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Synthesis of high purity ferroelectric materials for microwave applications and electromagnetic numerical characterization of complex materials

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U n iv e r s it y
of
C a l if o r n ia
Los Angeles
Synthesis of High Purity Ferroelectric Materials for Microwave
Applications and Electromagnetic Numerical Characterization of
Complex Materials
A dissertation submitted in partial satisfaction
of the requirements for the degree
Doctor of Philosophy in Electrical Engineering
|
by
|
Franco De Flaviis
Ii
1997
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i
UMI N u m b er:
9818034
UMI Microform 9818034
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UMI
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The dissertation of Franco De Flaviis is approved.
Tatsuo Itoh
to-. 4 .
W. G. Clark
O. M. Stafsudd Comrnitte^Co-Chair
N. G. Alexopoulos Committed Co-Chair
University of California, Los Angeles
1997
ii
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I
To my parents ,
i
1
Arnaldo and Fernanda De Flaviis
ii
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Table of Contents
List of figures
List of Tables
List of symbols
Acknowledgments
Vita
Abstract of the dissertation
Chapter 1 Ferroelectric Materials
1.1 Relation between dielectric constant and polarization
1.2 Dipolar polarization theory for static field
1.3 Dipolar polarization theory for time varying field
1.4 Ferroelectric theory
1.5 Structural origin of the ferroelectric state
1.6 Hysteresis
1.7 Effect of grain size on ferroelectric behavior
References Chapter 1
Chapter 2 Ferroelectric Material Synthesis by Sol-gel Technique
2.1 Synthesis of high purity ferroelectric materials by Sol-gel chemistry
2.2 Thin film and thin ceramic sample preparation
References Chapter 2
Chapter 3 Electrical Measurements of Ferroelectric Materials
3.1 Thin film electrical measurements
3.2 Thin ceramic low frequency electrical measurements
iv
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3.3 Thin ceramic high frequency electrical measurements
62
References Chapter 3
71
Chapter 4 Microwave Phase Shifter Design using Ferroelectric Materials
72
4.1 Thin ceramic tunable capacitor
72
4.2 Distributed thin ceramic BTO tunable microstrip line
79
4.3 Distribute thin ceramic BST tunable microstrip line
90
4.4 Power requirements and power handling consideration for microstrip
phase shifter
98
References Chapter 4
103
Chapter 5 Diaz Fitzgerald Time Domain Technique for the Solution of
Maxwell Equations in the Time Domain
104
3.1 Introduction
103
3.2 The Fitzgerald pulley and rubber-band model
106
3.3 Extension to dielectric lossy materials
110
3.4 Extension to Debye dielectric materials
111
3.3 Extension to frequency dependent magnetic materials
117
3.6 Eigenvalue problem
124
5.7 Scattering problem
126
5.8 Echo experiments
129
5.9 Comparison with other numerical techniques
133
References Chapter 5
139
Chapter 6 Electronically Steerable Beam Microstrip Antenna Array
6.1 Project Phases
142
142
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6.2 Single patch microstrip antenna design
144
6.3 Two patches antenna array design
1S2
6.4 Phase shifter design using D-FTD technique
158
6.5 Design of the RF feeding network
162
6.6 Test and measurements
169
References Chapter 6
173
Chapter 7 Conclusion
174
Appendix A Band pass Filter Design
176
Appendix B Vector Potential Formulation
181
Appendix C Vector Potential Formulation for Electric Debye Medium
183
Appendix D Vector Potential Formulation for Magnetic Debye Medium
185
Appendix E DC-Block design
187
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List of figures
Fig. 1.1.1
Dielectric material between parallel plates
2
Fig. 1.1.2
Frequency dependence of polarization mechanisms
4
Fig. 1.2.1
Bistable dipole model
7
Fig. 1.3.1
dipole flux in time varying regime
10
Fig. 1.3.2
Frequency variation of dielectric parameters.
13
Fig. 4.1.1
Curie-Weiss law
16
Fig. 1.5.1
Perovskite structure
16
Fig. 1.5.2
Atomic displacements as BaTi0 3 approaches Tc
19
Fig. 1.5.3
TiC>6 octahedra displacements in the ferroelectric transition
of BaTiC>3
19
Fig. 1.6.1
Hysteresis loop for polarization
20
Fig. 1.6.2
a) Domain microstructure without an applied field
b) Domain growth in direction of an applied field
Fig. 1.6.3
21
Effects of Temperature on BaTi0 3 hysteresis loop,
a-c ferroelectric state, d paraelectric state
22
Fig. 1.7.1
Ferroelectric behavior of different particle size of BaTi0 3
23
Fig. 2.1
Condensation of =Si-0-Si= bonds
25
Fig. 2.2
Additional linkage of =Si-OH groups
26
Fig. 2.1.1
Firing schedule for lead titanate (PbTi0 3 )
31
Fig. 2.1.2
Schematic diagram for the production of PbTi0 3
32
and Pbi-xCaxTi0 3
Fig. 2.1.3
Schematic diagram for the production of BaTi0 3 and
Bai.xSrxTi0 3 thin film and powder
vii
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Fig. 2.1.4
X-rays analysis obtained for our precursor for the
production of PbTi0 3
Fig. 2.1.5
33
X-rays analysis for single layer thin film of PbTi0 3 deposited
on alumina substrate (AI2O 3) and nicrome-60 (NiCr)
copper (Cu) contacts
Fig. 2.1.6
34
X-rays analysis for five layer thin film of PbTi0 3 deposited
on alumina substrate (AI2O 3) and nicrome-60 (NiCr) copper
(Cu) contacts
Fig. 2.1.7
34
X-rays analysis for thin film Pbi.xCaxTi0 3 on MgO substrate
with percentage of calcium varying from 0% to 40%
35
Fig. 2.1.8
X-rays analysis for BaTi0 3 ceramic sample
36
Fig. 2.1.9
X-rays analysis for Bao.9 Sro.iTi0 3 ceramic sample
36
Fig. 2.1.10
Typical thermogravimetric station
37
Fig. 2.1.11
Thermogravimetric analysis for PTO sample
38
Fig. 2.1.12
Thermogravimetric analysis for BTO sample
38
Fig. 2.1.13
Typical differential thermal analysis station
39
Fig. 2.1.14
Typical result of DTA on ferroelectric sample
39
Fig. 2.1.15
DTA result for BTO ceramic sample
40
Fig. 2.1.16
DTA result for PTO and PCT ceramic samples (Ca - 0%-40%)
40
Fig. 2.2.1
Schematic of multiphase FEM film due to metal diffusion.
41
Fig. 2.2.2
Parallel plate capacitor used to estimate the change of dielectric
constant versus bias: a) Three layer metallization utilized to
reduce the diffusion o f the copper into the film, b) Platinum
Fig. 3.1
metallization on MgO c) bulk ceramic with copper contacts
42
Typical P-E hysteresis loop and RF signal around bias DC field
47
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Fig. 3.2
dP/dE behavior for material having P-E hysteresis loop.
Fig. 3.1.1
Electrical measurement of capacitance vs bias voltage for
thin film PTO
Fig. 3.1.2
50
Electrical measurement o f conductance vs bias voltage for
thin film PTO
50
Fig. 3.1.3
Equivalent dielectric constant vs the amorphous layer thickness
Fig. 3.1.4
Electrical measurement of capacitance versus bias voltage for
five layer thin film PTO sample.
Fig. 3.1.5
54
54
Capacitance versus bias field for the thin film BTO sample on
MgO substrate
Fig. 3.2.1
53
Capacitance versus temperature for 4 layer thin film BTO on
MgO substrate
Fig. 3.1.7
52
Electrical measurement o f resistance versus bias voltage for
five layer thin film PTO sample.
Fig. 3.1.6
48
55
Schematic layout of the sample geometry and the setup to perform
the measurement of electrical parameters versus temperature
56
Fig. 3.2.2
Capacitance versus temperature, for the barium titanate sample
56
Fig. 3.2.3
Conductance versus temperature, for the barium titanate sample
57
Fig. 3.2.4
Frequency dependence o f capacitance for the BTO sample
58
Fig. 3.2.5
Frequency dependence o f conductance for the BTO sample
58
Fig. 3.2.6
High voltage set up for low frequency ceramic capacitor
measurement.
59
Fig. 3.2.7
Bias dependence of capacitance for the barium titanate sample
59
Fig. 3.2.8
Bias dependence of conductance for the barium titanate sample
60
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Fig. 3.2.9
!
for ferroelectric material having P-E hysteresis loop
60
Fig. 3.2.10
Capacitance versus bias voltage for BTO ceramic sample
61
Fig. 3.2.11
Conductivity versus bias voltage for BTO ceramic sample
61
Fig. 3.3.1
Schematic of the resonant cavity set-up for the measurement
of the ceramic samples
62
Fig. 3.3.2
Rectangular resonant cavity field distribution for the TE 101 mode
64
Fig. 3.3.3
Necessary steps to perform comparative resonant cavity
i
measurements
I
j
Losses due to a small RF signal near specific bias points
Fig. 3.3.4
i|
S 11 measurement obtained for the resonant cavity
with different BTO-BST samples
Fig. 4.1.1
68
Tunable phase shifter using lumped tunable capacitor as tuning
element
Fig. 4.1.2
66
72
Measurement set-up used to measure the scattering parameters
of the phase shifter
73
Fig. 4.1.3
Detail of the band pass filter used as insulator
74
Fig. 4.1.4
Usable bandwidth (flat one) after the insertion of the bias network 74
Fig. 4.1.5
Magnitude of reflection coefficient (Sh ) for bias and
unbias condition
75
Fig. 4.1.6
Phase of reflection coefficient (S 11) for bias and unbias condition
75
Fig. 4.1.7
Magnitude of transmission coefficient (S21) for bias and unbias
condition
Fig. 4.1.8
76
Phase of transmission coefficient (S21) for bias and unbias
condition
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76
Fig.4.1.9
,
obtained by single capacitors
77
Fig.4.1.10
Reflection based phase shifter
79
Fig. 4.2.1
Schematic structure of ferroelectric microstrip based phase shifter 79
Fig.4.2.2
Detailed layout o f planar ferroelectric phase shifter
Fig.4.2.3
Mounting schematic of the planar phase shifter on conventional
80
microstrip transmission line
82
Fig. 4.2.4
Matching circuit used for the phase shifter design
83
Fig. 4.2.5
Modified SMA tab-type for on chip FEM measurements
83
Fig. 4.2.6
Measured used to characterize the transition from coaxial
i
to microstrip
Fig. 4.2.7
j
!
Possible microwave network to combine the phase shift
84
Measured and modeled S u for the transition in Fig. 4.2.5
in the range 1-10 GHz
84
Fig. 4.2.8
Reflection coefficient for the FEM phase shifter at 1.85 GHz
86
Fig. 4.2.9
Phase measurement for the transmission coefficient for
ij
different bias conditions.
87
!
|
Fig. 4.2.10
Magnitude measurement for the transmission coefficient
for different bias conditions.
Fig. 4.2.11
Magnitude measurement for the reflection coefficient
for different bias conditions.
Fig. 4.2.12
87
88
Extracted behaviour of dielectric constant and loss vs. bias
for BTO
89
Fig. 4.2.13
DC power requirement for the BTO phase shifter
89
Fig.4.3.1
S21 phase measurement for the BST ceramic sample
91
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j
I
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Fig.4.3.2
RF phase shift versus applied bias field for the BST
ceramic sample
92
Fig.4.3.3
S 21
magnitude measurement for the
BST
ceramic sample
92
Fig.4.3.4
S 11
magnitude measurement for the
BST
ceramic sample
93
Fig.4.3.5
S parameter measurement for Vbias=OV and Vbias=250V
94
Fig.4.3.6
S 11 magnitude modeled vs. measurement for Vbias=100V
96
Fig.4.3.7
S 21 magnitude modeled vs. measurement for V b ia s = 1 0 0 V
96
Fig.4.3.8
S 21 phase modeled vs. measurement for V b ia s = 1 0 0 V
97
Fig.4.3.9
Effective dielectric constant and loss extracted values vs
applied bias field
97
Fig.4.4.1
DC Power requirement versus bias field for the BST phase shifter 99
Fig 5.2.1
Array of rigid pulleys connected by rubber bands
106
Fig.5.2.2
Action-reaction mechanism of propagation of motion
106
Fig.5.2.3
Pulley and rubber-bands represented as springs
107
Fig.5.2.4
Torque resulting form the composition of the four forces
108
Fig.5.3.1
Pulley immersed in a viscous fluid bath.
1 10
Fig.5.4.1
Mechanical model for single electrical Debye materials.
112
Fig.5.4.2
Coaxial arrangement of two Debye terms
114
Fig.5.5.1
Magnetic Debye material model: the two springs are connected
in series
118
Fig.5.5.2
Spring in series with a dissipative device.
Fig.5.5.3
Series representation of two magnetic Debye terms as oscillators
in series
Fig.5.6.1
118
121
Resonance frequencies for square resonator
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125
Fig.5.7.1
Comparison between computed and exact solution of the inner
field for a lossless dielectric cylinder, along a cut i=200
127
Fig.5.7.2
Comparison between computed and exact solution
128
Fig.5.7.3
Comparison between our model and FDTD
128
Fig.5.8.1
Reflection coefficient for air-water interface
130
Fig.5.8.2
Reflection coefficient for two-pole Debye material
131
Fig.5.8.3
Reflection coefficient for single magnetic Debye materials
131
Fig.5.8.4
Reflection coefficient for double magnetic Debye materials
132
Fig.5.8.5
Reflection coefficient for wide band absorbing material
133
Fig.5.9.1
Comparison of the field components location in the elementary
cell between 2-D FDTD and D-FTD
Fig.5.9.2
Field distribution and experimental geometry for the grid
dispersion numerical experiment
Fig.5.9.3
136
Spectrum content of the signal observed at the same observation
point for FDTD and D-FTD
Fig.5.9.5
135
(a) Time response of FDTD due to locally plane wave (b) Same
quantity for D-FTD formulation
Fig.5.9.4
134
137
(a) Phase response of FDTD plotted versus frequency (b) Same
quantity for D-FTD
137
Fig. 6.1.1
Steerable beam microstrip antenna system
144
Fig. 6.2.1
Schematic of a patch microstrip antenna
145
Fig. 6.2.2
First order equivalent model for microstrip patch antenna
146
Fig. 6.2.3
Patch antenna layout utilized in the full wave analysis
147
Fig. 6.2.4
Characteristic impedance versus line width for duroid RT5870
148
Fig. 6.2.5
Propagation constant versus line width for duroid RT5870
148
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I
|
Fig. 6.2.6
Quarter wave length versus line width for duroid RT5870
149
Fig. 6.2.7
Characteristic impedance versus frequency for duroid RT5870
149
Fig. 6.2.8
Propagation constant versus frequency for duroid RT5870
150
Fig. 6.2.9
Magnitude of the S u versus frequency for the patch antenna
150
Fig. 6.2.10
Phase of the S u versus frequency for the patch antenna
151
Fig. 6.2.11
Radiation pattern at <j)=0° (H-plane)
151
Fig. 6.2.12
Radiation pattern at <J>=90° (E-plane)
152
Fig. 6.3.1
Array of antenna
152
Fig. 6.3.2
Microstrip patch antenna
153
Fig. 6.3.3
Theoretical radiation pattern for two element microstrip antenna
154
Fig. 6.3.4
Geometry used to analyze the radiation pattern
155
Fig. 6.3.5
Radiation pattern in the $ plane for a difference of the feeding
i
!
arm length Al=0mm
Fig. 6.3.6
156
Radiation pattern in the $ plane for a difference of the feeding
arm length Al=12mm
|
Fig. 6.3.7
I
156
Radiation pattern in the $ plane for a difference of the feeding
arm length Al=18mm
!
Fig. 6.3.8
157
Radiation pattern in the 0 plane for a difference of the feeding
arm length Al=34mm
157
Fig. 6.4.1
Microstrip structure
159
Fig. 6.4.2
Time variation of at different positions along the direction
of propagation
160
Fig. 6.4.3
Efective dilectric constant of the BST microstrip
162
Fig. 6.5.1
Array feeding network design
163
Fig. 6.5.2
Smith chart for the design of the feeding network
164
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Fig. 6.5.3
Array feeding network design
165
Fig. 6.5.4
Array feeding network design
165
Fig. 6.5.5
Fig. 6.5.6
S parameters for different length of the dc-block
167
Sparameters of the feeding network under different bias
conditions
168
Fig. 6.5.7.
Bias network of the microstrip array system
169
Fig. 6 .6 .1
Measured reflection coefficient for the array system
170
Fig. 6.6.3
Radiation pattern measurement set-up
170
Fig. 6.6.4
Measured and computed radiation pattern for the microstrip array 171
Fig. 6.6.5
Measured Radiation pattern under different bias conditions
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171
List o f Tables
|
Table 3.1.1
Electrical measurement of thin film PTO ferroelectric sample
49
Table 3.1.2
Electrical measurement of five layer PTO sample
52
Table 3.3.1
Resonant cavity measurements for different size, same material
sample
67
Table 3.3.2
Resonant cavity accuracy measurements
68
Table 3.3.3
Loss tanS for different samples of BTO and BST material
69
Table 4.1.1
Comparison between different topologies to combine two phase
shifters
|
It
I
i
i
Table 4.3.1
Effect of wire bond length on the performance o f the phase shifter 95
Table 5.2.1
Equivalence between electrical and mechanical quantities in our
model.
Table 5.4.1
i
i
f
if
78
110
Equivalence between electrical and mechanical quantities in our
model.
Table 5.4.2
113
Equivalence between electrical and mechanical quantities in our
model.
Table 5.4.3
116
Equivalence between electrical and mechanical quantities in our
model.
Table 5.5.1
117
Equivalence between electrical and mechanical quantities in our
model.
Table 5.5.2
Table 6.5.1
122
Equivalence between electrical and mechanical quantities in our
model.
123
S parameter of the feeding network
168
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List of symbols
FEM
Ferroelectric material
BTO
Barium titanium oxide
BST
Barium modified strontium titanium oxide
ST
Strontium titanium oxide
PTO
Lead titanium oxide
PCT
Lead calcium modified titanium oxide
Tc
Curie temperature
DTA
Differential thermal analysis
TGA
Thermal gravimetric analysis
RF
Microwave signal
DC
Voltage component at zero frequency
FDTD
Finite difference time domain
D-FTD
Diaz Fitzgerald time domain
MPIE
Mixed potential integral equation
DFT
Discrete Fourier trasform
SLL
Side lobe level
MDS
Microwave Design System
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Aknowledgments
I would like to thank Dr. Nicolaos G. Alexopoulos for making this research project
enjoyable and successful through his patient guidance and both moral and financial
support.
My thanks also to Prof. Oscar M. Stafsudd for his precious guidance in the
synthesis of ferroelectric materials and for the assistance in the realization and testing
of the materials.
Special thanks to Dr. Rodolfo E. Diaz and to Mr. Massimo Noro for the
helpful discussions and criticism on the work on the numerical analysis.
I also express my gratitude to Professor Tatsuo Itho, Professor George W. Clark for
serving on my Ph.D. dissertation committee.
xviii
±
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Vita
October 25,1963
Bom Teramo, Italy
1990
Electronic Engineer Degree (Laurea)
Ancona University, Italy.
1990
State Engineer Certification
Italy.
1994
Master Electrical Engineering in Electromagnetics
University of California, Los Angeles
1994-1996
Teaching Assistant and Research Assistant
Depatment of Electrical Engineering
University of California, Los Angeles
PUBLICATIONS AND PRESENTATIONS
[1] F. De Flaviis, T. Rozzi, F. Moglie, A. Sgreccia, and A. Panzeri, “Accurate
Analysis and Design of Millimeter Wave Mixers,” IEEE Trans. Microwave Theory
Tech., vol. MTT-41, pp. 870-873, May 1993.
xix
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[2] F. De Flaviis, M. J. Tsai, S. Chang Wu, and N. G. Alexopoulos, “Optimization of
Microstrip Open End,” in IEEE Antennas and Propagation Intematinal Symposium.
Los Angeles pp. 1490-1493, June 18-23 1994.
[3] F. De Flaviis and S. A. Maas, “X-Band Doubly Balanced Resistive FET Mixer
with Very Low Intermodulation,” IEEE Trans. Microwave Theory Tech., vol. MTT43, pp. 457-460, February 1995.
[4] F. De Flaviis, J. G. Ho, N. G. Alexopoulos, and O. M. Stafsudd, “Ferroelectric
Materials for Microwave and Millimeter Wave Applications,” in SPEE The
international Society for Optical Engineering, Smart Structures and Materials. S.
Diego CA pp. 9-21, February 1995.
[5] F. De Flaviis, D. Chang, J. G. Ho, N. G. Alexopoulos, and O. M. Stafsudd,
“Ferroelectric Materials for Wireless Communications,” in COMCON 5 5th
International Conference on Advances in Communication and Control. Rithymnon,
Crete (Greece) June 26-30 1995.
[6 ] F. De Flaviis, M. Noro, and N. G. Alexopoulos, “Diaz-Fitzgerald Time Domain
(D-FTD) Method Applied to Dielectric Lossy Materials,” in ICEAA 95 International
Conference on Electromagnetics in Advanced Applications. Torino (Italy) pp. 309311, 12-15 September 1995.
[7] F. De Flaviis, D. Chang, N. G. Alexopoulos, and O. M. Stafsudd, “High Purity
Ferroelectric Materials by Sol-Gel Process for Microwave Applications,” in ICEAA
95 International Conference on Electromagnetics in Advanced Applications. Torino
(Italy) pp. 12-15 September 1995.
[8 ] F. De Flaviis, M. Noro, and N. G. Alexopoulos, “Diaz-Fitzgerald Time Domain
(D-FTD) Technique Applied to Electromagnetic Problems,” in IEEE MTT-S Int.
Microwave Symp. S. Francisco pp. 1047-1050, June 1996.
xx
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[9] F. De Flaviis, N. G. Alexopoulos, O. M. Stafsudd and D. Chang, “Ferroelectric
Materials for Microwave Applications,” in 1996 Int. Union of Radio Science
Boulder, Colorado pp. B/A/Dl-4, January 1996.
[10] F. De Flaviis, D. Chang, N. G. Alexopoulos, and O. M. Stafsudd, “High Purity
Ferroelectric Materials by sol-Gel Process for Microwave Applications,” in IEEE
MTT-S Int. Microwave Symp. S. Francisco pp. 99-102, June 1996.
[11] F. De Flaviis, N. G. Alexopoulos, and O. M. Stafsudd, “Planar Microwave
Integrated Phase Shifter Design with High Purity Ferroelectric Materials,” IEEE
Trans. Microwave Theory Tech., vol-54, pp.963-969 June 1997.
[12] M. J. Tsai, F. De Flaviis, O. Fordham, and N. G. Alexopoulos, “Modeling Planar
Arbitrarily-Shaped Microstrip Elements in multi-layered Media,” IEEE Trans.
Microwave Theory Tech., March 1997.
[13] F. De Flaviis, M. Noro, R.E. Diaz, and N.G. Alexopoulos, “Diaz-Fitzgerald
Time Domain Method Applied to Electric and Magnetic Debye Materials,” in
Applied Computational Electromagnetics ACES Symposium. Monterey California,
1997.
[14] F. De Flaviis and N. G. Alexopoulos “Ferroelectric Based Low Cost Steerable
Antenna System for Wireless Communications,” in COMCON 6 6 th International
Conference on Advances in Communication and Control. Corfu, Cipro (Greece) June
23-27 1997.
[15] F. De Flaviis, M. Noro, R.E. Diaz, and N.G. Alexopoulos, “Time Domain Vector
Potential Formulation for The Solution of Electromagnetic Problems” IEEE AP-S
Int. Symp. Montreal, Canada, July 1997.
[16] F. De Flaviis and N.G. Alexopoulos, “Low Loss Ferroelectric Based Phase
Shifter for High Power Antenna Scan Beam System” IEEE AP-S Int. Symp.
Montreal, Canada, July 1997.
xxi
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[17] F. De Flaviis, M. Noro and N.G. Alexopoulos, “Applications o f Time Domain
Vector Potential Formulation to 3-D Electromagnetic Problems” TSMMW’97
Topical Symposium on Millimeter Waves, Kanagawa, Japan July 1997.
[18] F. De Flaviis, M. Noro, R.E Diaz, and N.G. Alexopoulos, “The Diaz-Fitzgerald
Time Domain Model for the Solution of Electromagnetic Problems” NATO-ASI
Conference Samos Greece, July 26-August 5,1997.
xxu
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ABSTRACT O F THE DISSERTATION
Synthesis of High Purity Ferroelectric Materials for Microwave Applications and
Electromagnetic Numerical Characterization of Complex Materials
by
Franco De Flaviis
Doctor of Philosophy in Electrical Engineering
University of California, Los Angeles, 1997
Professor N. G. Alexopoulos, Co-Chair
Professor O. M. Stafsudd, Co-Chair
Ferroelectric materials (FEM) are very attractive because their dielectric constant can
be modulated under the effect of an externally applied electric field perpendicular to
the direction of propagation of a microwave signal. FEM may be particularly useful
for the development of a new family of planar phase shifters which operate up to Xband. The use of FEM in the microwave frequency range has been limited in the past
due to the high losses of these materials at microwave frequencies and due to the high
electric field necessary to bias the structure in order to obtain substantial dielectric
constant change. We demonstrate in this research how a significant reduction in
material losses is possible.
XX1U
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We achieve this by using a new sol-gel technique to produce FEM. We also
demonstrate how the use of thin ceramics reduces the required bias voltage needed to
change the property of the FEM, with almost no power consumption. As example of
application a tunable phase shifter and an array of microstrip patch antennas where
the beam is electronically scanned are constructed and tested.
For the design and analysis of the ferroelectric material, we decide to use numerical
techniques which rely on the time domain discretizzation of Maxwell equations. Time
domain techniques are easy to implement and capable to model complex structures
with minimal effort. On the other hand they require a lot more computational power
with respect to frequency domain techniques. We could use several other techniques
which have been proposed in the past for the solution of Maxwell's equations in the
time domain, such as the Finite Difference Time Domain (FDTD), which relies on the
discretization of Maxwell's equations in time, or the Transmission Line Model (TLM)
which is based on Huyghens principle. These techniques are attractive because of
their simplicity but are limited because they do not model effectively highly
dispersive and lossy materials such as ferroelectric ceramics. In this research a new
technique based on discretization of Maxwell's equations in the vector potential form
is presented. This new technique has the advantage of condensed node representation
for the field components, and offers an easy way to treat lossy or dispersive media.
xxiv
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C h apter 1
Ferrolectric Materials
Ferroelectric materials are nonlinear dielectrics having a dielectric constant which is a
function o f the electric field. The nonlinear behavior of these materials makes them
good candidates for the realization of advanced high frequency devices operating up
to the centimeter range. Ferroelectrics have been successfully employed in many
optical devices, but their application at microwave and millimeter wave frequencies
has been limited, mostly due to the high losses and to the large bias voltage required
to significantly change the electrical properties of the bulk material. However, today
there are several new techniques available to produce ferroelectric thin ceramics and
thin films which require only a medium or low bias voltage to change significantly
the dielectric constant. In addition, with the use of the sol-gel process ferroelectric
materials can be produced with acceptable loss characteristics. These processes open
the way for the development of a new family of planar devices which are compatible
with conventional microstrip circuits. These new devices can be integrated in an
existing microwave system.
From the material point of view, ferroelectric materials are a class of nonlinear
dielectric ceramics [ 1, 2 ], which present a reversible spontaneous alignment of
electric dipoles. Nonlinear dielectrics are an important class of crystalline ceramics
which can exhibit very large dielectric constant, due to spontaneous alignment or
polarization of electric dipoles. The spontaneous alignment o f electric dipoles result
in a crystallographic phase transformation below a critical temperature Tc. The
electric dipoles are ordered parallel to each other within the crystal in regions called
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domains. When an electric field is applied, the domains can switch from one direction
of spontaneous alignment to another. This gives rise to very large changes in
polarization and dielectric constant. Hence the name nonlinear dielectrics.
1.1 Relation between Dielectric Constant and Polarization
In order to give a good interpretation of ferroelectricity, it is important to introduce
some basic concepts which explain the relationship between dielectric constant and
polarization in a dielectric. Consider a parallel plate capacitor as shown in Fig. 1.1.1
area A
dielectric £
Fig.1.1.1 Dielectric material between parallel plates
The expression for the capacitance of such capacitor is given by the well known
expression
( 1. 1. 1)
Since the capacitance represents the ratio between the stored charge and the applied
voltage (C=Q/V), clearly the presence of the dielectric increases the ability of the
plates to store charges. The reason for this phenomenon is in the fact that the material
contains charged species which can be displaced in response to the field across the
material. There are four primary mechanisms of polarization in ceramics [3]. Each
2
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mechanism involves a short range motion of charge and contributes to the total
polarization o f the material. These mechanisms are:
1) Electronic polarization (Pe) due to the shift o f the valence electron cloud of the
ions within the material with respect to the positive nucleus. This mechanism
occurs at very high frequencies (order of 1015Hz) in the UV range.
2) Atomic polarization (Pa) occurring in the frequency range between 1012 to 1013 Hz
due to the displacement of positive and negative ions in a material with respect to
each other. In this case, a resonance absorption occurs at a frequency characteristic
of the separation between ions.
3) Dipole polarization ([Pj) due to the perturbation of the thermal motion of ionic or
molecular dipoles, producing a net dipolar orientation in the direction o f the
applied field. This polarization can be further divided in two mechanisms, the first
one involves molecules with a permanent dipole moment, which may be rotated
against an elastic restoring force about an equilibrium position. The second
mechanism of dipolar polarization involves the rotation of dipoles between two
equivalent equilibrium positions. It is the spontaneous alignment of dipoles in one
of the equilibrium positions which gives rise to the nonlinear polarization behavior
of ferroelectric materials. The first type of polarization occurs at a frequency of the
order of 10l 1 Hz, while the second one occurs in the frequency range between 103
to 106 Hz at room temperature.
4) Interfacial polarization (Pi) occurring in the frequency order of 10' 3 Hz. This is not
considered in this research.
For our formulation we will consider only the dipole orientation, since it is the only
one which is important in the microwave range. A graph which shows the effect on
3
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dielectric constant (s') and losses (tanS) due to the different polarizations versus
frequency is shown in Fig. 1.1.2
tan 5 A
log(f)
log(f)
Fig.1.1.2 Frequency dependence of polarization mechanisms
We can think of the total field displacement in a dielectric material (D) as the sum of
an external electric field E and the polarization P of the material
D = enE + P = e * E
(1.1.2)
Upon introduction of an absolute equivalent dielectric constant £* we can define
P = E ( e * - f 0) .
(1.1.3)
The corresponding equivalent relative dielectric constant is £r *=£*/£0 and so we can
write
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If we introduce an electrical susceptibility % 35
P
(1.1.5)
eq. (1.1.4) becomes
er*= x + l
( 1.1.6)
This is the relation between the dielectric constant and the total polarization in the
material. Now, if we desire to have a relationship between £* and the fundamental
polarizability of the charge mechanism contributing to the total polarization P, we
will need to consider a new model. Let us assume for instance that the total P can be
obtained as a summation of single dipoles (/if), so if Ni is the number of dipoles of
species i, P is given by
(1.1.7)
Since the average dipole moment of a charged particle is proportional to the local
electric field (E') which acts on the particle
( 1. 1.8)
where a, is the polarizability of the average dipole moment per unit local field
strength. Thus the total polarization is
P = N fr E
(1.1.9)
5
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For gases with little molecular interaction, the locally acting field E ' is the same as
the externally applied field E a. However, for insulating dielectric solids, the
polarization of the surrounding medium substantially affects the magnitude of the
local field. Mosotd was the first to derive the local field contribution by integrating
the normal component of the polarization vector over the surface of a spherical cavity
in the material. The obtained result is:
E' = Ea + P /3£„
(1.1.10)
and by using eq. (1.1.9)
< u ll)
•
Using the definition of P (eq. 1.1.3) and k* from eq. (1.1.11) we obtain
£ * -1
£*+2
1
NfX:
3£0
(1.1.12)
which is the classical Clausius-Mosotti equation. It describes the relationship between
the complex dielectric constant of material £*, the number of polarizable species A//,
and the polarizability of the species ctj.
Since there are four major classes of
polarizable species in ceramics and glasses, i. t . a ^ a ^ a ^ a , then
£ f ^ = 3 ^ [N .« . + * .« . + N A + *<«,]
(1113)
1.2 Dipolar polarization theory for static field
There are four primary mechanisms of polarization in ceramics. Each mechanism
involves a short-range motion of charge and contributes to the total polarization of the
material. The polarization mechanisms include: electronic polarization (Pe), atomic
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polarization (Pa), dipole polarization (Pd), and interfacial polarization (Pi). As
mentioned before we concentrate our attention on the dipolar polarization effect Let
consider a bistable dipole as shown in Fig. 1.2.1
@
nvo possiote
equilibrium position
Fig.1.2.1 Bistable dipole model
There is a random oscillation (due to temperature) of the ion from the equilibrium
position. The probability of a jump (p) between energy wells is exponentially related
to the temperature and the energy barrier V (without an external electric field)
p = A e-v,Kr
,
(1.2.1)
where T is the absolute temperature in °K and K is the Boltzman constant. If we now
apply an electric field the potential energy of the two sites will become different by
an amount
0i - 0 2 = e(bE) = ebEcosd
(1.2.2)
where b is the distance between the potential wells, 0 the angle between E and the
jump vector b. Thus this model is equivalent to a 180° rotation of the dipole with a
dipole moment
Vd = j z e b
(1.2.3)
7
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with z being the valence of the ion and e the electron charge. If we assume that there
are N bistable dipoles per unit volume, with N being small enough so these will not be
dipolar interaction and cosd=l for all dipoles, then pi=P2 without an applied field,
and V » K T . The jump probability from 1 to 2 can be written as
A, = * e x p [-^ ± M ] =
(1.2.4)
since the typical value for n is of the order 10' 18 esu, as long as EclO 5 esu (more than
breakdown voltage for most ceramics) This will result at fJE /K T « 1 , so we can
expand the exponential as
a 2 -5)
and obtain
a2 6 )
Also by using (1.2.1)
P „ = p (l-^ )
d-2-7)
is obtained. The probability of a jump of the ion in the opposite direction with respect
to the electric field will be consequently:
( l 2 -8)
Under equilibrium conditions the average charge population will not change. This
must result to
N\P\1 = NlPz\
d - 2'9)
8
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with N j and N 2 being the total number of charge carriers in each well. Eq. (1.2.9) can
also be rewritten using (1.2.7) and (1.2.8) as
or
( 1.2. 11)
Since the number of wells N occupied per unit volume is constant, the static
polarization per unit volume (Ps) is
V\ = ( N , - N t )fl
(1.2.12)
Using eq.(1.2.3) and because N=Ni+N 2 , we obtain
(L2B)
and
P
z1Ne1b 2
g = l + -4 7 = l+
e0E
4KT
(1.2.14)
1.3 Dipolar Polarization theory for Time Varying Field
In a time varying regime the change in number of dipoles in site 1 must be equal to
the outflow to site 2 minus the inflow from site 2 and vice versa, as shown in Fig.
1.3.1
9
t
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dNi/dt
dNlldt
Fig. 1.3.1 Dipole flux in time varying regime
thus
dNx _
dt
(1.3.1)
N\P\z +
dN,
—£ - = - N 1p2l+Nlpn
(1.3.2)
Upon subtracting (1.3.2) from (1.3.1) we obtain
dNx
dt
dN2
dt
~N\P\i+ ^zPzi + ^zPz\
^\P\z
(1.3.3)
or
~ \ ^ (~ dt Nl) =
(1.3.4)
“ N*P*
Using eqs. (1.2.7) and (1.2.8) we will have upon rearrangement
1 d(N { - N 2)
...
2----- ^
= ~P(Ni ~ ^ 2)+
U E ,XI
'
(1.3.5)
Because of eq. (1.2.12) and since N -N ]+ N 2 , eq. (1.3.5) becomes
ldPl
P
„U E
- p — +pN*—
2 dt p
p K KT
-
(1.3.6)
10
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Upon defining as relaxation time the quantity x=U2p, dipolar polarizability
a d = fjL2/K T , we have
T ^ + Pd = N a dE
at
(1.3.7)
where the subscript d has been added on the polarization vector to indicate the time
varying quantity due to polar polarization. The right-hand side o f eq. (1.3.7) has the
dimension of a polarization vector which we can identify as static polarization (Ps)
dP
+
dt
(1.3.8)
Before we solve this equation we need to introduce some simplification. Since a e and
ad occur very rapidly (10 *11 sec) a high frequency polarization contribution (P_) can
be defined as
P _ = P ,+ P a
•
(1.3.9)
So at frequencies higher then 10u Hz the dielectric constant will only depend on P..
Consequently at low frequency the static value of es can be expressed as
e-
1= 5 ..- * ? -
e fi
+( g » ~ Wk
( 1 3 10 )
e„E
’
and
( £ ,- e j £ o E = P,
(1.3.11)
To find the solution of (1.3.8) let us assume a steady state sinusoidal field as external
excitation E = E0ejoM, so eq. (1.3.8) becomes
r ^ f - + P d = ( e ,- e _ ) e 0E„e“-
(1.3.12)
11
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The general solution of this differential equation has form
Pd = €0e * E 0e'“ + ete~*
(1.3.13)
The coefficient /? can be obtained by substituting the general solution back into
eq. (1.3.12) and by splitting the resulting equation in two equations respectively for
the real and imaginary parts. This yields
(-T£,/J+ <•>-* = 0
(1.3.14.a)
(es - em)£0E - xe0e * jd E - e0e * E = 0
(1.3.14.b)
so from the first one we have (3 = l/r , while the second one gives
(££j 10 £oE
1+ jcm
Recombining the real and imaginary parts we obtain the final solution
PJ = g ,e -'', + T ~ — eoE0e'“
1+ JOJT
(1.3.16)
The first term on the right-hand side of this equation describes the time-dependent
decay of the dc charge on the capacitor. The second term describes the ac behavior o f
the polarization when a field of magnitude Eo and frequency Q) is applied.
Since the electronic and atomic polarizations are frequency independent in the range
o f our interest, they can be separated from the time dependent expression e* by
introducing
£ * -£ „ = - %
(1.3.17)
£0E
Now by using the definition of polarization we obtain for the imaginary part
12
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e
* _ e
-
( e ,- O g p E o
(l+ jc o t)e 0E0
(1.3.18)
1+ jcot
Since the general expression for a lossy dielectric is
e* = e’- je"
(1.3.19)
on separating the real and imaginary parts we obtain
(1.3.20)
1 + jo n
£ " = (£ ,- O :
cot
(1.3.21)
1 + co2t 2
and therefore the loss tangent is obtained as
c
( e - e m)cot
— 2- 2
£"
=—
(1.3.22)
These equations are the desired frequency-dependent relationship of the charging and
loss constants and the loss tangent of the dielectric material. These equations are
known as the Debye equations and they yield the graphical relationship shown in Fig.
1.3.2
tan5
1/2
Fig. 13.2 Frequency variation of dielectric parameters.
13
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1.4 Ferroelectric Theory
Ferroelectricity is the reversible spontaneous alignment of electric dipoles by their
mutual interaction [2]. Ferroelectricity occurs due to the local field E ' increasing in
proportion to the polarization which is increased by the aligning of dipoles in a
parallel array with the field. The alignment is spontaneous at a temperature Tc, where
the randomizing effect of thermal energy kT is overcome.
The defining equation for the onset of ferroelectricity follows from the definition of
electric polarization
P = ( e '- l) c 0E = N aE
(1.4.1)
E’ = E + 3cft
(1-4.2)
where
and k' is the relative dielectric constant. Thus
P = Na E + 3c,
= NaE +
NaP
3c„
(1.4.3)
Rearranging yields
P=
N aE
(1.4.4)
3c,
Since the electric susceptibility, %, is defined as
(1.4.5)
x = e - l= l j i
substitution of (1.4.4) into equation (1.4.5) yields
N a /e 0
N a)
13c,o y
(1.4.6)
14
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Equation (1.4.6) is called the Clausius-Mosotti which may be rearranged as
Na
e0
=
e -1
r
(1.4.7)
£+ 2
we can see from equation (1.4.6) that when
Na
3^o
>1
(14.8)
then P, X-> and k' must go to infinity. As we observed previously the orientation of a
dipole is inversely proportional to temperature:
a 0 = C /R T
(1.4.9)
where C is the Curie constant of a material. If we consider materials where
a 0 » a t + a a + a i, then a critical temperature Tc occurs when the following
condition is met:
r‘ = 3 * |
'
(1'4 10)
Below this temperature spontaneous polarization sets in and all the elementary
dipoles have the same orientation. Combining the above equations we have
^ =
=
(1.4.11)
1_ 3 ^
Equation (1.4.11) is also known as the Curie-Weiss law. From (1.4.11) and (1.4.10)
we obtain the behavior of the dielectric constant versus temperature
.. _ 3A ta/3 £0
*
Na
1
3e0
3 T JT _
Tc
1
T
3Te
t —T
c
15
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Fig. 1.4.1 shows the linear Curie-Weiss dependence of X~l on temperature above the
onset of ferroelectricity. At the Curie point there is a spontaneous alignment o f the
dipoles leading to a discontinuity in the temperature dependence.
Fig. 4.1.1 Curie-Weiss law
1.5 Structural Origin of the Ferroelectric State
i
The spontaneous alignment of dipoles which occurs at the onset of ferroelectricity is
often associated with a crystallographic phase change from a centrosymmetric,
1
!
nonpolar lattice to a noncentrosymmetric polar lattice. Barium dtanate is an excellent
example to illustrate the structural changes that occur when a crystal changes from
nonferroelectric (paraelectric) to a ferroelectric state. The Ti ions of BaTiC>3 are
surrounded by six oxygen ions in an octahedral configuration, as shown in Fig. 1.5.1
Barium
!0
Oxigen
Titanium
Fig. 1.5.1 Perovskite structure
16
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Since a regular TiOg octahedron has a center of symmetry, the six Ti-O dipole
moments cancel in antiparallel pairs. A net permanent moment of the octahedron can
result only by a unilateral displacement of the positively charged Ti4+ ion against its
negatively charged O 2 surroundings. Ferroelectricity requires the coupling of such
displacements and the dipole moments associated with the displacements. For TiC>2,
each oxygen ion has to be coupled to three Ti ions if each Ti is surrounded by six
oxygens. In rutile, brookite and anatase (three crystal modifications of Ti02) the T i06
octahedra are grouped in various compensating arrays by sharing two, three, and four
edges respectively with their neighbors. Consequently, all the Ti-O dipole moments
cancel and none of the Ti0 2 crystal forms are ferroelectric. However, in the ABO 3 or
BaTi0 3 (perovskite-like) structure, each oxygen has to be coupled to only two Ti
ions. Consequently, the TiOg octahedra in BaTi0 3 can be placed in identical
orientations, joined at their comers, and fixed in position by Ba ions. This gives the
opportunity for an effective additive coupling of the net dipolar moment o f each unit
cell. Thus in BaTi0 3 the Ba and O ions form a face cubic centered (FCC) lattice with
Ti ions fitting into octahedral interstices as visible in figure 1.5.1. The characteristic
feature of the Ba, Pb, and Sr titanates is that the large size of Ba, Pb and Sr ions
increases the size of the cell of the FCC Ba0 3 structure so that the Ti atom is at the
lower edge of stability in the octahedral interstices. There are consequently minimum
energy positions for the Ti atom which are off-center and can therefore give rise to
permanent electric dipoles. At high temperature T>TC, the thermal energy is sufficient
to allow the Ti atoms to move randomly from one position to another, so there is no
fixed asymmetry. The open octahedral site allows the Ti atom to develop a large
dipole moment in an applied field, but there is no spontaneous alignment of the
17
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dipoles. In this symmetric configuration the material is paraelectric, i.e. no net dipole
moment when E=0. When the temperature is lowered to below Tc, the position of the
Ti ion and the octahedral structure changes from cubic to tetragonal symmetry with
the Ti ion in an off-center position corresponding to a permanent dipole. These
dipoles are ordered, giving a domain structure with a net spontaneous polarization
within the domains. The crystallographic dimensions of the BaTi0 3 lattice change
with temperature, due to distortion of the TiOg octahedra as the temperature is
lowered from the high temperature cubic form. Because the distorted octahedra are
i
|
coupled together, there is a very large spontaneous polarization, giving rise to a large
dielectric constant and large temperature dependence of the dielectric constant.
|
The spontanous polarization is considerably stronger in the c-direction which results
i
in the larger dielectric constant in this orientation. Let us see how the ferroelectric
phase transformation occurs at Tc. As mentioned above at Tc BaTi0 3 is isotropic. The
j
Ti atoms are all in equilibrium positions in the center of their octahedra. Thermal
agitation produces strong fluctuations around the equilibrium position. An external
i
|
field will make the net moment nonzero by displacing the Ti atoms unilaterally.
However, in the absence of an external field the isotropic crystals are nonpolar. As Tc
is approached, there is an increase in probability that one of the TiOg octahedra will
be permanently polarized with a Ti+4 ion in an off-center position. How this occurs is
illustrated in Fig. 1.5.2.
18
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O
Oxygen
•
Titanium
£
Fig. 1.5.2 Atomic displacements as BaTiO; approaches Tc
If Ti* moves toward Oi the dipole moment Oi-A becomes stronger and the Oij-A
moment becomes weaker. Consequently, Oi moves toward A and away from On- B
and C follow the motion of A and D. E tends to follow suit because o f the coupling
between ions O ih and Oiv- They tend to move downward, repelled by Oi. The
magnitude of displacement of the Ti in its oxygen coordination is
oxygen displacements arc 0.03
A as shown in Fig.
0.12 A and
the
1.5.3
0.72/i
0.03A
o
•
•
Oxygen
Barium
Titanium
Fig. 1.53 TiC>6 octahedra displacements in the ferroelectric transition of BaTi0 3
19
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Thus a net dipole moment has been produced in the octahedron by the permanent
displacement of the Ti ion against its surrounding oxygen ions. The coupling between
neighboring octahedra increases the displacements and increase the internal field.
1.6 Hysteresis
The result of the spontaneous polarization of ferroelectric at Tc is the appearance of
very high k' and hysteresis loop for polarization. The hysteresis loop is due to the
presence of crystallographic domains within which there is complete alignment of
electric dipoles. At low field strengths in unpolarized material, the polarization P is
initially reversible and is nearly linear with the applied field, the slope gives k ', the
initial dielectric constant, as indicated in Figure 1.6.1 and equation 1.6.1 and 1.6.2.
Pk
Ps
£
Fig. 1.6.1 Hysteresis loop for polarization
The value k[ will be similar to k’ o f the cubic phase
P
E
( 1.6 . 1)
tan a = (fc- - l ) e 0
( 1.6 . 2 )
tancr
20
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where k' is the initial dielectric constant At higher field strengths, the polarization
increases considerably as a result of the switching of the ferroelectric domains. The
polarization switches so as to align with the applied field by means of domain
boundaries moving through the crystal as shown in Figure 1.6.2
a)
I
i
\
!
Fig. 1.6.2 a) Domain microstructure without an applied field b) Domain growth
in direction of an applied field
Figure 1.6.1 shows that at high field strengths, the change in polarization is small due
to polarization saturation; that is, all the domains of like orientation are aligned with
the field. Extrapolation of the high field E curve back to E=0 gives P s, the saturation
;
polarization, corresponding to the spontaneous polarization with all the dipoles
aligned in parallel. When the applied field continues to be applied at values greater
than required to achieve P s, the polarization continues to increase, but only
proportional to k[. This is because all o f the domains are oriented parallel to each
other. However, the individual TiC>6 polarizable units can continue to be distorted
increasing the unit polarization. This is an important contrast to ferromagnetic and
ferrimagnetic materials where application of a magnetic field greater than required for
Ms does not increase the net magnetic moment of the material. When E is cut off, P
does not go to zero but remains at finite value, called the remanent polarization, Pr.
This is due to the oriented domains being unable to return to their random state
21
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without an additional energy input by an oppositely directed field. The strength o f E
required to return P to zero is the coercive field Ec. There is a substantial effect of
temperature on the shape of the hysteresis loop. At low temperature, the loops
become fatter and Ec increases corresponding to a large energy required to reorient
domain walls; that is, the domain configuration is frozen in. As the temperature is
increased, Ec decreases until at Tc no hysteresis remain and e’ is single valued at a
value characteristic of the paraelectric phase as shown in Figure 1.6.3
a)
b)
d)
c)
Fig. 1.63 Effects of Temperature on BaTi0 3 hysteresis loop,
a-c ferroelectric state, d paraelectric state
1.7 Effect of Grain Size on Ferroelectric Behavior
Fig. 1.7.1 shows the ferroelectric transition of BaTiC>3 at 120°C for ultrafine particles.
When this is compared with single crystalline BaTi0 3 we can see some markedly
different behavior.
22
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1-2pm particles
6000
02 pm particles
2000
T(°C)
Fig. 1.7.1 Ferroelectric behavior of different particle size of BaTi0 3
For single crystalline material the transition is extremely sharp. In the case o f fine
particles (1 to 2 pm) the transition is gradual. This indicates that there is a relationship
between the size of the crystalline structure and the equilibrium position of titanium
ions in the polarized state. In ultrafine powders (0.2pm) there exists little or no
orientational relationship. Likewise, the increase in dielectric constant is much less
for ultrafine particles. This again shows the interrelationship of the microstructure and
the ferroelectric domains. The domain orientation in an ultrafine powder is random.
This randomization tends to broaden the ferroelectric transition, as seen in Figure
1.7.1.
23
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References Chapter 1
[1] Burfoot, Ferroelectrics, an Introduction to the Physical Principles. London:
Nostrand, 1967.
[2] M. E. Lines and A. M. Glass, Principles and Applications o f Ferroelectrics and
Related Materials. Oxford: Clarendon Press., 1977.
[3] R. Pepinsky, Physics o f Electronic Ceramics. New York: Dekker, 1972.
24
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C h apter 2
Ferroelectric Materials Synthesis by Sol-gel Technique
Broadly defined, sol-gel chemistry is a process which results in a fluid-to-gel
transition. It is a method for preparing ceramics through the hydrolysis and
polymerization o f metal alkoxides and metallo-organics [1,2]. Motivations for solgel processing include potentially higher purity and homogeneity, lower processing
temperatures than traditional glass or ceramic methods, and the possibility of making
thin films by dip- or spin-coating. The classic example of sol-gel synthesis is the
production of SiC>2 coatings by the hydrolysis and condensation of a silicon alkoxide.
The first steps o f this process are hydrolysis where the Si(OCH 3)4 react with water
molecules as shown in Fig.2.1, and linkage (Fig. 2.2). all of which occur in a solvent.
A number of factors can influence the gelation process, including temperature,
solution pH, solvent, and the alkyl group. As the hydrolysis and polycondensation
reactions form a sol (solid particles suspended in a liquid solvent), the solution can be
applied to a substrate if a Him is desired. The colloidal panicles resulting from
hydrolysis and condensation gradually link to form a connected network, and
eventually the mixture thickens to become a gel.
och3
oh
I
I
OCH 3 — Si — OCH 3 + 4 H20 —► OH— Si — OH + 4(CH 3OH)
O CH 3
oh
Fig. 2.1 Condensation of =Si-0-Si= bonds
25
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I
OH
I
I
I
OH— Si— OH O f r - Si— OH
OH
OH
OH
Si— OH
OH— S i— O -/- Si
OH
OH
Sol
OH— S i— OH OH— Si— OH
OH
OH
+ 6 Si-OH =
Fig. 2.2 Additional linkage of =Si-OH groups
The gel then is dried to remove solvent and reaction products from the pores. During
the drying process, the gel shrinks due to the loss of solvent volume. After the gel
network reaches sufficient strength, further heating removes liquid from the pores.
Finally, at even higher temperatures, the gel densities as the pores collapse, leaving a
purely inorganic film of Si0 2 - The steps to form a perovskite, which has the generic
formula ABO 3, where A and B are two different metals, are similar to those described
above for silica. However, because these materials contain two different metal ions, it
is important to be able to precisely control the stoichiometry to maintain this ratio on
a microscopic scale. Sol-gel synthesis offers the potential to establish this condition
by formation of double-metal-alkoxide precursor solutions. In this case, different
alkoxyl groups may be required for the two starting compounds in order that the
"oxygen bridge" -A-O-B- will form. Even if this structure does not form, if the
perovskite is the most stable structure it may still be produced by solid-state reactions
26
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at rather moderate temperatures because the necessary components are intimately
mixed in the gel.
2.1 Synthesis of High Purity Ferroelectric Materials by Sol-gel
Chemistry
In our laboratory we develop a new technique for the sol-gel synthesis o f lead
titanium oxide (PbTi0 3 or PTO), calcium modified lead titanium oxide (Pbj.
xCaxTi0 3 or PCT), barium titanium oxide (BaTi0 3 or BTO) and strontium modified
barium titanium oxide (Bai-xSrxTiC>3 or BST). This new technique [3-5] is based on a
process first developed by J.B. Blum and S.R. Gurkovich [6]. The complex alkoxide
precursor o f PTO is prepared dissolving lead acetate trihydrate, Pb(C 2H 302 >2*3 H 2 0 ,
in 2-methoxyethanol (2-MOE) at a concentration of 0.5 molar. Acetic anhydrite
(AAA) is then added at a molar ratio of 3:1 with respect to lead to react with the
3H 2O, producing acetic acid. Finally, titanium isopropoxide, Ti(OC 3H 7)4 , is added
into the solution to yeld a 1:1 molar ratio of lead to titanium. The PCT precursor is
produced by introducing calcium acetate monohydrate, Ca(C2H 3 0 2 ) • H2O, and lead
acetate in the desired molar ratio prior to the addition of acetic anidride acid (AAA).
The amount of AAA is adjusted accordingly to transform all the H 2O to acetic acid.
Only methanol is used for PCT as solvent due to the low solubility of calcium acetate.
Ultrasound is utilized to facilitate even mixture of the components at each mixing
stage. All procedures are carried out in room temperature in ambient air (no moister
sensitive), and the precursor appears to have shelf-life of about six months. This
presents a major improvement over the previous process which requires extensive
refluxing o f the solution above 100°C to remove water addition of nitric acid to
27
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achieve the desired pH and prevent hydrolysis. Barium titanate (BTO) powder and
thin films are made by synthesizing a sol-gel derived precursor. This precursor is
produced starting from barium hydroxide octahydrate Ba(OH )2 * 8 H2O which is first
dissolved in methanol at concentration of 0.1M. Titanium isopropoxide Ti(OC3H 7>4
is then added to the solution to yeld a 1:1 molar ratio o f barium titanium. All
procedures are followed at room temperature and in nitrogen atmosphere. BTO
powder can be made by calcining the precursor to 550°C. BTO films can be produced
by first spin-coating the precursor onto a substrate and subsequently firing the
substrate at 750°C. Similar procedure is used for production of BST, where the
Ba(OH )2 • 8H 2O is replaced by strontium hydroxide octahydrate Sr(OH)2 • 8H 2O
diluted in methanol and successively mixed with Titanium propoxide.
This new approach of producing BTO and BST ferroelectric material presents several
advantages, like high purity and absence of barium carbonate which are usually
difficult to remove with simply firing process, low firing temperature, low dielectric
loss, and very low cost. Ability of control on the grain size of the ceramic is important
to optimize electrical loss and dielectric constant of the ceramic. Our technique to
produce powder allows to synthesize ceramic powder having grain size ranging from
ljim up to 40pm [7].
As a practical example here follows the necessary steps to prepare the precursor for
PTO: Suppose we want to prepare a M (where M stands for Molarity) PbTi0 3 ,
precursor, starting from lead 2 -ethylhexanoate Pb(CgH 1502)2 and titanium
isopropoxide Ti(OC3H7)4 . The first step is the evaluation of the molecular weights of
the two compounds [8 ] which are respectively:
w 1={Pb(C8H i502)2}=l*{Pb}+16-{C}+30-{H}+4*{0}= 493.61 g
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.1.1)
w2={Ti(0C3H7)4}=l*{Ti}+4»{0}+12«{C}+28*{H}= 284.25 g
(2.1.2)
the corresponding density of the two compounds [8 ] is:
8i=density[Pb(CgHi502)2]=112 g/ml
52=density[Ti(OC 3H 7)4 ]=0.955 g/ml
(2.1.3)
(2.1.4)
Let us call yl and y2 the percentage of each of the two compounds dissolved in its
solvent (concentration) respectively. If [S]uters represents the amount o f solvent
added to the solution then:
________ Moles o f Compound (ft)________
[ P b C A O .l^ T iO .C u H ^ + p iU
'
'
■'
Since
[PbC,sHJ0O4]lti= ^ i
Yi
,
(2.1.6)
[ T iO .C u H ,,] ^ ^
Y2
(2.1.7)
the amount of solvent needed is obtained from (2.1.5) as
[S L „
r' \
yA
(2 . 1. 8 )
The total amount of prepared precursor results in the formula
[ '’L - P L + ^ + I S T A Y 2^2
(2.1.9)
The precursor preparation for lead dtanate oxide starts with titanium isopropoxide
(Ti(OC3H7)4), and lead 2-ethylhexanoate (Pb(C8His02)2)- The precursor is prepared
from an equimolar mixture of the two metal alkoxides. The titanium isopropoxide is
29
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in 95% isopropanol solution and has a density of 0.955 g/cc and a molar weight of
284.25 g/mole. The lead 2-ethylhexanoate is in a 55% solution of mineral spirit, has a
density of 1.12 g/cc and a molar weight of 493.6 g/mole to prepare 0.05 moles of
PTO we need an amount of Ti(OC3H 7)4 given by:
[Ti)(OC,H7),] = (M° lar " * * » » « * » > . 284.5 *0.05 =
Concentration
0.95
^
and an amount o f Pb(CsH 1502)2 given by
rn. , ~ TT „ , 1 (Molar weight x Moles) 493.6 x 0.05 ..
[Pb(C,H l! 0 ;); ] CoJ
---------------^ --------a i . i i ,
.
the respective volume will be
[Ti4(OC3H7)4] = Mass x Density =
= 0.01566 liters
[Pb(C,Hl50 2)2] = Mass x Density = y j y = 0.04006 liters
(2.1.12)
(2.1.13)
The liquid precursor volume prepared will be 0.05572 liters having 0.897 molarity.
The two ingredients are mixed in a dry flask, and stirred for over eight hours to insure
homogeneity. This process can be carried out in air. Afterwards, the solution is
diluted with an equal volume of 2 -propanol, resulting in a precursor solution with
0.449 molarity. This concentration is sufficiently dilute to prevent cracking o f the
film during the firing process. Although a small precipitation does occur, this solution
has a shelf life o f about six months. The deposition of the sol-gel is accomplished
using a photoresist spinner. The precursor is applied to the substrate from a syringe
which contains a 0.2 pm disposable filter. The substrate is spun at 2000 rpm for 50
30
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seconds. Multiple depositions can be carried out to obtain thicker films. Individual
layers are about 0.15 pm thick. A typical firing schedule for PbTiC>3 is shown in Fig.
2 . 1.1
60min
650
1
!
30min
400
100
30min
5 °Clmin
time
Fig. 2.1.1 Firing schedule for lead titanate (PbTi0 3 )
PTO is one of a large group of FEMs having the Perovskite structure (ABO3).
The low-temperature (ferroelectric) structure is tetragonal, and transforms to a cubic
phase above the Curie temperature o f 490°C [9]. The two preparation process
represented schematically for PTO/PCT and BTO/BST respectively in Fig. 2.1.2 and
Fig. 2.1.3 respectively. In order to verify the purity of our materials, we produce the
same powder by starting from the liquid precursor, by following the same fire
schedule used to produce thin film and carrying out a complete X-rays analysis for the
different type of powders and thin films.
31
with permission of the copyright owner. Further reproduction prohibited without permission.
Lead Acetate*3HrO
Acetic
anhydrite
2Me
Calcium Acetate «H20
Acetic
anhydrite
Methanol
by-product
Titanium Isopropoxide
Acetic acid
Titanium Isopropoxide
Precursor
multiple
2(Ca-0)=Ti=20R
(1 -X)*2(Pb-0)=Ti=20R
X*2(CaO)=Ti=20R
1-X
Spin/Dip Coal
Thinfilm
layers
Ceramic
Crystallize 700°C
!I
Crystalize at 700°C/2hr
Grain growth at 1150°C/8hr
Fig. 2.1.2 Schematic diagram for the production of PbTi0 3 and Pbi-xCaxTi0 3
Barium hydroxide
continuous stirring
T= ll3X xl2h
cooled down slowly
T=25‘C
Iodine
Brown clear solution
Clear colorless solution
2Me
Hydrazine added dropwise
Titanium isopropoxide
reflux 24 hat 80 X
vacuum distillation
o f solvent |—
G ear yellow solution (0.1M)
Gel precursor
White barium titanate powdei
Dissolved into 2Me (0.5M)
Fig. 2.1.3 Schematic diagram for the production of BaTi0 3 and Bai.xSrxTi0 3
thin film and powder
The use of X-rays to analyze the purity of a crystal structure is widely used in
material synthesis [10]. The basic idea is quite simple, an X-ray beam heats the
sample under test, and the scattering direction of the beam is recorded. The angle of
32
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the scattering direction will be intimately related to the sample lattice parameter.
Comparison of the scattering directions with tabulated values for known crystals [11]
determine if different species from the desired one are present A detailed description
on how to use X-rays analysis to determine the purity of the powder can be found in
[10]. Results for X-ray analysis on our sample for PTO powder obtained by the solgel route are shown in Fig.2.1.4
200
900
300
600
500
•100
-
QioJ
300 200
-
d
L
20
30
50
40
60
70
80
2 theta
Fig. 2.1.4 X-rays analysis obtained for our precursor for the production of PbTi0 3
Excellent agreement is observed between the X-rays result and the reference X-rays
card for PTO [11] for all the peaks positions, corresponding to the different
crystallografic orientations, except for a peak around 29*, this peak probably due to
the excess of one of the two ingredients it turns out that doesn't constitute a real
problem since it disappears after we fire the same film (as shown from the X-rays
obtained for a thin film of PTO in Fig. 2.1.5 and Fig. 2.1.6 respectively.)
33
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. ftl.3.
y J L u ijj
Fig. 2.1.5 X-rays analysis for single layer thin film o f PbTiO} deposited on alumina
substrate (AI2O 3) and nicrome-60 (NiCr) copper (Cu) contacts
1
20 0
1CO
'
'j
(/I
■■v
000
900
500
x
600
500
400
300
200
0
20
30
40
50
60
70
80
Fig. 2.1.6 X -rays analysis for five layer thin film o f PbTi0 3 deposited on alumina
substrate (AI2O 3) and nicrome-60 (NiCr) copper (Cu) contacts
34
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Other peaks are due to the alumina substrate and due to the metallization as marked in
the figures, but the one at 29* is not present.
Similar analysis is carried out for the PCT thin film deposited on MgO substrate, Xray analysis of the resulting powders showed crystallization of the perovskite phase
occurs immediately after the complete removal o f organic with no evidence of any
secondary phases. PCT lattice parameters obtained from X-ray analysis confirmed
the linear dependence on Ca doping concentration. This is clearly observable from the
shift of the peak in Fig. 2.1.7.
PK200]
PTO film on Pt/MgO
PTO w/ 30% Ca
film on Pt/MgO
[111]
PTO powder
sintered at 700 "C
10V.C*
20 •/. Ca
30 •/. Ca
50% C i
L l
20
30
40
SO
60
70
ae
2 0
Fig. 2.1.7 X-rays analysis for thin film Pbi-xCaxTiC>3on MgO substrate with
percentage of calcium varying from 0% to 40%
Results of X-ray analysis on BTO and BST ceramics are reported in Fig. 2.1.8 and
Fig. 2.1.9 respectively
35
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900
Relative
Intensity
800
700
500
in
500
400
300
Ml
4oo
9©1
200
<02
Z
oi
100
(210
20
30
50
40
60
2 theta
Fig. 2.1.8 X•rays analysis for BaTi0 3 ceramic sample
300
Relative
I ntensi ty
250
200
1 50
20»
10 0
0 0 2.
I ff
lo o
ooI
50
20
70
40
2 theta
Fig. 2.1.9 X-rays analysis for Bao.9Sro.iTi0 3 ceramic sample
36
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Again comparison with reference table [11] confirm the purity of our materials. To
ensure that all the organics are removed after firing, and also to set a lower limit in the
firing temperature, thermogravimetric analysis (TGA) and differential thermal
analysis (DTA) are performed in air. In the thermogravimetric analysis the precursor
is placed on a high sensitivity scale, and is heated slowly, in this manner when the
solvents leave the sample by evaporation, a change in weight versus temperature is
recorded. A typical setup is schematically shown in Fig. 2.1.10
PC station
heating plate
sw”Ple
x-y plotter
heating controller
and scale transducer
Fig. 2.1.10 Typical thermogravimetric station
This will allow to estimate the minimum temperature needed to remove all the solvent
from the sample. Results of TGA analysis for BTO and BST are shown in Fig. 2.1.11
and Fig. 2.1.12 respectively. All organic components of the precursors were removed
above 500°C for the BTO precursor and above 550°C for the PTO one.
37
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ir
rra n iA «nr ias
UK
O a tftM :
«
4 1 .2 4 4 0 m
T U M O C SC/M IN
ll/a
41
109
TGA
ru« PTDO
t109.01
J Mb
m m O a ta :
1 0 /0 0 /9 4
m e 47
100
09'
•0
79
I
T oaaoroturo ( * a
000
O tfiorol V2.2A
Fig. 2.1.11 Thermogravimetric analysis for PTO sample
S c a o l t: 0T0 O H IO • 2C0*C / 9
S lx r
9 0 .2 9 0 0 a g
Method: TGA 3*C/M IN 7 0 0 *C
C o aao n t: OS 30ML/MXN
ISO
TGA
F l i t : 0TQ0710T.0&
O paratar: CNANO
Run Oata: 07/19/09
1* 47
too
m
*
S
90
I
40
too
200
9 00
T aaoaratura (*C)
400
900
900
O a n a r a l v s .S A D y M n t
Fig. 2.1.12 Thermogravimetric analysis for BTO sample
38
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
In the differential thermal analysis (DTA) a ceramic sample is placed inside a box
capable of measuring the heat flow. When the heating of the sample is slowed down a
massive heat transfer will occur above the Curie temperature, due to the energy
j
required to change the state (from cubic face to tetragonal phase), and successively
|
heat exchange (of opposite direction) will occur during the cooling process. If the heat
transfer is recorded in time a clear indication of the Curie temperature (so indirectly
o f the purity of the sample since impurities change the curie temperature) can be
I
obtained. Schematic diagram of a DTA setup is shown in Fig. 2.1.13.
sample
/
PC station
x-y plotter
—1-------- L_
1*1
mu
=
heating controller
and sensors
l
I
Fig. 2.1.13 Typical differential thermal analysis station
The expected behavior of a ferroelectric sample is shown in Fig. 2.1.14
cool-down
Temperature
a
a:
Heat suck (•)
worm-up
Fig. 2.1.14 Typical result of DTA on ferroelectric sample
Results of DTA for the BTO sample is shown in Fig. 2.1.15
39
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0 . 0*
tfe
40
Tl
Fig. 2.1.15 DTA result for BTO ceramic sample
For the PTO and PCT samples the result of DTA are shown in Fig. 2.1.16
0 .1 5
PCT (0.1)
PTO
PCT (0.2)
0 . 10-
O
\X
»o-4
u.
4J
0 .0 5 -
PCT (0.3)
z■
PCT (0.4)
0 . 00-
-0 .0 5
100
200
400
T a a p a ra tu ra
500
800
("C)
Fig. 2.1.16 DTA result for PTO and PCT ceramic samples (Ca - 0%-40%)
The gradual increase of calcium from 10% up to 40% reduces the Curie temperature
from 490°C down to 130°C as expected.
40
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2.2 TSiin Film and Thin Ceramic Sample Preparation
Ferroelectric thin films and thin ceramics were successfully prepared by the sol-gel
route as described above, the thin film was deposited on planar substrates, while the
thin ceramic samples were obtained by cold press procedure. In order to estimate the
dielectric constant changes versus bias and versus temperature at low frequencies
(below 10 MHz), a parallel plate capacitor is build in both cases using the bottom
metallization as ground plane and the top contact (of circular shape) as seen in Fig.
2.2.2. For the thin film we use two different substrate types, i.e. MgO and AI2O 3. The
substrates were cleaned by conventional procedures prior to deposition o f the metal
which serves as ground contact. For MgO we use platinum as the ground contact,
deposited on the substrate with RF sputtering (see Fig. 2.2.2). This process is
expensive, due to the high melting point of platinum (above 2000°C) and since the
good lattice match of the MgO avoids metal diffusion, and reduces film cracks during
crystallization. To reduce the cost o f the sample, we propose the use o f alumina
substrate and standard metallization technique (metal evaporation). Unfortunately
most good conductors cannot be used with film because they diffuse during the firing
process into the film. A good compromise was found between conductivity and low
diffusion in the film with the use of Nirome-60. One problem related to the diffusion
of the metal into the film is the formation of a hybrid layer between the film and the
ohmic contacts shown in fig. 2 .2 .1
dl
2
d2
d=dl+d2
__dl + d2
e ■ £1 62
Fig. 2.2.1 Schematic of multiphase FEM film due to metal diffusion.
41
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This appears to be especially critical for film thickness in the sub-micrometer range.
Fig. 2.2.1 shows the effect of a low-permittivity interface layer on the total dielectric
constant In the example shown, if the respective permittivity of the high and low
phase are 1000 and 10, and if the overall thickness of the low permittivity interface
layer is 10% of the total film thickness, the measured relative permittivity is 92
corresponding to a net reduction o f 90%. The nature of the interphase may vary with
film material and substrate, but to optimize the film properties with respect to some
common parameter such as permittivity, it is necessary to minimize the interface layer
between electrodes and film. This hybrid layer lowers dramatically the equivalent
relative dielectric constant which can be measured, and also it introduces a series
resistance in the contact which limits the performance of the device at high
frequencies. We solve this problem by the use of a three layer metallization
constituted by NiCr-Cu-NiCr as shown in Fig. 2.2.2
copper
FEM film
nichiome
copper
nichiome ^
FEM film
Copper
Platinum
bulk ceramic
\
alumina
substrate
a)
—
Copper
^
MgO substrate
GND plane (Cu)
b)
C)
Fig. 2.2.2 Parallel plate capacitor used to estimate the change of dielectric
constant versus bias: a) Three layer metallization utilized to reduce
the diffusion of the copper into the film, b) Platinum metallization on MgO
c) bulk ceramic with copper contacts
The first layer of nicrome having thickness of about 0.1 |im guarantees the adhesion of
the metal to the substrate. The copper layer having thickness of 1.2 |im ensures low
contact resistance, and the second layer of nicrome prevents the diffusion o f the
42
■
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
copper into the film. The measurement of the thickness of the metallization was
carried out by using a DEK-TAK machine. Looking at the phase diagram for Cu-Cr
and Cu-Ni, no authentic is observed between them, so this kind o f structure
theoretically should stand up to 1084.67*C. In practice this is just an upper limit,
since in reality we do not have a uniform combination of the two metal, and this
temperature limit will be affected by the thickness of the metallization and by the
length o f time we heat the substrate. From our experiment we find that the this
metallization can stand up to 700*C for lhr. This result is very good since our PbTi0 3
film has Curie temperature of 490*C, so in order to crystallize will require a
temperature below 700*C. In this way we reduce the ohmic surface resistance by two
orders of magnitude. In fact we used to have for a single layer NiCr metallized plate
( r x P ) resistance (edge to edge) o f 50Q, or for three layer NiCr metallization a
resistance o f 20Q. With the copper layer the measured resistance is below 0.5Q.
Notice that the top contact can be applied using a standard evaporation technique
since the film is already crystallized, so no diffusion will occur. Deposition of the solgel coatings was accomplished with a photoresist spinner. The solution was applied to
the surface from a syringe which contained a 0.2 pm disposable filter. The sample
was spun at 2000 rpm for 60 seconds. Multiple deposition was carried out to increase
the thickness. Each layer was dried at 400°C in air atmosphere, each layer was
approximately 0.5 pm after drying. After the last film layer was deposited the film
was heated in two steps. In the first step, the sample was heated at 400°C for 30
minutes, for removal of residual organic and densification of the gel. In the second
step the sample was heated at 700°C to obtain crystallization. All the procedures were
carried out in air. Thin ceramics (thickness below 0.2 mm) are very attractive
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ferroelectric materials for microwave applications. One advantage of a thin ceramic,
with respect to a thin film, is thickness (between 50-200 ^m compared to 0.3-2 p.m
i
for the thin film). This allows low capacitance value(between 1-100 pF using a
geometry between 1-0.1 mm) compatible with microwave design. The high purity of
the compound is guaranteed by the sol-gel process, and the absence of diffusion
problems is guaranteed by the fact that all metallization is carried out almost at room
temperature after crystallization. This also allows the use o f electrical contacts with
low cost metals such as copper or silver instead of platinum. BTO ceramic sample
II
|
powder obtained from the sol-gel process is pressed into pellets at a pressure of
j
2000Kg/cm 2 prior to sintering at 1300°C (83% of his melting point in °K) for 1 hr.
After the firing process the sample is sanded down to 0.1 mm and the faces are
polished using diamond wheels. The ground plane and the microstrip line are
I
|
fabricated evaporating three layers of metal such as chromium-copper-gold under
|
vacuum to prevent oxidation. The chromium guarantees good cohesion to the
ceramic, the copper serves as a buffer layer and makes cohesion between the
i
|
nichrome and the gold, and the gold ensures electrical conduction. Similar procedure
is used to produce BST, PTO and PCT samples.
44
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I
Ii
References Chapter 2
[1] R. W. Jones, Fundamental Principles o f SOL-GEL Technology. The Institute of
Metals, 1989.
[2] Larry L. Hench and Jon K. West, “The Sol-Gel Process,” Chemical reviews, vol.
90,No.l, 1990.
[3] F. De Flaviis, J. G. Ho, N. G. Alexopoulos, and O. M. Stafsudd, “Ferroelectric
Materials for Microwave and Millimeter Wave Applications,” in SPIE The
international Society fo r Optical Engineering, Smart Structures and Materials. S.
Diego CA February 1995, pp. 9-21.
[4] F. De Flaviis, D. Chang, J. G. Ho, N. G. Alexopoulos, and O. M. Stafsudd,
“Ferroelectric Materials for Wireless Communications,” in COMCON 5 5th
International Conference on Advances in Communication and Control. Rithymnon,
Crete (Greece) June 26-30 1995, pp.
[5] F. De Flaviis, D. Chang, N. G. Alexopoulos, and O. M. stafsudd, “High Purity
Ferroelectric Materials by Sol-Gel Process for Microwave Applications,” in ICEAA
95 International Conference on Electromagnetics in Advanced Applications. Torino
(Italy) 12-15 September 1995, pp. 157-159.
[6 ] J. B. Baiun and S. R. Gurkovich, “Sol-Gel Derived PbTi03,” Material Science,
vol. 20, pp. 4479, 1985.
[7] F. De Flaviis, D. Chang, N. G. Alexopoulos, and O. M. stafsudd, “Ferroelectric
Materials for Microwave Applications,” in Union Radio Science. Boulder (CO)
January 1996 1996, pp. B/A/Dl-4.
[8 ] F.A. Cotton and G. Wilkinson, Advanced Inorganic Chemistry. New York: Jhon
Wiley & Sons, 1980.
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[9] T. Mitsui and S. Nomura, “Landolt-Bomstein: Numerical Data and Functional
Relationship in Science and Technology,” Ferrolelectrics and Related Substances,
vol. 16,1981.
[10] W. H. Bragg and W. L. Bragg, X rays and crystal structure. London: G. Bell,
1942.
[11] E. Howard and Swanson, “x-ray diffraction powder patterns,” Washington, DC:
U.S. Dept, o f Commerce National Bureau o f Standards, v o l., 1985.
{
;
i
46
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C h apter 3
Electrical Measurements of Ferroelectric Materials
Before we measure our devices, it is essential to understand what the dielectric
constant versus bias voltage theoretical behavior we expect. Consider a simple
parallel plate capacitor as shown in Fig. 3.2, having ferroelectric material as
dielectric, and suppose that the behavior o f P(E) is the typical hysteresis loop
behavior as shown in Fig. 3.1. Since to measure the dielectric constant o f our
capacitor we use DC bias field to bias the film plus a small RF signal source to detect
the capacitance, the situation will be as illustrated in Fig. 3.1
Pk
kysteresys loop due to
the RF signal
DC bias field (E=E*)
Fig. 3.1 Typical P-E hysteresis loop and RF signal around bias DC field
The total displacement of the field in the dielectric will be given by
D = e#E + P
(3.1)
Particularly at E=E* we will have
eE* = e,E ♦ + ?(£ * )
(3.2)
47
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If we expand P(E) around the bias point E*, we can write
AE +
P(E) = P(E*) + ^
d 2P
AE +■
(3.3)
*=£•
E=E*
and by retaining the first order expansion terms we can rewrite equation (3.2) as
e(E * +AE) = £0(E * +AE) + P(E*) +
dP
<*
AE
(3.4)
E=E*
and therefore from eq. (3.4) the following result is obtained
£ = £„ +
dz
(3.5)
E=E
Consequently assuming that P(E) exhibits hysteresis behavior, the dielectric constant
versus the bias field must be as shown in Fig. 3.2
dP
Fig. 3.2 dP/dE behavior for material having P-E hysteresis loop.
The different location of the two dielectric constant peaks is clearly due to the
hysteresis loop which presents different slope for positive and negative bias voltage
(see Fig. 3.1).
48
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3.1 Thin Film Electrical Measurements
In our first experiment a single layer film of PbTi0 3 on alumina substrate
having three layer metallization (NiCr-Cu-NiCr) was made, the film was dried at
100*C first for 30 minutes, then at 400*C for 30 min. and finally crystallized at 700*C
for lhr. The heat rate was 5*C/min as shown in Fig. 2.1.1 in Chapter 2.
In order to verify the consistency o f our electrical measurement at low frequency we
evaluate the ratio Ei/t for each of the four different size capacitors. Using the simple
parallel plate capacitor model which for a capacitor having thickness t and circular
plate diameter d provides the formula
£r.
t
(3.1.1)
eM dny
Averaging the result of the measurement for three different samples, for each size
capacitor, we obtain
Diameter
Cap.
d(mm)
C (nF)
Resistance
R (kft)
Ratio
£i/t
2
4.52
5.69
1.62 E 8
1
1.3
21
1.87 E 8
0.5
0.356
64.1
2.05 E 8
0.25
0.11
206
2.53 E 8
Table 3.1.1 Electrical measurement of thin film PTO ferroelectric sample
We observe a consistent value for the ratio £f/t, which also increases about 30% for
the smaller capacitor. This is very easy explainable if we consider that in the smaller
size capacitors we have more fringing field effects, which can be seen as apparent
increase of the dielectric constant. The measurement of the capacitance and of the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
resistance (by looking at the parallel circuit topology) versus the bias voltage
reported in Fig. 3.1.1 and Fig. 3.1.2
4 000 10 10
3.500 10'°
2.500 10'°
2.000 10 10
-24
-16
8
0
8
16
24
Vb(V)
!
;
Ii
Fig. 3.1.1 Electrical measurement of capacitance vs bias voltage for thin film PTO
7.000 I04
6.000 104
5.000 I04
3.000 I04
2.000 104
1.000 I04
0-24
-16
Vb(V)
Fig. 3.1.2 Electrical measurement o f conductance vs bias voltage for thin film PTO
50
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A change in the capacitance by biasing the substrate (consequently the dielectric
constant) of 37% with respect to the maximum value (obtained at zero bias) is
observed. Fig.3.1.2 also confirms the prediction o f our model regarding the
theoretical behavior of the dielectric constant versus bias reported in Fig. 3.2. One
problem which seems to be present in our device is the low dielectric constant
obtained. This result is easily explainable if we recognize the presence of an
amorphous layer of film between the ferroelectric film and the metal contact. Now if
we consider that this amorphous layer has a relative dielectric constant between 3-10,
the total equivalent dielectric constant for the sandwich will be given by
= (d, + d1)e,le,1
‘q dx£rl + cLi£rl
'
( 3 .,. 2)
where di and eri are the thickness and the dielectric constant of the amorphous layer,
while di and
of the film respectively. It is easy to verify, that even for a thin layer
of the amorphous material, the equivalent dielectric constant will drop dramatically as
shown in Fig. 3.1.3
51
with permission of the copyright owner. Further reproduction prohibited without permission.
d=lnm , e r1=300, er2=3-10, dl=0.002-0.25 tun
300
Cu
250
N iC r o iid e (e rl= 3 -t0 )
200
_ N iC r
-C o
'N i C r
U
150
100
50 -
el=I0
el=6
El=3
0
0.05
0.1
0.15
0.2
0.25
0.3
dl (jun)
Fig. 3.1.3 Equivalent dielectric constant vs the amorphous layer thickness
To confirm this assumption we made another sample using the same precursor, but
with five layers o f film instead of one. The electrical measurement is reported in
Table 3.1.2
Diameter
d (mm)
Cap.
c m
Resistance
R (kft)
Ratio
Ef/t
2
3.53
5.22
1.315 E8
1
0.993
18.4
1.48 E 8
0.5
0.275
64.7
1.64 E 8
0.25
0.0687
237
1.64 E 8
Table 3.1.2 Electrical measurement of five layer PTO sample
Higher value for the dielectric constant is observed. In fact if we assume that each
layer has the same thickness, since in the first case we only had one layer, and now
we have five, considering the capacitor having diameter d=0.5mm we have the result
52
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
e jt
erS/5 t
2.05-10*
1.64-10s
(3.1.3)
and therefore
(3.1.4)
This confirms our hypothesis on the amorphous layer. Also because the effective film
has higher thichness, higher modulability (about 45% against previous 37% respect
the highest value) is achieved as shown in Fig. 3.1.4 and Fig. 3.1.5. Electrical
measurement for a thin film BTO sample was also carried out. Fig. 3.1.6 shows the
thermal dependance of the capacitance for a 4 layer thin film of BTO deposited on
MgO substrate, having platinum as ground contact.
3.500 10 10
3.025 10 10
£
O
2.550 10 10
2.075 10-10
1.600 10-'°
-32
-24
0
-16
8
16
24
32
Vb(V)
Fig. 3.1.4 Electrical measurement o f capacitance versus bias voltage
for five layer thin film PTO sample.
53
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7.000 10*
6.000 I04
Rtfl)
5.000 I04
3.000 I04
2.000 104
1.000 10* •
-32
-24
-16
V(v)
Fig. 3.1.5 Electrical measurement of resistance versus bias voltage
for five layer thin film PTO sample.
0.12
10
220
0.10
ta n 4
180
0.09
160
0.08
140
0.07
120
0.06
100
0.05
80
0.04
60
0.03
40
0.02
20
0.01
50
10 0
150
250
200
300
350
0.00
400
T e m p e r a tu r e (cleg C)
Fig. 3.1.6 Capacitance versus temperature for 4 layer thin film
BTO on MgO substrate
54
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tan (5
200
We observe that near the Curie temperature the dielectric constant reaches a peak
value before starting to drop. The reason is because the peak, even though quite broad
compared to the ceramic case, is due to the ultrafine particle size of the thin film, as
was explained in Chapter 1, section 1.7.
Capacitance measurements versus bias at different frequencies for the same sample
are shown in Fig. 3.1.7.
10' *
10"
°
a
o
1kHz .
7
10kHz!
6
lOOkH.
1MHz
4MHz
4
2
1
0
1
2
Vbias (V)
Fig. 3.1.7 Capacitance versus bias field for the thin film BTO
sample on MgO substrate
We notice a modulability of 50% at the lower frequency. The independence of
modulability from frequency confirms the absence of metal diffusion into the film.
3.2 Thin Ceramic Low Frequency Electrical Measurements
BTO thin ceramic sample measurements of capacitance and conductance versus
temperature at 1 MHz were carried out by immersing the sample in a silicon oil bath,
55
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and by establishing the electrical contacts using silver paint. A computerized system
allow us to record the capacitance change versus temperature. The schematic layout
of the sample geometry and the setup are shown in Fig. 3.2.1.
CPU-Unit
A/D
converter
ceramic
HP-IB interface
Termocouple
transducer
HP Impedance
Analyzer
electrodes
sample
termocouple
sample
silicon oil bath
heating plate
IA W V V W V W V 1
D/A
converter
heating unit
Fig. 3.2.1 schematic layout of the sample geometry and the scrip to
perform the measurement of electrical parameters versus temperature
results of the measurements are reported in Fig. 3.2.2 and Fig. 3.2.3
VgaitalV. ( - 1 MHz. a-1mm . t-O.lmm, paralM cki
10 ' o
10
10 ' •
10
’ *
10 1<
10 11 0
50
1 00
1 SO
200
2 SO
T("C)
Fig. 3.2.2 Capacitance versus temperature, for the barium titanate sample
56
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Vgan-iV. (-1MHz. d-lm m . t-O.imm. parallel cfct
10 s
10
10
10
10
10
10
0
200
250
T(°C)
Fig. 3.23 Conductance versus temperature, for the barium titanate sample
A clear peak is observed close to the Curie temperature. This confirms that our
ceramic (at room temperature) is in the correct phase to exhibit ferroelectric
\
2
properties. Because of the high heating rate, (about 30°C/min) a shift from the Curie
|
temperature is observed. This is due to the different termal capacity of the sample and
of the thermocouple. More precise differential thermal analysis already confirms the
j
j
exact value of the Curie temperature. Also the same high peak is oserved around
50°C, (due to contact problems) for the dielectric constant and for losses. This is in
perfect agreement with the theory o f ferroelectric materials. The frequency
dependence of the capacitance and conductivity for zero bias is shown in Fig. 3.2.4
and Fig. 3.2.5
57
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Vgen=1 V, Vb=OV, d=1mm. t*0.1mm, paraJM cfct
10
10 t
10 *
10 t
0
0
200
400
600
800
1000
((kHz)
Fig. 3.2.4 Frequency dependence of capacitance for the BTO sample
Vgen=1 V. d=1mm, 1=0.1mm, parallel ckl
10
10
10
10
10
10
10
10
0
0
200
400
600
800
1000
((kHz)
Fig. 3.2.5 Frequency dependence of conductance for the BTO sample
The frequency independence of the capacitance versus frequency confirms the
absence of diffusion layers. Because the HP impedance analyzer only allows low bias
voltage, an external bias network was built to allow higher bias of the sample. Fig.
3.2.6 shows this set-up
58
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
DC blocking
capacitor
blocking
resistors
0-2000 V
DC power
supply
H FEM capacitor
under test
HP-impedance
analyzer
DC blocking
capacitor
blocking
resistors
Fig. 3.2.6 High voltage set up for low frequency ceramic capacitor measurement.
The high-voltage set up limits the operating frequency of the impedance analyzer
between 50kHz and 1 MHz. The results of the measurement for the capacitance and
conductivity for the ceramic sample at 1 MHz versus bias are shown in Fig.3.2.7 and
Fig. 3.2.8
Vgen=1 V, 1=1 MHz, d=1mm, 1=0.1mm, parallel cfct
i so
125
100
75
S0
2S
0
200
300
400
500
600
Vgen(V)
Fig. 3.2.7 Bias dependence of capacitance for the barium titanate sample
59
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Vgen=1V, f=1MHz. d=1mm. 1=0.1 mm, parallel ckt
83.33
— 66.87
33.33
16.67
600
6 00
Vgen(V)
Fig. 3.2.8 Bias dependence of conductance for the barium dtanate sample
200
j
j
300
400
High modulability is obtained, even by using a field below 20kV/cm (200V in the
j
sample having 0.1mm thickness) 40% modulability is achieved. Also substantial
t
reduction of the losses is obtained under bias condition. Notice the behavior of the
|
conductivity versus bias voltage is in perfect agreement with the hysteresis loop P-E
|
of a ferroelectric material. This will give low loss at low bias voltage (narrow
I
hysteresis loop) and low losses at high bias voltage (saturation region) as shown in
i
j
Fig. 3.2.9
high bias
medium bias
low bias
Fig. 3.2.9 losses due to a small RF signal near specific bias points
for ferroelectric material having P-E hysteresis loop
60
t
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Vgen=1V, f=lM H z, d*1m m . t*0.1mm, parallel ckt
1 50
1 20
90
8
uS-
60
3 0
0
•600
-400
0
- 200
200
400
600
V g « n (V )
Fig. 3.2.10 Capacitance versus bias voltage for BTO ceramic sample
Vgen>1V, f = l M H z , d = 1 m m , UO. I mm, parallel c kt
100
80
C=O
L
60
_>
sa
-a
c
o
o
40
20
0
-600
-400
-2 0 0
200
400
600
Vgen(V)
Fig. 3.2.11 Conductivity versus bias voltage for BTO ceramic sample,
61
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The presence of an hysteresis memory is clearly, shown in Fig. 3.2.10 and Fig. 3.2.11
were capacitance and conductance are plotted versus bias voltage, starting with a
sample which was never biased before.
3.3 Thin Ceramic High Frequency Electrical Measurements
To evaluate the property of the ceramic samples at higher frequencies resonant cavity
measurements were performed. The measurement employs an iris coupled reaction
type cavity, constructed from standard rectangular waveguide operating in the TEio
mode. A cylindrical sample holder made of Styrofoam is placed at the geometrical
center of the cavity. A small hole drilled on the upper broadside wall of the cavity
allows the sample to be inserted in to the sample holder, without disassembling the
cavity and the coupling iris. Also a movable short is used for fine frequency tuning,
and a movable stub helps to achieve the critical coupling. The utilized set-up utilized
is schematically shown in Fig. 3.3.1
to the network
analyzer
tunin8 movable
t
tuning movable
short
sample
" \
t
7
SMA-Waveguide
,
..
transition
circular coupling
aperture
I
I1 h
s
v_
7
styrefoam sample
holder
'■-----
--—
,
f
waveguide section AA‘
Fig. 3.3.1 Schematic o f the resonant cavity set-up for the measurement
o f the ceramic samples
The cavity is coupled to the waveguide through a small centered circular aperture
placed at a distance d from the short-circuited end as shown in Fig. 3.3.1. The
62
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
resonant frequency of such a cavity for the generic m,n,p mode can be expressed [1]
as
*(?) *(5)
corresponding to a propagation constant
(3.3.2)
For a high Q cavity we can approximate for the TEioi mode [1] the propagation
constant k0*=kioi and those quantities can be related to the propagation constant of the
feeding waveguide (Pio) operating in the TEio mode as:
Ao = A 2- | f f = f
(3.3.3)
Knoledge of Pio, kjo i, the cavity dimensions and the desired Q allow us to determine
the critical coupling factor a m [1] as
^ f d o
'
(3-3 4)
fyonJzPio-j
from which the hole diameter (h) can be calculated as
A=
(3.3.5)
In our set-up the waveguide used has dimensions a=72mm and b=34mm. So to obtain
a resonant frequency of 2.3 GHz for the TEioi we need from eq. (3.3.1) a depth
d= 150mm. This will correspond to a propagation constant k jo i-4 8 3 4 m which for a
63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Q factor between 500 and 5000 gives a hole size (eq. 3.3.5) between 12.1mm and
17.8mm. We select an aperture of 15mm, and the normalized field for the T E ioi
resonant mode is expressed therefore as:
(3.3.6)
where a,b,d, are the cavity dimensions as shown in Fig. 3.3.2.
sample
b
Fig. 3.3.2 Rectangular resonant cavity field distribution for the TEioi mode
So the field will be maximum in the center of the cavity where we will place our
sample under test. Because absolute measure ments for the dielectric constant and
especially for losses are difficult to perform, we decided to carry out relative
measurements, starting from samples with known properties and comparing them
with our samples. This procedure is based on the fact that for similar samples (in
shape and dielecric constant), the shift in the resonance frequency can be related to
the dielectric constant of the sample, while the Q of the cavity will be related to its
losses. More specifically, for a given resonant frequency fo and a given Qo
64
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
corresponding to the empty cavity, we relate the dielectric constant (real and
imaginary part) to the new resonance frequency with the sample in the cavity f s and
the new Qs using [2]
^ -1 = - ^ - A
V fs
(3.3.7)
€ " = -
A
wher
tj
Qo
is the empty factor defined as
J ltfd v
T] = ^f j--J|E | dv
(3.3.8)
in which Vc represent the volume of the empty cavity and Vs the volume o f the
sample. If we now suppose to have two samples, the first one of known properties
(e'r,£" Vr) and the second one of unknown properties (£ ',£ " Vs), using Eq. (3.3.7) we
can write:
e 'r - l
V, f , f o - f r
(3.3.9)
e7 _ f] r Qr Q0 - Q ,
K
IsQsQo-Qr
If the two samples have similar shape and their volume is small compared to the
cavity volume, we can approximate
J|E|2rfv
Vr
(3.3.10)
j \ E f d V ~ Vr
65
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Upon rearrangement of (3.3.9) we can rewrite:
£; = i + ( e; - i ) ! L / L J L
v
}vsf0- f r
(3.3.11)
p”— -<»K Qr Qo Q t
'
f . a a - a
Eq. (3.3.11) will allow us to determine the dielectric constant of the sample under
test, starting from the known sample parameter and the changes in the resonant
cavity. As an example Fig. 3.3.3 shows the necessary steps to perform this type of
measure.
empty cavity
reference sample
volume Vr
£r = e’r - j e ?
sample under test
volume Vs
freq.
3dB
3dB
3dB
a)
b)
c)
Fig. 3.33 Necessary steps to perform comparative resonant cavity measurements
As first step (a) we tune the cavity to achieve the highest Q at the desired resonance
frequency. This is done by using the movable short (to adjust the resonant frequency)
66
si
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and by consequendy moving the tuning stub to reach the critical coupling (see Fig.
3.3.1). After the critical coupling is achieved (the measurement of the S // is done
using a network analyzer) the Qo of the cavity can be estimated as shown in Fig.
3.3.3-a. The second step (b) consists of inserting the reference sample in the cavity,
and of estimating the change in the resonance frequency (going from fo to f r) and the
degradation of the quality factor (going from Qo to Qr). Finally, in the third step (c)
the unknown sample is inserted in the cavity (see Fig. 3.3.3-c) and again the new
resonance frequency f 5 and the new Qs are measured. All these value are substituted
into Eq. (3.3.11) and the desired parameters are extracted. It is important that before
we start the measurements we verify the accuracy of the assumptions we made
regarding the size of the sample. To evaluate the critical volume of the sample which
can be measured keeping good accuracy, four sample having identical electrical
properties, similar shape but different size. Table 3.3.1 shows the measured
parameters for those samples:
Sample
Vol.
(mm^)
S ll
(dB)
Res.freq.
fo
(MHz)
3 dB
Band. 4 f
(KHz)
Quality
factor
Q
190.0
-14.1
2295.75223
320.0
7177
122.0
-16.975
2296.06933
201.3
11409
84.3
-19.126
2296.21423
151.8
15129
46.5
-22.127
2296.34623
101.4
22649
emotv
-37.0
2296.70233
23.4
98149
Table 33.1 Resonant cavity measurements for different size,
same material sample
Measuring the Q factor for each of them, we can evaluate from Eq. 3.3.11 the ratio
V(/Vs. Using the smallest sample as reference, this ratio should be consistent (apart
67
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
from a constant factor) with the ratio calculated from the physical dimensions. This
will give us an upper limit on the maximum sample dimensions. Table 3.3.2 shows
the result of the above described measurement:
Samples
Volumes
(mm3)
ratio
46.5/84.3
0.5516
46.5/122.0
46.5/190
a a-a
Constant
factor
Error
%
0.6074
1.10116
0.3811
0.4384
1.15050
reference
-4.48
0.2447
0.2629
1.07437
2.43
QrQo-Qs
Table 3 3.2 Resonant cavity accuracy measurements
From Table 3.3.2 it is clear that even for the bigger sample dimension (190 mm3) the
error is confined in the 5% range. Result of compared cavity measurements for
commercial BTO versus our BTO, and BST with 10-20% of strontium, are shown in
Fig. 3.3.4 CH1 !»1SM lag
mas
6 38/
REF 0 as
Cor
E /E
-ii
Ba%8Sr01T.0.
CENTER
2
2 9 0 .3 7 0
490
MHz
SPAN
1 .9 0 0
000
MHZ
Fig. 3.3.4 S u measurement obtained for the resonant cavity with different samples
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
As observed our BTO ceramic has lower dielectric constant and lower loss compared
to the ceramic made of commercial powder. Also the doping of strontium (10% and
20% respectively) further reduces the loss. Complete list of extracted values for the
losses with different percentage of strontium and different fire schedules is reported in
Table 3.3.3
Sample
Heat treatment
Grain size Curie Temp. Losses
pm
°C
tanS
ST
F900°C lh
1
37
0.00364
ST
R1200°C lOh
3
37
0.00322
ST
R1500°C lOh
15
37
0.002957
ST
R 02 1500°C lOh
30
37
0.002957
BST(0.8/0.2) F02 1500°C lOh
BST(0.5/0.5) F02 1500°C lOh
30
30
105
218
0.005615
0.0304
BST(0.7/0.3)
F1375°C lOh
8
280
0.135
BST(0.8/0.2)
F1300°C lh
F1300°C lh
F1300°C lh
8
324
0.081
10
9
391
387
0.21
0.32
BTO
Comm. BTO
Table 3.3.3 Loss tan5 for different samples o f BTO and BST material
i
I
As noticeable our BTO has losses which are about 40% lower compared to
commercial BTO powder. Also, the introduction of strontium in the ceramic lowers
the losses, which goes from tan5=0.21 for pure barium down to tan5=0.00364 for
pure strontium [3]. Unfortunately, while introducing strontium, the Curie temperature
(Tc) also drops, so our sample will not exhibit ferroelectric properties at room
temperature (300 °K) if the Tc is below that value. So, for practical applications,
unless we have the capability to cool down our sample, we need to limit the amount
of strontium to 20% (corresponding to TC=391°K). We also notice, the different type
69
t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
of firing schedule affects the grain size of the sample. For example for strontium
titanate (ST) refiring the ceramic after first fired at 900°C for 1 hour (F900°C lh)
allows the grain size to grow, from lfim to 3|im. Refiring at 1200°C for 10 hours
(R1200°C lOh) increases the size up to 15|im. Also, further grain growth up to 30
fim, is obtained refiring in oxygen atmosphere (RC>2l5(X)0C lOh).
70
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
References Chapter 3
[1] R. E. Collin, Guided Waves Second edition. New Y ork:, 1991.
[2] H. M. Altschuler, Handbook o f Microwave Measurements. New York: Brooklyn
Polytechnic Press, 1963.
[3] F. De Flaviis, D. Chang, N. G. Alexopoulos, and O. M. Stafsudd, “High Purity
Ferroelectric Materials by sol-Gel Process for Microwave Applications,” in IEEE
MTT-S Int. Microwave Symp. S. Francisco June 1996, pp. 99-102.
71
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I
C h apter 4
Microwave Phase Shifter Design using
Ferroelectric Materials
In this section I will show all the steps necessary to efficiently use ferroelectric
materials, for the design of microwave tunable phase shifters. The first design will
involve a discrete type of phase shifter, where the phase shifting capability is obtained
from the change in the capacitance of a parallel plate capacitor made o f FEM
imbedded in a microstrip set-up. For the other types of design, I will use a continuous
type of phase shifter, where the phase shifting capabilities are obtained from the
change in the dielectric constant of the substrate of a microstrip transmission line
printed on FEM material.
4.1 Thin Ceramic Tunable Capacitor
In this first design, a lumped capacitor having BTO as dielectric is used as phase
shifter element, the capacitor is inserted through wire bonds in a microstrip
transmission line printed on allumina substrate as illustrated in Fig. 4.1.1
Allumina substrate
FEM capacitor
50£i microstrip
0.63 mm
SMA connector
wire bond
Fig. 4.1.1 Tunable phase shifter using lumped tunable capacitor
as tuning element
72
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
In order to measure the S-parameters of this circuit under different bias conditions,
one needs to have a suitable bias set-up which doesn't interfere with the microwave
signal and avoids voltage leaks in the system. A sketch of the set-up utilized for this
measurement is shown in Fig. 4.1.2.
calibration
planes
BPF
HP-8510
FEM,
BPF
Hi voltage
generator
Fig. 4.1.2 Measurement set-up used to measure the scattering
parameters of the phase shifter
Use o f a quarter-wave high impedance line and a quarter-wave low-impedance open
stup in the band pass filter (BPF), ensure isolation of the bias from the microwave
signal for a reasonably good bandwidth see Fig. 4.1.3. Additional resistors were used
(their value is not important since the circuit has almost no current flowing through)
for safety reasons, so that in the case of a bias voltage short (due to breakdown of the
ceramic), only limited current will flow through the circuit. The two bandpass filters
were centered around 2GHz. The filters where especially designed [1] to withstand
high static field, i.e. their comers were rounded and 2-mm high dielectric rigidity glue
was deposited in their air gap. A schematic layout of the filter and the X/4 isolator is
shown in Fig. 4.1.3, details on the filter design are reported in Appendix A. The
calibration of the network analyzer is done after the filters (see Fig. 4.1.2). Because of
73
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the presence of the filters and of the bias network, the usable calibrated range will be
smaller compared to the one preset on the network analyzer.
to the device
RT-Duroid substrate
1575
SMA connector
to the network
analizer
Fig. 4.1.3 Detail of the band pass filter used as insulator
The new range will be essentially dictated from the out of band isolation of the filter
and from the sensitivity of the network analyzer. This bandwidth can be obtained
directly from the network analyzer, connecting together the two ports. The result of
this test is shown in Fig. 4.1.4
.CHI l->e
lo g
MAS
S dB /
REF 0 dB
M
l;
.0 0 1 6
dB
0 0 0 . 1 1 0 0 OCO MHZ
C2
E/E MARKER
GHz
START
1 0 0 0 .0 0 0
000
MHZ
STOP 3
0 0 0 .0 0 0
000
MHZ
Fig. 4.1.4 Usable bandwidth (flat one) after the insertion of the bias network
74
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We observe we can still have accurate measurement with the bias network between
1.2 GHz and 2.5 GHz. The result from the measurement of the S parameters for the
circuit of Fig. 4.1.1 is reported in Fig. 4.1.5-4.1.8.
I^IC M
lo g
MAG
0 0 0 . MOO 0 < 0
GH2
C2
E /E
START
1 9 0 0 .0 0 0
000
M Hz
STO P
2
9 0 0 .0 0 0
000
MHz
Fig. 4.1.5 Magnitude of reflection coefficient (S u ) for bias and unbias condition
CH2
1 -1 C M
on«««
60
* /
REF 0
*
1 ;- 1 0 4 .2 6
*
0 0 0 . 11 0 0 0< 0
MHz
S O O .O O O
MHz
MARKER
ca
E/C
START
1 S O O .O O O
000
MHZ
STO P
2
000
Fig. 4.1.6 Phase of reflection coefficient (S u ) for bias and unbias condition
75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
R E F O dB
C2
E /E
MARI : e r
2 GHj
^= 5
STA RT
—
\
i
dB
H
X
X
fc -.B S S O
2 000.1IOO oc
0
1 dB /
M
1
1>
■
9 0 0 .0 0 0
000
I—
MHz
STO P 2
S O O .O O O
000
i
M Hz
Fig. 4.1.7 Magnitude of transmission coefficient (S21) for bias and unbias condition
CHI
l« 2 6 M
phddd
20
• /
REF
Xi
0
92.302
000.1100 0<0
MHZ
9 0 0 .0 0 0
MHZ
C2
E /E
GH2
s ta r t
1 9 0 0 .0 0 0
000
MHZ
STOP
2
000
Fig. 4.1.8 Phase of transmission coefficient (S21) for bias and unbias condition
76
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Almost 10° phase shift is obtained at 2 GHz, with a change in the magnitude o f the
transmission coefficient below 0.3 dB, and keeping the return loss below -20 dB for
biased and unbiased conditions without the aid of any matching circuit. With the use
j
of a simple model on Touchstone (EEsof). we model the circuit, and we estimate a
|
change o f 35% in the dielectric constant o f the capacitor. This confirms that full
modulability is preserved from 1MHz (see Chapter 3) up to 2.5 GHz. This
phenomenon is clear from Fig. 4.1.5 where no cange in the difference between the
phase o f the transmission coefficient is observed in the overall bandwidth. This
j
unique property o f these materials make them a candidate for extremly broad band
|
I
)
devices. Measuring the losses due to the connectors (which were not deembedded
during the calibration) and due to the microstrip we found an insertion loss o f 0.4 dB.
This clearly proves that this kind of configuration, because the small dimensions of
the device can be used efficiently, with insertion loss below 0.2 dB. The problem in
|
the use of FEM tunable capacitors as shifting elements is the fact that they will not be
j
capable to produce a phase shift larger than 90°. In this case the capacitor will operate
j
as an open circuit and no transmission will occur. So a clear limit in capacitor usable
phase shift is about 30°. The combination of two capacitors in a two rat-race
microwave network is shown in Fig.4.1.9.
bias
L
tunable
capacitor
from
previous stage
A/4 to next stage
tunable
capacitor
Fig.4.1.9 Possible microwave network to combine the phase shift of capacitors
77
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Capacitors are connected to the isolated arms of two rat-race devices which add the
total signal (and the phase shift) at the output stage. Based on a similar concept, an
isolated power splitter or circulators can be used instead of the rat-race. Theoretical
results simulated using the HP Microwave Design Software (MDS) are shown in
Table 4.1.1.
Configuration
S li
((JB)
Sn
(dB)
4 Phase(S21)
(dgg)
One capacitor
-6.31
-1.16
7.84
Two capacitors
-2.26
-3.91
12.4
One rat-race
-9.16
-0.59
7.56
Two rat-race
-18.66
-0.117
14.69
One pow. split.
-8.38
-0.683
7.71
Two pow. split.
-10.1
-0.451
12.32
Table 4.1.1 Comparison between different topologies to combine two phase shifters
Values are obtained for a capacitance change of about 30% of its nominal value,
measured for BTO capacitors at 2.4 GHz. The results reported for the Sj] and S 21
parameters are the worst case between the two possible bias conditions. Clearly, the
advantage o f the combined circulators, power splitter, and rat-race over the two series
capacitor is observed. Another possible solution in order to increase the total phase
shift by a factor two, consists of the use of an impedance transformer with a
circulator, as shown in Fig.4.1.10.
78
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
circulator
from
previous stage
\
r /*
impedance
transformer
reflection type
phase shifter
to next
Fig.4.1.10 Reflection based phase shifter
The impedance transformer (a X/4 transmission line for example) must be designed
such that the maximum advantage in terms of phase shift can be achieved, for a
minimum change of the tunable device.
O f course this design using lumped capacitors has many other limitations, but the
main propose was to test the tunability of the barium titanate at microwave frequency,
which result to be very good.
4.2 Distributed Thin Ceramic BTO Tunable Microstrip Line
In this design the use of BTO is as active substrate supporting a printed microstrip
line as shown in Fig. 4.2.1 (the matching circuit is not shown for clarity)
RF signal
output
bias
voltage
ferroelectric
material
RF signal
input
Fig. 4.2.1 Schematic structure of ferroelectric microstrip based phase shifter
79
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The phase-shift capability of the FEM results from the fact that if we are below the
Curie temperature, the dielectric constant of such a material can be modulated under
the effect o f an electric bias field. Particularly, if the electric field is applied
perpendicularly to the direction of propagation o f the electromagnetic signal, as
shown in Fig 4.2.1, the propagation constant (P=2jc/X) o f the signal will depend upon
the bias field [2] since $ = 2 k ^ et / \ and er(Y u.).
The total wave delay will become a function of the bias field, and therefore this will
produce a phase shift A$=A|$/, where / is the length of the line.
The reason why FEM hasn't been widely used for microwave applications todate is
mainly because of the large bias voltage required to change the dielectric constant,
typically a waveguide phase shifter based on FEM requires a bias voltage of 2kV [3]
and due to the high losses in the material. Use o f a new sol-gel technique for the
synthesis of high quality low loss ferroelectric materials, combined with the use o f a
thin ceramic structure, greatly reduces the insertion loss and the bias voltage.
A more detailed schematic layout of the electrically tunable microstrip based phase
shifter is shown in Fig.4.2.2.
connection
pad
ground
plane
t
ground
line
top-view
bottom-view
side-view
Fig.4.2.2 Detailed layout of planar ferroelectric phase shifter
As observed, the active part of the device consists o f a microstrip line overlapping the
ground plane. The ceramic disk thickness is 0.1mm, the microstrip line is 200[im
80
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
wide and the total length of the line is 7.8mm. These three quantities are the basic
parameters for the first phase shifter design. The total length o f the strip will
determine the maximum phase shift which can be obtained for a fixed change of the
propagation constant (A(3), associated with the maximum bias voltage applied. The
total phase shift (A<J>) is given by [2] A0 = 2
Thi s means that longer strips
will give larger phase shift for a fixed change in the propagation constant (A(3). In
practice it is very difficult to obtain good samples with diameter larger than 15mm, so
we compromise for a diameter of 10mm. The ratio between the microstrip line width
(w) and the substrate thickness (r) will determine the characteristic impedance o f the
phase shifter (for a given dielectric constant). Because the substrate thickness is of the
order of 0.1mm and the effective dielectric constant of the ceramic is on the order of
900, we need to choose w as small as possible in order to be able to match the circuit
with a 50Q system. Widths below 50pm are not very practical due to the associated
high ohmic resistance and due to difficulties in the fabrication process. For this
particular example a width of 200pm was choosen. The thickness of the substrate also
determines the required bias field to obtain a desired shift. For example BTO and
BST have a break down field strength of the order of 12MV/m and therefore they
require an electric field strength of 2500kV/m in order to show a pronounced change
in their dielectric constant. For this reason in order to have a bias voltage below
400V, a thickness of 0.15mm or less must be used. A substrate thinner than 0.1mm is
impractical because besides the fact that it is hard to manage, it also leads to a
microstrip line characteristic impedance which is too low to be useful.
The tested ceramic sample was prepared using powder BaTiOj, which was obtained
from the sol-gel process as described in Chapter 2. As observed in Fig.4.2.2, the two
extremes of the microstrip line have patches which facilitate the connection of the
81
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
device with the circuit These patches also allow multiple wire bonds in order to
reduce the parasitic inductance associated with the connection. To reduce the parasitic
capacitance associated with the larger patches, the ground plane of the sample extends
only under the strip as seen in Fig.2. The sample is mounted using wire bonds on the
two ends o f the 50Q microstrip line printed on alumina substrate. The ground contact
is established using multiple via-holes. Epoxy glue is used to hold the sample on the
circuit as visible in Fig.4.2.3.
microstrip
line
wire
act^ve
microstrip
FEM
substrate
epoxy
via-hole
alumina
substrate
ground plane
Fig.4.2 J Mounting schematic of the planar phase shifter on conventional
microstrip transmission line
If further reduction of parasitics due to the wire bond is desired, the wire bonds can
be eliminated by mounting the device upside down. An extensive set o f tests to
determine the effect of wire bonds on the device performance is carried out later in
this chapter.
A bias set-up similar to the one used to characterize the lumped
capacitor (see Fig. 4.1.2) is also used in this case, with the addition of a matching
circuit after the filters. The matching circuit is designed using a simple single stub
type of matching, placed at proper distance from the phase shifter as illustrated in
Fig. 4.2.4.
82
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
7C
]
0.63mm
wire bond
15 mm
Fig. 4.2.4 Matching circuit used for the phase shifter design
Of course the use of a stub to obtain the matching will limit the operative bandwidth
of the phase shifter, at the gain o f improved return loss. The matching circuit was
designed with an unbiased substrate (higher loss) in order to minimize the loss in the
worse case, while for a biased substrate a small of mismatch is acceptable since the
substrate loss becomes lower.
To design the matching circuit the following steps were taken:
1) build and characterize a special modified SMA connector capable to establish a
direct connection with the FEM substrate, at the operative frequency as shown in
Fig.4.2.5.
FEM microstrip
wire bond
FEM substrate
FEM ground plane
GND plane
Fig. 4.2.5 Modified SMA tab-type for on chip FEM measurements
The FEM is glued to the ground plane using silver paint, and the contacts with the
microstrip line are established through pressure wire bond (no soldering is used to
avoid discontinuities). This type o f contact can be fully modeled between 1-10 GHz
83
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
with a 5012 transmission line having length 5.5mm with a series inductance of InH.
Results of the measured and modeled transition in the above range when the transition
is terminated at a 5012 microchip resistor, as shown in Fig. 4.2.6, are reported in Fig.
4.2.7.
g
A
measurement section
5 5 mm
50Q chip resistor
ln H ,B
I
1
1
I
-o
I
50C2
equivalent
Uo'B '
Calibration section
Fig. 4.2.6 Measured used to characterize the transition from coaxial to microstrip
(ft
tn
1.0 GHz
staat
t .000 ooo 000 IM|
freq
10.0 GHz
■TO* 10.000 000
Measured
Modeled
Fig. 4.2.7 Measured and modeled S ii for the transition in Fig. 4.2.S
84
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Clearly the model is in perfect agreement up to 10 GHz and it ful-fills our need at 2
1
GHz. The schematic layout of the utilized matching circuit is shown in Fig. 4.2.7
i
j
2) measure the 5 parameters for the phase shifter in unbiased condition connected to
the modified SMA connector, and locate them on the admittance Smith chart as
seen in Fig. 4.2.8 (point A).
3) extract the parameters of the FEM microstrip line deembedding the measurement
set-up parameters (point B)
4) add the effect o f the bonding wire which are 1.5 mm long (point C)
5) increase the tranmission line lengths (LI) such that it hits the unit circle in the
admittance chart (point D)
6) add a open microstrip stub having length L2 so as to cancel the imaginary part of
the reflection coefficient (point E).
|
!
7) add two transmission lines having length X/2 at the two ends to establish the
external connection
85
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 4.2.8 Reflection coefficient for the FEM phase shifter at 1.85 GHz
Results of the measurement for the phase of the transmission coefficient (in the
frequency range between 1.35 GHz and 2.46 GHz a it reported in Fig. 4.2.9
86
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
200
150
100
$
■5
50
CM
CO
®
aat
0.
-C
-5 0
-100
-1 5 0
-200
1.75
1.8
1.85
1.9
Frequency (GHz)
Fig. 4.2.9 Phase measurement for the transmission coefficient
for different bias conditions.
More than 120° phase shift is achieved at 2.46 GHz with bias voltage of 400 V. The
insertion loss in the same frequency range are shown in Fig. 4.2.10
o
5
Vb=400V
-10
15
-20
-2 5
1 .75
1 .8
1.85
1 .9
Frequency (GHz)
Fig. 4.2.10 Magnitude measurement for the transmission coefficient
for different bias conditions.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
From Fig. 4.2.10 we reach a minimum insertion loss of 3.5 dB at 1.85 GHz, when the
device is under bias condition. We also notice a total change in the insertion loss of
the order 3 dB between the biased and unbiased condition. The corresponding return
loss is reported in Fig. 4.2.11
I
I
I
1
I
I
I
I
I
I
I
I
I
I
1--------- 1
vtt^bov
Vb=0V
CO
CD
"O
1.8
1.85
Frequency (GHz)
Fig. 4.2.11 Magnitude measurement for the reflection coefficient
for different bias conditions.
Notice the best match is obtained for unbiased condition (Sn=-37 dB), while lower
matching result under bias condition (Su= -22 dB). Starting from these measured
results it is possible by using proper modeling (which will include all the parasitics of
the circuit) to de-embed the substrate constitutive parameters (dielectric constant and
loss see section 4.3 for more detail) under bias condition. The result of this analysis
are reported in Fig. 4.2.12
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.22
900
0.2
c 800
0 .18
700
S 0 .1 6
i 600
0 .1 4
500
0. 12
|
1000
400
0
100
200
300
0
400
100
200
Vbias (V)
300
40 0
Vbias (V)
Fig. 4.2.12 Extracted behaviour of dielectric constant and loss vs. bias for BTO
Again as was observed a lower frequency and for the lumped capacitor model, a net
change o f relative effective dielectric constant of 35% is achieved (going from er=900
down do er=600) with corresponding reduction of loss o f 38%. The total power
absorbtion at the highest bias condition (400V) is of the order of 20 mW as reported
in the graph in Fig. 4.2.13.
2S
20
1 5
0.
O 10
Q
5
0
200
300
400
Vbiaa (V)
Fig. 4.2.13 DC power requirement for the BTO phase shifter
89
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4.3 Distribute Thin Ceramic BST Tunable Microstrip Line
Use of BTO thin ceramic microstrip illustrated in the previous chapter, shows the
feasibility of having a phase shifter with moderate insertion loss (between 6.5 dB and
3.5 dB for the unbiased and biased conditions respectively), and phase shifting
capabilities up to 120°. This design is also a good starting point, in that it can be
further improved using barium modified strontium titanate (BST) which, as was
shown in Chapter 3 (Table 3.3.3), has lower electric loss compared to BTO. In this
new design the focus will be to maximize the phase shifting capabilities, reducing at
the same time the variation of the insertion loss between the biased and unbiased
conditions. Reduction of the variation of the insertion loss can be more important than
the insertion loss itself for practical design, since a constant loss can be easily
recovered by a fixed gain amplifier.
The physical layout of the phase shifter is identical to the one shown in Fig.
4.2.3, and also all the bias and matching networks are similar. The ceramic disk was
also processed in a similar way, and has a thickness of 0.1mm, while the printed
transmission line is 50pm wide, and the total length of the line is 8mm. Use of
narrower microstrip line (with respect to the previous one which was 200pm) was
possible due to a better polishing process for the ceramic surfaces.
90
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
200
tsov
ISO
ar
200V
175V
100
1S0V
at
CNJ
CO
aUl
a
-C
a.
-5 0
100V
-100
50V
-150
0V
-200
1
1.5
2
Frequency (GHz)
2.5
3
Fig.4.3.1 S21 phase measurement for the BST ceramic sample
Having narrower microstrip will increase the characteristic impedance o f the
transmission line, allowing easier matching circuit. Measurements for the S 21 phase
(in the frequency range between 1 and 3 GHz) are shown in Fig.4.3.1 for selected bias
i
i
points. More than 160° phase shift is achieved at 2.43 GHz with bias voltage around
1
250V. The correspondence between phase shift and bias voltage is reported in
Fig.4.3.2.
91
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
160
14 0
120
8*
100
80
a01
CO
60
40
20
0
0
100
50
150
200
250
V b ia s <V>
Fig.4.3.2 RF phase shift versus applied bias field for the BST ceramic sample
Very little saturation is reached, since the correspondence between the phase shift and
the bias voltage is almost linear up to 2S0V (after an initial inertia observed for
Vbias<60V). This implies that further phase shift can be produced by biasing the
device harder. Measurement of insertion loss in the same frequency range is shown in
Fig.4.3.3.
o
1S0V
2
175V
'2 5 0 V
ffi
4
OJ
CO
O)
3
0V
2SV
50V
7SV10° V 125V)
6
8
10
1
1.5
2
Frequency (GHz)
2.5
3
Fig.4.33 S21 magnitude measurement for the BST ceramic sample
92
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Because of the way the matching circuit was designed we observe the magnitude of
S 21 having a total change smaller than 2 dB in the frequency range 1.6-3 GHz. This
makes the device particularly suitable for broad band operations. The insertion loss at
2.43 GHz is below 4 dB with no bias field, and reduces to 2.6 dB when a bias field of
250V is applied. The corresponding 5 /; measurements are reported in Fig.4.3.4.
175V
asov.
CO
01
a
2
-20
-2 5
-3 0
150V
75V
-3 5
100V
125V
-4 0
1
1.5
2
Frequency (GHz)
2.5
3
Fig.4.3.4 S u magnitude measurement for the BST ceramic sample
Particular attention was dedicated to the design of the matching circuit to minimize
the variation of insertion losses between biased and unbiased conditions. This goal
was successfully achieved with the observation that FEM losses decrease under bias
condition [4]. This property can be used by having a very well matched circuit in the
unbiased condition (S/y=-30 dB) and worse matching under biased condition (Sy/=11 dB). With this approach we reduce the maximum total change of insertion loss to
1.6dB [5]. This concept is essential for a good design of this type of phase shifter, and
is well illustrated in Fig.4.3.5, where the dashed line represents the S parameters for
the unbiased circuit while the solid line corresponds to the biased condition.
93
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
unbiased
-20
-25
-30
1
1.5
2
2.5
3
Frequency (GHz)
Fig.4.3.5 S parameter measurement for Vbias=0V and Vbias=250V
Another critical parameter for this design is the length of the wires which interconnect
the phase shifter with the rest of the circuit. These wires in fact introduce a little series
resistance, and an additional parasitic inductance in series with the microstrip line,
which at some critical value can seriously compromise the matching of the circuit.
Three different measurements were performed to find this critical length. The first
measurement uses multiple wire bonds (very low inductance) having length around
1mm on each side of the line; subsequently single wire bonds with increased length
from 1mm up to 2mm are tested. Results of this test are summarized in Table 4.3.1.
94
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
W in length estimated MaUhing Insertion loss
(mm)
inductance
(dB>
(JB)
(nH)
0.5
-10.98
-2.66
1
1
-14.8
-2.73
1.5
1.5
-8.58
-3.4
2
2
-3.4
-8.58
1 (<double)
Table 4.3.1 Effect of wire bond length on the performance of
the phase shifter
Clearly when the wire exceeds the length of 1mm (total 2mm on the two sides) the
performance degenerates. In this test the wires used were made of gold, and they have
a diameter of 0.0127 mm. Bigger diameter also helps to reduce the associated
parasitics. To extract the dielectric parameters, A first rough model of the phase
shifter can be designed using the HP Microwave Design Simulator. The software
allows to model most of the circuit parts, including the parasitic elements and
conductor ohmic resistance. Based on the implemented model and on the set of
measured data shown in Figs.4.3.1-4.3.4, we can extract the dielectric constant value
and losses for the device as a function of the bias voltage. This characterization is
quite simple since all the components in the circuit can be modeled accurately when
the software is used for such a low frequency. In this case the only unknown
parameters are the dielectric constant and the losses o f the FEM substrate.
Furthermore, the location of the S u ripples or the phase shift of S 21 is uniquely
related to the value of the effective dielectric constant of the substrate, while the value
of tan5 will determine the average slope of S 21 versus frequency. So once we select
the dielectric constant for the substrate based on the phase shift, we just need to vary
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!
I
I
the losses until we match the curves corresponding to the magnitude of S u and S2iAn example of parameter fitting at Vbias=100V is given in Figs. 4.3.6-4.3.8.
CO
■o
-20
CO
^
-3 0
-4 0
&
M easured
Simulated
-5 0
1
2
1.5
2.5
3
Frequency (GHz)
Fig.43.6 Si i magnitude modeled vs. measurement for Vbias=100V
o
1
a
M easured
Simulated
2
3
4
5
6
7
1
1.5
2
2.5
3
Frequency (GHz)
Fig.4.3.7 S 21 magnitude modeled vs. measurement for Vbias=100V
96
I
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200
M easured
150
a
Simulated
100
I
CM
CO
e
-5 0
a.
-1 0 0
-150
-200
1
1.5
3
2.5
2
Frequency (GHz)
Fig.43.8 S21 phase modeled vs. measurement for Vbias=100V
Results of the modeling procedure at 2.43GHz for all bias voltages conditions are
shown in Fig.4.3.9
700
0.09
600
0.085
500
0.08
E 400
300
S 0.075
200
0.07
100
0
50
100
150
200
0.065
250
0
w v>
50
100
150
200
250
v b ias< v >
Fig.4.3.9 Effective dielectric constant and loss extracted values vs applied
bias field
97
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Increase of bias voltage reduces the dielectric constant and losses in the material as
expected from theory. The lowest estimated tan8 =0.072 at 2.43GHz is achieved with
bias voltage o f 250V. Clearly in this last design several improvements have been
\
achieved compared to the first BTO phase shifter, among them a lower insertion loss
|
(from 3.5 dB to 2.6 dB), a lower variation of insertion loss (from 3 dB to 1.3 dB),
lower bias voltage (from 400V to 250V) and lower power consumption (from 20mW
to lmW see Section 4.4).
4.4 Power Requirements And Power Handling Consideration for
Microstrip Phase Shifters
Three main considerations need to be made regarding the power requirements of the
i
phase shifters we design:
|
1) A DC power required to maintain a fixed phase shift;
|
2) The instantaneous power required to change the phase shift from one condition to a
new condition;
\
3) The maximum microwave power which can flow in the device without altering its
functionality, either due to the occurrence of a nonlinear phenomenon or due to
heat dissipation.
The dc resistivity (p) measurements of Bao.8 Sro.2TiC>3 give a value of p= 4.3 107 Qm. Since the cross-sectional area for bias current flow in our sample is 5.03*10 '5 m2 ,
the corresponding dc resistance of the sample is 8.6*107Q. The dc voltage applied to
the phase shifter varies from zero to 250V, so the maximum power required will be
0.7211*10*3W. The above calculation shows that in order to maintain a fixed phase
the dc power requirements (less than lmW) are extremely small. Therefore, the bias
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power dissipated and hence the heating effects on the characteristics o f the
ferroelectric material arc negligible. Results of power measurement on the sample are
shown in Fig.4.4.1.
E
*
o
a.
o
o
0.5
0
5 0
100
150
200
250
Vb l . . ( V )
Fig.4.4.1 DC Power requirement versus bias field for the BST phase shifter.
The actual total power is slightly higher than the theoretical value, probably due to
other current leakage, e.g. in the circuit of the voltmeter or other measurement
instruments. Although it is not possible to give a quantitative estimate of the amount
of driver power dissipated in the ferroelectric, since a knowledge of the large signal
properties of ferroelectric materials is currently not available, some considerations are
possible: the phase shifter as viewed from the terminals of the driving source is
essentially a nonlinear parallel plate capacitor. It can be shown by an idealized
calculation [6] that the energy transferred in changing the voltage from zero to 2S0V,
thereby changing phase by 160°, is about 0.003 joules. The instantaneous current
requirements are a function of the speed with which the phase shifting is performed.
99
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Although the peak power capability of the driver is high, it is at least one order of
magnitude smaller than that required for comparable frequency and speed ferrite
phase shifters. One o f the advantages of these phase shifters is that very little of the
energy supplied by the driver to effect a rapid change in phase is dissipated in the
ferroelectric material. The bulk of the energy tranferred is used to change the energy
stored in the electrostatic field of the capacitor or mechanically stored in the ceramic
(these ferroelectric materials are also piezoelectric) [7]. The power handling
capability of the phase shifter would be limited by the following factors :
I
1) The RF voltage level which can be sustained without breakdown,
2) The tolerable temperature rise due to the attenuation of the transmitted RF power,
3) The material response to large amplitude RF signal.
In general in a microwave system employing microstrip lines, the coaxial connectors
i
I
rather than the microstrip set the ultimate limit to peak power. In our case since the
!
substrate is very thin we need to consider both possibilities to obtain the lower
breakdown voltage. The substrate used has a thickness of 0.1mm, and a maximum
|
breakdown voltage for pulsed signal of 8.5 MV/m. This corresponds to a maximum
tolerable voltage of 850V.
The coaxial connector sets a limit because of air
breakdown, (the breakdown electric field strength for dry air is 3MV/m and the
internal difference in the radii of a 50Q connector is approximately 1.5mm, thus the
breakdown voltage is 4500 V). So the lower limit is determined by the substrate, and
corresponds to 850V. With any transmission line having characteristic impedance Zo
and maximum breakdown voltage Vbk the peak power allowable is given by:
100
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In our case the characteristic impedance o f the microstrip line is below 10Q so the
maximum corresponding transmitted power is about 36kW. Although this is just a
theorethical value, it shows that the breakdown voltage is not a limiting factor, even
for a thickness of 0 .1mm.
The temperature rise due to conductor and dielectric losses is well treated in [8]. The
expression for the temperature rise above ambient is:
A-
0.2303/1
A / = -----------
5C/W
,
(4.4.2)
where ac and oy are the conductor and dielectric losses respectively (in decibels per
meter), wef f and Wgg(f) are the effective microstrip widths, and K is the thermal
conductivity of the substrate. If we consider a rise 0 above ambient, a microstrip line
could carry a maximum average power given by
(4.4.3)
For our sample we estimate AT=0.04°C/W and the characteristic impedance is below
10Q. If we consider a rise in the temperature of 0=10° above ambient, the total power
handled ranges up to lkW. The effects of large amplitude RF voltage on the
properties of ferroelectric materials are largely unknown, and would have to be
determined by experiment. Although only limited information is available, it is likely
that large RF fields will cause nonlinear behavior, e.g., harmonic generation and
mixing. In conclusion, the real limiting factors to power handling capability are the
101
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heat dissipation and the manifestation of nonlinear effects. No particular attention is
i
focused to these phenomena for powers below 500W.
i
i
!?
;
ii
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References Chapter 4
[1] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance
Matching Networks, and Coupling Structures. Deham MA: Artech House, 1980.
[2] R. E. Collin, Foundations fo r Microwave Engineering. McGraw-Hill International
Editions, 1966.
[3] D. C. Collier, “Ferroelectric Phase Shifters for Phased Array Radar Applications,”
in IEEE MTT-S Int. Microwave Symp. Digest. September 1992, pp. 199-201.
[4] T. Mitsui and S. Nomura, “Landolt-Bomstein: Numerical Data and Functional
Relationship in Science and Technology,” Ferrolelectrics and Related Substances,
vol. 16, 1981.
[5] F. De Flaviis, O.M. Stafsudd, and N.G. Alexopoulos, “Planar Microwave
Integrated Phase Shifter Design with High Purity Ferroelectric Materials,” IEEE
Trans. Microwave Theory Tech., vol. 45, June 1997.
[6] M. Chon and A. F. Eikenberg, “UHF Ferroelectric Phase Shifters Research,” in
Electric Communication Inc. Final Rept. on Contract No. AFI9(604)-8379. April 30
1962, pp.
[7] C. Kittel, Introduction to Solid State Physics. New York: John Wiley & Sons Inc.,
1986.
[8] K. C. Gupta, R. Garg, and I. J. Bahl, Microstrip Lines and Slotlines. Dedham MA:
Artech House, 1979.
103
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C h apter 5
Diaz Fitzgerald Time Domain Technique for the
Solution of Maxwell Equations in Complex Materials in
the Time Domain
Electromagnetic phenomena can be simulated by the dynamics of a mechanical system
as long as the Hamiltonian of the electromagnetic and the mechanical systems coincide.
In this thesis a generalization of G.F. FitzGerald's pulleys and rubber-bands
mechanical model for the interaction of electromagnetic waves with complex media
such as ferroelectric materials is presented. A direct analogy between the FitzGerald
model and the electric vector potential formulation, at each stage of the extension of the
original model is shown: each mechanical observable has a unique correspondence in
the vector potential formulation. This strict analogy allows further inductive
developments of the mechanical model and extends the pedagogical importance of the
original FitzGerald model. As a consequence very complex materials from the
electromagnetic point o f view, such as frequency dependent magneto dielectric
materials are easily understood and implemented with simple modifications in the
mechanical system. The condense node representation of the field in the vector potential
formulation results in lower grid dispersion compared to other numerical techniques
such as the Finite Difference Time Domain (FDTD). Several applications, such as
classical scattering problems from dielectric, magnetically permeable, dielectrically
lossy and Debye materials are described. The simulations are validated with comparison
to canonical solutions, or with FDTD calculations.
104
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5.1. Introduction
Mechanical analog models can provide an excellent time domain visualization tool for
propagation, scattering and radiation of electromagnetic waves in dispersive media. It
has been known since the late 19th century that electromagnetic phenomena can be
simulated by the dynamics of a mechanical system, as long as the Hamiltonians of the
mechanical and the electromagnetic systems coincide. It was shown by Diaz [1,2] that
George Francis FitzGerald's 188S [3] model of electromagnetic propagation leads to a
finite difference numerical formulation that is different from the conventional Finite
Difference Time Domain method (FDTD). This is because FDTD is based upon the
discretization of Maxwell's equation in the classical formulation, while the Diaz time
domain discretization o f FitzGerald's mechanical m odel coincides with the
discretization of Maxwell's equations in the vector potential formulation [4].
In this thesis the original model proposed by FitzGerald is presented and the model for
the treatment of complex materials including lossy and dispersive media. As a further
confirmation of the validity of the analogy, we present a set of canonical problems is
presented, including scattering from a cylinder, an eigenvalue problem, and an echo
experiment. All extensions of the model are continuously supported by analogies with
the vector potential formulation (see Appendices). It is also shown how the condense
node representation of the Held causes lower grid dispersion with respect to the
traditional Yee scheme of FDTD. For the two dimensional cases considered, our time
domain technique requires 38% less memory and, consequently, 38% less
computational time.
105
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5.2 The Fitzgerald Pulley And Rubber*Band Model
Consider the distributed discrete system of Figure 5.2.1. Pulleys of moment of
inertia / (</ are connected to each other through rubber bands of elasticity k.±^ j±^.
ltij.l/2
Inertia I
Angular velocity
‘•i.lflj
Tension
tight
elasticity k
■■ij-l/2
Fig 5.2.1 Array of rigid pulleys connected by rubber bands
If at time t=0 pulley (ij) is spun (Figure 5.2.2a), its rotation strains the four rubber
bands connecting it to its four neighbors.(Figure 5.2.2b). The total force of tension and
compression applied at the peripheries of the pulleys exert to each one an angular
acceleration a through Newton's second law: their angular velocity 0) increases by adt
after a time step dt (Figure 5.2.2c).
Oi+lj
OiJ-l
S
\
strained
b)
C)
Fig.5.2.2 Action-reaction mechanism of propagation of motion
a)
106
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Thus, an angular velocity pulse applied to the central pulley propagates outwards by
action and reaction to all pulleys of the system. FitzGerald showed that this behavior
mimics exactly generation and propagation of electromagnetic waves in Maxwell's curl
equations: the inertia of the pulley I represents the medium permeability (ji), and the
elastic constant k represents the medium inverse permittivity (1/e). FitzGerald further
noted that if the rubber bands are allowed to slip then we obtain a lossy dielectric
medium with the conductivity being inversely proportional to the coefficient of friction
between the rubber bands and the pulley. O f course, by Heaviside's duplex equations
[S] we can also choose to identify Ix £, k<*H\i according to Diaz [2]. Application of
Greenspan's approach [6] to this mechanical system yields a set of finite difference
equations as follows. Consider for the sake of simplicity the one dimensional system as
shown in Fig.5.2.3. Note that the elasticity of the rubber band is represented pictorially
as a spring with elastic constant k. Each pulley is connected to two springs as illustrated
in Figure 5.2.3, which contribute to the overall torque acting on it.
i-l
i-1
i
<
/+ /
Fig.5.2.3 Pulley and rubber-bands represented as springs
The generic i* pulley is subject to four forces due to the neighboring rubber bands:
107
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/?> =-*<‘>(9, - e M)a
/* » — * » ( « „ - « >
' S ’ — * S (» M - « >
As a consequence the resulting torque T[ acting on the i^1pulley, is given by
T, = (F™ + F™ ) a - ( F f >+ F g j a = t f ( e M - 0,) +
-9 .) .
(5.2.2)
see Fig.5.2.4. The acceleration of the i**1pulley after a time step At is equal to:
( i r l = kJr {e-
'
(5.2.3)
"*) ■
Ti
F . (P + F .(1)
i-l * i
F (2) + f (2)
i
i-l
Fig.5.2.4 Torque resulting form the composition of the four forces
Q) being the angular velocity at time step n. If now the diameter of the pulley has the
grid size, A x-a, the previous expression becomes after rearrangement
AM
^
A w y = (Ax)4 1 — M
A/ J.
/,
(
AM
“ i
AM
AM
n
“ |- I
Ax
Ax
(5.2.4)
since the inertia o f the pulley /,• with radius a, width h and density p can be written as
108
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If we take the limit Ar—*0 and Ax-»0, since I is proportional to Ax^ the continuum
differential equation of (5.2.4) together with the definition of angular velocity becomes:
(5.2.6)
The electrical analog of the same one dimensional Maxwell's curl equations is given in
great detail in Appendix B, and it leads to [7]:
dEt _ i f a T i a p '
dt
e d x y n dx
(5.2.7)
Note that in the derivation of eq. (5.2.7) there is no constraint on the space invariance
of the dielectric constant nor of the magnetic permeability. The extension to the two
dimensional case is straightforward and is omitted. A summary of the correspondence
between the mechanical, see eq.5.2.6, and the electrical quantities, see eq.5.2.7, is
presented in Table 5.2.1. In the mechanical model the presence of a dielectric medium
( e * £0) is taken into account by increasing the moment of inertia of the pulley; while
the presence of a magnetically permeable medium (/i * fiQ) is modeled by modifying
the elasticity of the rubber (the elastic constant of the spring) [8,9].
109
I
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Mechanical
Electrical
Ez
dEzIdt
Az
z-comp. electric field
(0
angular velocity
a
angular acceleration
-6
angle
Ez rate of change
z-comp. vector potential
E
permittivity
1/a2
n
permeability
l/ik a 2) rubber elasticity
At
mesh grid size
a
inertia of the pulley
pulley radius
Table 5.2.1 Equivalence between electrical and mechanical quantities in our model.
5.3 Extension To Dielectric Lossy Materials
The first extension of FitzGerald's model to mimic dielectric lossy materials consists of
i
i
the immersion of one or more pulleys in a bath of viscous fluid as shown in Figure
Viscous
fluid N
Fig.5.3.1 Pulley immersed in a viscous fluid bath.
The equations of motion of the immersed pulley are slightly modified in that the
effective tension on a pulley caused by the neighbors is modified by the addition of a
viscous damping term:
T,ff,cuv* ~ ^baxdt ~
(5.3.1)
t
110
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which translates into the equation for the acceleration
da> a2 d ( 2 <90^
1r =T * r
a2 e
(5.3.2)
It is shown in detail (see Appendix B) how to derive the vector potential equations for
the case of electric lossy materials starting from Maxwell's equations; we report here
only the final expression for the time derivative of the electric field in one dimension:
dE.
dt
1 d rl
e dx H dx
(5.3.3)
e
Comparison of eq. (5.3.2) to eq.(5.3.3) leads to the conclusion that the mechanical
quantity "viscous friction coefficient" J; corresponds to the conductivity at :
4
<x€
(5.3.4)
5.4 Extension To Debye Dielectric
Materials
In this section attention is focused on dispersive media whose behavior can be modeled
by a sum of Debye terms [10]. For a material characterized by a single Debye relaxation
we can write [11, 12]
D = e .E + P
,
(5.4.1)
where
dt
t
E --P
t
(5.4.2)
In eq. (5.4.2) et and £_ are the dielectric constants of zero (static) and "infinite"
frequency, and r is the relaxation time constant. The modification of the mechanical
111
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model takes the form of an additional weighted ring (of moment of inertia l n>) resting
on the pulley and connected to it through a coefficient of friction (Qfl)) as shown in
Figure 5.4.1.
rotation axis
to other
pulley
Fig.5.4.1 Mechanical model for single electrical Debye materials.
The time constant of the Debye pole t corresponds to the ratio I a)/Q (l>. Action and
reaction of the top pulley then simulates the storage and dissipation of polarization. The
new set of equations for this system can be derived from the original equations in the
one dimensional case, by adding an inertial reaction due to the top pulley which is
coupled to the bottom one through a friction coefficient.
dt
70) K l
*'
d(oi _ a 2 _
/51) dto?
dt
I, dt
dQl
— - — CO
dt
‘
(5.4.3)
Here co\ represents the angular velocity of the weighted ring, /•
the moment of
inertia and g f ’, the friction coefficient. The system of equations (5.4.3) can be
rewritten as:
112
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This extension of the FitzGerald model finds as exact analogy in the vector potential
formulation of Maxwell's Equations. In this case the polarization vector needs to be
introduced (we carry out a detailed derivation in Appendix C). Here we just report the
final equations in the one dimensional case:
T —
dE,
dt
dA* _
i T ~
■ ~xp'
t
dx^H dx ,
- E.
£
—£
(5.4.5)
c-
f
‘
Comparison of eq.(5.4.4) to eq.(5.4.5) clarifies that there are several other analogies
between the mechanical quantities and the electrical ones which are listed below:
Electrical
Mechanical
0)
pt K.es - e j
a
( e s - e^) / x
Qm
es ~
6 «.
IW/ a 2
I/*2
Table 5.4.1 Equivalence between electrical and mechanical quantities in our model.
113
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Extension to double Debye materials is straightforward. In fact mimic double Debye
terms can be simulated by adding two rings surmounting the original pulley arranged
concentrically as in Figure 5.4.2.
rotation axis
to other pulley
Fig.5.4.2 Coaxial arrangement of two Debye terms
Since the two rings are directly connected to the bottom pulley, they are independently
coupled to it though two different friction coefficients. The mechanics of the system is
enriched by two inertial reaction contributions due to the action and reaction of the top
two rings:
\d<o™
— =
dt
dm™
— — =
dt
a2$ 2),
-(0 )
/<2) K
'
a2Q(l),
— =7—(CO - c o l
/<” v ‘
‘ ]
a2$ l)
M
a M
* ""T
^
—
(5.4.6)
a 2Q(2)
= <0,
where the mechanical quantities referring to the two rings are indicated with the
superscript 1 and 2 respectively. It is useful to define a weighted average of the angular
114
x
t
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velocities o f the two rings, where the weighting factors are related to the relative
moment of inertia of the two rings:
(O =
/ (I)
I (2)
<ow +
0)(2)
(5.4.7)
Therefore the average angular acceleration becomes, with the aid of eq. (5.4.6):
a 20(2)
Q1
(5.4.8)
System (5.4.6) can be rewritten in the form:
dm
a2Q(l)
(c0 dt
7(1) + / <2)'
doa _ a \ r
dt
I ‘
dd
— = <u
— =
V »( 2)
o>(l)W a g
'
(c o - 0 ,m )
/ (1) + / (2)'
dm
'
(5.4.9)
The analogy with the vector potential formulation is possible also in this case, where
two partial polarizations Pjl) and P}2) and an "effective" polarization Pt = P™ + P™
have been defined. Each polarization obeys the differential equation
dpi*)
P.(m) + T.
(5.4.10)
~ d T =€p8^
so the total effect is simply given by the superposition principle, and we used the
notation ep = e J - e m, gn and r„ being the weight of each pole and its time constant,
respectively [10]. The equations for the vector potential formulation for double Debye
materials become:
115
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Each polarization corresponds to the angular velocity of one of the rings, while the total
polarization corresponds to the "average" angular velocity of the rings. The analogies
between the mechanical system and the electrical formulation are easily drawn and are
collected in Table 5.4.2.
Electrical
/(£*-£„)
fz V
£_)
P™l %(£,-£_)
^(Cj- £j / Tl
h <«r- e j t ' h
-O
Mechanical
©
©0)
©®
QO)
q
{2)
l (IV a2
l<2V a2
eoo
// a2
Table 5.4.2 Equivalence between electrical and mechanical quantities in our model.
Extension to n-pole Debye materials at this point can be carried out by induction. The
representation of the system is a collection of n rings placed on top o f a pulley. Each
ring is characterized by its own moment of inertia (/W) and is coupled to the bottom
pulley though a friction coefficient ( Q ^ ) independent of the other rings. Therefore we
identify in the model the n angular velocities of the rings with the n partial polarizations
116
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[13]. The relations between the mechanical and electrostatic quantities are collected in
Table 5.4.3.
Electrical
Mechanical
'J
<o
a <n>
' f >/
Q (n)
* •< * » -« J /r.
l<n)
s nj - ej - t 00 )
e so
I
Table 5.43 Equivalence between electrical and mechanical quantities in our model.
5.5 Extension To Frequency Dependent Magnetic Materials
Less common but still very interesting from both theoretical and engineering
point of view are dispersive magnetic materials. Again attention is restricted to materials
which can be modeled by a sum of Debye terms. A single Debye magnetic medium is
characterized by the following equation:
(5.5.1)
where
(5.5.2)
Here /z, and /t„ are the permeability at zero (static) and "infinite" frequency and t is
the relaxation time constants. In this case the mechanical model can be extended once
more to describe such materials by modifying the rubber bands elasticity to attribute
their elastic constants a Debye character. The modification takes the form of an
additional spring connected in series with the original one. The new spring is connected
117
t
r
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with a device immersed in a viscous fluid, so that the oscillation of the second spring is
damped according to the viscous coefficient of the fluid y, see Figure 5.5.1.
Fig.5.5.1 Magnetic Debye material model: the two springs are connected in series
We have already discussed how the elastic coefficient can be viewed as the reciprocal of
the magnetic permeability. Now we introduce the time constant of the magnetic Debye
material ra s corresponding to the ratio //£ /. The mechanical model needs at this point
to be further discussed. Consider a mechanical model where a spring is connected in
series as shown in Figure 5.5.2, to a dissipative device characterized by a damping
coefficient y.
y
Fig.5.5.2 Spring in series with a dissipative device.
118
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When the first spring (with elastic constant ko) is elongated by x0 , and the damped
oscillator allows for an elongation x /, the total elongation becomes Xeq=xo+xi. The
restoring force is constant along the direction of motion:
F = - * 0jc0 = - M
(5.5.3)
i
and can be written in a more compact way as F = - k t^xt)l, an "equivalent spring
constant" is defined as
*o
k, + y d /d t )
k„ + k, + y d / d t
where the time differential with respect to time is treated as an operator. Therefore the
restoring force becomes:
f — k0+kl
y V +y y^^A
/d>t x«r
"
'
(5-5 5 )
or, equivalently,
dF ___.
dt
0 dt
*o+*i
y
where the last equation is the differential equation for the restoring force. The total
elongation is given by the stretch of the rubber band between two neighboring pulleys
x„ = aA0 = a(0M - 0 .)
,
(5.5.7)
or in the continuum limit, as before, the new set of equations for the system is:
119
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In eqs. (5.5.8) F represents the restoring force, kQ and kt the elastic constant of the
springs and y the damping coefficient This extension of the FitzGerald model finds an
exact analogy in the vector potential formulation of the Maxwell's equations, as shown
in Appendix D, giving:
dHy
1 d f dAt '
dt
fim dty dx j
dEt _ 1 dHy
dt
€ dx
- F
ll,
1
dx
x"y
(5.5.9)
Comparison of eq.(5.5.8) to eq.(5.5.9) clarifies that there are several other analogies
between the mechanical quantities and the electrical ones, and this are collected in Table
5.5.1. It is clear that this model can be extended to double Debye magnetic relaxations
by simply adding a second damped oscillator connected in series to the existing ones as
shown in Figure (10). The first spring (with elastic constant ko) connected in series, as
shown in Figure 5.5.3, with two damped oscillators characterized by elastic constants
kj and *2. and damping coefficients Yl and J2-
120
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Fig.5.5 J Series representation of two magnetic Debye terms as oscillators in series
When the three springs suffer an elongation xo,xi and X2 , respectively the total
elongation is Xeq=xo+xi+X2. The "effective spring constant" becomes:
1
1
k = — +•
ko kl + yl d/dt
1
k2 + y2 d /d t)
(5.5.10)
Using the more compact notation kt - kt + y,— , the restoring force becomes:
at
F_
k^k,
k0kl +k0k2+klk2 Xtq
(5.5.11)
or, equivalently,
kJcF + kJ^F + kJijF = -k^kJijX^
(5.5.12)
which is the differential equation for the restoring force F. As before we take the total
elongation as the stretch of the rubber band between two neighboring pulleys in the
continuum limit The new set of equations for the system is:
121
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(yjz
- + {Yi+ r 2X * o + kt
+ foA +
+*1*2
=
= - a 2(kvklk2) ^ - a 2{kckl y 2 + k v k 2Yl ) ^ ^ - a 2( k v Y j 2) ^ ^
OX
of a t
cf a t
<faj
dir
dd
dir
(5.5.13)
a^dF
/ dr
= a>
where F represents the restoring force, kQ, k { and k2 the elastic constants of the springs
and Yj and Y2 the damping coefficients. The analogy with the vector potential
formulation [14], see Appendix D, is possible in this case as well:
d2/ /
dH
(M-*,*2) - ^ r + [ ( / * - +/*2h + ( / * - + ^ ) * 2] - ^ - + ( r . + Mi + t h ) n , =
. (5.5.14)
dEt _ 1 d //,
dr
£ dr
dA
dr
The analogies between the mechanical system and the electrical formulation are easily
drawn and are collected in Table 5.5.1.
Electrical
Mechanical
F
H >
k a2
O
kja2
1/ * , ( / * * - m J
l 'g£ns - n j
k2a 2
T j/ g y i n s - n j
y\°2
r 2 / S 2i n s - n j
y 2 a2
Table 5.5.1 Equivalence between electrical and mechanical quantities in our model.
122
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At this point the extension to multiple magnetic Debye materials folds out easily. The
|
model is built by combining in series several dumped oscillators. By induction we
derive the analogies shown in Table 5.5.2, are derived.
i
i
Electrical
Mechanical
■Hy
F
k
o
1/M
xn lg nU s - eJ
! / £ r_l ( £5 - £
an
kn
)
Table 5.5.2 Equivalence between electrical and mechanical quantities in our model.
The last extension of the FitzGerald mechanical model consists in the simultaneous
treatment of dielectric and magnetic Debye materials. These are extremely interesting
from both a theoretical and practical point of view because of the possible application in
the modeling o f particular absorbing materials. The mechanical model is built
combining the features needed to model the electric and magnetic frequency dependent
materials. Each pulley is surmounted by one (possibly many) weighted ring coupled to
the bottom supporting pulley though a friction coefficient. The action and reaction o f
the top rings simulates the energy storage of the polarization vector, and ensures the
electrical Debye character. On the other hand the frequency dependent magnetic
character is ensured by the peculiar elastic properties of the connecting rubber bands,
which can be schematically represented as an ideal spring connected in series with one
(possibly many) dumped oscillators. Using the many analogies derived before, it is not
needed to re-derive the equation of motion for the quite complex system, but merely
add to the basic equation of motion all the necessary ingredients in a heuristic fashion:
123
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dcof) _ a2Q
(5.5.15)
This extension of the FitzGerald model finds again as exact analogy in the vector
potential formulation of Maxwell's Equations, which can be derived by induction from
the two cases treated above. The same analogies between the mechanical quantities and
the electrical ones hold, as described before in detail. Proceeding along the same lines,
it is possible to build a generic complex material, characterized by multiple electric and
magnetic Debye poles, by simply adding the proper number of rings and dumped
oscillators to the simple mechanical model.
5.6 Eigenvalue Problem
As a validation of the two dimensional FitzGerald mechanical model a typical
eigenvalue problem is presented in this section, and the obtained numerical results are
compared with the theoretically predicted values. A rectangular resonator is excited with
an electric field pulse, and, after steady state is reached, by means of Discrete Fourier
Transform (DFT) [15] the resonant frequencies are extracted. These are compared with
the exact theoretical values for the m,n mode calculated from [16]:
(5.6.1)
124
J
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where a and b are the linear dimensions of the resonator and c is the speed of light. In
our experiment we use a square resonator (a=b). The space was discretized by using
33 x 33 cells; the dimension of each cell is 0.2 mm. A Gaussian pulse in shape Is
applied in the geometrical center of the resonator, the width of the pulse is 40 time steps
corresponding to 20 psec. The simulation is time stepped for a long enough time so that
a steady state is reached, typically 8,000 steps. DFT algorithm is used at a non zero
field point, away from the center and from the boundary to obtain the frequency
resonances of the square two dimensional cavity. The DFT analysis of the z-component
of the electric field was performed using 200 points to represent a span of 50 GHz
centered around 25 GHz to localize all the excited modes. The resonance frequencies
are shown in Fig.5.6.1.
0 .0 7
0 .0 6
^
is
0 .0 5
£
0 04
|
0 .0 3
0.02
0.01
0
1 1010 2 1010 3 1010 4 1010 5 1010 6 1010 7 1010 8 1010 9 1010
FREQUENCY (Hz)
Fig.5.6.1 Resonance frequencies for square resonator
Notice that the amplitude of each peak depends on the particular choice of the
observation point, care must be used to avoid null points. The frequency of the
125
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fundamental mode is determined experimentally as the position of the first maximum in
Fig.5.6.1, and it coincides within an error less than 0.1% with the analytical result
calculated from eq. (5.6.1). The same analysis was repeated for every other mode and
{
the error was always of the same order [17].
i
|
5.7 Scattering Problem
As a validation of the possibility of treating dielectric lossy materials, results o f a
numerical experiment for the internal electric field o f a uniform, circular dielectric
cylindrical scatterer are presented. The cylinder is assumed to be infinite in the zdirection. The incident radiation is a wave TM with respect to the cylinder symmetry
I
axis. Because there is no variation of either scatterer geometry or incident field in the zdirection, this problem may be treated as a 2-dimensional one. A two dimensional grid
!
of 400 by 300 mesh points is used. The cylinder axis is positioned at point (200,125).
Second order absorbing boundary conditions are used to truncate the grid, as shown by
Clayton [18]. Grid coordinates internal to the cylinder with radius 0.06m, are given by
(m - 200)2 + (n -1 2 5 )2 £ 202 and are related to the dielectric parameters. All the grid
points outside this grid are related to the free space parameters. The plane wave source
obtained as series of Gaussian pulses for the z-component of the field is activated along
a line at the mesh position m=100. The program is time-stepped for a long enough time
so that the plane wave is scattered from the cylinder and the scattered field reaches the
observation region. DFT algorithm is used to extract the information o f the field
distribution of the frequency component at 1.5 GHz. In the first simulation we use the
following parameters: e4 - 2.0eo, As=3mm, Af=0.5psec. Results are shown in Figure
5.7.1, which graphs the amplitude of the 1.5 GHz component of the field Ez measured
126
t
s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
inside the cylinder, along its diameter perpendicular to the incident wave, after 3,500
time steps.
2
a
Exact
Computed
1.5
1
0.5
observation region
0 —‘
-0.06
-0.04
-
0.02
0
0.02
0.04
0.06
Position (m)
Fig.5.7.1 Comparison between computed and exact solution of the inner
field for a lossless dielectric cylinder, along the cut i=200.
The exact solution, is calculated using the summed series technique as in Jones [19].
The computer solution locates the positions of all the maxima and minima of the
envelope of the electric field with error less then 0.3%. For the second example the
cylindrical scatterer has the same parameters, except that it is lossy with a relative
dielectric constant ed = 2.0eo and conductivity a e=0.0356 S/m. The result of this
simulation is reported in Fig. 5.7.2.
127
ri.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
a
Exact
Computed
observation region
N
Ui
■O
3o
C
8*
2
0.5
-0.04
-0.06
-
0.02
0
0.02
0.04
0.06
Position (m)
Fig.5.7.2 Comparison between computed and exact solution
An analogous simulation was run with the same geometrical parameters and the same
j
|
j
ii
I
pulse shape, but for a magnetic cylinder. The assumed relative magnetic permeability of
the cylinder is Hd = 2.0/zo. The result is shown in Figure 5.7.3 and the comparison is
______
made with Finite Difference Time Domain (FDTD) [20,21] calculations for the same
case.
0.5
a
D-FTD
FDTD
0.4
N
LU
9
0.3
■o
3
-0.06
•0.04
-
0.02
0
0.02
0.04
0.06
Position (m)
Fig.5.7.3 Comparison between our model and FDTD
128
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In all the considered cases computation times and memory requirement were identical
using FDTD and this technique [22].
5.8 Echo Experiments
Now a plane wave incident upon the flat infinite air-medium interface is
considered [4]. This geometry also allows a simple implementation of the analytic
solution. The one dimensional space consists of 1000 cells: 700 are used to model the
free space (air) and the remaining 300 are used for the complex material. Each cell
corresponds to a length of 0.1 mm and the time step is 0.25 psec. The incident wave is
a Gaussian pulse with maximum frequency of 200 GHz and width o f 20 time steps.
The pulse is launched at the cell position 300 and the DFT of the incident pulse is
performed at position m=310 for 300 time steps. This represents the spectrum of the
incident wave. The simulation is time stepped for a long enough time until the pulse
reaches the interface and is partially reflected. A second DFT analysis is performed on
the reflected pulse, accurately windowed, at position m- 600 for the same number of
time steps. This represents the spectrum of the reflected wave. The reflection coefficient
as a function of frequency is therefore calculated as the ratio of the two spectra. The
calculated reflection coefficient is compared to the corresponding analytical quantity
obtained in the frequency domain from the following relation:
|* M =
Hi-Ho
(5.8.1)
Vi + Vo
where
tjq
and rjt are the characteristic impedance of free space and the complex
medium respectively, and are given by
129
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( 5 -8 -3 )
In the first experiment the water-air interface is considered; the complex permittivity of
water can be approximated by a single order Debye relaxation. The following
parameters es = 81.0eo, £_ = 1.8f0, and xo=9.4»10*12 sec [10] have been used. In the
second experiment the reflection coefficient at the interface between air and a two pole
electric Debye material is studied, for which the following values: es = 100.0£o,
= 4.0eo, ti= 10-11 sec, t2=5.3«10*11 sec, gi=0.7 , g2=0.3 have been selected. The
poles have been chosen in such a way that the two relaxation times are well separated.
The results for the reflection coefficients are plotted in Fig. 5.8.1 and Fig.5.8.2
0.9
Analytical I-
- -a - - Computed j;
0.85
as
0.8
0.75
§
^
0.7
0.65
0.6
2 10
3 10
4 10
5 10
6 10
FREQUENCY (Hz)
Fig.5.8.1 Reflection coefficient for air-water interface
130
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0.9
Analytical
- - a - - C o m p u te d
0.85
0.8
0.75
0.7
0.65
0.6
0
1 1010
2 1010
3 1010
4 1010
5 1010
6 1010
FREQUENCY (Hz)
Fig.5.8.2 Reflection coefficient for two-pole Debye material
0.9
Analytical
- - a - - C o m p u ted
0.85
0.8
I
0.75
i
0.7
0.65
0.6
0
FREQUENCY (Hz)
Fig.5.8.3 Reflection coefficient for single magnetic Debye materials
131
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0.9
Analytical
- - a - - C om puted
0.85
0.8 — ^
sa
CJ
=3
§
i
0.75
0.7
0.65
0.6
0
FREQUENCY (Hz)
Fig.5.8.4 Reflection coefficient for double magnetic Debye materials
In the next two experiments we consider single and double magnetic Debye materials.
For the first experiment we have used
//,
= 81.0//o,
//_
= 1.8//0, and to = 9 . 4 « 1 0 '12 sec;
while for the second experiment we have used fi, = 100.0/zo, //_ = 4 . 0 / / o, t i = 1 0 - 11
sec, t2=5.3«10'n sec, gi=0.7 , g2=0.3. Results are shown in Figures 5.8.3 and
5.8.4, which compare the reflection coefficient calculated from the simulations to the
analytical results.
In the last experiment a complex medium characterized by an electric and magnetic
Debye relaxation is considered. The constitutive parameters of the medium are matched
in order to obtain a perfect absorbing material at all frequencies (Heavyside condition).
In particular
we have used
es = 81.0e#,
= 1.8e0, and xe=9.4« 1 0 ' 12
sec,//, =81.0//„, \im- 1.8/i„, and xm=9.4*10'12 sec, such that the ratio
V£(<»)
frequency independent and equal to rj0. The results reported in Figure 5.8.5 confirm
the prediction of no reflection as all frequencies, even though the calculation is affected
132
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by an error below 38 dB due to the finite cell size. In fact, higher precision is achieved
by decreasing the cell size up to the stability lim it
-35
•40
-55
-60
0
FREQUENCY (Hz)
Fig.5.8.5 Reflection coefficient for wide band absorbing material.
5.9 Comparison With Other Numerical Techniques
The key feature of the technique presented in this work, besides the pedagogical value
of the mechanical analogies, resides on the condensed node representation o f the field
components. Once space and time are discretized, in order to solve Maxwell's
Equations in the vector potential (VP) formulation, all the components of the quantities
involved in the model refer to the same location of the computational grid. This
property is common also to other established numerical techniques such as condensed
Transmission Line Matrix (TLM)[17][18], while differs substantially from the well
known Yee representation scheme exploited by FDTD[15]. Fig. 5.9.1 shows a plot of
the field components used in the two dimensional FDTD cell (Ez, Hx, Hy) and the
analogous components used in our vector potential formulation (Ez, Az).
133
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z
11
- y
x /
FDTD
D-FTD
Fig.5.9.1 Comparison of the field components location in the
elementary cell between 2-D FDTD and D-FTD.
Note that the FDTD components are dislocated along the sides of the cell, while the
vector potential components are both located in the center of the cell. As a consequence
of the fact that the field is condensed, our technique offers superior performance in
terms of grid dispersion with respect to the FDTD formulation, as it was already proven
in the literature for the analogous TLM. At the same time no price has to be paid in
terms o f memory requirements for all the cases considered, and this is an advantage
with respect to TLM. Furthermore, for the two dimensional case considered below only
two quantities need to be considered and stored in time (Ez, Az) as opposed to the
FDTD representation where three components need to be used (Ez, Hx, Hy). This
results in one third memory saving over FDTD, with no penalty in execution time and
algorithm complexity. In the following numerical experiment the cutoff frequency due
to grid dispersion in a computer experiment is evaluated and it is compared with the
same quantity evaluated in the analogous FDTD experiment. A locally plane wave tilted
45° with respect to the grid main exes is generated as shown in Fig. 5.9.2.
134
s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
^ ______________ 0.16m ________________ ^
Fig.5.9.2 Field distribution and experimental geometry for
the grid dispersion numerical experiment
The wave is generated by a Gaussian pulse 62.S fsec in width corresponding to a
maximum frequency content of 200 GHz. The wave front extends for 100 cells across
corresponding to 2.0 cm, and the observation point is placed along the perpendicular to
the wave front, 7.07 mm away from the wave source location. This geometry was
chosen so that the wave is almost plane when it leaves the observation point. The mesh
size is coarse enough such that grid dispersion will occur. As shown in [19], for a
wave propagating at 45° along the FDTD grid, the grid dispersion causes the wave
propagation velocity to fall to zero when As > 0.5A, where X is the wavelength of the
electromagnetic wave examined. The corresponding cutoff frequency for our particular
choice of As is frnax=150GHz. The simulation is time-stepped for a long enough time
so that the Gaussian pulse goes entirely past the observation point. The time responses
are recorded in both cases and are shown in Fig. 5.9.3a-b.
135
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0.8
0.8
D-FTD I
FDTD
0.6
0.4
0.4
0.2
-
-
0.2
0.2
-0.4
-0.4
0
2 .S
10 " 5 10 " 7.5 10 " 1 10"*1.25 1 0 10
time (sec)
0
2.5 10 " 5 10 " 7.5 10 " 1 10',a1.25 10’
time (sec)
a)
b)
Fig.5.9.3 (a) Time response of FDTD due to locally plane wave
(b) Same quantity for D-FTD formulation.
Due to grid dispersion, the original pulse is altered and develops a ringing tail. DFT
analysis gives the spectrum content which is shown in Fig. 5.9.4, where we plot the
magnitude of the two spectra is shown, and in Fig. 5.9.5a-b, where we plot the
corresponding phase versus frequency is graphed. Notice the sudden drop of the
magnitude at the cutoff frequency and the corresponding loss of phase linearity.
136
*
i
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0.035
D-FTD I
FDTD I
0.03
0.025
uj
■®ao
0.02
o
0.015
<0
2
0.01
0.005
0
5 10 10
1.5 101
1 10 '
2 10
11
Frequency (Hz)
Fig.5.9.4 Spectrum content of the signal observed at the same observation
point for FDTD and D-FTD.
fd td !
D-FTD
Phase E(f)
08
Q* -0.5
2
1
0
l
5 1 0 '° 1 10’ YS 10' ’2 10' 12.5 10' ’3 10’ ’
Frequency (Hz)
0
5I0’” ' ,0 ’ Ys
'2 4 5 ,0' ’3
Frequency (Hz)
’
Fig.5.9.5 (a) Phase response of FDTD plotted versus frequency (b) Same
quantity for D-FTD. Notice that the loss in linearity of the phase with frequency
marks the cutoff frequency.
137
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The observed FDTD cutoff frequency agrees with the predicted value o f ISOGHz,
while our technique exhibits the cutoff at 170GHz, corresponding to a 13%
improvement in bandwidth. The same bandwidth (170GHz) can be achieved by FDTD
i
if the mesh size is reduced to As = 8.5- 10~*m corresponding to a memory increase o f
38% and execution time increase of 38%.
138
i
i
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References Chapter 5
[1] R. E. Diaz, The Analitic Continuation Method fo r the Analysis and Design o f
Dispersive Materials. PhD Dissertation, University of California at Los Angeles,
1992.
[2] R. E. Diaz, “A Discrete FitzGerald Time Domain Method for Computational
Electromagnetics,” in International Conference on Electromagnetics in Aerospace
Advanced Applications (ICEAA). Politecnico di Torino, ITALY 1993, pp. 391-394.
[3] G. F. FitzGerald, Letter to Oliver Lodge, 3 Mar. 1894, in The Maxwellians.
Ithaca: Cornell University Press, 1991.
[4] C. A. Balaniis, Advanced Engineering Electromagnetics. New York: John Wiley
and Sons Inc., 1989.
[5] O. Heaviside, Electromagnetic Theory. New York: Chelsea Publishing Co.,
1971.
[6] D. Greenspan and L. F. Healt, “Supercomputer Simulation of the Modes of
Colliding Microdrops of Water,” J.PhysD, Appl. Pkys., vol. 24, pp. 2121-1123,
1991.
[7] L. Lapidus and G. F. Pinder, Numerical Solutions o f Partial Differential
Equations in Science and Engineering. New York: John Wiley and Sons, 1982.
[8] F. De Flaviis, M. Noro, and N. G. Alexopoulos, “Diaz-Fitzgerald Time Domain
(D-FTD) Method Applied to Dielectric Lossy Materials,” in ICEAA 95 International
Conference on Electromagnetics in Advanced Applications. Torino (Italy) 12-IS
September 1995, pp. 309-311.
139
j.
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[9] F. De Flaviis, M. Noro, R.E. Diaz, and N. G. Alexopoulos, “Diaz-Fitzgerald Time
Domain (D-FTD) Technique Applied to Electromagnetic Problems,” in IEEE MTT-S
Int. Microwave Symp. S. Francisco June 1996, pp. 1047-1050.
[10] K. Luebbers, The Finite Difference Time Domain Method fo r
Electromagnetics. Boca Raton, Florida: CRC Press, 1993.
[11] T. Kashiwa, N. Yoshida, and I. Fukay, “A Treatment by the Finite Difference
Time Domain Method of the Dispersive Characteristics Associated with Orientation
Polarization,” IEICE Transactions, vol. Vol. E-73, pp. 1326-1328, 1990.
[12] J. L. Yang, “Propagation in Linear Dispersive Media: Finite Difference Time
Domain Methodologies,” IEEE Transaction Antennas Propagation, vol. AEP-43,
pp. 422-426, 1995.
[13] F. De Flaviis, M. Noro, R. E. Diaz, and N. G. Alexopoulos, “Diaz-Fitzgerald
Time Domain Method Applied to Electric and Magnetic Debye Materials,” in Applied
Computational Electromagnetics, ACES Symposium. Monterey (CA) March 17-21
1997, pp. 781-788.
[14] F. De Flaviis, M. Noro, R. E. Diaz, and N. G. Alexopoulos, “Time Domain
Vector Potential Formulation for The Solution of Electromagnetic Problems,” in IEEE
AP-S Int. Symp. Montreal, Canada July 1997.
[15] C. M. Furse, S. P. Mathur, and O. P. Gandhi, “Improvements to the Finite
Difference Time Domain Method for calculating the Radar Cross Section of a perfectly
conducting target.,” IEEE Trans. Microwave Theory Tech., vol. MTT-38, pp. 919927, 1990.
[16] R. E. Collin, Foundations fo r Microwave Engineering. McGraw-Hill
International Editions, 1966.
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[17] F. De Flaviis, M. Noro, G. Franceschetd, and N. G. Alexopoulos, “Applications
of Time Domain Vector Potential Formulation to 3-D Electromagnetic Problems,” in
TSMMW '97 Topical Symposium on Millimeter Waves. Tokyo, Japan July 1997.
[18] R. Clayton and B. Engquist, “Absorbing Boundary Conditions for Acoustic and
Elastic Wave Equations,” Bulletin Seism. Soc. America, vol. 67, pp. 1529-1540,
1977.
[19] D. S. Jones, The Theory od Electromagnetics. New York: Macmillan, 1964.
[20] K. S. Yee, “Numerical Solutions of Initial Boundary Value Problems Involving
Maxwell's Equations in Isotropic Media,” IEEE Trans. Ant. Prop., vol. AP-14, pp.
302-307, 1966.
[21] A. Taflove and M. E. Brodwen, “Numerical Solution of Stady State
Electromagnetic Scattering Problems Using The Time Dependent Maxwell's
Equations,” IEEE Trans. Microwave Theory Tech., vol. MTT-23, pp. 623-630,
1975.
[22] F. De Flaviis, M. Noro, R. E. Diaz, and N. G. alexopoulos, “The Diaz-Fitzgerald
Time Domain Model for the Solution of Electromagnetic Problems,” in NATO-ANSI
Conference. Samos, Greece Agust 5 1997.
141
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C h a pt e r 6
t
»
I
Electronically Steerable Beam Microstrip Antenna Array
F
The possibility of having an electronically steerable antenna system is of significant
current importance for many communication technologies. Applications such as airport
9
traffic control, satellite tracking and in general radar systems emphasize the importance
of electronically steerable antennas. Due to the high cost and the large dimensions of
j
such systems, applications have been limited. This work proposes the use of thin
|
ceramic ferroelectric materials for the design of phase shifters for the realization of the
i
electronically scanned antenna system, as a means of overcoming the limitations of
I
current technology. A simple and novel design employing two ferroelectric phase
i
shifters in conjunction with two microstrip antennas operating at 2.1 GHz is presented,
and measurement results of reflection coefficient and radiation pattern are provided in
this work.
6.1. Project Phases
The realization of a steerable beam microstrip array system is quite complicated from
the material and design point of view. In order to overcome some of the difficulties, the
project have been divided in individual tasks, which can be summarized as follows:
1 - Establish the design goals, such as secondary beam side lobe level (SLL), return
loss, and maximum beam steer angle desired.
2 - Selection of materials which will be employed for the patch antennas, and for the
phase shifter.
142
_>
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i
3 - Design of single patch microstrip antenna using simplified model (such as resonant
transmission line model).
i
4 - Full wave analysis of microstrip transmission line, to determine the characteristic
j
impedance and propagation constant, versus frequency and versus physical
5■
dimension.
5 - Use of full wave analysis (Mixed potential integral equation MPFIE) to optimize the
performance of a single patch antenna.
6 - Use of array theory to estimate the radiation pattern main features for the two
element microstrip patch array system.
|
7 - Full wave analysis (MPFIE) and optimization to determine the radiation pattern of
I
the two element array microstrip patch system under different bias condition, and to
j
1
i
determine the maximum steerable angle.
8 - Full wave analysis to characterize the ferroelectric material phase shifter under
i
j
different bias condition, using the D-FTD technique to account for dispersion in the
ferroelectric material due to Deby relaxation.
9 - Design of the RF feeding network using commercial software package
(Touchstone), importing full wave analysis results horn points 7 and 8.
10 - Design of the dc bias network.
11 - Test and measurements of reflection coefficient and radiation pattern of the antenna
array under different bias conditions.
The steerable beam microstrip antenna system is schematically shown in Fig. 6.1.1
143
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
bias signals
bias network
RF signal
j
DC-block
'EM phasej
shifter J
DC-block
'EM phasi
shifter
patch
antenna 1
RF Feeding
circuit
patch
antenna 2
Fig. 6.1.1 Steerable beam microstrip antenna system
|
The microwave signal is matched to the source through the feeding circuit, the dc
!
blocks avoid dc leaks in the microwave circuitry, while the two phase shifters,
I
separately controlled through the bias networks provide different phase signal at the
two patche antennas. In the design we will try to keep the side lobe level below 10 dB,
this will allow a total beam scan of 40° as will be shown later in this chapter.
6.2. Single Patch Antenna Design
For the design of the system a duroid substrate is selected for the antennas and for the
microstrip lines. Duroid is often used for microstrip patch antenna design, because it
exhibits low loss, has low dielectric constant and can easily be manufactured in planar
form. The duroid selected is RT-5870, which has a nominal dielectric constant of 2.33,
tan5=0.0012 at 10GHz, and its thickness is 1.S75 mm. At first, the transmission line
model method [1] is used to design the antenna, this method is simple and enough
accurate enaugh for a first design iteration. The transverse dimension of the patch (w)
must be chosen to give a reasonable input impedance, while the longitudinal dimension
144
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(L) must be half effective wavelength at resonance (larger dimensions may allow
unwanted higher-order resonance's to occur in the patch at the design frequency). A
schematic of the patch is shown in Fig. 6.2.1
radiating
edges
patch
substrate
feed point
side view
top view
Fig. 6.2.1 Schematic of a patch microstrip antenna
The fundamental resonant frequency of the patch structure can be estimated as
/,=
c
1
2 ^ 7 L + 2Ala
(6 . 2 . 1)
where the effective dielectric constant of the substrate under the patch is given by
( 6 -2 -2>
and the length extension to compensate for the fringing fields at the patch edges is
OC_
= 0.412
(<V + 0 .3 ^ + 0.264)
(6.2.3)
( e „ - 0 .2 5 8 ^ + 0 .8 1 3 )
145
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At resonance the input impedance of the patch antenna is purely real. This resistance is
primarily due to radiation (as well as some power loss in the antenna metallic structure
and surface wave modes). Recall in the transmission line model, the patch consists of
two radiating edges (slot) connected by a wide microstrip line. The radiation resistance
of one slot can be approximated as
^ = 120Ao
'a
W
which is accurate for w « A, the input impedance at the edge of the patch consists as
first approximation of one slot in parallel with the impedance of the other slot (as
transformed through the section of microstrip line with length L, and width w). This is
shown schematically below:
L
Fig. 6.2.2 First order equivalent model for microstrip patch antenna
Using this model, the dimensions of the patch can be determined for the desired
resonant frequency and input impedance subject to the constraints discussed earlier.
Use of a commercial CAD simulator such as MDS or Touchstone, allows to model the
impedance of the patch, provided that the edge/slot radiation resistances are inserted
explicitly, as the mentioned CAD do not include radiation effects.
Using this first design iteration we obtain for the desired frequency (2.1 GHz) a patch
having a length of 43 mm and width of 64 mm. These two values were used as starting
parameters and were plugged in the full wave analysis code based on mixed potential
146
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
integral equation [2, 3] (MPIE) for a more accurate characterization. The geometry of
I
the analyzed patch with its correspondent sub-griding needed for the MPIE is shown in
i
Fig. 6.2.3.
Fig. 6.23 Patch antenna layout utilized in the full wave analysis
The layout also includes a quarter wave transformer to obtain matching with the 50Q
transmission line used to extract the reflection coefficient. The wavelength and
characteristic impedance versus physical dimensions for the duroid substrate were
obtained using full wave analysis, and are reported in Fig. 6.2.4 through Fig. 6.2.6.
147
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
f=2.1GHz. Duroid RT5870 (t=1.575mm er=2.33)
200
150
a
o
N
l
100
0
5
10
15
2 0
2 5
Width (mm)
Fig. 6.2.4 Characteristic impedance versus line width for duroid RTS870
f=2.1GHz, Duroid RT5870 (t=1.575mm er=2.33)
1.45
o
1.35
0
5
10
15
2 0
2 5
Width (mm)
Fig. 6.2.5 Propagation constant versus line width for duroid RT5870
148
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
f=2.1GHz, Duroid RT5870 (t=1.575mm er=2.33)
2 8
27
2 6
2 5
24
23
0
5
10
15
20
25
Width (mm)
Fig. 6.2.6 Quarter wave length versus line width for duroid RTS870
the same parameters are also plotted versus frequency in Fig. 6.2.7 and Fig. 6.2.8
w-4.7mm, subs Duroid RT5870 (t-1.575mm er-2.33).
53
52.5
5 2
51.5
5 1
50.5
50
49.5
4 9
0
1
2
3
4
5
6
7
8
Frequency (GHz)
Fig. 6.2.7 Characteristic impedance versus frequency for duroid RTS870
149
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wa4.7mm, subs Duroid RT5870 (U1 .575mm •r«2.33).
1.43
1.425
1.42
1.415
1.405
0
1
2
3
4
5
6
7
8
Frequency (GHz)
Fig. 6.2.8 Propagation constant versus frequency for duroid RT5870
Results of the return loss (S/y) in magnitude and phase obtained after the optimization
of the physical dimension of the patch antenna are reported in Fig. 6.2.9 and Fig.
6 . 2.10
CO
- 20
-25
-30
1.85
1.9
1.95
2
2.05
2.1
2.15
2.2
frequency (GHz)
Fig. 6.2.9 Magnitude of the Syy versus frequency for the patch antenna
150
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150
100
ai
•8
55
S
a
•C
a.
-50
-100
-150
1.85
1.9
1.95
2
2.05
2.1
2.15
2.2
2.25
frequency (GHz)
i
|
Fig. 6.2.10 Phase of the S n versus frequency for the patch antenna
The resonant frequency was centered at 2.1 GHz as desired, using a patch width of
66 mm and a length of 45.8 mm. The corresponding radiation pattern in the E and H
plane for the tangential (Ee) and transversal field component (E$) are reported in Fig.
6.2.11 and Fig. 6.2.12 respectively.
40'
Fig. 6.2.11 Radiation pattern at <|>=0o (H-plane)
151
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Fig. 6.2.12 Radiation pattern at $=90° (E-plane)
6.3. Two Patches Antenna Array Design
The two element array will be constituted by two patch microstrip antennas placed next
to each other. As before in order to calculate the radiation pattern when each of the two
elements fed with a signal having different phase, we can use standard array theory [1].
Let us consider the very general case at first where n elements are placed at arbitrary
distance with respect to each other (dn), fed with signals having different amplitude (/„)
and phase (a n), as indicated in Fig. 6.3.1
'
I
Il/ttl 12/02
lo/CXa
Fig. 6.3.1 Array of antenna
152
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For such an array we can estimate the magnitude of the vector position as
\K\ = \r\-d , cosy
(6.3.1)
so the corresponding array factor can be written as
•
*=l
(6.3.2)
For equispaced elements such that dn=nd the array factor will result in
A(Y) = eJM'£ r meJ[a--"*a*r]
(6.3.3)
If now we assume that the array element pattern for a microstrip patch antenna [4]
given by
Ea ~ cos
sind
(6.3.4)
where the angles are shown in Fig. 6.3.2,
Fig. 6.3.2 Microstrip patch antenna
153
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the total radiation pattern will be given by the product of Eq. 6.3.3 and 6.3.3. So the
magnitude of the radiated field versus 9 (since Y=n/2-9) can be obtained from:
D(0) = cos
sin0 I ^ / Jtcos(rt/W sin0-a(l)
1
r w
+ £ /„sin (/w W sin 0 -a,,) I I .
(6.3.5)
In this design only two patch antennas (n= 2) are considered printed on duroid
(Er=2.33), operating at 2.1 GHz, and spaced 0.7Xo center to center. With this set of
values the theoretical radiation pattern obtained from Eq. 6.3.5 is shown in Fig. 6.3.4.
(continuous line). Changing the phase of the signal feeding the two antennas, the
changes the radiation pattern can be observed.
10
0
10 dB
22
10
20
a
2
-30
-40
-50
-60
-100
5 0
-50
100
Angle (deg)
Fig. 6 .3 3 Theoretical radiation pattern for two element microstrip antenna
Particularly, it is seen that keeping the magnitude of the signal at the two antennas
constant, and increasing the phase difference, the maximum beam location shifts from
154
(
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0° to ±22° (only the positive shift is shown in the graph) while the secondary beam
i
amplitude is kept below lOdB. It is also noticed that further phase difference produces a
growth in the secondary side lobe level. Again with the aid of full wave analysis the
performance of the array was optimized at the resonance frequency. This analysis was
i
ji
conduced using the geometry shown in Fig. 6.3.4, where the phase difference was
i
simulated varying the length of the feeding arms of the antenna.
l
l
Fig. 6.3.4 Geometry used to analyze the radiation pattern
Results for the radiation pattern in the $ plane are reported in Fig. 6.3.S to Fig.6.3.8
155
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E-tf*u m pm* <?o'
30*
-to
•21
90 *
Radiation Pattam
Fig. 6.3.5 Radiation pattern in the 6 plane for a difference
of the feeding arm length Al=Omm
E-ttwua 0*m
—
E-sMaphUfo*
JO*
• 10'
60 *
•21
W
90*
Radtotion Pattam
Fig. 6.3.6 Radiation pattern in the <J>plane for a difference of the
feeding arm length Al=12mm
156
1
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•to-
60*
•21
90*
90*
Radatfon Pattam
Fig. 6.3.7 Radiation pattern in the $ plane for a difference of the
feeding arm length Al=18mm
xr
-10
. 60*
90 *
Radatfon Pattam
Fig. 6.3.8 Radiation pattern in the $ plane for a difference of the
feeding aim length Al=34mm
157
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As noticed before a maximum length difference of 12 mm is required in order to keep
the secondary side lobe below 10 dB. With this delay the radiation pattern is tilted of
about 20°, so a total steer of 40° is achieved. Using Fig.6.2.5, we can calculate the
phase shift corresponding to 12 mm transmission line delay is calculated. In fact at 2.1
GHz from Fig. 6.2.5 a PlPo-1-32, the amount of phase of the Al transmission line is
given by
A<p = pAl = ' -P' ]2 * A1 = 0.69rad = 39°
aJ c
(6.3.6)
so the phase shifter must be capable of a total phase change of 39°.
6.4. Phase Shifter Design using D-FTD Technique
The previous analysis shows that a total phase change of about 40° is required for the
phase shifter. In Chapter 4 it was observed that using an 8 mm transmission line
printed on BST substrate a maximum phase change o f 158° was obtained at 3 GHz.
This will certainly satisfy in the design requirements for the phased array. The fact that
lower phase shift is required, also allows reducion of the total length of the phase
shifter to 6 mm. Because ferroelectric materials are dielectric dispersive materials, a full
wave analysis which accounts for this phenomenon is necessary to fully characterize
the device. The use of the technique described in Chapter 5 is adopted to characterize
the microstrip transmission line in terms of effective dielectric constant versus
frequency under different bias conditions. These results are later implemented in the
Touchstone software to optimize the performance of the overall system.
Using the new time domain technique a typical microstrip on a BST substrate has been
investigated. Unlike other investigations, which look for the propagation velocity of the
158
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
field only, the time-domain approach actually finds the space-time distributions of the
fields everywhere within the finite difference space-time mesh. In many cases this
provides clear pictures and illuminating details of the field variations. At the same time,
the frequency-domain design data can also be easily obtained through Fourier transform
of the time-domain fields.
The microstrip structure used in the calculation is shown in Fig. 6.4.1
nyAs
magnetic wall >
w/2
metal strip
substrate
nzAs
nxAs
pulse
exitation plane
Fig. 6.4.1 Microstrip structure
Because of the symmetry of the problem, only half of the structure is considered. The
parameters used for the calculation are the following:
substrate thickness
H=0.1 mm
metal strip width
substrate dielectric constant
W=0.05 mm
£r=250 (soft bias)
metal strip thikness
t=0 mm
To accomodate the structural details of the microstrip, the mesh parameters used are as
follows:
159
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Space interval
As=5»10'5 m
time step
domain cells
At=k»As/c sec
nx=40, ny=80, nz=160
where c is the velocity of light in air and k is the constant restricted by the stability
criterion k = 0 J 15 in this calculation. A Gaussian pulse excitation is used at the front
surface, uniform under the strip with only the Ex component
(6.4.1)
where to=140«At and T=140»At; elsewhere on the front surface, Ex=Ey=0. The pulse
width in space is about 20As, which is wide enough to obtain good resolution. The
frequency spectrum of the pulse is from dc to about 700 GHz. Fig. 6.4.2 shows the
vertical electric field time variations at different positions along the propagation
direction.
0 .0
2 0 0 .0
SOO.O
0 4 0 .0
1120.0
1 4 0 0 .0
1600.0
TIME STEP
Fig. 6.4.2 Time variation at different positions along the direction o f propagation
The dispersive properties of the microstrip are quite obvious from the distortion of the
pulse as it travels away from the feeding point. The effective dielectric constant £reff(G))
160
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can be used to account for the dispersive characteristics of the microstrip. It is caculate
as follows: Take the Fourier transform of Ejrft) at two different positions (underneath
the center of the strip), with a separation of L, along the propagation direction:
Ex(n>, z = 0) =
J £ x(r,z = 0 )ejaMdt
(6.4.2)
Ex(o), z = L) =
J £ x(r,z = L)eJaMdt
(6.4.3)
Taking the ratio of Eq. 6.4.2 and Eq. 6.4.3, we can get the transfer function of this
section of microstrip is obtained as
(644)
=
£ x(©,z = 0)
where
Y(oj) = a(Q)) + jp((o)
(6.4.5)
The constant CrefK®) is defined through P(io) as
p(6)) = Q)^n0e0£nff(co)
(6.4.6)
or
=
(6.4.7)
(0 HqCq
Fig. 6.4.3 shows the result for the calculated effective dielectric constant.
161
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540
520
500
480
460
440
420
400
0
50
100
150
20 0
Frequency GHz
250
300
Fig. 6.4 J Efective dilectric constant of the BST microstrip
The variation of the characteristic impedance with frequency is also obtained through
the ratio V((o)H((D). Here I(co) is the Fourier transform of the current defined as the
loop integral of the magnetic Held around the metal strip. For V( qj), two kinds of
definitions are used, one is the line integral of vertical electric field under the whole
strip, the other is the averaged line integral of vertical electric field under the whole
strip. For a fairly large frequency range starting from dc, it turns out that the results of
the center line integral and averaged line integral are very close to each other, indicating
that the voltage uniqueness is well satisfied in that frequency range.
6.5. Design of the RF feeding network
The feed network is obviously a very important part o f the system, since must
guarantee constant power flow in the two patches under different phase delay provided
by the two phase shifters. The fact that at resonance (2.1 GHz) the patch antenna has a
pure real impedance, which is about 1300, is used as starting point for the design.
162
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With the aid of the Smith chart the feeding network can be designed to match the 50Q
impedance. Implementation of the microstrip feeding circuit into a commercial software
package, together with the S parameter model for the phase shifter obtained by the
numerical analysis (see section 6.4 in this chapter) and the model for the antenna
obtained using full wave analysis (see section 6.3 in this chapter) allow optimization of
antenna performance over bandwidth, also taking in to account the fringing field and
other second order effects. This optimization is carried out for fixed phase of the BST,
(soft bias), corresponding to higher loss and higher dielectric constant Further
optimization is done to allow the same power flow in the two patches, under different
bias condition. In this manner constant power is provided to the two patches, while
different phase is created by biasing the phase shifters. As observed in Fig. 6.5.1, a
50Q characteristic impedance line (width 4.7 mm) is used to connect the patch.
AnU m c
»•
Zi«Zo(2.6*-jO)
Zi»Zo(0.464-j0.193)
Zi»Zo(0.456+j0.368)
Fig. 6.5.1 Array feeding network design
Since at resonance the patch has an impedance which is pure real of 130(2, (point PI in
the Smith chart) the line (having length of 18 mm) will transform this impedance as
seen in the admittance Smith chart in Fig. 6.5.2 (point P2). At this point the line must
be connected to the phase shifter through a wire bond having an estimated length of 2
mm.
163
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IMPEDANCE OR ADMITTANCE COORDINATES
TO-VUY2
fe+wtor
Fig. 6.5.2 Smith chart for the design of the feeding network
This can be treated as a pure inductor having 2 nH inductance (notice at 2 GHz a thin
wire bond in air presents about InH/mm inductance), so the admittance is further
•v
shifted to point P3. Now we can use the data obtained from the simulation of the phase
shifter under different bias conditions to see how the characteristic impedance varies
164
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(points A-D). To move further in the design, we select the soft biased substrate
condition (point A) and from that we add the effect of a second wire bond as visible in
Fig. 6.5.3
Zi«Zo(0.146»j0_566)
Zi-Zo(0.26*j0.47)
Zi-Zo(0.192*j0J57)
Zi»Zo(0.102+j0.563)
(A)
(B)
(Q
(D)
Zi-Za<7-26»j4-37) (A)
Fig. 6.5.3 Array feeding network design
Use of a dc-block is required to insulate the bias field from the RF signal (details on the
dc-block design are given in Appendix E). The impedance moves (due to the dc-block)
on the Smith chart to point P4. Use of additional transmission line (having length of
41.5 mm) moves the admittance to point P5, corresponding on the admittance chart to
point P6 (since Y=l/Z). Now if two identical arms are connected in parallel, the total
admittance will be the sum of the two (Ytot=Yl+Y2, point P7). The admittance at point
P7 can further be transformed with a transmission line section (length 32.6 mm) as
visible in Fig.6.5.4.
4.
Zi«Zo<p.814+j207l)
Zi«Zo(l+jO)
Fig. 6.5.4 Array feeding network design
165
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
At point P8, the use of single open stub (length 30.5 mm) will remove the imaginary
part of the impedance, providing the final match (point P9).
At this point the circuit is implemented into the Touchstone design software. Also the
antenna impedance characteristic obtained from the full wave analysis and the phase
shifter characteristic are introduced in this software (as 5 parameters versus frequency).
The use of Touchstone allows optimization of the physical parameters o f the matching
network versus frequency (since designh on the Smith chart was done at single
frequency). If the matching network is considered as a three port circuit, the S 21 can be
evaluated as being proportional to the signal flowing into patch 1, and S 31 as being
proportional to the signal flowing into patch 2. After the dimensions were optimized to
maximize matching versus frequency, it is observed that as the length o f the dc-block is
changed it affects the frequency value where S21 results to be equal to S31 (same power
on the two patches), while a different bias condition is imposed. This concept is shown
in Fig. 6.5.5, where the S parameters are plotted for different length o f the dc-block
while the two phase shifters are biased under different conditions (one is biased at low
voltage and the other one is biased at higher voltage).
166
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
epl« 2 5 0 . ep2-450
S21 t S 3 t
point
S 2U S 31
S214S31
length * 17.6 mm
-20
2 10*
2.1 10®
2 .0 5 10*
2.15 109
Fiaquancy (GHz)
Fig. 6.5.5 S parameters for different length of the dc-block
It is observed for example that if a dc-block length of 18.6 mm (instead of 20.5 of the
original design) is selected, for a very narrow frequency band (around 2 GHz)
S2 1 —S3 1 . In other words, if the dc-block is used as originally designed, then for even
bias condition of the two phase shifters identical power flows, for quite a broad band
region (0.15 GHz). But this condition is not longer valid when one of the phase
shifters is biased harder. So by proper adjusting the dc-block length, the same power
can be obtained flowing into the antennas, for soft bias and hard bias condition (this
will only happen at single frequency). In this design a dc-block was selected having
length of 18.5 mm, so as to guarantee the same power flow under different bias
condition at 2.1 GHz. The simulated data obtained for the S parameters with this type
of dc-block are reported in Table 6.5.1
167
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
mag(S2i)
phase(S2i)
mag(S3i)
(dB)
(dB)
-7.46
-7.17
-7.39
-9.52
-7.4
-7.17
-7.46
-9.52
(deg)
(deg)
121°
152.7°
90°
66.8°
mag(Su;
AS
40
(dB)
(deg)
(dB)
0.06 31°
0
0°
0.07 -31°
0
0°
-35.1
-27.6
-35.1
-27.6
phase(S3i)
90°
152.7°
121°
66.8°
£r2
Ert
250 450
250 250
450 250
450 450
Table 6.5.1 S parameter of the feeding network
It is noticed that when both phase shifters are hard biased (corresponding to
£rl=€r2=250) the same power flows (S 2 i= S5/ , so AS=0), while when one is biased
hard (£r2=250) and the other one is soft biased (£r2=450) still comparable power is
flows (S 2 1 - - 7 3 9 dB ,Ssi =-7.46 dB, so AS=0.07 dB ). So a total phase difference of
31° can be obtained with almost no change in AS. This situation is aslo illustrated in
Fig. 6.5.6.
0
,
,
i
i
*
..............
;
_r
1■ r
S21.S31
vb1-250V
vb2-250v
1
-I
.j
...................................._ . . . . \ .........................j
-5
h ° _ _ _ _ _ ______ _—
s
5£
■
n
\
-V
^
^—
^ —
~
*0
\
- 10
\
IK
s.
A
.
-15
*
\
S31
______________ yb.V-450.V..................
vb2-250v
S21
vbl-450V :
vb2-250v :
.
2.05 10*
,
.
.
.
" —
it
-
-20 -- 2 10*
1
---------- — -j
.
.
2.1 10®
Frequency (GHz)
.
' '1
-i
—
"
J
.
i
:
2.15 10®
2.2 10^
Fig. 6.5.6 5 parameters of the feeding network under different bias conditions
Finally the schematic of the two antennas together with the dc bias network is shown in
Fig. 6.5.7
168
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Hi voltage
HP-8510
OHO
DC-btocks
RF input
bias 2
Fig. 6.5.7. Bias network of the microstrip array system
The bias network is designed using high impedance microstrip line (width 0.15 mm)
having a length which is a multiple of A/2, connected to a resistor (open circuit at RF)
so that at the beginning section it will correspond to an open circuit, so no RF will flow
trough. Clearly, the ground of the network analyzer and the voltage generator must be
connected together as was perviously explained in Chapter 4.
6.6. Test and Measurements
For this array prototype, two sets of measurements are performed. The first one
consists of measuring the return loss versus bias condition, while the antennas are
radiating in free space, and the second one consists in determining the power radiation
pattern of the array for different bias conditions. To execute these measurements, a free
space environment is needed to prevent the antenna interaction with scatterers. For the
S ii parameters measurement the set-up used is the one illustrated in Fig. 6.5.8. Results
of measured reflection coefficient for different bias conditions are reported in Fig. 6.6.1
169
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CO
□
aI
E
- 20
-2 5
Vb1«450V Vb2«<50V
V61-250V Vb2-450V
Vb1-450V Vb2«2S0V
-30
2.05
2
2.1
2.15
2.2
2.25
2.3
Frequency (GHz)
Fig. 6.6.1 Measured reflection coefficient for the array system
It is observed that the resonance frequency (2. IS GHz) agrees with the predicted one
(2.1 GHz) within an error of 2%. It is also noticed that a good symmetry of the
reflection coefficient is obtained under opposite bias conditions. For the radiation
pattern, a standard horn antenna connected to a spectrum analyzer was used in a set-up
as shown in Fig. 6.6.3
Microstrip patch
array
reciving horn
antenna
RF Generator
spectrum
Analyzer
Microwave
a m p lifie r
turn table
Fig. 6.6.3 Radiation pattern measurement set-up
170
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The radiating array was mounted on a turntable support, which allows to determine its
I
I
angle with respect to the receiving station. Results of the measured and simulated
radiation pattern for positive bias are reported in Fig. 6.6.4.
*3
I
I
-20
3J
Ct
V bit»«2S0
— - V b iata* 5 0
V biasO SO
A
V b iat» 2 S 0
O
V biaaO S O
O
Vbit t - * »
-3 0
PI
(tim uM ion)
(tim ulalion)
(tim ulalion)
(m a t l ucod)
( m a ttu r t d )
( m a ttu r t d )
-4 0
-100
-50
0
50
100
Angle (deg)
Fig. 6.6.4 Measured and computed radiation pattern for the microstrip array
0
5
I
£
$
10
15
-20
1e
o»
-3 0
« — VbiUB250 (nwuurad)
• - - Vbtat-450 (mMturod)
Vbias--450 (nwuurad)
-3 5
-4 0
-100
-50
50
1 00
Angle (deg)
Fig. 6.6.5 Measured Radiation pattern under different bias conditions
171
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Very good agreement is obtained, especially for small angles, while for larger angles,
some multiple reflection appears to occur, due to obstacles near the antenna in the
measurement enviroment available.
The total beam steer was 18.5° on each side (total 37°) instead of the 20° predicted. The
full scanned measured radiation pattern and the corresponding simulated one are
reported in Fig. 6.6.5
172
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References Chapter 6
[1] R. S. Elliott, Antenna Theory and Design.
[2] M. J. Tsai, F. De Flaviis, O. Fordham, and N. G. Alexopoulos, “Modeling Planar
Arbitrarily-shaped Microstrip Elements in multi-layered Media,” IEEE Trans.
Microwave Theory Tech., vol. 45, pp. 330-337, March 1997.
[3] F. De Flaviis, M. J. Tsai, S. Chang Wu, and N. G. Alexopoulos, “Optimization of
Microstrip Open End,” in IEEE Antennas and Propagation Internatinal Symposium.
Los Angeles June 18-23 1994, pp. 1490-1493.
[4] C. A. Balanis, Antenna Theory: Analysis and Design. New York: Wiley, 1982.
[5] R. Clayton and B. Engquist, “Absorbing Boundary Conditions for Acoustic and
Elastic Wave Equations,” Bulletin Seism. Soc. America, vol. 67, pp. 1529-1540,
1977.
173
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C h apter 7
j
[
Conclusion
A novel sol-gel technique for the production of low loss ferroelectric material for
'i
|
microwave applications has been developed in this research. New design for planar
i
|
phase shifter circuits has been presented and a design methodology to obtain low loss
broad band phase shifter operating at 2 GHz has been discussed and implemented.
Measured results show net improvement over existing ferroelectric phase shifters, in
S
t
|
terms of reduction of required bias voltage, broad band capability and reduction of loss.
Use of planar structure devices allows the integration of this new type of phase shifter
with conventional microwave circuits. Use of the phase shifter for the design of a novel
steerable beam microstrip patch antenna system has also been simulated and a prototype
was build and tested. Good agreement between measured and predicted results
confirms the validation of the adopted model.
A new time domain technique has been proposed for the characterization of complex
materials such as ferroelectric ceramics. The generalization o f the mechanical model
first proposed by FitzGerald, to account for different realistic materials has been fully
exploited. The advantage of the mechanical analogy resides on the ability to visualize
immediately the propagation mechanism of the different electromagnetic quantities and
their relation to ponderable media which can be modeled simply by modifying the
mechanical properties of the objects composing the medium. At all stages we have
supported several extensions of the original mechanical model with rigorous analogies
with the classical vector potential formulation. In addition to this nice mechanical
analogy, we note that our formulation of the problem resides essentially on the vector
174
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potential, rather than on the fields themselves. Clearly, this is not only a formal point,
since the condensed node representation leads to low grid dispersion and savings in
terms of memory and computational time requirements with respect to FDTD, for the
two dimensional cases considered. Furthermore, the simplicity of the resulting
equations must be stressed, because in the considered two-dimensional cases E and A
exhibit only one component each at variance of H. Various results have been presented
in this work to suggest a wide spectrum of possible engineering applications. We
presented scattering problems from classical objects composed of dielectric,
magnetically permeable, dielectrically lossy, Debye and absorbing materials, and
validated our results by comparison with rigorous frequency domain canonical
solutions, or FDTD calculations. The condensed node character of this time domain
formulation results in lower grid dispersion with respect to FDTD. Our analysis
concludes that, in order to obtain the same bandwidth, an increase in 38% in memory
size and an analogous increase in computational time is required for the FDTD two
dimensional case.
175
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A p p e n d ix A
Band-pass Filter Design
The procedure to design the microstrip band-pass filter used as insulator in the
measurement set-up is the well known Chebysev technique as in [1] in Chapter 4. The
specifications for the filter mask are the following:
center frequency fo=2.5 GHz
lower frequency f 2= l GHz
higher frequency f3=4GHz
lower frequency rejection fi=0.5 GHz
higher frequency rejection f4=4.5 GHz
and their meaning is illustrated in Fig. A.I. The order of the filter chosen is n=5, and
the ripple level L ar= 0.5 dB.
Attenuation
cur
frequency
Fig. A .l Band-pass filter mask
The filter will be printed on duroid substrate having relative dielectric constant of
2.33, and thickness of 1.575 mm. The length o f the resonant elements must be A/4,
176
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where X is the wavelength of the center frequency in the substrate. Our center
frequency is 2.5 GHz so it will result in
A / 4 = ■■/ f — =0.01637 m
4/oVer
(A.l)
From [1] (see Chapter 4) we obtain for the Chebysev response, for order n=5 and
Lar=0.5 dB the following normalized parameters:
g0=1.0
gi=1.7058
g2=1.2296
g3=2.5408
g4=1-2296
g5=1.7058
g6=1.0
From [11 we also have
(0 [ = 1
(A.2)
w=
(A.3)
/o
so we can evaluate the ratios Ji,i+i/Yo
~TT~ Yo \ 2g0g1
'
1
(A.4)
Si2 - py. 1 .-1.301
^0
-yjgig2
Yo
2(o[
■ k .--**
(A.5)
■1
=1.0664
(A.6)
1
=1.0664
(A.7)
Yo
2fi), -yjg3g4
Yo
2 0 >; y/gtg.
— L _ = 130 i
(A.8)
177
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from those ratios we can compute the even and odd impedance o f each coupled
section of the filter, using Eq. (A. 10) and (A. 11)
i+ ii±
( j i.i+ l '
i = 0 ...n
(A. 10)
i = 0 .. .n
(A .ll)
^0
2"
1
~Yo
1
_
From the odd and even impedance we can extract [1] the width and the separation for
each resonant element. The corresponding values are reported in Table A.I.
i
(Zoe)i4+i
(Zoo)i4+i
wy+i
(O)
52.6
(mm)
0
(O)
157
sy+i
(mm)
1.01
0.16
1
199.6
69.5
0.57
0.21
2
160
53.5
0.97
0.16
3
160
53.5
0.97
0.16
4
199.6
69.5
0.57
0.21
5
157
52.6
1.01
0.16
Table A.1 Odd and Even impedance table for the resonant elements
of the band-pass filter, and corresponding dimensions.
The filter layout will be as illustrated in Fig. A.2
178
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Wl
w.
w.
Fig. A.2 Bandpass microstrip filter printed on duroid substrate
After the design the filter response was simulated using the Microwave Design
System (MDS from Hewlett Packard) to take into account edge parasitics effects and
other higher order phenomena. Simulated response was compared with the measured
one. Fig.A.3 and Fig.A.4 show respectively the results of the simulation and
measurements for the transmission coefficient (S 2 1 ) of the filter. These results are in
very good agreement.
1.0 GHz
freq
3.0 GHz A
Fig. A.3 Simulated transmission coefficient for the bandpass filter
179
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chi
1 -2
lo g
m ag
a da/
re f
o da
i ;- .
2 2 9 .
100
4
iig
0C
a
da
MHZ
:2
:/£
start
l 000.000 000 MHz
S T O P 3 000.000 000 MHz
Fig. A.4 Measured transmission coefficient for the band-pass filter
180
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A ppen d ix B
Vector Potential Formulation
Consider Maxwell's equations in a dielectrically lossy source free region:
VxE=
(B l)
dt
Vx H = ^
(B2)
+ oE
(B3)
(B4)
at
V« D = 0
V «B = 0
From (B.4)
B = VxA,
(B.5)
and combining (B.l) and (B.5)
E=- f - V ,.
(B.6)
For a non dispersive material D = cE and (B.3) can be rearranged to read
V * E + Vl n e » E = 0
,
(B.7)
which can be combined with eq. (B.6) and rearranged to obtain
V20 - e n - r r + V In e • V 0 = —^ ? • A + q i ^
at
at
L
+ A • V In e
(B.8)
dt
Equation (B.8) couples $ and A ; but the two variables can be decoupled with the
o f the Lorentz-Gauge condition,
V« A = -£ /!-“ •
at
(B.9)
and for the particular geometry such that
181
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A « V ln e = 0
(S. 10)
which correspond to the condition o f space invariant permittivity, £=ConsL or
£ = e (x ,y), and A = \ z . If the analysis is restricted to the above case, taking the curl
of both sides of eq. (A5) and combining with eq. (A2),
/* ^ ® U o iu E = V ( V . A ) - V 2A - V / i x H
(B .ll)
is obtained. Therefore summarizing together eqs. (B.6), (B.9) and (B .ll) the result is
= V x —V x A - o E
n
dt
f= -E -V ,
V• A=
(B.12)
dt
For the two dimensional case, all the space derivatives in the z-direction disappear,
and therefore
£*l=
dt
_
1 ’ d_ f 1 d \ )
£ dx
dx
J
r\
^ y
^
ft).
— E.
dt
182
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(B.13)
A p pe n d ix C
Vector Potential Formulation for Electric Debye Medium
In the case o f an electric Debye material, the polarization in the time domain, is
defined as follows
eEt = e„Et + Pt
(C.l)
where the polarization satisfies the following differential equation:
dPt
P '+ r - £ = epEt .
(C.2)
Substituting eq. (C .l) and (C.3) into (B.13), and restricting attention to the lossless
case,
1 dfL = - L
eP dt
v^
dE, _
1 ' d_
dt
dx
f 19dxA')j + -dy\I f i ddy\ J.T
dAdt
1 dp,
£„ dt
(C.3)
-E .
is obtained. At this stage the derivation can be easily extended to double or multiple
electric Debye relaxations. This is outlined here for the case o f double Debye
materials, but the extension to multiple Debye structures is immediate.
The same definition for the polarization is used as in eq. (C .l), but now Pt
may be imagined as the sum of two polarizations Pfl) and /*(2), so that the "effective"
average polarization is obtained as Pt —P,0) + P™. This notation comes particularly
183
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useful because of the particular form of the dielectric constant in the time domain in
the case of two Debye relaxations. In fact the electric displacement can be written as:
£Et
e_Et + £p
gi
l + f xd /d t
(C.4)
l + T2d/dt
where g, + g2 = 1; or using the polarization
s
€p8l
_
rr
^p8l
,
1 + Tj d/dt
r _ n(l) . n(2)
1 + r 2 d /d t
~
(C.5)
where it is clear that each polarization can be thought as to act independently from the
other, and obeys the differential equation
* dt
—£PglEl.
(C.6)
Substitution of eq. (C.l) and (C.6) into (B.13), yields, for the lossless case,
1 dp, =
£p dt
dt
- I
l
( *«
p (1) _ C
Jpg1
1
____
£m d x { n dx )
' n (2)
-ill
N
*
J p82
d y { n dy
F
*>
J.
_L£.
£m dt
184
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(C.7)
A ppe n d ix D
Vector Potential Formulation for Magnetic Debye
Medium
In the case of a magnetic Debye material, the derivation is restricted here to the one
dimensional case; the extension to the two dimensional case is straight forward. In the
time domain a Debye permeability can be written as follows
uJH = u H +
H = u H +— ^ — H
y
y 1+ ttdd/dt
/d t y
* 1 + xd
/d t r
t d/dt
(D.l)
Upon introduction of the vector potential
B> -
IT
(D.2)
'
eq. (D .l) and (D.2) are combined, to yield a differential equation for Hy:
dHi _ __ 1 d
dt
fi„ d t dx
1 dAt
/ i . r dx
H
fimT y
(D.3)
1 dA,
In one dimension —
= - //„ , therefore the time evolution equation for the Electric
H dx
y
fields
dE, _
dt
Id
1 dA,
e dx U dx
(D.4)
can be replaced by the equations
H dx
dE, ^ 1 d
\
dt
e d x ' y'
(D.5)
185
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" here it is clear that the top equation is a differential equation for Hy. Combining eq.
(D.5) with eq. (D.3) gives the result:
dHy _
l 3 d A ,___ 1 dA,
dt
dt dx / x . r dx
dEt _ 1 dHy
dt
e dx
_ v
* ■ — 1E-
H,
(D.6)
The derivation o f the vector potential formulation is outlined here for the case of a
double magnetic Debye material. Extension to multiple Debye relaxations la
straightforward. The Debye permeability assumes the form
g,
J 4 -^ d/dt
M.
H
1+ XLd/dt
(D.7)
where g, + g2 = 1; with the introduction of the vector potential, this equation becomes
a differential equation for Hy. Following the same derivation outlined before, the
equations for double Debye materials are obtained as
+ / i 2)T, + (/z_ + M ,) t2] - ^ + ( / i . + / / 1 + / i 2) / / r =
dA
,
xd d \
(
^d
dA.
(D.8)
dEt _ 1 dHy
dt
e dx
^ — £
dt
where //, =
186
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A p p e n d ix E
DC-Block Design
The function of the dc-block is to guarantee that microwave signal flows through,
while dc components are rejected. It is in principle acting like a capacitor. O f course
at microwave frequencies it is difficult to use lumped components, so this function is
achieved trough two coupled lines as shown in Fig. E.1
0 6 mm
I
0 2 mm
, 0 .6 mm
ss:
2 0 -5mm
Fig. E .l dc-b!cck layout
The length of the coupled lines must be X/4 wavelength in the substrate, while the
width and the space between the line is determined optimizing at the desired
frequency. Results of the optimization (using Touchstone) are reported in Fig. E.2
m
•o
-
-.0
E
-3 0
-4 0
S11 (dB)
S21 (dB)
-5 0 —
1 10*
1.5 109
2 10®
2.5 i t f
3 10®
Frequency (Hz)
Fig. E.2 5 parameters for the 2.1 GHz dc-block
187
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