# 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i UMI N u m b er: 9818034 UMI Microform 9818034 Copyright 1998, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I To my parents , i 1 Arnaldo and Fernanda De Flaviis ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 permission of the copyright owner. Further reproduction prohibited without permission. 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 viii permission of the copyright owner. Further reproduction prohibited without permission. 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 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 xi j I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 xu Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 xv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 xvi ! I I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 ± Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [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 ! Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. . 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ! 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. heat dissipation and the manifestation of nonlinear effects. No particular attention is i focused to these phenomena for powers below 500W. i i !? ; ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. /?> =-*<‘>(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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 ( Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. " 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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