modes and its effect on a circuits'
performance will be discussed. In the first section, the accuracy of the full-wave method of
moment (MOM) procedure developed in Chapter 2 will be verified. Two groups of circuits
are fabricated, enclosed in a brass cavity, and measured. The measured results are
compared to those obtained with the MOM procedure. Following this, the MOM procedure
is used to examine some of the fundamental aspects of resonant mode coupling. In
addition, methods of reducing this coupling will also be investigated.
60
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3.2
Experimental Verification of the MOM for Circuits in Resonant Enclosures
3.2.1
Introduction
To verify the accuracy of the full-wave method of moment (MOM) procedure
developed in Chapter 2, two groups of circuits are fabricated and measured while enclosed
in a brass cavity with at least one resonant mode occurring in the operating bandwidth. The
effect of adding loss to the enclosure is also examined. The measured results are compared
to those obtained with the MOM procedure. The first group consists of a large gap (~ Xg)
in a transmission line. The second group consists of a shunt open circuit stub attached to a
transmission line. In addition, the second group was measured in two different size
enclosures. The smaller enclosure has one resonant mode, while the larger cavity has five
resonant modes.
The circuits were fabricated on a 1.27 mm thick Duroid 6010 substrate. Duroid
6010 has a relative permittivity o f 10.5 ± 0.25 and a loss tangent o f 0.0023. The circuits
were enclosed in two different size cavities milled out of solid brass. The first cavity,
cavity A, has the following inner dimensions: a = 15 mm, b - 2 4 mm and c = 12.7 mm
(Figure 3.1). An enclosure o f this size has only one resonant mode in the 9-12 GHz band.
This mode, the T M no, was theoretically determined to occur at 10.8129 GHz and has a Q
of 4196 (see Chapter 2). The second cavity, cavity B, has the following inner dimensions:
a = 30 mm, b = 48 mm and c = 10 mm. An enclosure of this size has five resonant modes
in the 9-13 GHz band. The resonant frequency and Q for each mode of the high Q
enclosure is listed in Table 3.1.
61
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T
c
Figure 3.1
Basic geometry of the brass cavity with the following inner dimensions: a, b
and c.
62
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Table 3.1
The theoretical resonant frequency and Q for each mode in cavity B.
Mode
TM130
TM]40
TM210
TM220
TM230
fr (GHz)
9.7043+10.0089
11.9517+i0.0227
9.5793+i0.0084
10.6661+10.0132
12.1289+i0.0245
Q
5480
2634
5692
4049
2478
Attaching the substrate to the inside of the cavity proved to be very difficult. First,
the substrate had to be carefully cut to size in order to fit into the cavity. Cutting the
substrate too small will leave a large of a gap between the sides of the substrate and cavity.
If the substrate is cut too large it will not lay flat on the bottom of the cavity. This trimming
could result in the circuit not being positioned exactly in the desired location within the
enclosure. The exact location of the circuit in the enclosure was difficult to determine to
within ± 1 mm. In order to insure that a proper connection is achieved between the ground
plane of the substrate and cavity, the cavity is heated on a hot plate and the solder is
allowed to flow while applying pressure to the substrate.
In the experiments that follow, the measured data was obtained using a Hewlett
Packard 8510 network analyzer. A 50-Ohm system is assumed throughout. S M A coaxto
microstrip launches (Omni Spectra OSM 2052-1215-00) were utilized to connect the
circuits inside the cavities to the network analyzer.
3.2.2
Large Gap
As a first example, consider a transmission line with a large gap in the center as
shown in Figure 3.2. A large gap in a transmission line is chosen because of its simplicity
63
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b
Figure 3.2
Geometry of a transmission line with a gap in the center. The transmission
line, located at y c, has a width o f w = 1.4 mm and a gap o f g = 10.5 mm.
64
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and because the ideal response has a very small transmission coefficient (IS211)- However,
it will be shown that the cavity resonances can significantly increase the transmission
coefficient of the circuit. The transmission line, located at yc, has a width of w = 1.4 mm
and a gap o f g = 10.5 mm. The circuit is enclosed in cavity A which was previously
determined to have one resonant mode in the band 9-12 GHz. Since this cavity contains
only a small amount of loss it will be referred to as the high Q cavity A.
Figure 3.3 compares the calculated and measured transmission coefficient (IS2 1 O of
the large gap for two different locations, yc = 12 mm and 5 mm, in the high Q cavity A.
The MOM simulation was performed using 56 expansion functions. When the circuit is
located aty c = 12 mm, agreement between the calculated and measured results is very good
across the entire bandwidth except for a slight shift in the peak of IS2 1 I as shown in Figure
3.3a. When the circuit is located aty c = 5 mm, agreement between the calculated and
measured results also exhibits a slight shift in the peak o f IS2 1 I as shown in Figure 3.3b.
However, above 11.25 GHz the two curves start to deviate from one another. This
discrepancy, as large as 4 dB at 13 GHz, may be due to the circuit not being positioned
exactly aty c = 5 mm. It was mentioned above that the exact location of the circuit in the
enclosure was difficult to determine to within ± 1 mm. Assuming that the circuit is located
0.35 mm closer to the center of the enclosure is not unreasonable under these
circumstances. Therefore, the MOM simulation was repeated with the circuit located at y c
= 5.35 mm instead of y c = 5 mm with the results shown in Figure 3.4. Above 11.7 GHz
the two curves start to deviate from one another. However, moving the circuit from y c = 5
mm to yc = 5.35 mm resulted in better agreement between the simulated and measured
results. It also indicates the sensitivity of the measurement.
To reduce the effect of the resonant mode, a 1.27 mm thick microwave absorbing
layer, characterized by Br = 60(1-70.12) and |ir = 7.3(l-7'0.3), was attached to the cover of
the enclosure. The characterization of the absorbing material is described in Appendix C.
65
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 12 mm
-10
oa
■o
w
-20
MOM
Measured
-30
-40
9
11
10
12
F (G H z)
(a)
Figure 3.3
Comparison of the calculated and measured transmission coefficient of the
large gap in the high Q cavity A for two different locations: (a) y c = 12 mm
and (b) yc = 5 mm.
Continued, next page.
66
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High Q
Yc = 5 mm
•20
MOM
Measured
-40
9
10
11
12
F (G H z)
(b)
Figure 3.3
Continued.
67
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 5.35 mm
o
-10
-20
-30
MOM
Measured
-40
9
11
10
12
F (G H z)
Figure 3.4
Comparison o f the calculated and measured transmission coefficient o f the
large gap located at y c = 5 mm in the high Q cavity A. The MOM
simulation was performed with the circuit located at y c = 5.35 mm.
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Cavity A with this absorbing layer attached to the cover again has only one resonant mode,
the T M no, which was theoretically found to occur at 10.888 + y'0.285 GHz and has a Q of
19. Since the Q of this cavity is very small, it will be referred to as the low Q cavity A.
Figure 3.5 compares the calculated and measured transmission coefficient (IS2 1 O of
the large gap for two different locations, yc = 12 mm and 5 mm, in the low Q cavity A.
When the circuit is located a ty c = 12 mm, Figure 3.5a shows that agreement between the
calculated and measured results is excellent across the entire bandwidth. When the circuit
is located aty c = 5 mm, Figure 3.5b shows that the calculated and measured results again
exhibits about a 1 dB deviation across the entire bandwidth. As noted above, the
sensitivity was tested by repeating the MOM simulation with the circuit located at yc = 5.35
mm instead of y c = 5 mm. Figure 3.6 compares the calculated and measured transmission
coefficient (IS2 1 O of the large gap. Agreement between the simulated and measured results
is excellent.
Power lost to the package can be very significant for low Q enclosures. The
fraction of the incident power lost to the enclosure is defined to be
T = i W 4 , s = l - I S „ l 2 - I S 2 ll2
(3.1)
In Figure 3.7, 77 is plotted for two different locations of the circuit, y c = 12 mm and y c = 5
mm. The figure shows that when the circuit is located at y c = 12 mm a significant amount
of power is lost to the enclosure in vicinity of 11 GHz. Repositioning the circuit in the
enclosure from yc = 12 mm to y c = 5 mm reduces the power lost to the package by a factor
2 at 11 GHz.
To summarize, the large gap was located at two different positions in cavity A, y c =
12 mm and 5 mm. The resonant mode of the high Q enclosure has a drastic effect on IS2 1 1
of the circuit located at either position. Both circuits have a large transmission coefficient
(IS2 1 I - 0 dB) in the vicinity of 10.8 GHz. Next, a lossy dielectric layer was attached to
the cover of the enclosure. When the circuit is located a ty c = 12 mm in the low Q
69
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Low Q - Dielectric Cover
Yc = 12 mm
-10
/•“k
aa
T3
-20
'
-30
MOM
Measured
-40
9
11
10
12
F (G H z)
(a)
Figure 3.5
Comparison of the calculated and measured transmission coefficient of the
large gap in the low Q cavity A for two different locations: (a) y c = 12 mm
and (b) yc = 5 mm.
Continued, next page.
70
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Low Q - Dielectric Cover
Yc = 5 mm
-10
-20
* MOM
' Measured
-30
-40
9
11
10
12
F (G Hz)
(b)
Figure 3.5
Continued.
71
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Q - Dielectric Cover
Yc = 5.35 mm
-10
/-V
a
TS
-20
N
V.1
- MOM
■ Measured
-30
-40
9
10
11
12
F (G H z)
Figure 3.6
Comparison of the calculated and measured transmission coefficient of the
large gap located at yc = 5 mm in the low Q cavity A. The MOM simulation
was performed with the circuit located aty c = 5.35 mm.
72
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Q - Dielectric Cover
1.0
0.8
u
B
’ Y c= 12 mm
• Yc = 5 mm
0.6
C
A
M
i
0.4
0.2
0.0
9
11
10
12
F (G H z)
Figure 3.7
Computed rj for the large gap in the low Q cavity A for two different
locations: y c = 12 mm and y c = 5 mm.
73
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
enclosure, the effect of the resonant mode is much smaller than in the high Q enclosure.
Repositioning the circuit from y c = 12 mm to yc = 5 mm results in a 7 dB improvement in
the isolation over most of the bandwidth. In addition to a smaller transmission coefficient,
the circuit located atyc = 5 mm has less power lost to the package than the circuit located at
yc = 12 mm.
3.2.3
Shunt Stub
As a second example, consider a transmission line with a single shunt open circuit
stub attached as shown in Figure 3.8. The stub, located at xc = a ll , has a length L = 1.9
mm and is attached to a transmission line, located atyc, of width w = 1.4 mm. A shunt
stub is chosen because it is a typical building block used in more complicated circuits. The
ideal theoretical response for this circuit has a single null in IS2 1 I at 11.1 GHz as shown in
Figure 3.9. The ideal response was obtained by enclosing the circuit in cavity with no
resonant modes below 13 GHz. The shunt stub was measured in the enclosures cavity A
and cavity B.
3.2.3.1 Cavity A
To examine the effect that one mode has on the response of the shunt stub, the
circuit is located in cavity A at two different positions (yc = 12 mm and 5 mm). Figure
3.10 compares the calculated and measured transmission coefficient (IS2 1 O of the circuit in
the high Q cavity A. The MOM simulation was performed using 123 expansion functions.
When the circuit is located aty c = 12 mm, the response of the stub predicted by the MOM
has the same characteristics and is qualitatively similar to the measured results (Figure
3.10a). The first null in IS2 1 I is shifted down in frequency by about 0.2 GHz. When the
74
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
b
Figure 3.8
Geometry of a transmission line with a single shunt open circuit stub
attached. The stub, located at x c {all), has a length L = 1.9 mm and is
attached to a transmission line, located aty c, of width w = 1.4 mm.
75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Id eal
o
-10
-20
-30
-40
9
11
10
12
F (G H z)
Figure 3.9
Ideal theoretical response of the shunt open circuit stub.
76
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 12 mm
IS21I (dB)
-10
-20
-30
- MOM
• Measured
-40
9
11
10
12
F (GH z)
(a)
Figure 3.10
Comparison of the calculated and measured transmission coefficient of the
shunt stub in the high Q cavity A for two different locations: (a) yc = 12 mm
and (b) y c = 5 mm.
Continued, next page.
77
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 5 mm
-10
09
■O
-20
M
cc
'
-30
MOM
Measured
-40
9
10
11
12
F (G H z)
(b)
Figure 3.10
Continued.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
circuit is located at yc = 5 mm, agreement between the calculated and measured results also
exhibits a slight shift of the null in IS2 1 I (Figure 3.10b). In addition to the frequency shift
o f IS2 1 I, the simulated response does not exhibit the rapid variation of IS2 1 I in the vicinity
of 10.9 GHz that the measured response does. The discrepancy between the simulated and
measured results may be due to one or more of the following:
1) The uncertainty in circuit dimensions.
2) The uncertainty in dielectric constant
3) The uncertainty in determining the exact location of the circuit in the
enclosure.
To determine if the desired circuit dimensions were correct, a careful measurement
to within ± 0.025 mm, for each circuit was performed. For the circuit located a ty c = 12
mm, the actual length of the stub was measured to be 1.8 mm instead of 1.9 mm. If the
length of the stub was altered in the etching of the circuit one would also expect for other
the circuit dimensions to be effected. However, all of the other dimensions were
unchanged. Careful examination of the photographic mask used in the etching process,
revealed that the error in the length of the stub occurred in the making of the mask and not
in the etching. For the circuit located at yc = 5 mm, this error was not made in the
photographic mask and all circuit dimensions were measured to be correct. Figure 3.11
compares the calculated and measured transmission coefficient (IS2 1 O of the shunt stub
circuit located aty c = 12 mm in the high Q cavity A. The calculated response shown in
Figure 3.11 was performed for a shunt stub of length 1.8 mm located a ty c = 12 mm. The
response of the stub predicted by the MOM has the same characteristics and is qualitatively
similar to the measured results. However, the second null in IS2 1 I is shifted up in
frequency by about 0.1 GHz. Comparing Figure 3.11 to 3.10a shows that reducing the
length of stub in the simulation gives slightly better results overall.
79
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 12 mm
-10
ap)
iiz s i
-20
-
" MOM
• Measured
-30
^0
9
11
10
12
F (G H z)
Figure 3.11
Comparison of the calculated and measured transmission coefficient of the
shunt stub circuit of length L - 1.9 mm located aty c = 12 mm in the high Q
cavity A.
80
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
As previously stated, the second source of error may be due to the uncertainty in the
permittivity. The manufacture's quoted value for the permittivity of Duroid 6010 is 10.5 ±
0.25. Therefore, the MOM simulations were repeated with the permittivity varying from
10.25 to 10.75. The optimal value of the permittivity was determined to be 10.75;
however, this resulted in only a slight improvement.
In the previous example (large gap), it was shown that the MOM simulations are
very sensitive as to the exact location o f the circuit in the enclosure. It has been determined
that the best agreement between the simulated and measured results occurs when the
location of the circuit in the enclosure is slightly different than the stated value of yc (12 —>
11.45 mm and 5 —» 5.35 mm). A slight improvement was shown by relocating the circuit
Using the knowledge gained from examining the three sources of errors, the MOM
simulations were repeated. For the circuit positioned at y c = 12 mm, the MOM simulation
was performed with yc = 11.45 mm, L = 1.8 mm and s r =10.75. As shown in Figure
3 .12a, agreement between the calculated and measured results is very good. For the circuit
positioned a iy c = 5 mm, the MOM simulation was performed with yc = 5.35 mm, L = 1.9
mm and £r =10.75. Figure 3.12b shows that agreement between the simulated and
measured results is very good. The MOM solution no longer exhibits a slight shift of the
null in IS2ll and it does exhibit the rapid variation of IS2 1 I in the vicinity of 10.9 GHz.
In the previous section it was shown that attaching a 1.27 mm thick microwave
absorbing layer to the cover of the enclosure can reduce the effect of the resonant mode.
Figure 3.13 compares the calculated and measured transmission coefficient (IS2 1 O of the
shunt stub circuit for two different locations, yc = 12 mm and 5 mm, in the low Q cavity A.
For the circuit positioned aty c = 12 mm, the MOM simulation was performed with y c =
11.45 mm, L = 1.8 mm and er =10.75. As shown in Figure 3 .13a, agreement between the
simulated and measured results is very good except in the vicinity of the first null, where
the simulated response is much deeper. For the circuit positioned at y c = 5 mm, the MOM
81
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 11.45 mm
IS21I (dB)
-10
-20
'
-30
MOM
Measured
-10
9
11
10
12
F (GHz)
(a)
Figure 3.12
Comparison of the calculated and measured transmission coefficient of the
shunt stub in the high Q cavity A for two different locations: (a) yc = 12 mm
and (b) yc = 5 mm. For the circuit positioned at yc = 12 mm, the MOM
simulation was performed with yc = 11.45 mm, L = 1.8 mm and er
=10.75. For the circuit positioned at yc = 5 mm, the MOM simulation was
performed with yc = 5.35 mm, L = 1.9 mm and er =10.75.
Continued, next page.
82
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 5.35 mm
-10
-
-20
'
-30
MOM
Measured
-40
9
11
10
12
F (G H z)
(b)
Figure 3.12
Continued.
83
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Q - Dielectric Cover
Yc = 11.45 mm
-10
-20
MOM
Measured
-30
-40
9
10
11
12
F (G H z)
(a)
Figure 3.13
Comparison of the calculated and measured transmission coefficient of the
shunt stub in the low Q cavity A for two different locations: (a) y c = 12 mm
and (b) y c = 5 mm.
Continued, next page.
84
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Q - Dielectric Cover
Yc = 5.35 mm
-10
aa
-20
</}
- MOM
■ Measured
-30
-40
9
11
10
12
F (G H z)
(b)
Figure 3.13
Continued.
85
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
simulation was performed with y c = 5.35 mm, L = 1.9 mm and &r =10.75. Figure 3.13b
shows that agreement between the simulated and measured results is very good except in
the vicinity of the null, where the simulated response is again much deeper. This
discrepancy may be due to the finite conductivity of the microstrip line not being included
in the MOM calculation.
Power lost to the package can be very significant for low Q enclosures. In Figure
3.14, 77 is plotted for two different locations of the circuit, y c = 12 mm and y c = 5 mm.
The MOM simulation was performed with L = 1.9 mm and er =10.5. The figure shows
that when the circuit is located at yc = 12 mm a significant amount of power is lost to the
enclosure from 10.5 GHz to 11.75 GHz. Repositioning the circuit in the enclosure from
yc = 12 mm to yc = 5 mm significantly reduces the power lost to the package.
In order to verify Figure 3.14, reflection measurements were made for the circuit
located aty c = 12 mm and 5 mm. Combining the reflection measurements with the
transmission measurements done earlier, the fraction of incident power lost to the enclosure
was determined. Figures 3.15a and 3.15b plots 77 calculated from these measurements and
the MOM simulation for the circuit located at yc = 12 mm and yc = 5 mm, respectively.
For the circuit positioned at y c = 12 mm, the MOM simulation was performed with y c =
11.45 mm, L = 1.8 mm and £r =10.75. For the circuit positioned at y c = 5 mm, the MOM
simulation was performed with y c = 5.35 mm, L = 1.9 mm and er =10.75. W hen the
circuit is positioned atyc = 12 mm, agreement between the MOM simulation and the
measurements is very good. When the circuit is positioned at yc = 5 mm, agreement
between the MOM simulation and the measurements is very good up to 10.8 GHz. Above
10.8 GHz the two curves start to deviate from another. Since 77 is relatively small and
there is a large standing wave current, this discrepancy may again be due to the finite
conductivity of the microstrip line not being included in the MOM calculation.
86
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Q - Dielectric
Cover
1.0
0.8
u
s
(A
o
eu
■ Yc = 12 mm
' Yc = 5 nun
0.6
C/3
0.4
0.2
0.0
9
10
11
12
F (G H z)
Figure 3.14
Computed T] for the shunt stub in the low Q cavity A for two different
locations: y c = 12 mm and yc = 5 mm. The MOM simulation was
performed with L = 1.9 mm and er =10.5
87
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Q - Dielectric Cover
Yc = 11.45 mm
1.0
Ploss/Pinc
0.8
- MOM
’ Measured
0.6
0.4
0.2
-
0.0
9
11
10
12
F (G H z)
(a)
Figure 3.15
Comparison of the calculated and measured tj for the shunt stub in the low
Q cavity A for two different locations: yc = 12 mm and y c = 5 mm.
Continued, next page.
88
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Low Q - Dielectric Cover
Yc = 5.35 mm
1.0
0.8
w
s
2
V
i
VI
£
- MOM
• Measured
0.6
0.4
0.2
0.0
9
10
11
12
F (G H z)
(b)
Figure 3.15
Continued.
89
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To summarize, when the shunt open circuit stub circuit was located aty c = 12 mm
in the high Q cavity A, the resonant mode had a drastic effect on IS2 1 I. Repositioning the
circuit from yc = 12 mm to yc = 5 mm significantly reduced the effect of the resonant
mode. Next, a lossy dielectric layer was attached to the cover of the enclosure. For the
circuit located at yc = 12 mm in the low Q enclosure, the effect of the resonant mode was
much smaller than in the high Q enclosure. However, the resonant mode of the enclosure
still had a large effect on IS2 1 I. The circuit located atyc = 5 mm had very little power lost
to the package in addition to a transmission coefficient similar to the ideal response.
3.2.3.2 Cavity B
To examine the effect that several resonant modes have on the response o f the shunt
stub, the circuit is located in cavity B aty c = 24 mm. Figure 3.16 compares the calculated
and measured transmission coefficient (IS2 1 I). The MOM simulation was performed using
179 expansion functions. Very good agreement is obtained except for minor discrepancies
in the vicinity of the resonant modes.
To reduce the effect of the resonant modes two different materials were used to
lower the Q of the enclosure. The first material employed is a microwave absorbing layer
attached to the cover of the enclosure. The absorbing layer is 0.762 mm thick with a
permittivity of er = 60(l-_/0.12) and permeability of (ir = 7.3(l-y'0.3). Cavity B with this
absorbing layer attached to the cover will be referred to as the low 2-dielectric cover cavity.
The resonant frequency and Q for each mode of the low Q-dielectric cover is listed in Table
3.2.
90
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 24 mm
IS21I (dB)
-10
-20
- MOM
• Measured
-30
^0
9
10
11
12
13
F (GH z)
Figure 3.16
Comparison of the calculated and measured transmission coefficient of the
shunt stub in the high Q cavity B.
91
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 3.2
The resonant frequency and Q for each mode in the low 2-dielectric cover cavity B.
Mode
TM 140
TM230
f r (GHz)
11.845 l+j0.5578
12.0632+i0.5730
Q
1 0 .6
10.5
Figure 3.17 compares the calculated and measured transmission coefficient (IS2 1 O
of the shunt stub circuit in the low 2-dielectric cover cavity B. The figure shows that
reasonable agreement is obtained between the calculated and measured results.
The second material employed to reduce the effect of the resonant modes is a doped
silicon layer attached to the cover of the enclosure. The silicon layer is 16 mil (0.4064 mm)
thick with a resistivity o f 1.45 £2-cm. The resistivity of the silicon wafer was measured
using the 4 point probe method. Cavity B with this doped silicon layer attached to the
cover will be referred to as the low 2 -S i cover cavity. The resonant frequency and Q for
each mode of the low 2 -S i cover cavity is listed in Table 3.3.
Table 3.3
The resonant frequency of each mode in the low 2-S i cover cavity B.
Mode
TM 130
TM 140
TM 210
TM 220
TM230
fr (GHz)
9.5383+i0.0438
11.8080+i0.0477
9.4145+i0.0423
10.5052+10.0460
11.9908+i0.0451
Q
109.0
123.9
111.4
114.3
132.8
92
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Q - Dielectric Cover
Yc = 24 mm
IS21I (dB)
-10
-20
- MOM
• Measured
-30
-40
9
10
11
12
13
F (G H z)
Figure 3.17
Comparison of the calculated and measured transmission coefficient of the
shunt stub in the low (7-dielectric cover cavity B.
93
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Figure 3.18 compares the calculated and measured transmission coefficient (IS2 1 I)
of the shunt stub circuit in the low g-S i cover cavity B. The figure shows that good
agreement is obtained between the calculated and measured results.
As mentioned previously, 77 is a very important figure-of-merit for circuits
enclosed in low g packages. Figures 3.19 plots 77 for the circuit enclosed in the dielectric
and Si cover cavities. A significant amount of power for the circuit enclosed in the low g dielectric cover cavity. When the circuit is enclosed in the low g-Si cover cavity a
significant amount of power is lost to the enclosure in the vicinity of the resonant modes
only.
In order to verify Figure 3.19, reflection measurements were made for the circuit
enclosed in both low g cavities. Combining the reflection measurements with the
transmission measurements, the fraction of power lost to the enclosure was determined.
Figure 3.20 plots 77 calculated from these measurements and the MOM simulation. When
the circuit is enclosed in the low g dielectric cover cavity B, the agreement between the
simulated and measured results is satisfactory as shown in Figure 3.20a. Figure 3.20b
shows that when the circuit is enclosed in the low g-Si cover cavity B reasonable
agreement exists between the simulated and measured results.
To summarize, when the circuit is enclosed in the high g cavity B the resonant
mode of the enclosure had a significant effect on iS2il. A similar effect on IS2 1 I was also
observed when the circuit was enclosed in cavity A. The lossy dielectric layer appeared to
significantly reduce the effect of the resonant modes on the transmission coefficient.
However, examining 77, shows that a significant amount of power is lost to the enclosure
over a large portion of the bandwidth. Examining 77 for the doped silicon layer, shows
that a significant amount of power is lost to the enclosure in the vicinity of the resonant
modes only. However, the resonant modes of the enclosure still has a large effect on IS2 1 I-
94
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Low Q - Si cover
Yc = 24 mm
-10
/■“•s
as
■o
-20
ce
'
-30
MOM
Measured
-40
9
10
11
12
13
F (G H z)
Figure 3.18
Comparison o f the calculated and measured transmission coefficient of the
shunt stub in the low <2-Si cover cavity B.
95
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Q - Yc = 24 mm
1.0
Dielectric cover
Si cover
0.8
u
e
0*
W
£
0.6
0.4
0.2
0.0
9
10
11
12
13
F (GH z)
Figure 3.19
Computed r\ for the shunt stub enclosed in the dielectric and Si cover
cavities.
96
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Q - Dielectric cover
Yc = 24 mm
1.0
- MOM
• Measured
0.8
u
s
0.6
0.4
0.2
0.0
9
10
11
12
13
F (G H z)
(a)
Figure 3.20
Comparison of the calculated and measured 77 for the shunt stub enclosed in
the: (a) low Q dielectric cover cavity B and (b) low Q-Si cover cavity B.
Continued, next page.
97
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Low Q - Si cover
Yc = 24 mm
1.0
0.8
MOM
Measured
0.6
0.4
0.2
0.0
9
10
11
12
13
F (G H z)
(b)
Figure 3.20
Continued.
98
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3.3
Parasitic Coupling to Resonant Modes
In the previous section it was observed, experimentally and theoretically, that the
location of a circuit in an enclosure can effect the coupling of power to a resonant mode. In
order to gain a better understanding of this phenomenon, the effect that resonant mode
coupling has a discontinuity in a circuit on will be investigated. At a discontinuity, power
is radiated in the form of surface waves. The surface waves propagate parallel to the
substrate away from the circuit over a wide range of angles [1], [2]. This radiated power is
reflected at the side walls of the enclosure which can then recombine with the circuit at
another location. This effect can be catastrophic because it creates additional links between
parts of a circuit that were not meant to be coupled to one another [1], [2], [3]. In addition,
the reflected power will modify the standing wave current which will in turn modify the
power radiated in the form of surface waves and so on. At frequencies in the vicinity of a
package resonance, the fields that comprise the resonant mode become extremely large.
The larger the Q of the enclosure, the larger the fields associated with a resonance become.
Therefore, the resonant mode can drastically alter the standing wave current of the circuit
which can result in a drastic change in the scattering parameters of the circuit.
An important quantity for characterizing microwave circuits in a package is the
amount o f power lost to a resonant mode. The power lost to a resonant mode can be
expressed as:
C ° de = - | R e / j £ rmode(x,y)-7;(x,y)d!xrfy
(3.2)
where /)™°de is the power lost to the resonant mode, £ rmode is the tangential component of
the resonant mode electric field and Js is the surface current on the microstrip line. The
surface current on the microstrip line is determined by the full-wave MOM procedure
developed in Chapter 2.
99
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In a typical enclosure housing an MMIC chip, the cover height is low enough that
only the TM nmfl modes are resonant over the frequency of operation. For this reason,
only the TM mode will be considered in the following analysis. However, the analysis
could easily be applied to the TE mode. For a TM mode, the tangential component of the
resonant mode electric field is given by
E ? odeU y ) = E ™ (x ,y ) = x E ™ (x ,y ) + y E ™ (x ,y )
(3.3)
Substituting equation (3.3) into equation (3.2) yields
n m o d e pT M
pTM , pTM
H oss
~ H oss ~ r x
+ ry
W -* )
where
P™ = “
Re j j E ™ (x ,y )J * (x ,y )d x d y
(3.5a)
P ™ = - j R e JJ E ™ (x,y)Jy (x ,y )d x d y
(3.5b)
Equation (3.4) describes the total power lost to a resonant mode and equation (3.5)
describes the power lost to a resonant mode via the components of the resonant mode field.
However, equations (3.4)and (3.5) do not describe what regions o f a circuit couple more
strongly to the resonant mode.In addition, a circuit in a high Q enclosure will only lose a
small amount o f power to a resonant mode even though the resonant mode has a drastic
effect on the circuit Therefore, an additional measure o f resonant mode coupling is
needed. Equation (3.5) can be rewritten as:
P ™ = jp ™ ( x ) d x
(3.6a)
P ™ = ]p ™ (y )d y
(3.6b)
P™ (x) = - i - R e J E™{x,y)J*x (x ,y )d y
(3.7a)
P ™ 0 0 = ~ R e J E™(.x,y)J*y {x,y)d x
(3.7b)
where
In Chapter 2 it was shown that the tangential component of the electric field can be
expressed as:
100
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E ™ (jc, y) = A ™ cos(kxnx)sin(kymy)
(3.8a)
E ™ (x,y) = A$m sin(kxnx)cos(kymy)
(3.8b)
where
A ™ = ~V b ^ t ^ kxnJx{kxn' kym) +
)}
A ™ =
(3.9a)
( 3 -9 b )
( k xrt ’ k ym ) —
(x, y) c o s ik ^ x ) sin (kymy) d xd y
(3.1 Oa)
(x, y) sin(kx„x) cos (kymy) d xd y
(3.1 Ob)
s
J y ( k xn ’ k ym )
s
n and m correspond to the indices of the resonant mode and YM is defined by equation
(2.43). Inspection of equations (3.7), (3.8), (3.9) and (3.10) indicates that p ™ (x ) and/or
p ™ (y) can reduced by locating areas of high current in areas of low electric field and visaversa.
For example, consider a circuit where the current is predominantly x-directed and
centered at y = yc, it can be shown that J x ( k ^ , kym) is given by
Jx(kxn’kym) = Fx (kxn, kym) sin(kymyc)
(3.11)
Substituting equations (3.8a), (3.9a) and (3.11) into equation (3.7a), p ™ (x) can be
written as:
P ™ M - (p ™ )max sin2{kymyc)
(3.12)
Thus p ™ (x) can be reduced by changing y c- Note, reducing p ™ (x ) will ultimately
result in a reductionin P™ . Positioning the circuit in the center of the box will result in a
maximum power loss for the T M n o (kyi = n/b) resonance and a minimum power loss for
the TM 120 (ky2 = 2iz/b) resonance.
To develop a better understanding of resonant mode coupling, the two circuits
discussed in the previous section will be re-examined. The first circuit consists o f a large
gap (~ Xg) in a transmission line. The second group consists of a shunt open circuit stub
101
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
attached to a transmission line. Both circuits are enclosed in cavity A which was described
in the previous section.
3.3.1
Large Qap
As a first example, again consider the transmission line with a large gap in the
center as shown in Figure 3.2. The transmission line, located at yc, has a width o f w = 1.4
mm and a gap of g = 10.5 mm. The circuit is located in cavity A at two different positions
(yc = 12 mm and 5 mm).
The ideal response for the large gap is a very small transmission coefficient. As
result, the ideal current on the strip should exhibit a large standing wave on the first strip
and the current on the second strip should be very small. Also, the dominate current on the
circuit is x directed. Thus, power is radiated into the TM n o mode via the x component of
the electric field. Inspection of equations (3.6a), (3.7a) and (3.8a) indicates that moving
the circuit from yc = 12 mm to yc = 5 mm should reduce p ™ (x) and P ™ .
Figure 3.21 summarizes the computed transmission coefficient (IS2 1 Oof the large
gap for two different locations, yc = 12 mm and 5 mm, in the high Q cavity A. The
resonant mode of the high Q enclosure has a drastic effect on IS2 1 I of the circuit located at
either position. Both locations result in a large transmission coefficient (IS2 1 1~ 0 dB) in
the vicinity of 10.8 GHz. To gain a better understanding on how the coupling of power to
the TM i 10 mode effects the operation of the large gap, the current on the two strips at/ =
10.8129 GHz is illustrated in Figure 3.22. Note that the first strip extends from x = 0 mm
to x = 2.25 mm and the second strip extends from x = 12.75 mm to x = 15 mm. The
figure shows that a large and nearly equal current is on both strips for the circuit located in
either position. Figure 3.23 plots p ™ (x), the power lost per unit length to the T M i 10
mode via the x component of the electric field, at/ = 10.8129 GHz. If p ™ (x) is positive,
102
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High Q
ca
■B
■20
Yc = 12 mm
Yc = 5 mm
-40
9
10
11
12
F (G H z)
Figure 3.21
Summary of the calculated transmission coefficient of the large gap in the
high Q cavity A for two different locations: yc = 12 mm and y c = 5 mm.
103
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High Q
20
15
10
Yc= 12 mm
Yc = 5 mm
5
0
0
10
5
15
x (mm)
Figure 3.22
Current, a t / = 10.8129 GHz, on the two strips of the large gap in the high
Q cavity A.
High Q
500
300
Yc = 12mm
Yc = 5 mm
'©
100
X
C
ar
>
-100
E
-300
-500
0
10
5
15
x (mm)
Figure 3.23
Power lost per unit length to the T M i io mode via the x component of the
electric field a t / = 10.8129 GHz for the large gap in the high Q cavity A.
104
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
this indicates that power is being radiated from the T M i 10 mode and if p ™ (x) is
negative, this indicates that power is being absorbed into the T M i 10 mode. The figure
TM
TJUt
shows that p x (x) on the first strip is nearly equal and opposite in magnitude to p x (x)
Tldf
TKA
on the second strip. Thus, Px which is equal to the area under p x (x) is approximately
zero. Inspection of equations (3.6a), (3.7a) and (3.8a) indicates that moving the circuit
TKA
TKA
from yc = 12 mm to yc = 5 mm will reduce px (x) and Px
. Since the circuit is being
operating at the resonant frequency of the high Q cavity A, the TM i io mode electric field is
nearly infinite. Therefore, a reduction in the coupling of power to the T M i io mode is not
observed. However, for frequencies in the vicinity o f the resonant frequency, the
reduction in the coupling of power to the TM i io mode is due to the unloading of the Q of
the resonator.
To reduce the effect of the resonant mode, the large gap is enclosed in the low Q
cavity A. Figure 3.24 summarizes the computed transmission coefficient (IS2 1 Oof the
large gap for two different locations, y c = 12 mm and 5 mm, in the low Q cavity A.
Repositioning the circuit from yc = 12 mm to yc = 5 mm results in a 7 dB improvement in
IS2 1 I over most of the bandwidth. The current on the two strips at/ = 10.8129 GHz is
illustrated in Figure 3.25. The figure shows that when the circuit is located at yc = 12 mm,
the standing wave current on the second strip is larger than if the circuit is located at yc = 5
mm. Therefore, when the circuit is located at yc = 5 mm, the standing wave current on the
circuit is closer to the ideal current. Figure 3.26 plots p ™ (x) a t / = 10.8129 GHz. As
predicted by equations (3.6a), (3.7a) and (3.8a), moving the circuit from y c = 12 mm to yc
= 5 mm reduced p ™ (x) and PXM. An exact calculation of P ™ shown in Table 3.4
verifies this. The third column in Table 3.4 is the fraction of the incident power lost to the
enclosure, /}oss//^nc, including all fields.
105
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Low Q - Dielectric Cover
-10
n
-20
-30
" Yc = 12 mm
’ Yc = 5 mm
-40
9
10
11
12
F (G H z)
Figure 3.24
Summary of the calculated transmission coefficient of the large gap in the
low Q cavity A for two different locations: y c = 12 mm and yc = 5 mm.
106
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Q
20
15
10
Yc = 12 mm
Yc = 5 mm
5
0
0
10
5
15
x (mm)
Figure 3.25
Current, at/ = 10.8129 GHz, on the two strips of the large gap in the low Q
cavity A.
Low Q
500
r'
300
Yc= 12 mm
Yc = 5mm
rl
100
-
©
X
gS
-100 -
-300
-500
0
10
15
x (mm)
Figure 3.26
Power lost per unit length to the TM i io mode via the x component o f the
electric field at/ = 10.8129 GHz for the large gap in the low Q cavity A.
107
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 3.4
The ratio o f the T M n o mode powers to the incident power for the large gap.
y c (mm)
5
12
p T M /p
1x
/ 1 me
0.210
0.366
F \oss! Fine
0.229
0.395
3.3.2 Shunt Stub
As a second example, again consider the transmission line with a single shunt open
circuit stub attached as shown in Figure 3.8. The stub, located at xc = a ll, has a length L =
1.9 mm and is attached to a transmission line, located at yc, of width w = 1.4 mm. The
shunt stub is located in cavity A at two different positions (yc = 12 mm and 5 mm).
The ideal response for this circuit has a single null in IS2 1 I at 11.1 GHz as shown in
Figure 3.9. The dominate currents on the transmission line and on the stub are,
respectively, x and y-directed as illustrated in Figure 3.27. Thus they radiate power into
the T M i 10 mode via different field components. As a result, moving the circuit will
change the power radiated from the stub in a different way than it changes the power
radiated from the transmission line. Since the current on the transmission is predominately
x-directed and centered a ty c, equation (3.10a) can be written as
Fx (^xn ’kym) sm(&.y;j,yc)
The current on the stub is predominately y-directed and centered at x c\ thus, equation
(3 .10b) can be written as
Jy(kxn,kym ) —Fy (kxn, kym) sin(/rj,nxc)
Examining Figure 3.27 shows that for the majority the of circuit the current is x-directed;
therefore, the dominate mechanism for power being radiated into the T M i 1 0 mode is via
108
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Ideal
25
20
15
10
5
0
0
15
10
5
x (mm)
(a)
Ideal
30
25
20
15
10
5
0
0.0
0.5
1.0
1.5
2.0
y-yl (mm)
(b)
Figure 3.27
Ideal currents on the: (a) transmission line and (b) stub.
109
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E ™ . Thus, locating areas of high current along the transmission line in areas o f low E ™
and visa-versa will have a significant effect on the coupling of power to the TM i 10 mode.
Inspection of equations (3.6a), (3.7a) and (3.8a) indicates that moving the transmission
line from yc = 12 mm to yc = 5 mm will reduce p ™ (x) and P jM. On the other hand, the
same shift will only have a small effect on p ™ (y) and P ™ . The net power lost to the
T M i io mode, P ™ plus P ™ , decreases as the circuit is moved toward the side wall.
Figure 3.28 summarizes the computed transmission coefficient (IS2 1 I) of the shunt
stub circuit for two different locations, yc = 12 mm and 5 mm, in the high Q cavity A. The
figure shows that when the shunt open circuit stub circuit was located aty c = 12 mm in the
high Q cavity A, the resonant mode had a drastic effect on IS2 1 I- Repositioning the circuit
from y c = 12 mm to y c = 5 mm significantly reduced the effect of the resonant mode. To
gain a better understanding on how the coupling of power to the TM i 1 0 mode effects the
operation of the shunt stub, the current on the circuit a t / = 11.1 GHz is illustrated in Figure
3.29. Figure 3.29a shows that when the circuit is located at yc = 12 mm, the standing
wave current on the transmission line is significantly different than the ideal current When
the circuit is located aty c = 5 mm, the standing wave current on the transmission line is
nearly identical to the ideal current. A similar effect is observed for the standing wave
current on the stub as illustrated in Figure 3.29b. Figure 3.30 plots p ™ (x), the power
lost per unit length along the transmission line to the TM i 1 0 mode via the x component of
the electric field, at/ = 11.1 GHz, for the circuit located at y c = 12 mm. The figure shows
that the region in the vicinity of x = 2.5 mm is dominated by power being radiated into the
T M n o mode and the region in the vicinity of x = 12.5 mm is dominated by power being
absorbed from the TM i 1 0 mode. This occurs because a large current standing wave peak
is present at that these points and because the x-directed mode field is reasonably large
there. The figure also shows that the area under p ™ (x ), P ™ , is approximately zero.
p ™ (x ) for the circuit located at yc = 5 mm and p ™ (y ) for both locations are not shown
110
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High Q
IS21I (dB
-10
-20
-30 ‘ Yc = 12 mm
• Yc = 5 mm
-40
9
11
10
12
F (G H z)
Figure 3.28
Summary of the calculated transmission coefficient of the shunt stub in the
high Q cavity A for two different locations: yc = 12 mm and yc = 5 mm.
Ill
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 12 min
Yc = 5 mm
0
5
10
15
x (m m )
(a)
High Q
Yc = 12 mm
Yc = 5 mm
1.0
y -y l
(m m )
(b)
Figure 3.29
Currents, a t / = 11.1 GHz, on the: (a) transmission line and (b) stub. The
circuit is enclosed in the high Q cavity A.
112
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High Q
250
£
£
Yc = 12 mm
150
o
X
c
H
a.
-50
-150
-250
0
5
10
15
x (mm)
Figure 3.30
Power lost per unit length along the transmission line to the T M i 10 mode
via the x component of the electric field a t / = 11.1 G H z . The circuit is
enclosed in the high Q cavity A.
113
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because they are approximately zero. Equations (3.6), (3.7) and (3.8) predict that moving
the circuit from y c - 12 mm to yc = 5 mm will reduce the coupling of power to the TM i io
mode. Since the circuit is not being operating at the resonant frequency o f the high Q
cavity, this reduction is observed in 1S21I and in the standing wave current on the circuit.
As in the previous section, a 1.27 mm thick microwave absorbing layer was
attached to the cover of the enclosure to reduce the effect of the resonant mode. Figure
3.31 summarizes the computed transmission coefficient (IS2 1 O of the shunt stub circuit for
two different locations, y c = 12 mm and 5 mm, in the low Q cavity A. For the circuit
located at yc = 12 mm in the low Q enclosure, the effect of the resonant mode was much
smaller than in the high Q enclosure. However, the resonant mode of the enclosure still
had a large effect on IS2 1 I. The current on the circuit at/ = 11.1 GHz is illustrated in
Figure 3.32. The figure shows that when the circuit is located at yc = 12 mm, the standing
wave current on the transmission line and the stub is much different than the ideal current.
Again, when the circuit is located at yc = 5 mm, the standing wave current on the
transmission line and the stub is nearly identical to the ideal current. Figure 3.33a plots
'T'JJf
Px (x ) , the power lost per unit length along the transmission line to the TM i 1 0 mode via
the x component of the electric field, a t / = 11.1 GHz. The figure shows that the region in
the vicinity of x = 2.5 mm is dominated by power being radiated into the T M i 1 0 mode and
the region in the vicinity of x = 12.5 mm is dominated by power being absorbed from the
T'Uf
T M i 1 0 mode. The figure also shows that the area under p x (x) is not zero for the circuit
located aty c = 12 mm, indicating net power lost to the T M i 1 0 mode. For the circuit
T'ljt
located at yc = 5 mm, px (jc) is approximately zero along the transmission line. Figure
'T'JJf
___
3.33b plots p y (y ), the power lost per unit length along the stub to the TM i 1 0 mode via
the y component of the electric field, a t / = 11.1 GHz. The figure shows that for either
location of the circuit, power is being radiated into the T M i 10 mode. The figure also
shows that the area under p ™ (y) for the circuit located at y c = 12 mm is larger than the
114
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Low Q - Dielectric Cover
-10
as
"O
-20
-30 -
■ Yc = 12 mm
• Yc = 5 mm
-40
9
10
11
12
F (G H z)
Figure 3.31
Summary of the calculated transmission coefficient of the shunt stub in the
low Q cavity A for two different locations: yc = 12 mm and y c = 5 mm.
115
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Low Q
25
20
Yc = 12 mm
Yc = 5 mm
15
10
5
0
0
5
10
15
x (mm)
(a)
Low Q
25
<
E
10
Yc= 12 mm
Yc = 5 mm
0.0
0.5
1.0
2.0
y-yl (mm)
(b)
Figure 3.32
Currents, a t / = 11.1 GHz, on the: (a) transmission line and (b) stub. The
circuit is enclosed in the low Q cavity A.
116
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Low Q
250
i
s
£
150
Yc = 12 mm
Yc = 5 mm
•250
0
5
15
10
x (mm)
(a)
Low Q
25
i
£
£
20
«*>
©
Yc = 12 mm
Yc = 5 mm
X
O
0.0
0.5
1.0
1.5
2.0
y-yl (mm)
(b)
Figure 3.33
Power lost per unit length to the TM i io mode via the x component of the
electric field, at/ = 11.1 GHz, (a) along the transmission line and (b) along
the stub. The circuit is enclosed in the low Q cavity A.
117
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circuit located at yc = 5 mm. Equations (3.6a), (3.7a) and (3.8a) predict that moving the
transmission line from y c = 12 mm to yc = 5 mm will reduce P ™ . An exact calculation of
P ™ shown in Table 3.5 verifies this. In addition, the same shift also decreased P ™ .
The fifth column in Table 3.5 is the fraction of the incident power lost to the enclosure,
P\oss/Pinc ’ including all fields. Note that when the circuit strongly couples to the T M no
mode, the total loss is almost equal to P ™ plus P ™ . For weak coupling the total loss is
not dominated by the resonant mode loss.
Table 3.5
The ratio of the TM i io mode powers to the incident power for the stub.
yc (mm)
5
12
p1 x™ lIp.
r mc
-0.0166
0.436
pl y™ /I 1pme
0.0173
0.0316
p1 ™ II 1pme
0.007
0.468
P \oss! Pine
0.077
0.516
The analysis shows that package resonances can have a very significant effect on
circuit operation even at frequencies which are not very close to resonance. These effects,
can be reduced by including lossy material in the enclosure.
A further reduction in the coupling of power to resonant modes can be obtained by
repositioning the circuit Locating areas of high current in areas of low electric field in the
enclosure reduces the power lost to these resonant modes. This principle would find most
application in moderately sized enclosures where the resonant frequencies are not closely
spaced and the mode field structure is relatively simple.
118
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3.4
Conclusion
In this chapter the full-wave method of moment (MOM) procedure developed in
Chapter 2 was experimentally verified. Two groups o f circuits were fabricated, enclosed in
a brass cavity with at least one resonant mode occurring in the operating bandwidth and
measured. The effect of adding loss to the enclosure was also examined. The measured
results for the above circuits were compared to those obtained with the MOM procedure.
Agreement between the measured and calculated results were reasonable for all circuits.
After verifying the MOM procedure developed in Chapter 2, a discussion on the
coupling of power to resonant modes was presented. It was shown that the coupling of
power to resonant modes can be reduced by repositioning the circuit. Locating areas of
high current in areas of low electric field in the enclosure reduces the power lost to these
resonant modes. This principle would find most application in moderately sized enclosures
where the resonant frequencies are not closely spaced and the mode field structure is
relatively simple.
119
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Chapter 3 References
[1]
R.H. Jansen, "Hybrid Mode Analysis of End Effects of Planar Microwave and
M illimeter Wave Transmission Lines," Proc. Inst. Elec. Eng., Vol. 128, pt. H. pp.
77-86, April 1981.
[2]
J. Boukamp and R.H. Jansen, "The High Frequency Behavior of Microstrip Open
Ends in Microwave Integrated Circuits Including Energy Leakage," 14th European
Microwave Conf.Proc., pp. 142-147, 1984.
[3]
R.W. Jackson and D.M. Pozar, "Full-Wave Analysis of Microstrip Open-End and
Gap Discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT-33, pp.
1036-1042, October 1985.
[4]
P.B. Katehi and N.G. Alexopoulos, "Frequency-Dependent Characteristics of
Microstrip Discontinuities in Millimeter-Wave Integrated Circuits," IEEE Trans.
Microwave Theory Tech., Vol. MTT-33, pp. 1029-1035, October 1985.
[5]
R.H. Jansen, "The Spectral-Domain Approach for Microwave Integrated Circuits,"
IEEE Trans. Microwave Theory Tech., Vol. MTT-33, pp. 1043-1056, October
1985.
[6]
J.C. Rautio and R.F Harrington, "An Electromagnetic Time-Harmonic Analysis of
Shielded Microstrip Circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT35, pp. 726-730, August 1987.
[7]
R.H. Jansen, "Modular Source-Type 3D Analysis of Scattering Parameters for
General Discontinuities, Components and Coupling Effects in (M)MICs,” 17th
European Microwave Conf. Proc., pp. 427-432, 1987.
[8]
J.R. Mosig, "Arbitrarily Shaped Microstrip Structures and their Analysis with a
Mixed Potential Integral Equation," IEEE Trans. Microwave Theory Tech., Vol.
MTT-36, pp. 314-323, February 1988.
120
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[9]
N.H.L. Koster and R.H. Jansen, "The Microstrip Step Discontinuity: A Revised
Description," IEEE Trans. Microwave Theory Tech., Vol. MTT-34, pp. 213-223,
February 1986.
[10]
W.P. Harokopus and P.B. Katehi, "Characteristics of Microstrip Discontinuities on
Multilayer Dielectric Substrates Including Radiation Losses," IEEE Trans.
Microwave Theory Tech., Vol. MTT-37, pp. 2058-2066, December 1989.
[11]
L.P. Dunleavy and P.B. Katehi, "A Generalized Method for Analyzing Shielded
Thin Microstrip Discontinuities," IEEE Trans. Microwave Theory Tech., Vol.
MTT-37, pp. 1758-1766, December 1988.
[ 12]
R.W. Jackson, "Full-Wave, Finite Element Analysis of Irregular Microstrip
Discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT-37, pp. 81-89,
January 1989.
[13]
H.Y. Yang and N.G. Alexopoulos, "A Dynamic Model for Microstrip-Slotline
Transition on Related Structures," IEEE Trans. Microwave Theory Tech., Vol.
MTT-36, pp. 286-293, February 1988.
121
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CHAPTER 4
A SIM PLE CIRCUIT M ODEL FOR RESO NANT M ODE
COUPLING IN PACKAGED M M IC s
4.1
Introduction
A variety of full-wave techniques have been developed to analyze MMIC circuits in
an enclosure [1], [2], [3], [4], A full-wave analysis is a numerically rigorous method
which includes parasitic coupling to resonant modes, but the cost of this accuracy is
increased CPU time and complexity. Consequently a simpler approach is required.
Toward that end, Jansen and Wiemer [5] have developed a simple circuit theory
model to describe coupling of circuit junctions via an enclosure resonance. Their model
appears to be based on the assumption that the resonant mode coupling occurs only at a
discontinuity.
In contrast, Lewin [6] has shown that the interaction of a circuit with space wave
radiation and surface waves can occur at more than a guide wavelength from a
discontinuity. Although this phenomena is for a circuit with no cover plate and side walls,
it was shown in Chapter 3 that a similar effect occurs for a circuit in an enclosure. These
results suggest that resonant mode coupling can be modeled in a way which does not
strictly tie it to discontinuities.
In this chapter a simple circuit model is developed to accurately describe resonant
mode coupling between circuit elements within an MMIC enclosure. This model is easily
implemented on commercially available CAD packages and requires several orders of
magnitude less CPU time than a full-wave technique.
122
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4.2
Development of the Circuit Model
To implement the resonant mode circuit model, a circuit is divided into several
segments. In the center of each segment, the primaiy of a coupling transformer is inserted.
The secondaries of all the transformers for a particular mode are all connected in shunt with
a single shunt RLC tank circuit. In an enclosure with multiple modes, a separate set of
transformers and a tank circuit are required for each mode. The turns ratio of each
transformer is a function of the corresponding segment's location within the package, its
orientation, and the mode being modeled.
To illustrate the use of the resonant mode coupling circuit consider a small length of
a microstrip transmission line. The transmission line is divided into N segments of length
Axi (i = 1,..., N) and a coupling transformer is inserted in the center of each segment
(marked by a circle). The resulting circuit is shown in Figure 4.1. Assuming there is only
one resonant mode, the one tank circuit shown is the only one needed. The characteristics
of the tank circuit are independent of the number of transformers. An important
characteristic of this model is that all circuit components R, L, C and n can be determined
analytically as described in the next section.
4.2.1
Circuit Mutual Impedance
For the circuit model in Figure 4.1, the mutual impedance,
, between the z'th
and jth current elements of the transmission line can be written as:
’m odel
(4.1)
where Z fjr is the mutual impedance between the two circuit elements if there were no
resonant mode coupling present, n/ is the turns ratio for the z'th coupling transformer and
123
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Transmission line
(-*; ’ yi)
H
♦
(•^i+l’^i+l)
I
»
|— A xi - *|*
h
Axi+1— |
(a )
Resonant Mode Coupling
Circuit Model
resonant
circuit
A «/2
..
A */2
Aj.j+i/2
coupling
transformer
#i
A„-+1/2
coupling
transformer
# i+ l
(b)
Figure 4.1
Schematic of (a) a small length of a microstrip transmission line and (b) the
resulting circuit model.
124
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Yr is the admittance o f the RLC tank circuit. The turns ratio,«/, is defined as the ratio of
the number o f turns in the primary to the number of turns in the secondary. The resonant
frequency, co0, is given by:
o>02 = ^
(4.2)
Analytic expressions for the turns ratio of the coupling transformers and the R, L,
and C components of the tank circuit are determined via a comparison between the rigorous
full wave mutual impedance,
4.2.2
and the circuit model mutual impedance, Z™odel.
Full-wave Mutual Impedance
For the structure illustrated in Figure 4.2, the full-wave mutual impedance between
a u directed current element located at
2 *)
and a v directed current element located at
(X j,y j,zk ) is given by (see Chapter 2):
(z " l r ( z ™ l j + (z % l j
(43)
where
m
°° N ° ° *
,
= X X
,
“2
Q m ( k v l ‘y J J A . k , M j ( k x„,kym-)
(4.4)
M “ jV“
(z » )„ = X
X ^
f
X" ^ K e ^ „ , k ym) j ^ k „ , k „ ) j vj(kx„ k ym)
Jui^xn'kym) ~ a Jui(x,y)Tu(x ,y )d x d y
Si
(4.5)
(4.6)
Qtw = 7 xw
(4.7)
Yw = Y $ + Y i& l)
(4.8)
Tx (x,y) = cos(kxnx)sm (kymy)
Ty (x,y) = sin(kxnx)cos(kymy)
“
[Km
Xu =x
[*ym
if« = X
125
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11^*1111111111
Figure 4.2
Geometry of the MMIC package used to determine the full-wave mutual
impedance between a u directed current element located at (x/,y;,z*) and a v
directed current element located at ( x j,y j,z k).
126
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
uv
+1
if u = v
-1
if u & v
ro.5
k=
£k ~ [ 1 .0
o
k*0
W = E oxM
u ,v = x ,y
Yu# and Y ^ l) are given by equations (2.44) and (2.45), respectively.
If the current elements in the above equations are assumed to be infinitesimal
elements such that, Jui(x,y) = AUI<5(xI )5(yI ), equation (4.6) can be written as:
(4.9)
J u i ^ k y m ) = A uTu(x i^yi )
Combining equations (4.4) and (4.5) with (4.9), (z ™ ).. and
).. can now be written
as:
4
■ y tifa tU x j.] '] ) ]
a t*
<4.10)
In a typical enclosure housing an MMIC chip, the cover height is low enough that
only the TMnmO modes are resonant over the frequency of operation. Therefore, only the
terms of the summations in equation (4.10) that correspond to a resonant TMnmO mode
will significantly contribute to the mutual impedance. For this reason, only the TMnmO
modes will be considered in the modeling procedure developed below. However, this
modeling procedure could easily be applied to a TMnml or TEnml ( 1 = 1 , 2 , ...) modes.
Near a resonance of the enclosure, the term in the summation
(z™).. that
corresponds to the resonant mode (TMnmO) is much larger than the remaining terms.
Therefore,
can be written as:
(4.12)
127
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and the resonant part of ( z ™
where (-Z™).. is the non-resonant part of
.,
( Z ™ ) f 5, is given by:
(z™)** =Ae’al"
Qr^kx,,kym^ , lTM,yi)\K,Uxi,yl)} (4.13)
An expression can now be written for the full-wave mutual impedance by substituting
equation (4.13) into equation (4.3)
( 4 - 14>
where the non-iesonant part o f ( z ^ j .. is given by
(zn?-(isr£+{i2\
and the resonant part of { z ™
<4.i5)
is given by:
y( z ^ r =(z™f5
(4.16)
In Chapter 2 it was noted that there are an infinite number of poles and zeros of
Qtm which alternate along the co axis. A function that exhibits these characteristics can be
approximated by a rational fraction of the form [7]:
n
_ A
m
where 0 ^ 0 )3 ,
( 6 ) 2 - CO!2 ) ( 6 ) 2 - - 6/ n)32? )\ .-.(. 6( <v>2
? 2 -_ t/'ii?
a L . !. )'I
77tf ^ ^ 2 _ m | ) ( © 2 _ G)2y_^(02 _
and 0 , 6 )2 >®4 >'” >®2 n- 2 ^
(4.17)
die complex zeros and poles of Qtm ->
respectively. For frequencies in the vicinity of the pole that corresponds to the TMnmO
resonant mode, cor, QTM can be approximated by the following [7]
n
~j2C0
Tm
(co2 -c o l -jlcocofj
0
3
II
3
I dco )
(4.18)
1
where the enclosure is assumed to have a small to moderate amount of loss and
cor = co'r + jco" =co0 + jco"
(4.19)
It should be emphasized that when the enclosure contains loss, the resonant frequency
becomes complex. The solution for cor will be discussed shortly.
128
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4.2.3
Comparison of the Mutual Impedances
In the limit as A ui becomes small, equations (4.1) and (4.14) are approximately
equal. By comparing like terms in each equation one may write:
( 7 f w \ n r _ 7 cir
V m hj
*
(4.20)
(4.21)
where n; and Yr can be identified as:
n‘= ^ ^ r [AMXi’yi)]
4 £„£m kxn [Axi cos(kxnXi) sin(fcyTOy,-)]
ab k„
for an x directed current
sin(/:x„xi) cos(kymyi)j
for an y directed current
(4.22)
W
2) = Q™ {[0))
0)
= 1 +^ 2 .
R J 0 )K
(4.23)
Substituting equation (4.18) into the right side of equation (4.23) and solving fori? and C
yields:
i
( 2 vM
..\
2
dco
R=
(4.24)
0)=0)Q
1
2co"C
(4.25)
The inductance, L, of the tank circuit is given by equation (4.2). Calculating R with
equation (4.25) requires that the imaginary part of the complex resonant frequency, co", be
determined. An alternative expression for R may be derived by equating equation (4.7) and
the left side o f equation (4.23) with co = C0o. Solving for R yields
R = Qtm (°>o)
(4-26)
In summary equations (4.22), (4.24) and (4.26) determine n/, C and R in terms of
the circuit location and the electrical characteristics of the enclosure.
129
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4.3
Numerical Evaluation of Circuit Components
The initial step in the numerical evaluation of the circuit components is to determine
the resonant frequencies of the enclosure. The zeros of
(equation 4.8) correspond to
the resonant frequencies of the cavity. For a lossless cavity, YM is purely imaginary and
av is purely real. The zeros of YM can be located by searching for the frequency at which
Im (I^ ) = 0. If loss is present in the cavity, YM is complex and the resonant frequency Ct>r
is also complex. The zeros of YM are located by searching for a complex co where the real
and imaginary parts of YM simultaneously equal zero. Searching for the real co, C0o , where
Im(yM) = 0 may result in a solution which is not an actual zero of YM (i.e. R e (J ^ ) ^ 0 ).
However, for cavities with small to moderate loss, the real part of (Or can be determined
with good accuracy from co0 if co0 satisfies both of the following conditions (see Chapter
2):
(4.27)
(4.28)
After finding the real frequency, C0q, under these conditions, the components of the
RLC tank circuit can be determined by equations (4.2), (4.24) and (4.26).
For example, consider a cavity of the following dimensions: a = 15 mm, b = 24
mm and c = 12.7 mm. The substrate thickness is d \ = 1.27 mm and the relative
permittivity is £r i = 10.5(1/0.0023). In the band 9-12 GHz, there is one real frequency
for which Im(TM) = 0. The resonant frequency (f0 ) and Q of this enclosure have been
determined to be 10.8129 GHz and 10^, respectively. Substituting f 0 into equations (4.2),
(4.24) and (4.26) yields R = 0.1 M fi, C = 0.615 pF and L = 0.352 nH. Solving for the
complex resonant frequency,//-, yields 10.8129/0.00129 GHz. Comparing the real
resonant frequency, f 0, to the real part of the complex resonant frequency,//-, shows that
excellent agreement is obtained for high Q enclosures.
130
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Next, a dielectric layer of thickness ch, = 0.381 mm and relative permittivity of er3
= 11.9(1-7*25) is attached to the cover of the enclosure. The resonant frequency (f0 ) and Q
of this enclosure have been determined to be 10.7417 GHz and 123, respectively.
Substituting f 0 into equations (4.2), (4.24) and (4.26) yields R = 2.786 KQ, C = 0.655
pF and L = 0.335 nH. Solving for the complex resonant frequency, f r, in the conventional
manner yields 10.738 l-y‘0.0439 GHz. Comparing the real resonant frequency, f Q, to the
real part of the complex resonant frequency, f r, shows that good agreement is obtained for
Q's on the order of 100.
Lastly, the dielectric layer attached to the cover of the enclosure is replaced with a
microwave absorbing layer of thickness ds = 1.27 mm, relative permittivity of £r 3 = 60(1y0.12) and relative permeability of M-/-3 = 7.3(1-j0.3). In the band 9-12 GHz, there is one
real frequency for which Im (J^ ) = 0. The resonant frequency (fQ ) and Q of this
enclosure have been determined to be 11.0467 GHz and 19, respectively. Substituting/0
into equations (4.2), (4.24) and (4.26) yields R = 443 Q , C = 0.383 pF and L = 0.542
nH. Solving for the complex resonant frequency, f r, in the conventional manner yields
10.8883-y0.2849. Comparing the real resonant frequency, f 0, to the real part of the
complex resonant frequency, f r, shows that they differ by about 1.4%. Since finding the
real resonant frequency is much easier than finding the complex resonant frequency and the
error in obtaining the resonant frequency in this manner is not too large, this method will be
employed for low Q enclosures also.
After determining the tank circuit components for each resonant mode of the
enclosure, the circuit is divided into segments of length Aw-. In the center of each segment,
the primary of a coupling transformer is inserted. The turns ratio,«/, for a coupling
transformer located at (x/,y/) is given by equation (4.14). The secondaries of all the
transformers for a particular mode are connected in shunt to a RLC tank circuit. For
131
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
optimal results, a sufficient number of segments should be used so that the length of each
segment, AH-, is less than a quarter of a guided wavelength.
4.4
Results
To verify the accuracy of the proposed circuit model, the following circuits are
analyzed:
1. Small gap in a transmission line
2. Shunt open circuit stub
3. Stub in a larger enclosure
4 . Coupled line bandpass filter
The scattering parameters for each circuit are determined using the proposed circuit model
and compared to the results using the conventional MOM outlined in Chapters 2 and 3.
4.4.1
Small Gap
To examine the validity of using the circuit model to simulate a circuit operating in
the millimeter-wave band, consider a transmission line with a gap in the center. The
transmission line, located at y c = 1-55 mm, has a width of w = 0.1 mm and a gap of g =
0.1 mm. The circuit is enclosed in a cavity of the following dimensions: a = 3.2 mm, b =
3.1 mm and c - 0.6 mm. The substrate thickness is d \ = 0.1 mm and the relative
permittivity is er l = 12.9(l-j0.0016). An enclosure of this size has only one resonant
mode, the T M n o (61.471 GHz), in the band 50-70 GHz. The lumped elements R, C and
L were found to be 0.163 M£2, 0.333 pF and 0.0201 nH.
132
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The circuit is divided into 8 segments as illustrated in Figure 4.3. Eight
transformers are inserted with turns ratios listed in Table 4.1. The gap discontinuity itself
is modeled using the GAP element in a typical CAD program.
Table 4.1
Transformer turns ratios for the transmission lines on both sides of the gap.
1
1
2
3
4
5
6
7
8
xi (mm)
0.000
0.440
0.880
1.325
1.875
2.320
2.760
3.200
yi (mm)
1.55
1.55
1.55
1.55
1.55
1.55
1.55
1.55
A xi (mm)
0.22
0.44
0.44
0.45
0.45
0.44
0.44
0.22
ni (TMno)
9.72x10-2
1.90x10-1
1.37x10-1
1.53x10-2
-1.53x10-^
-1.37x10-1
-1.90x10-1
-9.72x10-2
Figure 4.4 compares the predicted transmission response of the small gap in the
high Q package using the circuit model versus the conventional MOM. The conventional
MOM is performed using 70 expansion functions. Over the entire bandwidth, the
difference between the circuit model and the conventional MOM is less than 3 dB. The
circuit model very accurately predicts the resonant mode coupling at the package resonance.
This example showed the use of the circuit model for a simple circuit enclosed in a
cavity with one resonant mode. In addition, this example also showed that the circuit
model can be used for circuits operating in the millimeter-wave band.
133
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T
-H
w
Figure 4.3
A xi
h -
Geometry of a transmission line with a gap in the center. The transmission
line, located a ty c = 1.55 mm (b/2), has a width of w = 0.1 mm and a gap
o f g = 0.1 mm.
134
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Gap - High Q
- MOM
‘ Model
-10
-20
-30
r*
CO
-40
-50
-60
50
55
60
65
70
F (G H z)
Figure 4.4
Comparison of the predicted transmission response o f the small gap in the
high Q package using the circuit model versus the conventional MOM.
135
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.4.2 Shunt Stub
As a second example, the circuit model will be used to simulate a transmission line
with a single shunt open circuit stub attached. A shunt stub is chosen because it is a typical
building block used in more complicated circuits. The stub, located at x c = 7.5 mm (a/2),
has a length L = 1.9 mm and is attached to a transmission, located at yc = 12 mm (b/2), of
width w = 1.4 mm. The stub is enclosed in a cavity of the following dimensions: a = 15
mm, b - 24 mm and c = 12.7 mm. The substrate thickness is d \ = 1.27 mm and the
relative permittivity is £r l = 10.5(l-j0.0023). An enclosure of this size has only one
resonant mode, the T M n o (10.8129 GHz), in the band 9-12 GHz. The lumped elements
R, C and 7. were found to be 0.1 M£2, 0.615 pF and 0.352 nH.
The transmission line on either side o f the stub is divided into 4 segments with 4
transformers and the stub is divided into 2 segments with 2 transformers as illustrated in
Figure 4.5. The turns ratio of the transformers for the transmission line are listed in Table
4.2 and the stub in Table 4.3. The Tee-junction itself is modeled using the Tee element in a
typical CAD program.
Table 4.2
Transformer turns ratios for the transmission lines on both sides of the stub.
1
1
2
3
4
5
6
/
8
xi (mm)
0.00
2.44
4.88
6.80
8.20
10.12
12.56
15.00
yi (mm)
12
12
12
12
12
12
12
12
A xi (mm)
1.22
2.44
2.44
1.40
1.40
2.44
2.44
1.22
ni CIMiio)
l.lO x lO -i
1.90x10-1
1.11x10-1
1.83x10-4
-1.83x10-2
-1.11x10-1
-1.90x10-1
-1.10x10-1
136
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i
» '0111^
iT *
-H
Figure 4.5
a
*;
i
T
h -
w
Geometry of a transmission line with a single shunt open circuit stub
attached. The stub, located at xc (a/2), has a length L = 1.9 mm and is
attached to a transmission line, located a ty c (b/2), of width w = 1.4 mm.
137
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Table 4.3
Transformer turns ratios for the stub.
1
9
10
xi (mm)
7.5
7.5
yi (mm)
12.7
14.0
A xi (mm)
0.70
1.55
ni (TMno)
-3.58x10-3
-1.32x10-3
Figure 4.6 compares the predicted transmission response of the stub in the high Q
package using the circuit model versus the conventional MOM. The conventional MOM is
performed using 123 expansion functions. Below 11.5 GHz, the agreement between the
circuit model and the conventional MOM is good. The difference between the circuit model
and the conventional MOM is no more than 3 dB. Above 11.5 GHz, the agreement
between the two methods deteriorates. A major source of the discrepancy between the
circuit model and the conventional MOM may be due to the coaxial to microstrip transition.
The MOM solution employs special basis functions to better simulate the coaxial to
microstrip transition, whereas the circuit model does n o t The ideal response for this circuit
with no enclosure has a single null in IS2 1 I at 11.1 GHz. However, the resonant mode of
the enclosure has a drastic effect on the circuit's performance and the circuit model does in
fact simulate this effect qualitatively.
To reduce the effect of the resonant mode, a microwave absorbing layer is attached
to the cover of the enclosure. The thickness of this layer is d$ = 1.27 mm, the relative
permittivity is e r 3 = 60(1-/). 12) and the relative permeability is M7 3 = 7.3(l-/).3). This
cavity was also analyzed in the previous section and was found to have one resonant mode,
the T M n o (11.13 GHz). The lumped elements R, C and L were found to be 443 £2,0.383
pF and 0.542 nH.
138
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High Q
Yc = 12 mm
SQ
■S3
CO
MOM
Model
-40
9
10
11
12
F (GH z)
Figure 4.6
Comparison of the predicted transmission response of the stub in the high Q
package using the circuit model versus the conventional MOM.
139
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 4.7 compares the predicted transmission response of the stub in the low Q
enclosure using the proposed circuit model versus the conventional MOM. Below 11.5
GHz, the agreement between the circuit model and the conventional MOM is good. The
difference between the circuit model and the conventional MOM is no more than 3 dB.
Above 11.5 GHz, the agreement between the two methods deteriorates. Again a major
source of the discrepancy between the circuit model and the conventional MOM may be due
to the coaxial to microstrip transition.
This example showed the use of the circuit model for a typical circuit enclosed in a
cavity with one resonant mode. The circuit model was very easy to implement on a
commercially available CAD package and gives reasonably good agreement with a more
rigorous analysis even in cases where a low Q resonance is involved.
4.4.3 Shunt Stub in a Larger Box
Next, the circuit model is used to simulate the shunt stub enclosed in a cavity with
several resonant modes. The shunt stub, of the previous example, is enclosed in a cavity
with the following dimensions: a = 30 mm, b = 48 mm and c = 10.0 mm. An enclosure of
this size has 5 resonant modes in the 9 to 13 GHz bandwidth. The resonant frequencies,
Q's and the lumped elements for each mode are listed in Table 4.4. In this new enclosure,
the stub is located atx c = 15 mm (a/2) and the transmission line is located a ty c = 24 mm
(b/2).
The transmission line on either side of the stub is divided into 8 segments and the
stub is divided into 2 segments. Since the transmission line is located aty c = 24 mm (b/2)
the turns ratios for the TM220 and TM 1 4 0 transformers are zero. Therefore, the
transmission line on either side of the stub requires only 24 transformers instead of 40
140
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Low Q - Dielectric Cover
Yc = 12 mm
-10
/—N
aa
TJ
S21
-20
—
v
-30
MOM
Model
^0
9
10
11
12
F (G H z)
Figure 4.7
Comparison o f the predicted transmission response of the stub in the low Q
enclosure using the proposed circuit model versus the conventional MOM.
141
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 4.4
The lumped elements of the resonant circuits for each mode in the high Q package.
Mode
TM130
TM140
TM210
TM220
TM23Q
fo (GHz)
Q
i?(M O )
C (pF )
L (nH)
9.7043
11.9517
9.5793
10.6661
12.1289
5.48x103
2.63x103
5.69x103
4.05x103
2.48x103
0.105
0.092
0.106
0.100
0.091
0.853
0.380
0.892
0.604
0.357
0.315
0.467
0.310
0.369
0.483
transformers. Since the stub is located at xc = 15 mm (a/2) the turns ratios for the TM 2 1 0 .
TM 2 2 O and TM 2 3 0 transformers are zero. Therefore, the stub requires only 4
transformers instead o f 10 transformers. The turns ratio of the transformers for the
transmission line are listed in Table 4.5 and the stub in Table 4.6. The Tee-junction itself is
modeled using the Tee element in a typical CAD program.
Figure 4.8 compares the predicted transmission response of the stub in the high Q
package using the circuit model versus the conventional MOM. The conventional MOM is
performed using 179 expansion functions. Agreement between the circuit model and the
conventional MOM is good except in the vicinity o f 12.2 GHz.
To reduce the effect of the resonant modes, a 16 mil (0.4064 mm) thick silicon
layer with a resistivity of 1.45 Q-cm is attached to the cover of the enclosure. The resonant
frequencies, Q's and the lumped elements of the resonant circuits for each mode are listed
in Table 4.7.
Figure 4.9 compares the predicted transmission response of the stub in the low Q
package using the circuit model versus the conventional MOM. Agreement between the
circuit model and the conventional MOM is good.
142
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Q - Si cover
Yc = 24 mm
-10
-20
* MOM
Model
-30
^0
9
10
11
12
13
F (G H z)
Figure 4.8
Comparison of the predicted transmission response of the stub in the larger
high Q package using the circuit model versus the conventional MOM.
143
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 24 mm
-10
sa
IS21I
T3
-20
MOM
Model
-30
-40
9
10
11
12
13
F (G H z)
Figure 4.9
Comparison of the predicted transmission response of the stub in the larger
low Q package using the circuit model versus the conventional MOM.
144
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 4.5
Transformer turns ratios for the transmission lines on both sides of the stub.
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
xi (mm)
0.0
2.2
4.4
6.6
8.8
11.0
12.85
14.3
15.7
17.15
19.0
21.2
23.4
25.6
27.8
30.0
yi (mm)
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
A xi (mm)
1.1
2.2
2.2
2.2
2.2
2.2
1.5
1.4
1.4
1.5
2.2
2.2
2.2
2.2
2.2
1.1
ni (TM130)
-2.73x10-2
-5.31x10-2
-4.89x10-2
-4.20x10-2
-3.30x10-2
-2.22x10-2
-8.31x10-3
-2.54x10-3
2.54x10-3
8.31x10-3
2.22x10-2
3.30x10-2
4.20x10-2
4.89x10-2
5.31x10-2
2.73x10-2
ni (TM210 )
5.53x10-2
9.91x10-2
6.69x10-2
2.07x10-2
-2.98x10-2
-7.41x10-2
-6.79x10-2
-6.97x10-2
-6.97x10-2
-6.79x10-2
-7.41x10-2
-2.98x10-2
2.07x10-2
6.69x10-2
9.91x10-2
5.53x10-2
ni (TM230 )
-4.23x10-2
-7.58x10-2
-5.11x10-2
-1.59x10-2
2.27x10-2
5.66x10-2
5.19x10-2
5.33x10-2
5.33x10-2
5.19x10-2
5.66x10-2
2.27x10-2
-1.59x10-2
-5.11x10-2
-7.58x10-2
-4.23x10-2
Table 4.6
Transformer turns ratios for the stub.
1
17
18
xi (mm)
15.0
15.0
yi (mm)
24.7
26.0
ni CIM130)
8.92x10-3
2.14x10-2
A xi (mm)
1.4
1.2
ni CIM140)
6.74x10-2
5.09x10-2
Table 4.7
The lumped elements of the resonant circuits for each mode in the low Q package.
Mode
TM130
TM140
TM210
TM220
TM230
fo (GHz)
9.5419
11.8119
9.4191
10.5051
11.9960
R (k£2)
1.914
4.072
1.953
2.719
4.884
Q
109
127
115
121
145
C (pF)
0.952
0.421
0.992
0.671
0.393
L (nH)
0.292
0.481
0.288
0.341
0.448
145
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
This example showed the use of the circuit model for a circuit enclosed in a cavity
with several resonant modes. Since there are several resonant modes, a large number of
transformers are needed to model this circuit. Thus, the circuit model was very tedious to
implement on a commercially available CAD package.
4.4.4 Band Pass Filter
To examine the use of the circuit model to simulate a circuit containing coupled
lines, consider a two resonator coupled line bandpass filter. The width and length of the
resonators are w = 0.64 mm L = 5.0 mm, respectively. The spacing of the resonators are
5] = 0.13 mm and S2 = 0.64 mm. The circuit is enclosed in a cavity of the following
dimensions: a = 24 mm, b = 15 mm and c = 6.35 mm. The substrate thickness is d \ =
0.635 mm and the relative permittivity is £r l = 10.5(l-y‘0.0023). The resonant
frequencies, Q's and the lumped elements of the resonant circuits for each mode are listed
in Table 4.8.
Table 4.8
The lumped elements of the resonant circuits for each mode in the high Q package.
Mode
TMno
TM210
So (GHz)
11.1654
15.0319
Q
2.01x104
1.22x104
R (MQ)
0.115
0.116
C (pF )
2.499
1.111
L (nH)
0.081
0.101
The circuit is divided into 14 segments with 28 transformers as illustrated in Figure
4.10. The turns ratio of the transformers for the bandpass filter are listed in Table 4.9.
The coupled line sections are modeled using the CPL element in a typical CAD program.
146
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T
i
L
a
~
r
XI
W
Figure 4.10
Geometry of a two resonator coupled line bandpass filter. The width and
length of the resonators are w = 0.64 mm L = 5.0 mm, respectively. The
spacing of the resonators are s i = 0.13 mm and S2 = 0.64 mm.
147
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 4.11 compares the predicted transmission response of the bandpass filter in
the high Q package using the circuit model versus the conventional MOM. The
conventional MOM is performed using 104 expansion functions. The response of the
bandpass filter predicted by the circuit model and the conventional MOM have similar
shapes, qualitatively speaking. Between 7 and 15 GHz, the difference is no more than 10
dB; however, a significant discrepancy exists at the lower and higher frequencies. The
discrepancy may be attributed to the coupled line circuit model in the CAD program.
Dunleavy [8] has also observed a discrepancy between a full wave method and CAD
packages when modeling a bandpass filter in an enclosure.
Table 4.9
Transformer turns ratios for the bandpass filter.
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
x[ (mm)
yi (mm)
0.0
2.0
4.5
7.0
9.5
9.5
12.0
12.0
14.5
14.5
17.0
19.5
22.0
24.0
6.09
6.09
6.09
6.09
6.09
6.86
6.86
8.14
8.14
8.91
8.91
8.91
8.91
8.91
A xi (mm)
1.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
1.5
ni (TMno)
4.01x10-4
1.29x10-1
1.11x10-1
8.13x10-4
4.30x10-4
4.45x10-2
0
0
-4.45x10-2
-4.3x10-4
-8.13x10-4
-i.iixio-i
-1.29x10-1
-4.01x10-4
ni (TM210 )
5.91x10-4
1.70x10-1
7.53x10-4
-5.1x10-2
-1.56x10-1
-1.62x10-1
-2.04x10-1
-2.04x10-1
-1.62x10-1
-1.56x10-2
-5.1x10-4
7.53x10-2
1.70x10-4
5.91x10-4
148
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High Q
-10
-30
-40
MOM
Model
-50
-60
6
8
10
12
14
16
F (G H z)
Figure 4.11
Comparison of the predicted transmission response of the bandpass filter in
the high Q package using the circuit model versus die conventional MOM.
149
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This example showed the use of the circuit model for a more complicated
microwave circuit The circuit was enclosed in a cavity with two resonant modes, one
occurring in the passband and the other in the stopband.
4.5
Conclusion
A circuit model has been developed that models resonant mode coupling. It has
good accuracy for enclosures with a Q of over 100 and is useful for Q's as low as 20. For
lower Q's, the full wave spectral summation, which serves as the starting point for this
model, no longer has a single spectral term which produces a dominant resonance.
The circuit model can be used on commercially available software packages.
Simple analytical expressions for the entire model are easily evaluated. This is a very
attractive feature of the circuit model for implementation into a CAD package. In addition,
it requires several orders of magnitude less CPU time than the MOM. However, the circuit
model for an MMIC circuit in a large enclosure (one with more than a few resonances) may
require a large number of transformers. The large number of transformers makes the
implementation of the circuit model into a CAD package very tedious. Consequently, this
circuit model may be best suited for MMIC circuits in moderately sized enclosures. A
potential solution to this inconvenience is an automated procedure for inputting the
transformers.
To test the feasibility of such an automated procedure, a simple FORTRAN
program was written to calculate the response of a circuit in an enclosure. The program
uses discontinuity models {e.g. a TEE element) from a commercially available CAD
package which, if needed, are read into the program as data files in the form of sparameters. The program automatically determines the number of resonances, thus the
number of transformers per location, the characteristics of each transformer and the
150
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components o f each resonant circuit Using the automated procedure significantly reduced
the time and complexity o f entering the model into a CAD package; which allows for the
utilization o f the circuit model for any size enclosure.
151
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Chapter 4 References
[1]
Rautio, J.C. and Harrington, R.F., "An electromagnetic time-harmonic analysis of
shielded microstrip circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT-35,
pp. 726-730, August 1987.
[2]
Dunleavy, L.P. and Katehi, P.B., "A generalized method for analyzing shielded
thin microstrip discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT37, pp. 1758-1766, December 1988.
[3]
Hill, A. and Tripathi, V.K., "An efficient algorithm for three-dimensional analysis
of passive microstrip components and discontinuities for microwave and millimeterwave integrated circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT-39,
pp. 83-91, January 1991.
[4]
Jansen, R.H. and Sauer, J., "High-speed 3D electromagnetic simulation for
MIC/MMIC cad using the spectral operator expansion (soe) technique," 1991 IEEE
MTT-S Digest, pp. 1087-1090, 1991.
[5]
R.H. Jansen and L. Wiemer, "Full-wave theory based development of mm-wave
circuit models for microstrip open end, gap, step, bend and tee," IEEE MTT-S Int.
Microwave Symp. Dig., pp. 779-782, June 1989.
[6]
L. Lewin, "Spurious radiation from microstrip," Proc. IEE, Vol. 125, No. 7, pp.
633-642, July 1978.
[7]
Beringer, R. "Resonant cavities as microwave circuit elements". In Montgomery,
C.G., Dicke, R.H., and Purcell, E.M. (eds) Principles of microwave circuits. MIT
Radiation Laboratory Series, Vol. 8., McGraw-Hill Book Company, Inc., New
York, pp. 207-239, 1948.
[8]
Dunleavy, L.P., "Discontinuity characterization in shielded microstrip: a theoretical
and experimental study," Ph.D. Thesis, University of Michigan, 1988.
152
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CHAPTER 5
ANALYSIS O F M M ICs IN RESONANT ENCLOSURES W ITH
THE DIAKOPTIC M ETHOD
5.1
Introduction
In the procedure developed in Chapters 2 and 3, circuits are divided into small cells.
The current is expanded over each cell using a set of known basis functions. The unknown
current coefficients are then determined by the method of moments (MOM). For a small
circuit, such as a Tee-junction, the number of unknowns is on the order of a few hundred.
With larger circuits, the number of unknowns is on the order of a few thousand and for an
entire MMIC chip, the number of unknowns can exceed tens of thousands.
The majority of analysis time using the MOM can attributed to two sources: the time
required to fill the matrix (fill time) and the time required to solve the linear system of
equations (solve time). The solve time represents a large portion of the analysis time when
the number of unknowns is very large. A simpler approach that reduces the solve time will
be developed in this chapter.
In order to reduce the solve time, Goubau et al [1] developed the diakoptic method.
The diakoptic method and the modified diakoptic method [2] were originally developed to
analyze complex multi-element antennas. Until very recently [3] the diakoptic method had
not been applied to MMIC problems and even now has not been reported in sufficient detail
for an assessment of it to be made.
In what follows, the diakoptic method is used to analyze MMICs in resonant
enclosures. Results obtained using the diakoptic method agree well with those obtained
using the MOM (Chapters 2 and 3) for a few simple circuits. However, a significant
discrepancy exists between the two methods for more complicated circuits. A new filtering
technique is presented which significantly reduces this discrepancy.
153
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5.2
Diakoptic Method
5.2.1
Theory
In this section, the diakoptic method is reviewed. Due to the similarity of the
diakoptic and modified diakoptic methods, only the modified diakoptic theory will be
discussed. To avoid confusion with the new method developed later in this chapter, we
will refer to the modified diakoptic method simply as the diakoptic method.
The first step in the diakoptic theory is to divide (diakopt) a circuit into smaller
elements. However, the size of each element is not nearly as small as the cells used in the
conventional MOM. As a result of diakopting a circuit, artificial ports are introduced. At
the terminals of each port, a voltage and current can be defined. For example, consider the
center fed dipole shown in Figure 5.1a. The port for the original structure is designated as
port i. The dipole is divided or diakopted into 4 elements as shown in Figure 5.1b. The
diakopted circuit has 3 ports. Port 2 is the port for the original dipole (port i), while the
other 2 ports are artificially introduced. The two elements meeting at the Ath port will be
referred to as structure k. For example, structure 2 is made up of elements B and C which
meet at port 2.
The second step in the diakoptic method is the characterization of the diakopted
circuit by an impedance matrix. If the diakopted circuit has N DK ports, the relationship
between the port currents and the port voltages is given by:
(5.1)
/=i
where V[DK and l f K are the voltage and current at the /th port and
/i=o
The elements of the impedance matrix are determined by [4], [5]:
154
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Port i
(a)
D
B
Port 1
Port 2
Port 3
(b)
Figure 5.1
Schematic of (a) center fed dipole and (b) the resulting diakopted circuit.
155
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z kiK = - J J [ ^ ( ^ , y ) ] r • JkDK(x ,y )d x d y
(5.3)
sk
Jk
is the normalized current distribution on the entire circuit due to an excitation at port k
with the other ports open circuited. E ? K is the electric field generated by the normalized
current distribution, J p K, and can be represented by:
E ? K(x,y) = £
' t ^
!Ln x , y ^ Q i k xn,kym) ■J ? K(kxn,kym)\
(5.4)
m=0n=0
where
JlDK(kxn,kym) = j \ T ( x , y ) J lDK(x ,y )d x d y
T(x,y) =
Sl
c o ^ k xnx)sm (kymy )
0
0
s in ( ^ nx)cos(/:>TO>-)
Q(kxn,kym) =
X xQ xx& xn ’kym )
^Q xy^xn ^yJ
yxQyx(kxrl ’kym )
yyQyy(kxn,kym) j
The elements of the impedance matrix, calculated by equation (5.3), are stationary about the
exact current; therefore, a first-order error in the current will produce only a second-order
error in Z ^ K .
The excitation of each element is generated by two different sources. The first
source is the current entering the element at its terminals and is referred to as current
coupling. The second source of excitation is called field coupling. Field coupling is
generated by the currents on all the other elements. If the length of structure k is below
resonance, current coupling is more dominant than field coupling; therefore, the current on
the two elements of structure k is much larger than on all the other elements.
Since current coupling dominates field coupling and the expression for Zk[
is
variational, the current distribution on the structure being excited can be approximated by
assuming the remaining elements are not present In Figure 5.1 for example, J®K is
approximated by exciting elements A and B at port 1 and assuming the currents on C and D
are zero. After approximately determining the normalized current distributions, JtDK (1=1,
156
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N d k ), under isolated-structure conditions, the diakoptic impedance matrix is
determined by equation (5.3). Equation (5.1) is solved for the port currents, l f K,
assuming all the port voltages, VjPK, are zero except the actual driven port, port 2 in Figure
5.1.
5.2.2
The Relationship Between the Diakoptic and Moment Methods
The use of the diakoptic method to analyze a circuit can be thought of as a double
application of the moment method [5], [6]. In the first application of the moment method,
the current on the &th structure,
, is expanded as follows:
= x X 4 f ^ )U ,y ) + y
i=i
j =i
W )
(5.5)
where the number of x-directed and y-directed expansion function on the kth structure are
and
, respectively. Since the size of a "structure" is much smaller than the entire
circuit, the number of expansion functions needed is small. Following Galerkin's method,
the electric field integral equation (EFTE) is tested with
and
which results in a set
of algebraic equations for the unknown current coefficients, such that:
y(A’) = Z (k)j(.k)
where the elements of the excitation vector are given by:
n-'i rr
(V
v V = j E ^ , y - > J pi(X,y^
{
0
at a port
and a typical element of the impedance matrix is given by:
M °° N ~
(4 * 1 =
J
I I
m = 0n= 0
a
^ Q Pq^ k ym) j f { k m ,kym) J ^ \ k m ,kym)
(5.7)
aD
p ,q = x o r y
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After determining the unknown current coefficients,
and i f f , a new expansion
function or "super mode", Jj?K, is formed from equation (5.5). This super mode can be
represented as:
3t“ W
Mik)
Myk)
i=l
j =1
=
=7
lk
(5.8)
T)K
where lj? is the port current and
/(* )
(5.9a)
lk
/(*)
<
’ =7f e
lk
(5.9b)
This procedure is repeated until a super mode is found for each structure. For a diakopted
circuit with N DK ports, N DK super modes are required.
In the second application of the moment method, the surface current on the entire
circuit is expanded into a set of super modes as follows:
n dk
J s(x ,y )= J , l F Kj P Kix ,y )
z=i
(5.10)
Following Galerkin's method, the electric field integral equation (EFIE) is tested with jj?K
which results in a set of algebraic equations for the unknown current coefficients:
WD K = Z DKI DK
(5n)
with the elements of Z DK determined by substituting equation (5.4) and equation (5.3):
oo
©o
& =I
(5.12)
m = 0n=0
a°
where the superscript T refers to the transpose of a matrix and
M[k)
(5-(5)
i= l
y=l
The elements of the excitation vector are given by:
sk
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The expression for Z $ K can be simplified by substituting equation (5.13) into (5.12):
M °° N °°
.
z™= X 1 = ab^n^m‘
{ ^ k\ kxn^m)[Qxxikxn^ yJF^l\k ^ ,k yJ + Q ^{k^kym) l f \ k xn,kym)\
(5.14)
+ F ^ ' \ k xn , k y m '^Q yX(Jcxn,k y m ')F^ ^(k xn >k ym )
Q yy(k x n ’k ym )
^(^OT’^ym)]j'
where
M(xk)
<5-15a>
i= l
M(yk)
Fik\ k m ,k ,m) = j , w V j « j \ k m ,kym)
(5.15b)
1=1
Substituting equation (5.15) into (5.14), Z $ K can also be written as:
\f(k) fij(l)
zSK=
1=1 j =1
f ^ k ) ftf(l)
±
i
u
+x x
i= l
J
7=1
7
xx
1=1 7=1
J
+x x
(5.16)
i= l 7=1
where
(z"0M) =
7
7Vf~ at~
X J d- ^ ^ Q pq( k „ ,k ym) j f { k „ , k ym) j " h k xn,k ,m)
m = 0n= 0
(5.17)
ab
p , q = x o ry
Equations (5.14) and (5.16) represent two different ways of calculating the
diakoptic impedance matrix. The disadvantage of using equation (5.14) is that the series
can not be summed using the technique developed in Chapter 2, since the coefficients W^P
and W^P are frequency dependent. The series is equation (5.16) can be summed using the
technique developed in Chapter 2. However, equation (5.14) will probably be more
efficient in terms of the number of operations needed to fill the diakoptic impedance matrix
and thus lead to lower CPU times if a technique can be applied to enhance the convergence
of the series in equation (5.14). Recently, S. Singh and R. Singh [7] have used Shank's
159
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transform to enhance the convergence of a free-space periodic Green's function.
Application of Shank’s transform to summations in equation (5.14) should make it more
attractive to use. For this research, however, equation (5.16) is used to calculate the
diakoptic impedance matrix.
As described previously, the diakoptic method above is a two step moment method
process. In the first step, N DK super modes are formulated for the circuit. The number of
operations that are required to form all N DK super modes is on the order o f l/6 (N DKM ^,),
where M ^ is the number of expansion functions that make up the kth super mode. One
operation is defined as one multiplication and one addition. For simplicity it is assumed
that
= M ^, for each of the pieces of the diakopted circuit. In the second step, the
super modes are used to expand the current on the entire circuit The total number of
operations required using the diakoptic method is approximately l/6 ^N DKMly +
operations; whereas the conventional MOM requires l/6 (A ^ 0Af) operations, where N MOM
represents the number of unknowns.
For example, consider a circuit that requires 1000 subsections when using the
conventional MOM. Assume that the circuit is diakopted into 100 structures with each
structure consisting of 20 subsections. Solving the linear system using the conventional
MOM requires approximately 1.7x10^ operations, whereas the diakoptic method requires
3.3x10^ operations. Using the diakoptic method can result in the dramatic reduction of
analysis time.
The example given above serves as a rough comparison between the diakoptic
method and the conventional MOM. Numerous factors will effect runtimes such as the
number of diakoptic ports or the number of subsections used to expand the current on a
diakoptic structure. In order to determine the number of operations required for the
solution of the linear system of equations, it was assumed that a direct solver such as
Gauss's elimination was used. However, for large matrices, an iterative solver (e.g.
160
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conjugate gradient method) would likely be used; this may reduce the number of operations
required.
5.2.3
Results
The diakoptic method described above is used to analyze three microstrip circuits.
The first two circuits are a half-wave dipole on a GaAs substrate and a quartz substrate,
respectively. The third circuit is a single shunt open circuit stub attached midway between
the input and output connectors.
5.2.3.1 Dipole on GaAs
As a first example, consider a center fed dipole o f length, L = 1.55 mm and width
of, w = 0.2 mm. The dipole is located at xc - 7.0 mm and yc = 2.85 mm in a cavity
(Figure 5.2) of the following dimensions: a = 10 mm, b = 6 mm and c = 0.5 mm. The
substrate thickness is d \ = 0.1 mm and the relative permittivity is er l = 12.9(l-j0.0016).
An enclosure of this size has three resonant modes in the band 25-45 GHz.
The dipole is diakopted into 4 elements with 3 ports as illustrated in Figure 5.1.
Figure 5.3 compares the computed input reactance of the dipole in the high Q package
using the diakoptic method versus the input reactance computed using the conventional
MOM. The conventional MOM is performed using 11 expansion functions while the
diakoptic method uses 3 super modes. The results obtained using the diakoptic method are
offset from the conventional MOM by no more than 5 £2 over the whole bandwidth.
161
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y
t
a
k — L
x
Figure 5.2
Geometry of a center fed dipole located at x c = 7.0 mm and y c = 2.85 mm.
162
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D ipole
50
30
MOM
Diakoptic
10
10
■30
•50
25
30
35
40
45
F (G H z)
Figure 5.3
Comparison of the computed input reactance of the dipole in the high Q
package using the diakoptic method versus the input reactance computed
using the conventional MOM.
163
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5.2.3.2 Dipole on Quartz
Next, consider a center fed dipole o f length, L = 1.319 mm and width of, w = 0.24
mm. The dipole is located at x c - 7.0 mm and y c = 9.4 mm in a cavity (Figure 5.2) of the
following dimensions: a = 14 mm, b = 18.8 mm and c = 0.635 mm. The substrate
thickness is d\ = 0.127 mm and the relative permittivity is £/-l = 4.5(l-j0.0001). An
enclosure of this size has seven resonant modes in the band 58-62 GHz.
The dipole is diakopted into 4 elements with 3 ports as illustrated in Figure 5.1.
Figure 5.4 compares the predicted input reactance of the dipole in the high Q package using
the diakoptic method versus the conventional MOM. The conventional MOM is performed
using 11 expansion functions while the diakoptic method uses 3 super modes. The results
obtained using the diakoptic method are offset from the conventional MOM by about 8 Q
over the whole bandwidth.
5.2.3.3
Shunt Stub
As a third example illustrating the application of the diakoptic method, consider a
transmission line with a single shunt open circuit stub attached. A shunt stub is chosen
because it is a typical building block used in more complicated circuits. The stub, located at
x c = 7.5 mm (a/2), has a length L = 1.9 mm and is attached to a transmission, located at yc
= 12 mm (b/2), of width w = 1.4 mm. The stub is enclosed in a cavity of the following
dimensions: a = 15 mm, b = 24 mm and c = 12.7 mm. The substrate thickness is d \ =
1.27 mm and the relative permittivity is er l = 10.5(l-j0.0023). An enclosure of this size
has only one resonant mode, the T M n o (10.8129 GHz), in the band 9-12 GHz.
The circuit is diakopted into 15 elements with 16 ports as illustrated in Figure 5.5.
Figure 5.6 compares the predicted transmission response of the stub in the high Q package
164
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D ipole
50
MOM
Diakoptic
25
0
58
60
61
62
F (GHz)
Figure 5.4
Comparison of the predicted input reactance of the dipole in the high Q
package using the diakoptic method versus the conventional MOM.
165
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i
J
l
T
w
Figure 5.5
Geometry of a transmission line with a single shunt open circuit stub
attached. The stub, located atx c (a/2), has a length L = 1.9 mm and is
attached to a transmission line, located aty c (b/2), of width w = 1.4 mm.
166
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High Q
Yc = 12 mm
-10
ea
■o
w
-20
-30
^0
- MOM
* Diakoptic
9
10
11
12
F (G H z)
Figure 5.6
Comparison o f the predicted transmission response of the stub in the hig
package using the diakoptic method versus the conventional MOM.
167
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using the diakoptic method versus the conventional MOM. The conventional MOM is
performed using 123 expansion functions while the diakoptic method uses 16 super
modes. Agreement between the diakoptic method and the conventional MOM is very poor.
5.3
Enhanced Diakoptic Method
In the previous section, the diakoptic method was applied to a shunt open circuit
stub enclosed in a conducting package. A significant discrepancy existed between the
results obtained using the conventional MOM (Chapters 2 and 3) and the diakoptic method.
Jackson [8] has also observed a significant discrepancy between the results obtained using
the conventional MOM [9] and the diakoptic method for circuits with no enclosures.
Howard and Chow [3], [10] indicated that they have also encountered a problem when
applying the diakoptic method. They reported that this problem can be avoided by
removing the source fringe effects; however, the details of this removal were not given.
The accuracy and efficiency of the diakoptic method depends upon the quality of the
super mode expansion functions, how well they resemble the final currents on the entire
structure. In this section, spectral filtering will be investigated as a way of improving the
super mode quality.
5.3.1
Theory of Spectral Filtering
The diakoptic method has been previously described as a two step moment method
process. In the first step, N DK super modes are formulated for the circuit. In the second
step, the super modes are used to expand the current on the entire circuit
When solving for the super modes, jjPK ( k = l , ..., N DK), in step one, the current
on the M i structure is expanded in a set of known basis functions with coefficients I (fc).
168
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As described in section 5.2.2 these coefficients are determined by solving equation (5.6).
-»ni7
Jk , is then formed from equation (5.5). However, experience has shown that removing
the lower order spectral terms (small kp ) in equation (5.7) will produce a superior super
mode. The removal of the lower order spectral terms can be thought as passing the
components of equation (5.7) through a high pass spectral filter. This high pass filtering
can be expressed by the following relationship:
Z (k)---------------------------- >Z(fc)
High pass spectral filter
(5.18)
Instead of equation (5.6), the unknown current coefficient on the fcth structure,
I (i), is now determined by solving the following equation:
y ( k ) = 2 l(k)1(k)
(5 .19)
A typical element of Z ^ is given by:
Af°° N °° _ 4
4 =2
( *%
(5.20)
m = 0n= N '
where
+1
“ jo
m<Mc
m >Mc
and Nc and M c are defined as the cutoff mode numbers. Although numerical values for the
cutoff mode numbers have not yet been defined, it will be shown shortly that N c and Mc
are large. Since N c and M c are large, kp will be also be large. From the discussion in
Chapter 2 and Appendix B, the spectral Green’s Function for large kp can be represented
as:
Qpqikm ,kym) =
i^ k y j
As a result, equation (5.20) can be expressed as:
M °° N °°
.
(3 ? )„ ' I I --- 7f-<$
3
m = 0n= N'
(5.21)
a0
In the second application of the moment method, the surface current on the entire
circuit is expanded into a set of super modes, J[DK (I = 1 ,..., N DK). The unknown
169
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current coefficients of the super inodes, 1DK, are determined by solving equation (5.11).
Note, when calculating the elements of the Z DK with equation (5.12), the total spectrum is
used.
In summary, the super modes are determined using the higher order spectral terms
as described by equation (5.21). The super modes are used to expand the current on the
entire circuit When calculating the elements of the diakoptic impedance matrix, the total
spectrum is used as described by equation (5.12).
5.3.2
Determination of the Cutoff Mode Numbers
Before determining the cutoff mode numbers, Nc and Mc, it will be convenient to
define cutoff wave numbers as:
C5'22a)
(5.22b)
where
Pc =
and
kc
(5-23)
is a frequency independent constant. Substituting equation (5.23) into equations
(5.22a) and (5.22b) yields the cutoff mode numbers, N c and M c:
NC=^ 2 x
M c = bK£ko
(5'24a)
V2 tt
To examine the effect of varying
kc
has on the solution obtained using the
enhanced diakoptic method, again consider the dipole on 0.125 mm thick quartz. The
dipole is diakopted into 4 elements with 3 ports as illustrated in Figure 5.1. Figure 5.7
compares the predicted input reactance of the dipole in a high Q package using the enhanced
diakoptic method for different values of k c versus the conventional MOM. The
170
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D ipole
1
CA
E
js
O
*
Mrn t
X
•
fT
MOM
I f -----■— Kc=7.625
Ef
0
Kr=8.625
f #
-----• —
kc=9.625
----- °— Kr=10.625
Fi
l[
-----*— v r = 11.625
.
.1
-5
58
59
60
61
62
F (G H z)
Figure 5.7
Comparison of the predicted input reactance of the dipole in a high Q
package using the enhanced diakoptic method for different values of
versus the conventional MOM.
171
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
conventional MOM is performed using 11 expansion functions. Below 59 GHz,
agreement between the enhanced diakoptic method with
kc
= 9.625 and 10.625 and the
conventional MOM is excellent. Above 59 GHz, the enhanced diakoptic method with
kc
=
9.625 gives the same values as the conventional MOM. The enhanced diakoptic method
with
Kc
= 8.625 and 10.625 deviates from the conventional MOM by no than 1 Cl. Based
on these observations, the optimum value of Kc for 0.125 mm quartz is 9.625. For
comparison, the same calculation using an unfiltered supermode resulted in an offset from
the conventional MOM by about 8 £2 over the whole bandwidth. Figure 5.8 summarizes
the comparison of the computed input reactance of the dipole using the enhanced diakoptic
method with the optimum value of
Kc
(9.625) versus the conventional MOM.
The above procedure is repeated for a dipole printed on different substrates. Table
5.1 lists the optimum kc for these substrates.
Table 5.1
Kc
for a few representative substrates.
S u b s tra te
£r
T h ick n ess (m m )
Kc
Duroid 6010
Duroid 6010
Quartz
GaAs
10.5
10.5
4.5
12.9
1.27
0.635
0.127
0.1
13.5
14.25
9.625
15.75
Unfortunately,
kc
in Table 5.1 is different for different substrates. In order for the
enhanced diakoptic method to be a useful tool for the simulation of microstrip circuits, a
procedure must be prescribed for finding
kc
for a particular substrate. Experience indicates
that an error function ec defined as
172
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D ipole
50
'
25
MOM
Enhanced Diakoptic
X ll
0
-25
-5 0
58
59
60
61
62
F (G H z)
Figure 5.8
Comparison of the computed input reactance of the dipole using the
enhanced diakoptic method with the optimum value of kc (9.625) versus
the conventional MOM.
173
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
has a numerical value of 0.03 when
optimum value o f
kc
0.03. Substituting
kc
is optimum. Thus, in order to determine the
for a particular substrate,
kc
kc
in equation (5.25) is varied until £c =
into equation (5.24) yields the cutoff mode numbers, N c and M c.
One important feature of
Therefore, once
kc
kc
is that it is not dependent on the circuit topology.
is determined for a given substrate it does not need to be recalculated
for different circuits.
5.3.3
Results
The enhanced diakoptic method described above is used to analyze five microstrip
circuits. The circuit types are as follows:
1. Shunt open circuit stub
2. Stub in a larger enclosure
3. Coupled line bandpass filter
4. Small gap in a transmission line
5. 60 GHz shunt stub
5.3.3.1 Shunt Stub
As the first example using the enhanced diakoptic method, again consider the shunt
open circuit stub discussed in the previous section. The circuit is diakopted into 15
elements with 16 ports as illustrated in Figure 5.5. Figure 5.9 compares the predicted
transmission response of the stub in the high Q package using the enhanced diakoptic
174
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 12 mm
IS21I (dB)
-10
-20
-30
'
MOM
Enhanced Diakoptic
-40
9
10
11
F (G H z)
Figure 5.9
Comparison o f the predicted transmission response of the stub in the high Q
package using the enhanced diakoptic method ( k c = 13.5) versus the
conventional MOM.
175
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
method ( kc = 13.5) versus the conventional MOM. Agreement between the enhanced
diakoptic method and the conventional MOM is excellent The ideal response for a single
shunt open circuit stub is a single null in IS2 1 I. However, the resonant mode of the
enclosure has a drastic effect on the circuit's performance and the enhanced diakoptic
method does in fact simulate this effect very well.
To reduce the effect of the resonant mode, a microwave absorbing layer is attached
to the cover of the enclosure. The thickness of this layer is J 3 = 1.27 mm, the relative
permittivity is e r 3 = 60(l-_/0.12) and the relative permeability is Mr3 = 7.3(1-j'0.3).
Figure 5.10 compares the predicted transmission response of the stub in the low Q
package using the enhanced diakoptic method versus the conventional MOM. The results
obtained using the enhanced diakoptic method deviate from the conventional MOM by no
more than a 1 dB over the whole bandwidth.
The shunt stub described above, the conventional MOM was performed using
N mom = 123 expansion functions while the diakoptic method used N DK = 16 super
modes. The total number of expansion functions used to create the &th super mode is
where k = 1,.., N DK.
is different for each super mode because basis
functions of varying size and type (e.g. functions that satisfy the edge conditions) are used
in different regions of the circuit. The average number of expansion functions used to
j Ndk
create the super modes was M ^ vg^ = ------------------- = 14.4. Solving the linear system of
N d k k= i
equations using the conventional MOM required approximately N ^ qm ~ 1/ 6(
3x l0 3
A
=
operations, whereas the enhanced diakoptic method required
ndkV
=1
/
6
j
= 3 x l ° 4 operations.
This example showed the use of the enhanced diakoptic method for a typical circuit
enclosed in a cavity with one resonant mode. Compared to the MOM, the enhanced
diakoptic method resulted in a reduction of the number operations needed to solve for the
176
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Q - Dielectric Cover
Yc = 12 mm
-10
IS21
-20
MOM
Enhanced Diakoptic
-30
-40
9
10
11
12
F (G H z)
Figure 5.10
Comparison of the predicted transmission response of the stub in the low Q
package using the enhanced diakoptic method versus the conventional
MOM.
177
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
unknown current distribution by a factor o f 10. The enhanced diakoptic method gives
excellent agreement with the MOM even in cases where a low Q resonance is involved.
5 .33.2 Shunt Stub in Larger Box
Next, the enhanced diakoptic method will be used to simulate the shunt stub
enclosed in a cavity with several resonant modes. The shunt stub, of the previous
example, is enclosed in a cavity with the following dimensions: a = 30 mm, b = 48 mm and
c = 10.0 mm. An enclosure of this size has five resonant modes in the 9 to 13 GHz
bandwidth. In this new enclosure, the stub is located at x c = 15 mm (a/2) and the
transmission line is located at y c = 24 mm (b/2).
The circuit is diakopted into 25 elements with 26 ports. Figure 5.11 compares the
predicted transmission response of the stub in the high Q package using the enhanced
diakoptic method ( kc = 13.5) versus the conventional MOM. Agreement between the
enhanced diakoptic method and the conventional MOM is very good.
To reduce the effect of the resonant modes, a 16 mil (0.4064 mm) thick silicon
layer with a resistivity of 1.45 O-cm is attached to the cover of the enclosure. Figure 5.12
compares the predicted transmission response of the stub in the low Q package using the
enhanced diakoptic method versus the conventional MOM. Again, the agreement between
the enhanced diakoptic method and the conventional MOM is very good.
Simulating the shunt stub in the larger box with the conventional MOM required
N Mom = 179 expansion functions while the enhanced diakoptic method used N DK = 26
super modes. The average number of expansion functions used to create the super modes
was
= 12.8. Solving the linear system of equations using the conventional MOM
required approximately Njfiom ~ 9.5x105 operations, whereas the enhanced diakoptic
method required N ^ LV ~ 3 .2 x l0 4 operations.
178
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
Yc = 24 mm
IS21I ((IB)
-10
-20
-30
'
MOM
Enhanced Diakoptic
-40
9
10
11
12
13
F (G H z)
Figure 5.11
Comparison of the predicted transmission response of the stub in the larger
high Q package using the enhanced diakoptic method ( Kc = 13.5) versus the
conventional MOM.
179
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Q - Si cover
Yc = 24 mm
IS21I (dB
-10
-20
-30
- MOM
• Enhanced Diakoptic
-40
9
10
11
12
13
F (G H z)
Figure 5.12
Comparison of the predicted transmission response of the stub in the larger
low Q package using the enhanced diakoptic method versus the
conventional MOM.
180
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This example showed the use of the circuit model for a circuit enclosed in a cavity
with several resonant modes. Compared to the MOM, the enhanced diakoptic method
resulted in a reduction of the number operations needed to solve for the unknown current
distribution by a factor of 30. Agreement between the enhanced diakoptic method and the
MOM was excellent even in the vicinity of the resonant modes.
5.3.3.3 Coupled Line Bandpass Filter
To examine the use of the enhanced diakoptic method to simulate a circuit
containing coupled lines, consider a two resonator coupled line bandpass filter. The width
and length of the resonators are w = 0.64 mm L = 5.0 mm, respectively. The spacing of
the resonators are s i = 0.13 mm and S2 = 0.64 mm. The circuit is enclosed in a cavity of
the following dimensions: a = 24 mm, b = 15 mm and c = 6.35 mm. The substrate
thickness is d \ = 0.635 mm and the relative permittivity is £r l = 10.5(l-y'0.0023). An
enclosure of this size has two resonant modes in the band 6-16 GHz.
The circuit is diakopted into 24 elements with 22 ports as illustrated in Figure 5.13.
Figure 5.14 compares the predicted transmission response of the bandpass filter in the high
Q package using the enhanced diakoptic method versus the conventional MOM. The
results obtained using the enhanced diakoptic method deviate from the conventional MOM
by no more than a 2.5 dB over the most of the bandwidth except less than 7 GHz.
To reduce the effect of the resonant modes, a 16 mil (0.4064 mm) thick silicon
layer with a resistivity of 1.45 Q-cm is attached to the cover of the enclosure. Figure 5.15
compares the predicted transmission response of the bandpass filter in the low Q package
using the enhanced diakoptic method
( kc
= 14.25) versus the conventional MOM.
Agreement between the enhanced diakoptic method and the conventional MOM is excellent.
181
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 5.13
Geometry of a two resonator coupled line bandpass filter. The width and
length of the resonators are w = 0.64 mm L —5.0 mm, respectively. The
spacing of the resonators are s \ = 0.13 mm and S2 = 0.64 mm.
182
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
-10
-20
-30
-40
MOM
Enhanced Diakoptic
-50
-60
6
8
10
12
14
16
F (G H z)
Figure 5.14
Comparison of the predicted transmission response o f the bandpass filter
the high Q package using the enhanced diakoptic method versus the
conventional MOM.
183
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Q - Si cover
o
-10
-20
-30
-40
MOM
Enhanced Diakoptic
-50
-60
6
8
10
12
14
16
F (GHz)
Figure 5.15
Comparison of the predicted transmission response of the bandpass filter
the low Q package using the enhanced diakoptic method ( k c = 14.25)
versus the conventional MOM.
184
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The results obtained using the enhanced diakoptic method deviate from the conventional
MOM by no more than a 2.5 dB over the most of the bandwidth except less than 7 GHz.
For the coupled line bandpass filter, the conventional MOM was performed using
N MOm = 104 expansion functions while the enhanced diakoptic method used N DK = 22
super modes. The average number of expansion functions used to create the super modes
was M ^ vg) = 7.9. Solving the linear system of equations using the conventional MOM
required approximately N mqm ~ 1-8x10^ operations, whereas the enhanced diakoptic
method required N ^ LV ~ 1.8x10^ operations.
This example showed the use of the enhanced diakoptic method for a more
complicated microwave circuit The circuit was enclosed in a cavity with two resonant
modes, one occurring in the passband and the other in the stopband. Compared to the
MOM, the enhanced diakoptic method resulted in a reduction of the number operations
needed to solve for the unknown current distribution by a factor of 100.
5.3.3.4 Small Gap
To examine the validity of using the enhanced diakoptic method to simulate a circuit
operating in the millimeter-wave band, consider a transmission line with a gap in the center.
The transmission line, located a ty c = 1-55 mm, has a width of w = 0.1 mm and a gap of g
= 0.1 mm. The circuit is enclosed in a cavity of the following dimensions: a = 3.2 mm, b
= 3.1 mm and c = 0.6 mm. The substrate thickness is d\ = 0.1 mm and the relative
permittivity is £r l = 12.9(l-j0.0016). An enclosure of this size has only one resonant
mode in the band 50-70 GHz.
The circuit is diakopted into 14 elements with 14 ports as illustrated in Figure 5.16.
Figure 5.17 compares the predicted transmission response of the small gap in the high Q
package using the enhanced diakoptic method versus the conventional MOM. Agreement
185
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A
A
---1-----f
Jr
b
J_... 1----i----1----1
t
-M s
W
3>c
Figure 5.16
Geometry of a transmission line with a gap in the center. The transmission
line, located at y c = 1.55 mm (b/2), has a width of w = 0.1 mm and a gap
of g = 0.1 mm.
186
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Gap - High Q
o
MOM
Enhanced Diakoptic
IS21I (dB)
-10
-20
-30
■40
-50
-60
50
55
60
65
70
F (GH z)
Figure 5.17
Comparison of the predicted transmission response of the small gap in the
high Q package using the enhanced diakoptic method ( k c = 15.75) versus
the conventional MOM.
187
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
between the enhanced diakoptic method ( kc = 15.75) and the conventional MOM is
excellent above 60 GHz. Below 60 GHz, the difference between the enhanced diakoptic
method and the conventional MOM is about 2.5 dB at the -30 dB level.
Simulating the small gap with the conventional MOM required N MOM = 70
expansion functions while the enhanced diakoptic method used N DK = 14 super modes.
The average number of expansion functions used to create the super modes was
=
8.7. Solving the linear system o f equations using the conventional MOM required
approximately
= 5.7x10^ operations, whereas the enhanced diakoptic method
required N ^ v = 1.5x10^ operations.
This example showed the use of the enhanced diakoptic method for a simple circuit
operating in the millimeter-wave band. In addition, the circuit was enclosed in a cavity
with one resonant mode. Compared to the MOM, the enhanced diakoptic method resulted
in a reduction of the number operations needed to solve for the unknown current
distribution by a factor of 370.
5.3.3.S 60 GHz Shunt Stub
The final example using the enhanced diakoptic method is a transmission line with a
single shunt open circuit stub attached operating in the millimeter-wave band. The shunt
stub is a more practical circuit than the small gap from the previous example. The stub,
located a tx c = 7.0 mm (a/2), has a length L = 0.62 mm and is attached to a transmission,
located at yc = 9.4 mm (bl2), o f width w = 0.28 mm. The circuit is enclosed in a cavity of
the following dimensions: a = 14 mm, b = 18.8 mm and c = 0.635 mm. The substrate
thickness is d \ = 0.127 mm and the relative permittivity is £r l = 4.5(l-j0.0001). An
enclosure of this size has seven resonant modes in the band 58-62 GHz. Since this cavity
contains only a small amount of loss it will be referred to as the high Q package.
188
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The circuit is diakopted into 47 elements with 48 ports. Figure 5.18 compares the
predicted transmission response of the stub in the high Q package using the enhanced
diakoptic method ( k c = 9.625) versus the conventional MOM. Agreement between the
enhanced diakoptic method and the conventional MOM is veiy good except around 59.5
and 62 GHz where the difference is about 2.5 dB.
For the circuit described above, the conventional MOM was performed using
N MOm = 213 expansion functions while the enhanced diakoptic method used N DK = 48
super modes. The average number of expansion functions used to create the super modes
was
= 7.8. Solving the linear system of equations using the conventional MOM
required approximately N mqm = 1-6x10^ operations, whereas the enhanced diakoptic
method required N ^ v ~ 4.3x10^ operations.
This example showed the use of the enhanced diakoptic method for a more
complicated circuit operating in the millimeter-wave band. In addition, the circuit was
enclosed in a cavity with several resonant modes. Compared to the MOM, the enhanced
diakoptic method resulted in a reduction of the number operations needed to solve for the
unknown current distribution by a factor of 370.
5.4
Conclusion
In this chapter the diakoptic method was used to analyze a complex MMICs in an
enclosures. In order to reduce the number of operations required to analyze complex
circuits, Goubau eta l [1] developed the diakoptic m ethod Results obtained using the
diakoptic method agreed well with those obtained using the MOM for a few simple circuits.
However, a significant discrepancy existed between the two methods for more complicated
circuits. A new filtering technique was presented in this chapter which significantly
reduced this discrepancy. The new technique was called the enhanced diakoptic method.
189
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Q
o
IS21I (dB )
-10
-20
-30
-40
MOM
Enhanced Diakoptic
-50
58
59
60
61
62
F (G H z)
Figure 5.18
Comparison of the predicted transmission response of the stub in the high Q
package using the enhanced diakoptic method ( k c = 9.625) versus the
conventional MOM.
190
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
As described above, the main advantage of using the enhanced diakoptic method
instead o f the conventional MOM is the reduction in the number of operations required to
analyze a complex circuit. Table 5.2 summarizes the number o f operations, N ^ qm
, needed to solve the linear system o f equations associated with the conventional
MOM and the enhanced diakoptic method, respectively. Compared to the MOM, the
enhanced diakoptic method resulted in a reduction o f the number operations needed to solve
for the unknown current distribution by a factor of 10 to 370. It is expected that for larger
circuits this reduction will be even greater.
Table 5.2
Comparison of the operation counts for the MOM and enhanced diakoptic methods.
w s o l v / n solv
n mom / n dk
Circuit
N m om
n s° l v
^ MOM
N dk
Stub
123
3 x l0 5
16
14.4
3 x l0 4
10
Stub in larger box
179
9 .5 x l0 5
26
12.8
3 .2 x l0 4
30
Bandpass filter
104
1.8x10s
22
7.9
1.8x10s
100
Small gap
70
5.7X104
14
8.7
1 .5 x l0 2
370
60 GHz stub
213
1.6x10s
48
7.8
4.3x10s
370
<
V5)
n solv
n dk
No previous mention has been made of the CPU time required to analyze a circuit
employing the MOM or the enhanced diakoptic method. It was previously stated that the
majority o f analysis time using the MOM can attributed to two sources: the fill time and the
solve time. For the conventional MOM, filling the matrix requires approximately
operations and solving the matrix requires approximately N f^ M operations where
191
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
n mom
= N ^ M a N mom and N s^om = V 6 ( n \ io m )- N A and M A are defined in Chapter 2.
The times required to perform these two operations are T^o.m and T ^ om where
Tmom = g N moM' Tmqm = <
xN mom and a is a machine dependent constant For a small
circuits such as those presented in this chapter, Tm$ m < T^/o m - Thus, employing the
diakoptic method to reduce Tmom f°r a small circuit will reduce the total solution time
(T mom + Tm o m ) only by a small amount. For example, consider the stub in the larger
box. Employing the enhanced diakoptic method reduced the solve time by a factor of 28.
However, the solve time represented only 6% o f the total solution time. Since all of the
circuits presented in this chapter were relatively small in terms of the number of unknowns,
the overall savings in total CPU time was typically around 5 to 10%. For large circuits, the
solution of the linear system of equations represents a large portion of the analysis time.
Therefore, employing the diakoptic method will result in a large reduction in the total CPU
time.
192
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Chapter 5 References
[1]
Goubau, G., Puri, N.N. and Schwering, F.K., "Diakoptic theory for multielement
antennas," IEEE Transactions on Antennas and Propagation, Vol. AP-30, No. 1,
pp. 15-26, January 1982.
[2]
Schwering, F, Puri, N.N. and Butler, C.M., "Modified diakoptic theory of
antennas," IEEE Transactions on Antennas and Propagation, Vol. AP-34, No. 11,
pp. 1273-1281, November 1986.
[3]
Howard, G.E. and Chow, Y.L., "A high level compiler for the electromagnetic
modeling of complex circuits by geometrical partitioning," IEEE MTT-S
International Microwave Symposium Digest, pp. 1095-1098,1991.
[4]
Harrington, R.F., Time harmonic electromagnetic fields. McGraw-Hill Book
Company, New York, 1961.
[5]
Butler, C.M., "Diakoptic theory and the moment method," 1990 IEEE AP-S
International Symposium Digest, pp. 72-75,1990.
[6]
Howard, G.E. and Chow, Y.L., "Diakoptic theory for microstripline structures,"
1990 IEEE AP-S International Symposium Digest, pp. 1079-1082,1990.
[7]
Singh, S. and Singh, R., "Efficient computation of the free-space periodic Green's
function," IEEE Transactions on Microwave Theory and Techniques, Vol. 39, No.
7, pp. 1226-1229, July 1991.
[8]
Jackson, R.W., Personal Communication.
[9]
Jackson, R.W., "Full-wave, finite element analysis of irregular microstrip
discontinuities," IEEE Transactions on Microwave Theory and Techniques, Vol.
37, No. 1, pp. 81-89, January 1989.
[10]
Howard, G.E. and Chow, Y.L., "Removal of source fringe effects in the diakoptic
theory," 1991 IEEE AP-S International Symposium Digest, p. 99, 1991.
193
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CH APTER 6
C O N C LU SIO N
In this dissertation the analysis of an MMIC in an enclosure was presented. Several
new techniques were developed and the results obtained were compared to a full-wave
method of moments (MOM) analysis. In view of its importance, the full-wave analysis of
packaged MMICs was discussed and a review of the MOM was given at the beginning of
the dissertation. The spectral Green's function was derived using an equivalent
transmission line. In addition, an acceleration technique for the efficient evaluation of
MOM impedance matrices was presented and a special basis function arrangement was
developed to better model the coaxial to microstrip transition. Also, a detailed discussion
of locating the resonances of a dielectric loaded cavity was presented. To verify the
accuracy of the full-wave MOM procedure several circuits were fabricated and measured
while enclosed in a brass cavity. Agreement between the measured and calculated results
was reasonable for all circuits.
The first goal of this dissertation was to examine some of the fundamental aspects
of resonant mode coupling. Using a full-wave analysis, it was shown that package
resonances can have a very significant effect on circuit operation even at frequencies which
are not very close to resonance. The addition of loss to an enclosure reduced resonant
mode coupling, but often did not totally eliminate it. A further reduction in the coupling of
power to resonant modes was obtained by repositioning the circuit Locating areas of high
current in areas of low electric field in the enclosure reduces the power lost to these
resonant modes. However, layout modifications such as these are best applied to a
relatively simple circuit consisting of only a few discontinuities in a moderately sized
enclosure. Since most current MMICs contain more than a few discontinuities,
repositioning the circuit in an enclosure is sometimes not practical.
194
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The next two goals of this dissertation were to develop alternatives to the MOM for
analyzing a circuit in an enclosure. A simpler alternative is o f interest because a full-wave
analysis, although rigorous, is also very complex to implement. In addition, the MOM is
difficult to use for circuit diagnosis or to develop an intuitive understanding of package
enhanced coupling.
Thus, a second goal of this dissertation was to develop a circuit model to describe
resonant mode coupling for use on commercially available CAD packages. The model
which was developed has good accuracy for enclosures with a Q o f over 100 and is useful
for Q's as low as 20. Simple analytical expressions for the entire model are easily
evaluated, making this is a very attractive feature of the circuit model for implementation
into a CAD package. In addition, it requires several orders of magnitude less CPU time
than the MOM. However, implementing the circuit model into a CAD package for a
complicated MMIC in a large enclosure is very tedious. Consequently, this circuit model
may be best suited for MMIC circuits in moderately sized enclosures.
A solution to this inconvenience is an automated procedure for implementing the
circuit model. To test the feasibility of such an automated procedure, a simple CAD
program was written. Using the automated procedure significantly reduced the time and
complexity of entering the model into a CAD package; which allows for the utilization of
the circuit model for any size enclosure.
Although the circuit model has good accuracy and requires several orders of
magnitude less CPU time than the MOM, it does have several limitations. The first
limitation, implementing the circuit model, was discussed above. The second limitation is
that accuracy of the circuit model is dependent on the availability of discontinuity models
(e.g. a TEE element) in CAD package. Some discontinuity models, such as the proximity
coupling between two bends, do not exist in CAD packages. In addition, as the size of
MMIC circuits increase they become too complicated to analyze using a straightforward
195
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full-wave approach. A full-wave analysis of a typical MMIC of moderate complexity may
require the solution of a large system of equations (1000 x 1000 matrix) to accurately
model the complete circuit The need for new ways to analyze MMICs in an enclosure is
therefore apparent Therefore, a second alternative to the MOM is sought for more
complicated MMICs.
The third goal of this dissertation was to develop a method to analyze a complex
MMIC circuit in an enclosure. The diakoptic method [1], [2] was employed in order to
achieve this goal. Results obtained using the diakoptic method agreed well with those
obtained using the MOM for a few simple circuits. However, a significant discrepancy
existed between the two methods for more complicated circuits. A new filtering technique
was developed which significandy reduced this discrepancy. The new technique was
called the enhanced diakoptic method.
The main advantage of using the enhanced diakoptic method compared to the
conventional MOM is that the enhanced diakoptic method will result in a reduction of the
CPU time. The majority of analysis time using the MOM can be attributed to two sources:
the time required to fill the matrix (fill time) and the time required to solve the linear system
of equations (solve time). For circuits of small complexity, the matrix fill time will be
much larger than the solve time. For circuits of moderate complexity, the matrix fill time
and solve time are approximately of the same order and for circuits of large complexity, the
matrix solve time will be much larger than the fill time. For the circuits presented in this
dissertation, the enhanced diakoptic method resulted in a reduction of the solve time by a
factor of 10 to 370 when compared to the MOM. However, the total CPU time required
was only reduced by about 10%. Since all of the circuits presented were relatively simple,
the majority of CPU time was dominated by the fill time. It is was shown that for more
complex circuits, the total CPU time will also be reduced.
196
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A desirable extension of the enhanced diakoptic method would be to develop a more
efficient technique for filling the matrix [3]. This would reduce the fill time for a circuit of
any complexity. After developing an efficient "fill" technique, the next logical step will be
to incorporate the enhanced diakoptic method into an existing a commercially available
MOM based electromagnetic simulator. The computer program developed in this
dissertation for the enhanced diakoptic method utilized many of the subroutines used in the
MOM computer program. The enhanced diakoptic method computer program also required
several new routines; however, they were generally very simple to im plem ent Therefore,
this author is of the opinion that a straightforward implementation of the enhanced diakoptic
method into a current electromagnetic simulator could easily be obtained.
197
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Chapter 6 References
[1]
Goubau, G., Puri, N.N. and Schwering, F.K., "Diakoptic theory for multielement
antennas," IEEE Transactions on Antennas and Propagation, Vol. AP-30, No. 1,
pp. 15-26, January 1982.
[2]
Schwering, F, Puri, N.N. and Butler, C.M., "Modified diakoptic theory of
antennas," IEEE Transactions on Antennas and Propagation, Vol. AP-34, No. 11,
pp. 1273-1281, November 1986.
[3]
Singh, S. and Singh, R., "Efficient computation of the free-space periodic Green's
function," IEEE Transactions on Microwave Theory and Techniques, Vol. 39, No.
7, pp. 1226-1229, July 1991.
198
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A P PE N D IX A
D ER IV A TIO N O F dYv /da)
In this appendix, an analytic expression for dYv /dco is developed. In order to
formulate an analytic expression for dYv Jdco, it will be convenient to express the solution
as.
S l - U S ul
dco vc dk0
(A.1)
y _ y(k) . y ( * + l )
XU ~ XLU + XRU
(A.2)
where
CO= VckQ
vc = velocity of light in free space
U = E o rM
Yj^j and Y $ y l'i are given by equations (2.44) and (2.45), respectively. Substituting
equations (2.44) and (2.45) into equation (A.2) and differentiating with respect to k0
yields:
dYv _ d Y $ )
dkQ
dk0
d YRU
$ 7 l)
dkQ
(A.3)
where
d yLU _ YLU dYrU + y{k)Mk)
*0 ~y%
TU t o 0 + T tu G lu
-\y{k+l)
y(k+1) 3y(A+l)
dXRU _ XRU dXTU
, v ( * + l) / - ( A + l)
RU
~ ^ - Y ^ - - d k ~ '
TU
(k) OILU
M ir "
“
dkp
rrr
v(k-1) s u W
dkn
(A.4)
(A.5)
■*£,[/
ljLU
\^TU cos &k + JY& ^ sin
199
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(A.6)
rJ k+1)
RU ~
-
U 2)
^ +i){[>^+i)]2- [ 4 i +2)f } + ^ +1) ^ Rdkn
~
ypfc+2)
ZRU
dkg
(A.7)
[ l $ +1) cos 0M + j Y & 2) sin 0M ]
p(i) _ Sri^rihdj
(A. 8 )
hi
d^TM _ ~ h y(i)
dkQ
*o& ™
Z-2
dY$
#
7 *7 2- 1*TE
dkQ
k0kZi
200
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(A.9)
(A. 10)
A PPEN D IX B
LARG E ARGUM ENT SPECTRAL G R E E N ’S FU N C TIO N
In this appendix, an asymptotic form of the spectral Green’s function for large
values of kp is developed. When kp » [(£r/O m ax]*o» where (er^r)max = m ax(£„//n ),
the following approximations can be made
kz i ~ - j k p
for i =
(B .l)
tan 6t = tan (kzidi) = —j tanh(/:p4 )
(B.2)
As a result of equations (B .l) and (B.2), the large argument spectral Green's
function derived in this appendix is equivalent to the quasi-static Green's function.
Substituting equation (B.2) into (2.44) and (2.45) yields:
|«>
r„,Ki(.>
tv
1(0
J S 1 (0
for i = l,.„,A:
(B.3)
for i = k + l,...,N
(B.4)
J(:> + \ y & ](,_1) tanh(^pfifj)
[Y g^ + lY & ftm h jkpd;)
TU
where U = E or M and
i(0 _ j£ rik0 _ j2 jrfs ri
TM
'
1(0
TE
1
no kp
-jk p
Vrikono
nokpvc
- j k pvc
(B.5)
(B.6)
27tfHriT]0
vc = velocity of light in free space
The quasi-static driving point admittance at z = z/c for the equivalent circuit shown
in Figure 2.2 is given by:
yv
isl(fc)
S = \ i lu
,( * + ! )
(B.7)
Examining equations (B.3) - (B.6) it is easily seen that Y$s is proportional to
frequency and Y ^s is inversely proportional to frequency; therefore, Y$s ( f ) and Y$s ( f )
can be written as:
201
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(B.8a)
JR
y^s( f ) = j Y i s(fR)
(B.8b)
where f R is a reference frequency.
The frequency dependence of the asymptotic spectral Green's function can be found
by substituting equation (B.8) into (2.57) such that:
kxn QtM ^ r ) ,
Qgs(kxn,kym) =n QS(k
k
k
\-
iZxy \ Kxn’*ym J ~ '*lyx \ Kx r f Kym> ~
(B.9a)
K fR/f
f/fR
V*P
Q n f lf e )
kxnkym f Qt m Wr)
,2
r,f
I
Q t£ W r )
f/fR
f
(B.9b)
if
f R/ f
kyn Qtm Wr ) , ^xn QtE (fR)
k
f R/ f
\ Kp f / f R
(B.9c)
where
o S (/)= -
Y$s ( f )
l
< 2 f(/) = YgS{ f )
(B.lOa)
(B.lOb)
202
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A PPEN D IX C
THE SC ATTER ING M ATRIX
In this appendix, the relationship between the augmented y-matrix and the scattering
matrix (s-matrix) is developed. To formulate the solution of the s-matrix for a network
with Np ports, it will be convenient to express the solution in terms of wave quantities.
The two quantities needed are the incident and reflected waves. The normalized incident,
a, and reflected, b, waves are defined as [1], [2]:
a = ^r-V2[Z + r]I
(C.l)
b = -|r _1/2[Z -r ]I
(C.2)
where
Zoi
r=
(C.3)
Z0N„
and Z is the open circuit impedance matrix. The reflected wave is related to the incident
wave by:
b = Sa
(C.4)
where S is defined as the scattering matrix or the s-matrix. Substituting equations (C.l)
and (C.2) into (C.4) and solving for S yields:
S = r -1/2[Z -r][Z + r ] - y /2
= r_1/2[Z + r - 2r][Z + r]_1r1/2
= r-1^2{ u - 2r[Z + r]-1}r^2
(C-5)
S = U - 2 r V2[Z + r]_1r1/2
The matrix, [ Z + r ] , is the open circuit impedance matrix of the augmented network. The
inverse. [Z + r]-1, is the short circuit admittance matrix of the augmented network, which
is refened to as Ya. The augmented network is defined as a network that combines in
203
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series the original network and the terminating impedance of each port [3]. In terms of Ya ,
the s-matrix is given by:
S = U - r 1/2Yar 1/2
204
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(C.6)
Appendix C References
[1]
Gonzalez, G., Microwave transistor amplifiers. Prentice-Hall, Inc., New Jersey,
Chapter 1, 1984.
[2]
Ha, T.T., Solid-state microwave amplifier design. John W iley & Sons, Inc., New
York, Chapter 2, 1981.
[3]
Balabanian, N. and Bickart, T.A., Electrical network theory. Robert E. Krieger
Publishing Company, Florida, Chapter 8, 1983.
205
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A PPE N D IX D
THE ELECTRICAL CHARACTERISTICS OF THE
D IELECTR IC ABSO R BER
When measuring the performance of the circuits discussed in Chapter 3, a lossy
dielectric material (absorber) was attached to the cover of the enclosure in an effort to
reduce the effects of the resonant modes. However, the electrical characteristics of the
absorber were not known. In this appendix, a simple circuit will be used to characterize the
lossy dielectric material. In order to determine the electrical properties of the absorber, the
relative permittivity and permeability of the absorber were varied until the agreement
between the simulated and measured results were excellent
The simple circuit consists of a transmission line with a large gap in the center as
shown in Figure D .l. The transmission line, located a ty c = 12 mm, has a width o f w =
1.4 mm and a gap of g = 10.5 mm. The circuit is enclosed in cavity A which was
determined in Chapter 3 to have one resonant mode in the band 9-12 GHz. Since this
cavity contains only a small amount of loss it will be referred to as the high Q cavity A
The optimum relative permittivity and permeability of this absorber were determined to be
60(l-y'0.12) and 7.3(l-;'0.3), respectively. Figure D.2 compares the calculated using the
optimum parameters of the absorber to the measured transmission coefficient (IS2 1 I) of the
large gap. Agreement between the measured and calculated results is excellent
206
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b
W
4
Figure D .l
Geometry of a transmission line with a gap in the center. The transmission
line, located atyc = 12 mm (b/2), has a width of w = 1.4 mm and a gap of g
= 10.5 mm.
207
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Low Q - Dielectric Cover
Yc = 12 mm
-10
CQ
-20
- MOM
* Measured
-30
-40
9
11
10
12
F (GHz)
Figure D.2
Comparison of the calculated and measured transmission coefficient of the
large gap located at y c = 12 mm in the low Q cavity A.
208
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/
BIBLIO G R A PH Y
Balabanian, N, and Bickart, T.A. Electrical Network Theory. Florida: Robert E.
Frieger Publishing Company, 1983.
Balanis, Constantine, A. Advanced Engineering Electromagnetics. New York: John
Wiley & Sons, 1989.
Collin, Robert E. Field Theory of Guided Waves. New York: IEEE Press, 1991.
Gonzalez, G. Microwave Transistor Amplifiers. New Jersey: Prentice Hall Inc, 1984.
Ha, T.T. Solid -State Microwave Amplifier design. New York: John Wiley & Sons,
1981.
Harrington, Roger F. Field Computation by Moment Methods. Florida: Robert E.
Frieger Publishing Company, 1968.
Itoh, Tatsuo. Numerical Techniques for Microwave and Millimeter-Wave Passive
Structures. New York: John W iley & Sons, 1989.
209
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